procesamiento de señales de georadar:...

Di r ecci ó n:Di r ecci ó n: Biblioteca Central Dr. Luis F. Leloir, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires. Intendente Güiraldes 2160 - C1428EGA - Tel. (++54 +11) 4789-9293

Co nta cto :Co nta cto : digital@bl.fcen.uba.ar

Tesis Doctoral

Procesamiento de señales deProcesamiento de señales degeoradar: implementación delgeoradar: implementación del

método de arreglos sintéticos demétodo de arreglos sintéticos deantenas emisorasantenas emisoras

Cedrina, Lorena Valeria

Este documento forma parte de la colección de tesis doctorales y de maestría de la BibliotecaCentral Dr. Luis Federico Leloir, disponible en digital.bl.fcen.uba.ar. Su utilización debe seracompañada por la cita bibliográfica con reconocimiento de la fuente.

This document is part of the doctoral theses collection of the Central Library Dr. Luis FedericoLeloir, available in digital.bl.fcen.uba.ar. It should be used accompanied by the correspondingcitation acknowledging the source.

Cita tipo APA:

Cedrina, Lorena Valeria. (2010). Procesamiento de señales de georadar: implementación delmétodo de arreglos sintéticos de antenas emisoras. Facultad de Ciencias Exactas y Naturales.Universidad de Buenos Aires.

Cita tipo Chicago:

Cedrina, Lorena Valeria. "Procesamiento de señales de georadar: implementación del método dearreglos sintéticos de antenas emisoras". Facultad de Ciencias Exactas y Naturales.Universidad de Buenos Aires. 2010.

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t ♥s ①ts ② trs

♣rt♠♥t♦ ís

Pr♦s♠♥t♦ ñs ♦rr

♠♣♠♥tó♥ ét♦♦ rr♦s ♥tét♦s

♥t♥s ♠s♦rs

ss ♣rs♥t ♣r ♦♣tr tít♦ ♦t♦r ❯♥rs

♥♦s rs ♥ ár ♥s íss

♦r♥ ❱r r♥

rt♦r ss r ♥ s

rt♦r sst♥t r ést♦r ♦♥♦♠♦

♦♥sr♦ st♦s r rt rrr♦

r r♦ r♣♦ ♦ís ♣ ② ♠♥t

♥♦s rs ♣t♠r

Pr♦s♠♥t♦ ñs ♦rr♠♣♠♥tó♥ ét♦♦ rr♦s

♥tét♦s ♥t♥s ♠s♦rs

♣r♥♣ ♦t♦ st ss s ♥tr♦r ♠ét♦♦ rr♦s s♥té

t♦s ♠s♦rs ♦rr ② srr♦r s té♥s ② rr♠♥ts áss

♣r r t s ♠♣♠♥tó♥ ♦s ♠ét♦♦s ss ♦rtr s♠

♣ ♣r♠t♥ rs♦r ♠② rss st♦♥s ①♣r♠♥ts ♥ ♣rtr

♦s st♦s r③♦s ♥ ③♦♥ P♦ ♥♦ ♣r♠tr♦♥ rtr③r

♠♥t strtrs rq♦ós ♦♥ ♦ ♦♥trst ♦♥ ♠♦ r♥

♥t ♥ ♥s st♦♥s s sñs s♦♥ ♠② és ② s ♥sr♦ t③r

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s ♠t♦♦♦ís ss ♦rtr ♠út♣ ts ♦♠♦ ♣♥t♦ ♠♦

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q s ♦r♥ ♥♦ s♦♥ s♥ts

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♦rr ♥ q s ♦♥sr♥ rr♦s ♥t♥s ♣r ♦t♥r ③♦♥s

♠♥s ♠ás strs ♦♥♥tr♥♦ ♥rí s♦r ♦s ♥♦s ♥trés

② s♠♥②é♥♦ ♥ s ③♦♥s ♣rérs ♥r♦rs sñs s♥rs

♥t st ♠t♦♦♦í ♦rtr ♠út♣ s ♦r ♠♦rr ró♥

♥t♥s sñ ♣r♠r rs♣t♦ ♦trs sñs s♥rs sí

♦♠♦ s ♦♥t♥ tr ♦s rst♦s ♦t♥♦s ♥ ♣rs♥t tr♦

♠♦strr♦♥ ♥♦s s♥t♦s ♥ ♥t♦ ♣ó♥ st ♥♦

♠ét♦♦ ♥ ♦rr ♣r str rt♦rs q ♥r♥ s♦♠♥t sñs

♠② t♥s ② ♣r ♠♦rr ♥ ♦r♠ st sñs ♦r♥s ♥ st♥t♦s

♦t♦s ♦ q ♦♥stt② ♥ ♣s♦ ♥♠♥t ♣r ♦♥t♥r ♦♥ srr♦♦

♠ét♦♦

Prs s ♦rr Pr♦s♠♥t♦ rrr♠s rr♦s s♥té

t♦s ♦rtr ♠út♣ ét♦♦ ♣r rtr③ó♥ strtrs

r♦♥ P♥trt♥ r ♥ Pr♦ss♥♠♣♠♥tt♦♥ ♦ t ②♥tt ♠ttr rr②

♠♥ ♦t ♦ ts ss s t♦ ♥tr♦ t s②♥tt ♠ttr

rr② ♠t♦ ♥ t♦ ♦♣ t s t♦♦s ♥ t♥qs ♦r ts ♠♣♠♥

tt♦♥ ❯s s♥ ♦st ♠t♦s ♦ s♦♥ ♠♥② r♥t ①♣r♠♥t

stt♦♥s ♥ ♣rtr sts ♥ t r ♦ P♦ ♥♦ ♦ q

t② rtr③♥ r♦♦ strtrs t ♦ ♦♥trst rs♣t t♦ t

srr♦♥♥ ♥r♦♥♠♥t ♥ s♦♠ stt♦♥s s♥s r r② ♥ t♥

♦tr ♠t♦♦♦s t ♠t♣ ♦st t♦ s t♦ ♠♣r♦ ts

st② t t s ♠t♦st ♠t♦♦♦s s s ♦♠♠♦♥ ♠♣♦♥t

s♦ ♠tt♦♥s s♥ ♥ s♦♠ ss t ♠♣r♦♠♥ts r ♥♦t s♥t

♦r r rtr③t♦♥ ♦ t trt

♥ ts ♦♥t①t t s②♥tt ♠ttr rr② ♠t♦ s ♣r♦♣♦s s ♠

t♦ ♠♣♦②s ♠ttr rr②s t♦ ♥rr♦ t ♠♥t♦♥ r ts ♦♥♥tr

t♥ t ♥r② ♦♥ t trts ♦ ♥trst ♥ r♥ t ♥ ♣r♣r rs

♥rt s♦♥r② s♥s r♦ ts ♠t♦st ♠t♦♦♦②

♥rs t s♥ t♦ ♥♦s rt♦ ♥ t tr ♦r♥ ♦ t ♥ts ts

s♥♥t② ♠♣r♦♥ t ♥t♦♥ ♦ r② s♥s ♥ t rs♦t♦♥

♦ s♥s ♦r♥t t r♥t ♦ts

②♦rs r♦♥ P♥trt♥ r rr♠ ♣r♦ss♥ ②♥tt

rr②s t♣ ♦st t♦ ♦r rtr③♥ strtrs

❮♥ ♥r

♥tr♦ó♥

♦♥t♥♦ ss

ét♦♦ ♦rr

q♣♦s ♦rr

Pr♦♣ó♥ sñs ♦rr

t♦♦♦í ♦rtr s♠♣

ñs ♣r♦s ♣♦r st♥t♦s t♣♦s rt♦rs

♠♣♦ s♦♥♦ ♦rtr s♠♣

Pr♦s♠♥t♦ ás♦ ♣r t♦s ♦rtr s♠♣

t♦♦♦í ♠ró♥ sñs

t♦♦♦ís ♦rtr ♠út♣ ♦♠♠♦♥ ♣♦♥t

♣ó♥ ♦s ♠ét♦♦s ss ♥ ♦rr

t♦ rq♦ó♦ P♦ ♥♦

♦③ó♥ ♥ strtr ♣rs

tó♥ ss ♥trrs

ét♦♦ ♦rtr s♠♣

♦♦

ét♦♦ ♦rtr ♠út♣ P

ét♦♦ rr♦s s♥tét♦s ♠s♦rs ♦rr

rr♦s ♠s♦rs

ét♦♦ rr♦s s♥tét♦s ♠s♦rs ♦rr

rtr③ó♥ ♠♣♦ rr♦

s♣st ♠ét♦♦ ♥ s♦s ás♦s

Pr♦s♠♥t♦ ♣r ♠♦rr sñs ♦♠♣s

♣ó♥ ♠ét♦♦ s♦s ♦♥ ♥♦s ♠út♣s

t♦s s♠♦s

t♦s ①♣r♠♥ts

rtr③ó♥ ♣rs t♣

♣ó♥ sñs ♣r♦s ♣♦r ss

ó♥ ♠ét♦♦

♥ó♥ ♥♦rs

♦♠♣ró♥ ♥tr ♦s ♠ét♦♦s ② P

t♦s s♠♦s

t♦ ♣qñ♦

t♦r ①t♥s♦

♦♥s♦♥s

s♠♥ rst♦s ② ♦♥s♦♥s

trs í♥s tr♦

♠ó♥ ♥♠ér rrr♠s

rr♦s ♥t♥s ♥ ♥ ♠♦ ♥♦r♠

Pr♦r♠ rr②

Pr♦r♠ ♥t♦r♥♦

r♠♥t♦s

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♥tr♦ó♥

♠ét♦♦ ♦rr s ♥ té♥ ♣r♦s♣ó♥ ♥♦ ♥s t r

s♦ó♥ q ♣r♠t ♥str s ♣s s♣r♦rs s♦ ♥ t

♦♥stt② ♥♦ ♦s ♠ét♦♦s ♠ás t③♦s ♥ ♥st♦♥s ♦íss ♦♥

♣♦♥s ♥ rss árs ♦♠♦ ♣♦r ♠♣♦ ♦♦í ❬❪ ❬❪ ❬❪ rq♦♦

í ❬❪ ❬❪ ❬❪ ♥♥rí ❬❪ ❬❪ ❬❪ ❬❪ ② ♠♦♥t♦r♦ strrá♥ ❬❪

❬❪ ❬❪ ♥ ♠ét♦♦ ♦rr r♦♥ ♣♥trt♥ rr P s ♠t♥

♣s♦s tr♦♠♥ét♦s s s♣r s♦ ♦s s s ♣r♦♣♥

trés ♠s♠♦ ♦♥ s r♥ ② rrt♥ ♥ s sss ♥trss ♥

s♣r s♦ s rstr ♠♣t sñ r ♦ r♦

t♦♦ st ♣r♦s♦ ♦♠♦ ♥ó♥ t♠♣♦ ♦ s ♥③♥ ♦s t♠♣♦s

trá♥st♦ ♣r ♦s ♣s♦s q ♥ rrs♦ sí ♦♠♦ ss ♠♣ts ② ss

♥ ♣rtr s s ♦♥♦ ♦ ♣r♦♣ó♥ ♥ ♦s ♠♦s s ♣♦s

st♠r s ♣r♦♥s s ♥trss ♣rtr ♦s t♠♣♦s

trá♥st♦

♦s q♣♦s rr ♣r ♣r♦s♣ó♥ ♦ís s♠♥t t③♥ r

♥s ♥tr ③ ② ③ ♥ r♥s ♠②♦rs ♦♥ts

♦♥ ♠♥♦rs ♣r♦♥ ♠♦r rs♦ó♥ ♥r♠♥t♦ s♦ró♥ ②

s♣rsó♥ ♥ s sñs tr♥s♠ts r♥ ♣r♦♥ ♣♥tró♥

♥t♦ ♠②♦r s r♥ ♥t♥ ♠♥♦s ♣r♦♥ s ③♦♥ ♣♦s

sr st r♥♦ ♣r♦♥s q s ♣♥ rtr③r ♣♥

♦♠♣♦só♥ ♠tr③ ♦♥ s ♥ ♦s ♦t♦s ♦ strt♦s

♥trés ♣♦r ♠♣♦ ♦♥t♥♦ rs ♠ s♦ t

② s rtrísts s ♥t♥s rr ♥ ♣♦♥s ♦íss s

t ♥③r ♣♥tr♦♥s s ♥s ♥s ♥tí♠tr♦s ♣r s

r♥s ♠ás ts st ♥s ♥s ♠tr♦s ♣r s r♥s

♠ás s

♥ ♠②♦rí s ♣♦♥s ♦rr st♥ ♥tr s ♥t♥s

♠s♦r ② r♣t♦r ♦st s ♠♥t♥ ♦♥st♥t r♥t s♦♥♦ ♠t♦♦

♦í ♦rtr s♠♣ ♦ s♥ ♦st ② ♣r ♥t♥s s s♣③

♦ r♦ í♥s ♣rs q s str②♥ ♥ trr♥♦ ♣r rr ♦♠♣

t♠♥t ár st♦ ♦♥ st ♠t♦♦♦í s ♣♥ ♥③r r♥s

♣♦r♦♥s s♦ ♥ t♠♣♦s rt♠♥t ♦rt♦s ② ♦♥ ♥

♥tó♥ s strtrs ♥trrs ❬❪ ❬❪ ♥ ♣rtr qs

ó♥ ♥ rs ♦♥ t ♥s í♥s ♣rs ♣ r ♠á♥s ♦s

♥♦s r♥ ❬❪ ❬❪

♥ ♠r♦ s ♥t♥s ♦rr t♥♥ rt ♠t ♦♥ ♦

♥♦s ♠♥ó♥ st♥t ♠♣♦s ❬❪ ❬❪ ♦♠♦ ♦♥s♥ ♥ ró♥

♠♣♦rt♥t ♥rí tr♥s♠t s ♣r r ♠♥♦ ♠s♦r r

t♦r r♣t♦r r♥♦ s ♣♦ss tó♥ s♣♠♥t ♥

s♦s ♦♥ ró♥ sñ r♦ ♦♠♦ ♦rr ♣♦r ♠♣♦ ♥ ♦♥♦♥s

t s♦ró♥ ♦ ♦♥trst ♦ ♥♦s ♣r♦♥♦s st ♣ér ♥rí

r t ♠t♦♦♦í ♦rr ♥♦ s t③♥ ♥t♥s

♠s♦rs ② r♣t♦rs ú♥s ② st♥ ♥tr s

tó♥ ♥tr♣rtó♥ sñs ♦t♦s ♥trr♦s s ♣♥ ♠

♦rr t③♥♦ ♠ét♦♦s ♦rtr ♠út♣ ♦♠♦ ♣♦r ♠♣♦ ♦s ♠ét♦♦s

♣♥t♦ ♠♦ ♦♠ú♥ ♦♠♠♦♥ ♠♣♦♥t P ❬❪ ❬❪ ❲ ♥

rt♦♥ ♥ rrt♦♥ ❬❪ ❱ ♠♣t rt♦♥ t ♦st ❬❪ ❬❪

♦ ❱ ♠♣t rt♦♥ t ♥ ❬❪

♥ s ♠t♦♦♦ís qsó♥ ♦rtr ♠út♣ s ♥t♥s ♠

s♦r ② r♣t♦r s ♦♦♥ s♦r s♣r ♦♥ ♥ s♣♦só♥ ♣rtr

♣r ♥♦ ♦s st♥t♦s ♠ét♦♦s ② s rí st♥ ♥tr s

st ♠♥r s ①♠♥ r♣ts s ♠s♠ ♣♦ró♥ s♦ ♣r♦

♠♥t♦ s r♣t ♥ ♣♦só♥ s♦ ♥str ♦ s ♣r♦s♥ ♦s

t♦s qr♦s ♣r♦r♥♦ ♦t♥r sñs ♠ás ♥t♥ss ② rs q ♥♦

s t③♥ ♠t♦♦♦ís ♦rtr s♠♣ ♥ ♥r qsó♥ ♦♥

♦rtr ♠út♣ s ♣ ♥ ♥ú♠r♦ ♣qñ♦ ♣♥t♦s ♣r ♦t♥r

♦ ♣r♦♣ó♥ ② ♣r♦♥ s ♣r♥♣s ♥trss ♦

r♦ ♥s ♣♦s í♥s ♦ ♠② rr ③ ♦ r♦ í♥s ♣rs ♣r

♦t♥r ♠♣♦ ♦s ② ♣r ♦rr ♥ ♠♣ ♦♠♣t♦ ss♦

♦s ♠ét♦♦s ♦rtr ♠út♣ s♥ r♦r③r s sñs ♣r♠rs ♥

ró♥ r♦ ② s sñs s♥rs ♠♦r♥♦ ♣♥tró♥ ② ♦

r♥ tr s sñs ♦t♥s t♦r ♠t♥t ♠ás ♠♣♦rt♥t

st♦s ♠ét♦♦s s q rqr♥ t♠♣♦s r♦s qsó♥ ② ♣r♦s♠♥t♦

❯♥ ♠♥♦ tr♥t♦ ♣r ♠♦rr s sñs ♦♥sst ♥ ♦♥♥trr

♥rí s♣♦♥ s♦r ♦s ♥♦s ♥trés ♠♥t ♥r♠♥t♦

rt ♦s ♠♣♦s tr♥s♠t♦s ♣♦r q♣♦ st ♥r♠♥t♦ s

♣ ♦rr t③♥♦ ♥ r♣♦ ♥t♥s ♠s♦rs r♥s ♥tr sí s q

♦r♠♥ ♥ rr♦ ♥ st s♦ s ♣♥ s♦♥r s r♦♥s s

st♥ ② ♠♣t ♥tr s ♥t♥s ♠♥r ♥♦str ♠♥t

♦s ♠♣♦s tr♥s♠t♦s ② rr♦s ♥♦ ♦s ♥♦s P♦r ♦tr♦

♦ s ♣♦s ♦t♥r rst♦s s♠rs ♦♥ ♥ ú♥♦ ♠s♦r ♦♦á♥♦♦

ss♠♥t ♥ s ♣♦s♦♥s q rí♥ ♦♣r s ♦♠♣♦♥♥ts rs

rr♦ ② ♦ s♣r♣♦♥♥♦ ♦s ♠♣♦s ♦s rstr♦s ♥s ♣r

s♥tt③r rs♣st t♦t rr♦

♥ ár ♦rr s ♥ r③♦ st♥t♦s t♣♦s st♦s q

♥♦r♥ ♦♥♥t♦s ♠s♦rs ♦ r♣t♦rs ♥ ♥ r♥ ♣rt ♦s s♦s

s t③♥ ♥s ♥t♥s q s♣r♥ ♦ r♥ ♥ ♦r♠ ♦♥st

② t♦♠át ♦♥ ♦t♦ ♠♥♠③r ♦s t♠♣♦s ♣r♦s♣ó♥ t♥t♦

♣r s♦♥♦ árs ①t♥ss ♦♠♦ ♣r ♦t♥r srs t♦s ♦♥ t

♥s s♣ ❬❪ ❬❪ P♦r ♠♣♦ ♥ t♦ t ❬❪ s s ♥ ♦♥♥t♦

♥t♥s ♦r♠♦ ♣♦r ♥♦ ♠s♦rs ② ♥♦ r♣t♦rs q sr s♣rs

s♥♠♥t ♣r♠t qrr t♦s ♥ ♦♥ró♥ P

P♦r s ♦ sí♥tss ♠♣♦s s ♥ ♠t♦♦♦í st♥t t③

♥ ár ♦ís sís♠ ❬❪ ❬❪ ② ♥ rrs ér♦s ♦s ♥ trr

♦♥s ② stéts ❬❪ ❬❪ P♦r ♦♥trr♦ ♥ ár ♦rr ①st ♥

♥t ♣♦♥s rt♠♥t ss rs♣t♦ ♦♥ ♥ ♣r

♣r♦♣♦ró♥ rst♦s ♣r♠♥rs ♥ s♣ ♣ ♠♥♦♥rs

tr♦ s♥ t ❬❪ q♥s ♥ srr♦♦ ♥ q♣♦ q

♥r ♦♥s ♣r♦①♠♠♥t ♣♥s ♣r♦r♥♦ ♣r♦r ♠t♦♦♦ís

♣r♦s♠♥t♦ t♦s ♣rtr s♥tét ♠r♠♥t s t ❬❪

sr♥ ♣r♦r t♦ ♥ ♦♥♦♠♥t♦s ♥③♦ ♥ ár

rrs ér♦s ♣r s ♣tó♥ sst♠s ♦rr ss♣♥♦s ② ♥

♣rtr s♦ rr♦s ♥s ♠s♦rs ♦ r♣t♦rs ♥ ❬❪

♥ ♣rs♥t♦ ♥ ♠t♦♦♦í ♣r ♣♦♥r ♥ ♥ú♠r♦ ♦♥s ♦♠♣♦♥♥

ts ♥ s ♥ s♦ ♥ ♠♦♦ ♣s ♦r③♦♥ts s♣♦♥♥♦ rts

rtrísts ♥ s sñs ② t ❬❪ ♦♥sr♥ ♥ ♣r♦♠♥t♦

♣rtr s♥tét ♦♥ ♦t♦ ♦t♥r ♠á♥s ♠♥s ♥t♣rs♦

♥s s♣rs ♥♠♥t t③ ② Prr♦ ❬❪ r③♥ ♥ ♥áss s♦r

♥♥ ♦s ♣rá♠tr♦s ♥ rr♦ ♥t♥s ♥ú♠r♦ ♥ts

s♣ró♥ ② s♣③♠♥t♦ t♠♣♦r rt♦ ♥tr ♥ts ♥ ♦s ♣tr♦♥s

ró♥ ♦s t♦rs ♠str♥ ♥s ♠♦rs ♦t♥s ♥ sñs

rt♦rs ♣♥♦s ② ♣♦ ♥♥ó♥ rs♣t♦ s sñs ♦rtr

s♠♣

♥q ♦s st♦s t♦s ♠str♥ ♥♦s ♥s ♠t♦♦ó♦s r

♦♥♦s ♦♥ sñs P qrs ♦♥ ♠út♣s ♥t♥s ♦s ♠s♠♦s ♥♦

♥ ♥t ♥ ♠t♦♦♦í ♥r ② ♥♠♥t q ♣r♠t ♥ ♣

ó♥ sst♠át ♦s rr♦s s♥tét♦s ♥ ár P♦r ♠♣♦ tr♦

③ ② Prr♦ ❬❪ s ♠t ♣tr♦♥s ró♥ ♥ í♦ ② ♥ ró♥

♠♣♦ ♥♦ ♥ st ♥áss ♣r♠t ♦t♥r ♥s rtrísts

♥rs ♣tró♥ ró♥ st♦s rst♦s s♦♥ t ♠t ②

q ♦s st♦s ♦ís♦s t♠♥t s r③♥ ♥ ♠♦s ♦♠♣♦s ♦♥

♥♦s str♦s ♥ t♦♦ r♥♦ st♥s s ♠♣♦ r♥♦ st

♠♣♦ ♥♦ s♠s♠♦ ♥♦ s trt♥ s♦s ♦♥ ♥♦s ♠ás ♥rs ♦♥

t♠ñ♦ ♦r♠ ② ♦r♥tó♥ rtrr ♦s q ♣r♥ ♠② r♥t♠♥t

♥ ♦s st♦s ♦ís♦s

♦t♦ st ss s srr♦r ♠♣♠♥tr ♠ét♦♦ rr♦s

s♥tét♦s ♠s♦rs ♦rr ② srr♦r s té♥s ② ♦s ♣r♦r♠s

♦♠♣t♦♥s ♥sr♦s ♣r ♣ó♥ ♠ét♦♦ ② ♣r♦s♠♥t♦

♦s t♦s

Pr st♦ s ♥ sr ♦s ♣s♦s q s sr♥ ♦♥t♥ó♥

r rtr③ó♥ strtrs ♦♥ ró♥ sñ r♦

s♥♦ ♦s ♠ét♦♦s ♠ás ss ♦st ♦ ② ♠ró♥ t♦s sí

♦♠♦ ♠♦♦ ♣r ♥tr ♥♦s ② ♠ét♦♦s s③ó♥

♥③r ♦s ♠♣♦s ♥r♦s ♣♦r rr♦s ♣♦♦s ♦s s♦r

s♣r s♦ ♣♦r ♠♦ s♠♦♥s ♥♠érs ♦♠♦ ♥ó♥

♦s ♣rá♠tr♦s rr♦ ♠ás r♥ts ② st♥ ♥tr

rr♦ ② ♥♦ ♥ ♦t♥r ♣ts q ♣r♠t♥ sñr rr

♦s ② ♠t♦♦♦ís ♣r♦♣♦s ♣r ♥r♠♥tr rt ♦s

♠♣♦s tr♥s♠t♦s ♠♥r q ♦s ♥♦s s♥ ♠♥♦s

♠♥t

str rs♣st ♠ét♦♦ rr♦s s♥tét♦s ♣♦r ♠♦

♠♦♦s ♥♠ér♦s

srr♦r ó♦s ♦♠♣t♦♥s q t♥ ♣ó♥ ♠ét♦

♦ rr♦s s♥tét♦s srr♦r ♠ás ♥t♦r♥♦s rá♦s q s♠

♣q♥ t③ó♥ ♦s ó♦s

r s ♠♦rs ♦t♥s ♦♥ ♠ét♦♦ t♥t♦ tt ♦♠♦

♥ttt♠♥t ♦♠♣rr ♦s rst♦s ♦t♥♦s ♦♥ ♦s ♦tr♦s

♠ét♦♦s ♦rtr s♠♣ ② ♠út♣

♣r ♠ét♦♦ s♦s ①♣r♠♥ts ♥ ♦s q ♦s s♦♥♦s r③

♦s ♦♥ ♥ ú♥♦ ♠s♦r rr♦♥ rst♦s ♣♦♦ stst♦r♦s

♦♥t♥♦ ss

♦♥t♥ó♥ ♣rs♥t ♥tr♦ó♥ ♣ít♦ ♥ ♣ít♦

st ss s ♣rs♥t♥ ♦s ♥♠♥t♦s tór♦s ② ♦s ♣r♥♣♦s ♥♦

♥♠♥t♦ ♠ét♦♦ ♦rr sr♥ s ♠t♦♦♦ís ♦rtr

s♠♣ ② ♠út♣ ② ♠t♦♦♦í ♠ró♥ t♦s

♥ ♣ít♦ s ♣rs♥t ♥ sr tr♦s r③♦s ♥ st♦

rq♦ó♦ P♦ ♥♦ ♦♥ s ♣♥ s té♥s ♠ró♥ P

② ♠♦♦ ♣r ♠♦rr ♥tr♣rtó♥ ♥ s♦s ①♣r♠♥ts ♦♥

ró♥ sñ r♦

♥ ♣ít♦ s ♣rs♥t ♠ét♦♦ rr♦s s♥tét♦s ♠s♦rs

♦rr ♣rt ♦s ♥♠♥t♦s t♦rí rr♦s ♥t♥s

t♣♦ ♣♦♦ ♥ í♦ ② ♦ s st ♠♣♦ rr♦ ♥ ♥ s♠s♣♦

♠str ♣ó♥ ♠ét♦♦ s♦s s♠♣s ♦♥ t♦s s♠♦s ② s

♣rs♥t ♥ ♠t♦♦♦í ♣r ♠♦rr ♦♥t♥ tr s sñs

♥ ♣ít♦ s ♠str♥ ♦s rst♦s ♣ó♥ ♠ét♦♦

st♥t♦s s♦s ♦♠♣♦s ♦♥ sñs ♠út♣s t♥t♦ s♥tét♦s ♦♠♦ ①♣

r♠♥ts

♥ ♣ít♦ s ú♥ ♥ttt♠♥t s ♠♦rs ♦t♥s ♥

♣ó♥ ♠ét♦♦ ♣♦r ♠♦ rs♦s ♥♦rs

♠♥ ♦♠♣r♥ ♦s rst♦s ♦t♥♦s ♦♥ ♠ét♦♦ rr♦s s♥té

t♦s ♠s♦rs ♦rr ② ♦♥ ♦s ♠ét♦♦s ② P ♦♠♣ró♥

s r③ t♥t♦ ♦♥ t♦s s♠♦s ② ♦♠♦ ①♣r♠♥ts

♥ ♣ít♦ s rs♠♥ ♦s rst♦s ♦t♥♦s ♥ ss s ♣r

s♥t♥ s ♦♥s♦♥s ♥rs ② s sr♥ s í♥s tr♦ tr♦

♥ ♦s ♣é♥s ② s ♥ ♦s ts s s♠♦♥s ♥♠érs

r③s ② s ♣rs♥t♥ ♦♥t♥♦s tór♦s q ♦♠♣♠♥t♥ ♦♥t♥♦

♥♠♥t ♥ ♦s ♣é♥s ② s sr♥ ♦s ♣r♦r♠s ♦♠♣

t♦♥s srr♦♦s ♥ ♠r♦ st ss ♥ st♦s ♣r♦r♠s s♦♥

♣rt ♠♣♦rt♥t s ♦♥t♥♦ ♦s ♠s♠♦s ♥ s♦ ♦♦♦s ♥ ♣é♥s

♣r ♦t♥r ♥ ♠②♦r ♦♥t♥ ♥ srr♦♦ st ss

♣ít♦

ét♦♦ ♦rr

♥ st ♣ít♦ s sr♥ ♦s ♣r♥♣♦s ♥♠♥ts ♠ét♦♦

♦rr ♦s ♦♠♣♦♥♥ts ás♦s q♣♦ ② s ♣rs♥t♥ s ♠t♦♦♦ís

qsó♥ ② ♣r♦s♠♥t♦ ♠ás t③s

q♣♦s ♦rr

♦s ♦♠♣♦♥♥ts ♣r♥♣s ♥ q♣♦ ♦rr s♦♥ sst♠ ♠

s♦r sst♠ r♣t♦r ② ♥ ♦♥tr♦ ② rstr♦ r ♥

♦♥tr♦ ♥t ♦♥ ♥ ♥r♦r sñs ♥ sst♠ rstr♦ ② ♣r♠t

s♦♥r ♦s ♣rá♠tr♦s ♠só♥ ② r♣ó♥ s sñs ♥r♦r

♥í ♥ sñ étr ♥t♥ ♠s♦r ② ♦r♥ q ést ♣r♦③ ♣s♦s

tr♦♠♥ét♦s q s tr♥s♠t♥ s♦ ♦s ♣s♦s s ♣r♦♣♥

r sq♠ ♦s ♦♠♣♦♥♥ts ♥ q♣♦ ♦

♥trtú♥ ♦♥ s st♥ts s♦♥t♥s ♥ ss♦ ♥t♥ r♣

t♦r ♣t sts sñs s tr♥s♦r♠ ♥ sñs étrs ② s rstr ♥

♥ ♦♥tr♦ ♥ ♥ó♥ t♠♣♦ ♥ ♦s♦♥s ♥ ♦♥tr♦ t

♥ ♥♦r♣♦r ♥ ♣♥t ♣r s③ó♥ ♦s t♦s ② ♥t ♦♥

♣ r③r ♥ ♣r♦s♠♥t♦ ♣r♠♥r ♦s t♦s ♥ s♠tá♥♦

♦♥ s qsó♥ ♥ ♥♦s s♦s st♦ ♣r♠t ♦t♥r ♥ ♥tr♣rtó♥

♣r♠♥r ♦s t♦s ♣r♦ ♥ ♥r ♣r♦s♠♥t♦ ♥tr♣rtó♥ s

r③♥ ♥ ♥ t♣ ♣♦str♦r ♦ r tr♥sr♦ ♦s ♠s♠♦s ♥

♦♠♣t♦r

r q♣♦ rr ♥ ♦♥ró♥ ♦rtr s♠

♦s q♣♦s ♦rr ♣♥ t♥r ♠♦♥t ♠♦♥♦stát♦ ♥ q s

♥t♥s ♠s♦r ② r♣t♦r s ♥ ♥ ♠s♠ ♥ ♦ ♠♦♥t stá

t♦ ♥ q s ♥ ♥ ♥s s♣rs st út♠♦ ♣r♠t ♣r

♣r♦♠♥t♦s qsó♥ t♥t♦ ♦rtr s♠♣ ♦♠♦ ♦rtr

♠út♣ ♥ r s ♠str ♥ q♣♦ ♠♦♥t stát♦ ♦♥

r♦ ♣r qrr t♦s ♦♥ ♥ s♣ró♥ ♦♥st♥t ♥tr s ♥t♥s

♠s♦r ② r♣t♦r

Pr♦♣ó♥ sñs ♦rr

♥ ♥r ♠ét♦♦ ♦rr s ♣ ♥ ♠♦s ♥trs ♦s ás

rst♥ s♥t♠♥t ♦♠♣♦s ♦♠♦ ♣r q ♣r♦♠ tr♦♠♥ét

♦ ♦♠♣t♦ ♣ sr rst♦ ♥ ♦r♠ ♥ít ♥ ♠r♦ ♦♥sr♥♦

♠♦♦s ♥ít♦s s♠♣s s ♣♦s ♦t♥r ♥ ♥ ♣r♦①♠ó♥

♥s s rtrísts ♣r♥♣s s sñs q s ♣r♦♣♥ ♥

ss♦ ♥trtú♥ ♦♥ s ♥trss ♥trrs

♣r♦♣ó♥ ♦♥s tr♦♠♥éts ♥ ♥ ♠♦ ♠tr s ♣

srr ♣♦r ♠♦ s ♦♥s ① ❬❪

~∇× ~E = −∂ ~B

~∇× ~H = ~J +∂ ~D

~∇ · ~D = q

~∇ · ~B = 0

♦♥ ~E s ♥t♥s ♠♣♦ étr♦ ~B s ♥s ♦ ♠

♥ét♦ ~H ♥ ♥t♥s ♠♣♦ ♠♥ét♦ ~D s t♦r s♣③♠♥t♦

étr♦ ~J s t♦r ♥s ♦rr♥t étr q s ♥s

r étr ② t s t♠♣♦

s r♦♥s ♦♥sttts ② ② ♠ sr♥ rs♣st

♦s ♠♦s ♠trs ♥ ♣rs♥ ♠♣♦s tr♦♠♥ét♦s ♥ ♠♦s

♥s sótr♦♣♦s s ♣♥ ①♣rsr ♦♠♦ ❬❪

~J = σ ~E

~D = ǫ ~E

~B = µ ~H

♦♥ ǫ σ ② µ s♦♥ srs ② r♣rs♥t♥ ♣r♠t ♦♥t

② ♣r♠ ♠♥ét rs♣t♠♥t

s ♦♥sr ♠♦♦ s♠♣ ♥ ♦♥ ♣♥ ♠♦♥♦r♦♠át

r♥ ♥r ω s ♣♦s ♦t♥r ♥ st♠ó♥ ♦

♣r♦♣ó♥ ♦♥ ♥ s♦ ♠♦s ♦♥ s ♣érs σωǫ

♦♥ s ♣r♦♣ ♦♥ ♦ s ❬❪

v≈ 1√µǫ

♦♥ ǫ = ǫ0ǫr ② µ = µ0µr ♦♥ ǫ0 ② µ0 ♣r♠t ② ♣r♠

♠♥ét í♦ rs♣t♠♥t ② µr ② ǫr ♦s ♦rs rt♦s ♦rrs♣♦♥

♥ts Pr ♥ ♠♦ ♥♦ ♠♥ét♦ s ♣ ①♣rsr ♦♠♦

v =c√ǫr

♦♥ c = 1/√µ0ǫ0 s ♦ ③ ♥ í♦ ♦♥t ♦♥

♥ ♠♦ s ♣ ①♣rsr ♥ tér♠♥♦s ♦ ♣r♦♣ó♥

♦♥ f s r♥ f = ω/2π s ♦♥s ② ♥♥

ró♥ ♥tr ♦s ♣r♥♣s ♣rá♠tr♦s q sr♥ ♣r♦♣ó♥

sñ ② s ♣r♦♣s tr♦♠♥éts ♠♦ ♥ t s

♠str♥ ♦rs tí♣♦s ♣r♠t rt ♦♥t ② ♦

♣r♦♣ó♥ ♥ ♠♦ ♦t♥ ♣rtr ♣r rs♦s ♠trs

❬❪

♥ ♥r ♠t♦♦♦í ♦rr rst ♠②♦r t ♥ ♠

trs ♦♥ s ♣érs ② q ♥ s♦s s♦s s sñs ♣♥tr♥ ♥

r♥ ♣r♦♥ ♥ ♣rát t♠é♥ s ♥♥tr♥ ♥t♦r♥♦s ♥ ♦s q

sts ♦♥♦♥s s ♣érs ♥♦ s ♠♥t♥♥ ♣♦r ♠♣♦ ♥ s♦

s♦s r♦s♦s ♦ ♦♥ strrá♥ s ♣♦r ♦ q ♣♥tró♥

♥ ♦s s ♠t

♥♦ sñ ♠t ♥③ sr ♠♦♦♥s ♦♠♦ ♦♥s♥

s ♥tró♥ ♦♥ ♠♦ q s♦♥ ♣r♦t♦ st♥t♦s ♣r♦s♦s ís♦s

♦ q ♠t♦♦♦í ♦rr s s ♥ tó♥ sñs r

s ♥ s ♥trss ss♦ ♥ó♠♥♦ r①ó♥ ♦♥stt②

♠♥s♠♦ ♥tró♥ ♣r♥♣ ♥tr sñ ② ♦s ♥♦s ♥ st♦ s

♦♥sr ♠♦♦ s♠♣ ♥ q ♥ ♦♥ ♣♥ ♥ s♦r ♥ ♥trs

♣♥ r s ♣♦s ♦t♥r ♦♥s q ♥♥ s ♠♣ts

s sñs q rst♥ ♦♠♦ ♥♦♥s ♦s ♣rá♠tr♦s ♦s ♠♦s

♠♦s ♦s ♥trs ♦s ♦♥ts ♥tr s ♠♣ts r

♥♥t R ② ♥t s ♠♣ts tr♥s♠t ♥♥t T stá♥ ♦s

♣♦r s ♦♥s ② s ♠♣♦ étr♦ s ♣r♦ ♣♥♦

♣r♦♣ó♥ ♠♦♦ tr♥srs♦ ♠♥ét♦ ② ② s ♠♣♦

étr♦ s ♣r♣♥r ♣♥♦ ♣r♦♣ó♥ ♠♦♦ tr♥srs♦ étr♦

θi s á♥♦ ♥♥ ② θt s á♥♦ tr♥s♠só♥ ❬❪

RTM =v2cos(θi)− v1cos(θt)

v2cos(θi) + v1cos(θt)

TTM =2v1cos(θi)

Pr♠t ❱♦ ♦♥t

rt ♣r♦♣ó♥

ǫr v ♠♥s σ ♠♠

r♥t♦ s♦

r♥ s

r ú♠

♦ r♥♦s♦ s♦

♦ r♥♦s♦ ú♠♦

♦ r♦s♦ s♦

♦ r♦s♦ ú♠♦

♦ ♦♥♦

③ str

strt♦ tr ♦

❱♦rs tí♣♦s ♣r♠t rt ② ♦♥t

rs♦s ♠trs ♦ ♣r♦♣ó♥ s

♣rtr t③♥♦ ♦s ♦rs ǫr ❬❪

r ♥ ♥♥t s♦r ♥ ♥trs ♣♥ ♣r ♦s

s♦s ♥ ♦s q ♠♣♦ étr♦ ~E s ♣r♦ ♣♥♦

♥♥ ♠♦♦ tr♥srs♦ ♠♥ét♦ ② ♣r♣♥r

♣♥♦ ♥♥ ♠♦♦ tr♥srs♦ étr♦

RTE =v1cos(θi)− v2cos(θt)

TTE =2v1cos(θi)

♥♦ ♠♥♦r s ♦♥trst q ♣rs♥t♥ s ♦s ♣r♦♣ó♥

♥ ♥trs sñ r s ♠ás é ② tr♥s♠t ♠ás ♥t♥s

P♦r st r③ó♥ s ♥ ♥ s♣r s♦♥t♥ ♥♦ ② ♦♥trst

s♥t s ♠② ♣♦s q sst♠ ♦rr ♥♦ tt sñ r

② ♥ ♦♥s♥ q ♥♦ s ♦r ♥tr ♥trs ♦rrs♣♦♥♥t

♠♥trs q s ♦♥trst s r♥ sst♠ ♦rr ttr

♦♥ ♥trs ♥q ♣♦s♠♥t ♥♦ ② s♥t ♥t♥s

tr♥s♠t ♦♠♦ ♣r ttr ♦trs ♥trss ♠ás ♣r♦♥s

♠②♦rí ♦s q♣♦s ♦rr s sñ♥ ♣r ♠tr ♣s♦s ♦♥

①t♥s♦♥s t♠♣♦rs ♣qñs ♥ ró♥ ♣r♣♥r s♣r

s♦ ② ♠♦♦ q ♦s ♠s♠♦s ♥♦ s ①t♥♥ ♥ s♣♦ ♠ás á ♥

ró♥ ♦r♠ ó♥ ♦♥ ♥ á♥♦ ♣rtr 90♦ ♣r♦①♠♠♥t

♠ q ♣s♦ s ♣r♦♣ ♥ s♦ ♠s♠♦ ♠♥ árs s

③ ♦♥ ♠♥♦r ♥t♥s ② q ♥s ♥rí ♣♦r ♥

s♣r s♠♥② ♠♥tr s♣r t♦t r♥t ♦♥s q s

♣r♦♣ ♦♠♦ ♦♥s♥ r♥ ♦♠étr ♠♣♦

♥♦ ♥ sñ s r ♥ ♥ ♦t♦ ♣r♦♥♦ rrs r♣t♦r ♦♥

♠♥♦r ♥rí ♣♦r ♥ ár q s s r ♥ ♥ ♦t♦ é♥t♦ ♣r♦

♠ás s♣r ♦t♦ ♥ q s ♣r♦ r①ó♥ s s♥

t♠♥t ♣r♦♥♦ sñ ♣♦rí ♥♦ sr tt ♣♦r sst♠ r♣t♦r

♦ s ♥t♥s ♣♦r ♥ ár ♥♦♠♥ ♣r♦♥

♣♥tró♥ ♠á①♠ ♣r♦♥ q s ♣ ttr ♥ rt♦r

st ♣♥ t♥t♦ r♥ ♦♠étr sñ ♦♠♦ ♦♥trst

♥ s sss s♦♥t♥s ♠♥♦♥♦ ♥tr♦r♠♥t

sñ sr ♠ás ♥ ♣ér ♥rí ♠ q s ♣r♦♣

trés ♠♦ ♠tr s♦ró♥ ♦♠♦ ♦♥s♥

tr♥s♦r♠ó♥ ♣rt s ♥rí tr♦♠♥ét ♥ ♦r tr♠♥

♣r♥♣♠♥t ♣♦r ♦♥t étr ♠♦ ♥♦ ♠②♦r

st♥ r♦rr sñ ♠②♦r s s♠♥ó♥ s ♥t♥s ♦

s♦ró♥ r♦ s♦ró♥ ♠♥t ♣r r♥s ♠②♦rs s

ñ ② ♣♥ s rtrísts ♠♦ ♥ ♣rtr ♠

♠②♦r r♦ ♠ ♠②♦r s♦ró♥

s sñs ♦rr ♥♦r♠♠♥t s tr♥s♠t♥ trés ♠♦s

♦♠♣♦s q ♣rs♥t♥ t♦♥s ♦ tr♦♥s ♥ ss ♣r♦♣s

tr♦♠♥éts sts s♦♥t♥s s♣rs♥ ♥rí ♥r♥♦ s

ñs s♥rs ♥♦ ss ② ♦♥tr②♥♦ t♥ó♥ ② ♦r♠ó♥

sñ t♦t ♦♠♦ s sñs s♥rs ♣r♥ ♥ ♦s rstr♦s s♣r

♣sts s sñs ♥trés ♥ ♠♦s s♦s s rt♠♥t ♠♣♦s

st♥r s sñs ♥trés ♦ s ró♥ ♠♣t ♦♥

♥t♦r♥♦ ♦ s ♦♥t♥

t♦♦♦í ♦rtr s♠♣

♥ s ♣♦♥s ♠ás ♦♠♥s ♦rr s ♥t♥s s trs♥

♥ts s♦r s♣r s♦ ♦ r♦ í♥ s♦♥♦ st ♠

t♦♦♦í s ♠ ♦rtr s♠♣ ② q ♣r ♣♦só♥ s♦r

í♥ s♦♥♦ s qr ♥ ú♥ tr③ ♠♣t r ♥ ♥ó♥

t♠♣♦ ♥ ♠r♦ ♥ ♣♦♥s s♣s s t③♥ ♦♥r♦♥s

♦rtr ♠út♣ ♥ s q ♣r ♣♦só♥ s♦r í♥ s♦♥♦ s

qr♥ rs tr③s ♣r st♥ts ♣♦s♦♥s ♠s♦r ② r♣t♦r ♥

r s ♣ r ♥ ♠♣♦ ó♠♦ s ♦t♥ ♥ ♣r ♦rtr

s♠♣ s ♦♥s ② ♦rrs♣♦♥♥ st ♦♥ró♥ ♦♥

s ♥t♥s t♥♥ ♦r♥tó♥ ♣r♣♥r í♥ s♦♥♦ ♠♦♦

❬❪

r t♦♦♦í ♣r ♦t♥só♥ ♥ ♣r ♦r

tr s♠♣

r s ♥ ♠♣♦ ♥ tr③ ♥ s ♣ r q ♥

♠♥t ♥tr t = 0 ② t = 2,5 ♥s s r ♥ ♣s♦ r♥ ♠♣t q s

♥♦♠♥ sñ rt stá ♦r♠♦ ♣♦r s♣r♣♦só♥ sñ ér

q s ♣r♦♣ ♥ r ♦ ③ ♥ í♦ ♣r♦①♠♠♥

t ② sñ trrstr q s ♣r♦♣ trés strt♦ ♠ás s♣r

s♦ ♦ ♣r♦♣ó♥ ♥ ♠♦ ♥ r s ♠str

♥ sq♠ ♦s ♠♥♦s r♦rr♦s ♣♦r sts sñs sñ rt t♥

♠♣t ♠② s♣r♦r s q ♣rs♥t♥ s sñs q ♣r♦♥♥ r

①♦♥s ♥ ♦s ♦t♦s ♥trr♦s ♣♦r ♦ q t♥ tó♥ ♦t♦s

♠② s♣rs ♥ s♦ r ②s r①♦♥s ♦rr♥ ♣r

t < 2,5 ♥s

♦♥♥t♦ s tr③s qrs ♦ r♦ ♥ í♥ s♦♥♦

r ♠♣♦ ♥ tr③ s ♠str♥ ♦rs ♥♦r♠

③♦s ♠♣t

r ♠♥♦s r♦rr♦s ♣♦r s sñs rt ér ②

trrstr ② ♣♦r sñ r

r ♠♣♦ ♥ rrr♠ ♣r s♦ ♥ q

s♣ró♥ ♥tr s ♥t♥s ♠s♦r ② r♣t♦r s ♥ ♦st

s♦r s♣r s s ♠ ♣r ♦ só♥ rt t♦s ♥♦ s

tr③s ♥ ♦ ♣r s r♣rs♥t♥ ♥ ♥ rá♦ ♦♥ ♦r③♦♥t

r♣rs♥t ♦♦r♥ s♣ ♠ s♦r s♣r s♦

rt r♣rs♥t t♠♣♦ tr♥srr♦ ♣rtr ♠só♥ ♣s♦ ②

♣♦r ♠♦ ♥ s ♦♦rs s r♣rs♥t ♥t♥s sñ

s ♦t♥ ♥ rá♦ ♥♦♠♥♦ rrr♠ ♥ ♠♣♦ s ♣ r ♥

♠t♦♦♦í ♦rtr s♠♣ rqr t♠♣♦s ♦rt♦s qsó♥

② ♣r♦s♠♥t♦ ② ♥ ♠♦s s♦s ♣r♠t ♦t♥r ♥♦r♠ó♥ s♥t

ss♦ ♥ ♠r♦ ①st♥ s♦s í tó♥ ♦♠♦ ♣♦r ♠♣♦

♥♦ ♠tr q ♦♠♣♦♥ ♠tr③ ss♦ ♣rs♥t ♥ t♥ó♥

♠② ♥♦ ró♥ sñ r♦ s ♠s♦ t ♦ ♥♦

s strtrs tts rqr♥ ♥ ♠♦r rs♦ó♥ ♥ s♦s s♦s s

♦♥♥♥t t③r ♥ ♠t♦♦♦í ♦rtr ♠út♣ ♦♠♦ s rá ♠ás

ñs ♣r♦s ♣♦r st♥t♦s t♣♦s

rt♦rs

♥♦ s ♦♥sr♥ st♥t♦s t♣♦s ♥♦s ♥trr♦s s ♦t♥♥

sñs ♦♥ rtrísts ♣rtrs q ♣♥♥ ♦♠trí ② ♣r♦

♣s ♥♦ ② s ♣r♦♣s ♠♦ ♥ q s ♥♥tr

♥ ♥r ♥♦ s♠♣r s ♦sr ♥ ♥t ♥tr ♦r♠ ♥♦ ②

s ♣♦s♦♥s ♦s t♠♣♦s ♥ rrr♠ ♦♠♦ s rá

♦♥t♥ó♥

❯t③♥♦ s ②s ó♣t ♦♠étr s ♣♦s ♦rr ♥ só♥ s♠♣

rst♦ ♥tró♥ sñ ♦♥ st♥t♦s ♥♦s ♥ s♦

♥ ♥trs ♣♥ r t♠♣♦ s ♣ ①♣rsr ♦♠♦

tv =2z0 cos(α)

v− 2 sen(α)

♦♥ z0 s ♣r♦♥ q ♥trs ♦rt z α s á♥♦ q

♦r♠ ♥trs ♦♥ ró♥ ♦r③♦♥t v s ♦ ♣r♦♣ó♥

♥ ♠♦ q s ♥♥tr rr ♥trs P♦r s♠♣ s ♦♥sr

♥ ♦♥ró♥ ♥ q s ♣♦s♦♥s s ♥t♥s ♠s♦r ② r♣t♦r

♦♥♥ ♥ xpm

♦♠♦ s ♣ r ♥ r sñ q s ♦t♥ ♥♦

♦♥♥t♦ ♠s♦r r♣t♦r s ♥♥tr ♥ ♥ tr♠♥♦ ♣♥t♦ s♦r

s♣r ♥♦ ♦rrs♣♦♥ r①ó♥ ♥ ♣♥t♦ ♣♥♦ q stá ♦

♦♥♥t♦ ♠s♦r r♣t♦r ♦♥strr ♥ rrr♠ ♦♥ s ♠♣ts ♥

♥ó♥ t♠♣♦ s ♦t♥ ♥ r♣rs♥tó♥ q s ♥ r s♠t

♦♥ ♦r♠ s♣r rt♦r stá st♦rs♦♥ ♣s s♠♥t♦

♥ r s ♦rrs♣♦♥ ♦♥ s♠♥t♦ ♥ ② ♠♦s

stá♥ s♣③♦s ♥tr sí sñ ♦t♥ stá s♣③ ♦♥ rs♣t♦

rt♦r q ♦r♥ ② ♣rs♥t ♥ ♥♥ó♥ st♥t ♥♥ó♥

rt♦r s α ♥ rrr♠ s ♦sr ♥ sñ ♦♥ ♥♥ó♥ αp

♣♦r ó♥

tan(αp) =2 sen(α)

♦s t♦s s♣③♠♥t♦ ♥♥ó♥ s♦♥ ♠ás ♥♦t♦r♦s ♣r ♥♥♦♥s

r♥s P♦r ♦tr♦ ♦ s ♥♥ó♥ rt♦r s ♣qñ ♦s t♠♣♦s

tv rstr♦s ♣♦r r♣t♦r ♣r ♦s ♣s♦s r♦s s ♥♥

♦♥ s ♣r♦♥s p ♦s rt♦rs sú♥ ①♣rsó♥

p =tvv

t♦r 12s q t♠♣♦ q sñ tr ♥ r♦rrr st♥

♥tr sst♠ ♦rr ② rt♦r s ♠t t♠♣♦ t♦t

r ①ó♥ ♥ ♥ ♥trs ♣♥ ♦♥ ♥♥ó♥

α ② rrr♠ rtríst♦

ó♥ ♣r♠t ♦t♥r ♥ ♣r♠r st♠ó♥ ♣r♦♥

♦s rt♦rs ♣r ♣♦só♥ s♦r s♣r ♦♥♦ ♦

♦♥ ♠s♠♦ ♣r♦♠♥t♦ q ♣r ♥ ♥trs ♣♥ s ♣♦s s

t♠r ♦s t♠♣♦s ♣r ♥ ♦t♦ ♣qñ♦ ♥ r s

♠str ♥ sq♠ ♥ ♦t♦ ♥trr♦ ♥ ♥ ♠♦ rtr③♦ ♣♦r

♥ ♦ ♣r♦♣ó♥ v t♠♣♦ s ♣ ①♣rsr ♣♦r

♠♦ s♥t ó♥

t2v(2z0/v)2

− (x0 − xpm)2

z20= 1

♦♥ z0 ② x0 s♦♥ ♣r♦♥ ② ♣♦só♥ ♦t♦ rs♣t♠♥t

xpm s ♣♦só♥ ♦♥♥t♦ ♥t♥s ♠s♦r r♣t♦r ♥ r

s ♠str ♥ ♠♣♦ tí♣♦ sñ q ♥r ♥ ♦t♦ ♣qñ♦ ést

♣rs♥t ♥ ♦r♠ q ♥♦ r s♠t ♦♥ ♦t♦ ♥ r s

♣ ♦srr q sñ s tt ♥ ♣♦s♦♥s ♠② s ♥♦

st♦ s ♦♥s♥ q ♦s ♠♣♦s ♠t♦s ♣♦r ♥t♥ t♥♥ ♥

♣rtr ♥r r♥ ♦ q ♣r♠t ♠♥r tr♠♥t ♦t♦

♥t♥s sñ s♠♥② ♠♥tr st♥ ♥tr ♦♥♥t♦

♠s♦r r♣t♦r ② ♥♦ ♦♠♦ ♦♥s♥ ♦s st♥t♦s ♥ó♠♥♦s

♥♠r♦s ♥ só♥

t♠♥t s ♠ rt♦rs ♦s ♦t♦s q ♣r♦♥ rs♣sts

♦♠♦ q s ♠str ♥ r ♠♥trs q ♦s ♦t♦s q

r rt♦r ♥trr♦ ♥ ♥ ♠♦ ♥♦r♠ ②

rrr♠ rtríst♦

r t♦ ♥trr♦ q ♣rs♥t ♦rs ② r

rr♠ rtríst♦ s s sñ♥ ♦s érts s

♣ér♦s

♣r♦♥ rs♣sts t♣♦ q s ♠str ♥ r s ♦s ♠

rt♦rs

♥ ♣rát á♥♦ s ♣rs♥t♥ ♥ rrr♠ sñs rt♦rs

s t r③r ♥ st♠ó♥ ♦ ♣r♦♣ó♥ sñ

♥ ♠♦ t③♥♦ ó♥ Pr ♦ ♦♥♦♦s ♦s ♦rs

tv ② xpm s s♣r♣♦♥ rrr♠ ♣ér♦ ♦t♥ ♣rtr

ó♥ ② s ♥ ♦s ♦rs ♣♦só♥ ♥♦ x0 ② z0 ②

♦ ♣r♦♣ó♥ v ♠♥r ♦t♥r ♠♦r st ♣♦s

♥tr ♣ér♦ ② sñ

② q ♦♥srr q s ♥ ♥ ♣rát s trt ♦♥ ♣r♦♠s ♠ás

♦♠♣♦s q ♦s ♦♥sr♦s ♥ ♦s ♣árr♦s ♥tr♦rs s sñs ♦♥ ♦r

♠ ❱ ♥rt s♦♥ rtríst ♦t♦s ♣qñ♦s ♦♠♣r♦s ♦♥

♦♥t ♦♥ sñ st t♣♦ sñs t♠é♥ s ♦sr ♥ ♦s

s♦s q ♥♦ ♣rs♥t ♦rs ♦ s♥s r♣t♦s ♦♠♦ s ♠str ♥

r ♦♥ s ♥r♥ sñs t♣♦ ♣ér♦s ró♥ ♥

♥ s sq♥s ♦t♦ rt♥r s s sñ♥ ♦s érts

s ♣ér♦s

♠♣♦ s♦♥♦ ♦rtr s♠♣

♥ st só♥ s ♣rs♥t ♥ ♠♣♦ ás♦ ♥tr♣rtó♥ t♦s

♦rtr s♠♣ s ♠♦♥s s r③r♦♥ ♥ ♥ ③♦♥ ♥ q s s♣♦♥í

①st♥ ñ♦s P❱ q tr♥s♣♦rt♥ ♦♥ ♣r♦♣óst♦

♦③r♦s s♣sr♦♥ í♥s s♦♥♦ ♥ ró♥ ♣r♣♥r

s♣st ♣r ♦s ñ♦s ② s qrr♦♥ t♦s ♦ r♦ s í♥s ♦♥

s♣ró♥ ♦♥st♥t 0,25 ♠ ♥tr s ♥t♥s ♠s♦r ② r♣t♦r ♦st

♦♥st♥t t③ó sst♠ rr ♥s♦rs ♦tr Ps

P ② ♥t♥s ♦♥ r♥ ♥tr ♠só♥ ③

♥ r s ♠str ♥ rrr♠ ♦t♥♦ ♣rtr ♦s t♦s

♥ s í♥s s♦♥♦ t r♦♥ rstr♦s ♣♦r sst♠ ♦r

r ♦♠♦ s ♣ ♦srr ♥ r s ♣♦s ♥tr sñ rt

♣r ♦♠♦ ♥s ♦r③♦♥ts ♥ ♣rt s♣r♦r rrr♠

♥tr t = 3 ♥s ② t = 12 ♥s sñ ♦t♦ ♥trr♦ s ♣ ♦srr ♦♥

t ♦♥ ért ♥ (x; t) = (−0,1 ♠; 16 ♥s) ♣r♦①♠♠♥t s ♥

r rr♠ ♦t♥♦ ♦♥ ♥ ♦♥ró♥ ♦

rtr s♠♣ ♦♥ ♦st 0,25 ♠ ♥ts ♣r s st♥ts

té♥s ♣r♦s♠♥t♦ ♥ ó♥ ért

sñ ró♥ ♦r♥ ♥ ♦t♦ ♥trr♦

♦♥ ♥ ♥ r

Pr ♦srr ♦♥ r sñ ♦t♦ ♥trr♦ s ♥sr♦ ♣r

st♥ts té♥s ♣r♦s♠♥t♦ ♦s t♦s ♠♥r ♠♦rr

♠♥ ésts té♥s s sr♥ ♥ só♥ ♥ r s

♠str rrr♠ ♦t♥♦ ♦ ♣r ♦ ♦rró♥ ♦r♥

t♠♣♦ ♠♥ó♥ sñ rt ② ♥♥ ♦s t♦s r

♣ ♦srr ♥ sñ tí♣ ró♥ ♥r ♣♦r ♦t♦ ♦♥

ért ♥ (x; t) = (−0,1 ♠; 12 ♥s) ♣r♦①♠♠♥t ♥ ③♦♥ ♥tr t = 0

♥s ② t = 10 ♥s s ♣♦s ♦srr ♥ ♥t sñs s♣r♣sts ②

q s trt ♥ s♦ r♥♦ ♠② ♥tr♥♦ ♥ q s ♣ ♥♦♥trr

♥ r♥ ♥t ♦t♦s ♣qñ♦s

Pr st♠r ♣r♦♥ ñ♦ ② ♦ ♣r♦♣ó♥ ♥

♠♦ s s♣r♣♦♥ ♥ rrr♠ rá♦ ♥ ♣ér♦ r

② s ♥ ss ♣rá♠tr♦s ♠♥r ♦t♥r ♠♦r st ♣♦s

r rr♠ ♦t♥♦ ♦♥ ♥ ♦♥ró♥ ♦

rtr s♠♣ ♦♥ ♦st 0,25 ♠ s st ♥ ♣ér♦

sñ ♣r♦ ♣♦r ♦t♦ ♥trr♦

r ♦t♦ ③♦♥ s♦♥♦ ♦ ①r ♣r

①♣♦♥r ♦t♦ ♥trr♦

♥tr r ② sñ ♦♠♦ s ①♣ó ♥ só♥ ♦♠♦ rst♦ s

♦t♥ ♥ ♣r♦♥ (0,38±0,12) ♠ ② ♥ ♦ ♣r♦♣ó♥

(0,07± 0,01) ♠♥s st ♦r ♦ s rtríst♦ s♦s ♦♥ t♦

r♦ ♠ r t só♥ ♦♠♦ s ♣ ♦srr

♥ r ♣r♦♥ ♣♥tró♥ s ♠t ② q ♥♦ s ♦t♥♥

sñs ♥t♥s ♣r ♣r t♠♣♦s ♠②♦rs q 16 ♥s

♥♠♥t s ♣ r ♥ r ♦t♦ ♦ sr ①♣st♦

st s ♥♦♥tr ♥ ♣r♦♥ 0,35 ♠ t♠é♥ s ♣♦s ♦srr

♣rs♥ ♥ r♥ ♥t ♦t♦s ♣qñ♦s ♥trr♦s t ♦♠♦

r ♣r♦ ♣rtr ♥áss ♦s t♦s ♦rr

Pr♦s♠♥t♦ ás♦ ♣r t♦s ♦r

tr s♠♣

♦♠♦ s ♦sr ♥ r só♥ ♥tr♦r s sñs ♦s

♦t♦s ♥trr♦s ♥♦ s ♣♥ st♥r ♦♥ r ♥ ♦s t♦s qr♦s

♦ st♥t♦s t♦rs ♦♠♦ ♣rs♥ sñs s♥rs ② ♣ér

♥t♥s q sr sñ ♣r♦♣rs ♥ ♠♦ P♦r st r③ó♥ ♦

qrr ♦s t♦s ② ♦r♥r♦s ♥ ♥ rrr♠ s ♥sr♦ ♣r rt♦s

♠ét♦♦s ♣r♦s♠♥t♦ ♦♥ ♣r♦♣óst♦ ♠♦rr s sñs ♥trés ②

s♠♥r ♦ ♠♥r s sñs s♥rs ♣r sí ♣♦r ♦t♥r ♥♦r♠ó♥

r♥t ♥tr♣rtr ♦rrt♠♥t ♦s t♦s t♣♦ ♣r♦s♠♥t♦

q s r ♣♥rá ♣♦r ♠♣♦ ♦s t♦s ♦r♥s ②

♦s rqr♠♥t♦s ♣r ♥tr♣rtó♥

♦♠♦ ♠♣♦ s rt♦♠♥ ♦s t♦s sr♣t♦s ♥ só♥ ♥ r

s r♣t r q ♠str ♥ rrr♠ ♦t♥♦ ♣rtr

♦s t♦s t ♦♠♦ s♦♥ rstr♦s ♣♦r sst♠ ♦rr ♥ st r

♦ ú♥♦ q s ♦sr ♦♥ r s sñ rt ♣r ♦♠♦

♥s ♦r③♦♥ts ♦ r♦ t♦♦ rrr♠ ♥ r s

♠str rá♦ tr③ q ♦rrs♣♦♥ x = 0 ♠ ♥ sñ

rt ♦♣ ♥tr♦ t = (3, 12) ♥s ♣r♦①♠♠♥t

t♠♥t ♥ ♠t♦♦♦í ♦rr ♠s♦r ② r♣t♦r s

♥ ♦rt st♥ r ♠s♦r ♦s ♠♣♦s t♥♥ ♦♠♣♦♥♥ts

r♥ s♦s ♦♥ ♠♣♦s tr♦stát♦s ♥t♦s q s

♦sr ♥ ♦s t♦s ♦♠♦ ♥ ró♥ t♠♣♦r ♥t ♥ ♠♦

sñ st t♦ s ♦♥♦ ♦♠♦ ♦ ② s ♣ r ♥ r ♦♠♦

♥ s♣③♠♥t♦ ♥ ♠♦ sñ ♣ s♣r♠r rst♥♦

ss♦s s♠♥t♦s tr③ ss ♦rs ♠♦s ♦ ♣♥♦ ♥ tr♦

t♠♣♦r ♣s t♦s sñ ♣r♦s♦ q s ♦♥♦ ♦♠♦ ♦ ❬❪

rst♦ ♣r st ♣r♦♠♥t♦ ♦s t♦s r s

r t♦s ♦t♥♦s ♥ts ♣r s st♥ts té

♥s ♣r♦s♠♥t♦ rrr♠ ② tr③ ♦rrs♣♦♥♥t

x = 0 ♠

r ♣ ♦ ♦s t♦s ♦t♥♦s rrr

♠ ② tr③ ♦rrs♣♦♥♥t x = 0 ♠

♠str ♥ r ♦♠♣r♥♦ s rs ② s ♣ r

♦rró♥ q s ♦r ♥ ♥ ♠♦ sñ

s♥t ♣s♦ ♦♥sst ♥ ♥r ♦r♥ t♠♣♦ t♦s s tr③s

② r♦ ♥ t = 0 ♥s ♦r♥ t♠♣♦ ♣ rr ♥ tr③ ♦tr

♣♦r rss r③♦♥s ♦♠♦ ♣♦r ♠♣♦ t♦♥s ♥ s♣r♦ ♣s♦

♠t♦♦♦í tí♣ ♦♥sst ♥ sr ♣r♠r ♣♦ ♠á①♠♦ ♦ ♠í♥♠♦

♣r♠r tr③ rrr♠ st s ♦r♥ ♥ sñ

rt ♥t♥ r♣t♦r ♦ s s♣③♥ t♦s s tr③s ♥ ♦r♠

♦♥♥t ♠♥r q ♣♦ s♦♥♦ ♥ ♣r♠r tr③ ♦♥ ♦♥

t = 0 ♥s r rs ② ♥ ♥s st♦♥s ♣ ♣rs♥trs

♥ s♣③♠♥t♦ ♦r♥ t♠♣♦s q rí ♦ r♦ rrr♠s

♥ s♦s s♦s ♥ ♥sr♦ ♣r ♣r♦♠♥t♦ sr♣t♦ ♥

s tr③s ♣♦r s♣r♦

♥ ♦s rrr♠s s♥ ♣r♦sr ♥♦r♠♠♥t s ♣rs♥t♥ ♥s ♦r

r r③ ♦rró♥ ♦r♥ t♠♣♦s

rrr♠ ② tr③ ♦rrs♣♦♥♥t x = 0 ♠

③♦♥ts t ♥t♥s q t♠♥t ♥♦ ♣r♠t♥ r s sñs

♥trés ésts s ♣♥ ♦srr ♥ r ♥tr t = −5 ♥s ② t = 5

♥s ♣r♦①♠♠♥t sts sñs s♦♥ ♦♥s♥ ♣r♥♣♠♥t

r♣ó♥ sñ rt ♣r♦♥♥t ♠s♦r ♣r♦ t♠é♥ s ♣♥

♦r♥r ♥ r①♦♥s ♣r♦s ♥ ♦t♦s ♦ ♣rs♦♥s s ♣♦r ♥

♠ s♦ ♣♦r ♠♣♦ ♥ t♦ ♦ ♦♣r♦r sst♠ ♦rr

t♠♥t ♣r r♠♦r sts ♥s ♦r③♦♥ts s ♦t♥ ♥ tr③

♣r♦♠♦ ♣rtr t♦s s tr③s q ♦♠♣♦♥♥ rrr

♠ ② s rst ♥ s tr③s rrr♠ st ♣r♦♠♥t♦

♠♥ t♦s s sñs ♦r③♦♥ts ♦ s ♦r③♦♥ts s r ♦s ♥t♦s

q stá♥ ♣rs♥ts ♥ ♥ r♥ ♥t tr③s ♥②♥♦ s sñs

rt♦rs ♥trr♦s ❯♥ tr♥t ♦♥sst ♥ r tr③ ♣r♦♠♦

♦♥sr♥♦ só♦ ♥ ♥t♥ ♥tr♦ rrr♠ ♦♥ r♥♦s ♣♦só♥

② t♠♣♦s ♣r♦♣♦s ♥ ♠♣♦ s r♠ sñ rt ♥tr♦ ♥

♥t♥ ♥ s t = −5 ♥s st t = 5 ♥s q s ①t♥ s x = −2

♠ st x = 2 ♠ st♦ ♣r♠t ♦♥srr s sñs s ♦r③♦♥ts q s

♣r♥ ♣rs♥tr ♥ t♠♣♦s ♠②♦rs t = 5 ♥s rst♦ s ♣ r ♥

s sñs ♣r♦♥♥ts ♥♦s ♣r♦♥♦s t♥♥ ♥ ♠♣t ♠♦

♠♥♦r q s q ♣r♦♥♥ ♥♦s s♣rs ② q ésts s t♥ú♥

rá♣♠♥t ♣r♦♣rs ♥ ss♦ r só♥ Pr s③r

s♠tá♥♠♥t s sñs ♣r♦♥♥ts st♥ts ♣r♦♥s s ♥

sr♦ r s ♠♣ts ♣♥♦ ♥ ♥ó♥ ♥♥ r♥t ♥

t♠♣♦ ♣r♦s♦ ♦♥sst ♥ ♠♥tr st♠♥t ♥t♥s ♦s

t♦s ♠♥r t q ♣r ♦s st♥t♦s t♠♣♦s ♥t♥s

r ♠♥ sñ rt rrr♠ ② tr③

♦rrs♣♦♥♥t x = 0 ♠

r st♦ ♣r ♥♥ rrr♠ ②

tr③ ♦rrs♣♦♥♥t x = 0 ♠

s s♠r Pr st♦ s ♣ ♥ ♥ó♥ ♠♣ó♥ ♣rtr♠♥

♥ ♥ t♦ ♥t♥ t♠♣♦r ♦♠♦ ♣ó♥ ♥♥ s ♥

♣r♦s♦ ♥♦ ♥ ♦tr♦s ♣r♦s♦s tr♦ ♥♦ ♣r♦♥ ♠s♠♦ rst♦

sr ♣♦s ♥ts ♦ s♣és ♣r ♥♥ rrr♠ rst♥

t ♣r ♥♥ ♦s t♦s r s ♠str ♥ r

sñ ró♥ ♦t♦ ♥trr♦ s ♣ r ♦♥ ért ♥

(x; t) = (−0,1 ♠; 12 ♥s

♥ ♥♦s s♦s s ♣♥ té♥s ♠ás ♦♠♣s ♣r♦s♠♥t♦ ♦♠♦

♣♦r ♠♣♦ st♥t♦s t♣♦s tr♦s s♣s ② t♠♣♦rs ♦♥♦ó♥

② ♦rró♥ t♦♣♦rí ♥ ♦s s♦s ♥ ♦s q sñ ♣rs♥t

ró♥ sñ r♦ s ♥sr♦ ♣r ♠t♦♦♦ís ♣r ♠♦rr ♠♥

♦♠♦ s rá ♠ás ♥t s♦♥s ② ② ♣ít♦

t♦♦♦í ♠ró♥ sñs

s sñs ♦r♥s ♥ ♦t♦s ♥trr♦s ♣r♥ ♥ rrr♠

♦♥ st♥t ♦r♠ ♣♦só♥ ♥♥ó♥ q ♦s ♦t♦s rs só♥

♦ q t ♥tr♣rtó♥ ♣r♥♣♠♥t ♥ s♦s q ♣rs♥t♥ r♥

s ♥ts sñs s♣r♣sts ♥tr sí ♦ ♣♥♥ts ♣r♦♥♥s

♥tr♣rtó♥ s ♣ tr s s ♦rr rrr♠ ♣r ♦t♥r ♥

♠♥ ♥♦ ♦s rt♦rs ♦rrt♠♥t ♦s ♥ s♣♦ st

♣r♦s♦ s ♥♦♠♥ ♠ró♥ ② ♦♠♣r♥ ♥♦q ② ♣♦s♦♥♠♥t♦

s sñs r①ó♥ ♥♦q ♦♣s s r♦♥s ♠①♠③♥♦

♠♣t ♠♥trs q ♣♦s♦♥♠♥t♦ ♦③ ♦s ♥t♦s ♦rrt♠♥t ♦

q rst r♥t ♥♦ ①st♥ rt♦rs ♦♥ ♣♥♥ts ♣r♦♥♥s ②

r♦♥s trs s♥ts ♥ ♦ ♣r♦♣ó♥ ♦♠♦ rs

t♦ s ♦t♥ ♥ ♠♥ ♠ás rst ♦s ♦t♦s ♥trr♦s ❬❪ ①st♥

rss ♠t♦♦♦ís ♠ró♥ ts ♦♠♦ ♠ró♥ r♦ s♠

♣ró ♠ró♥ ♥rs ♥ t♠♣♦ rrs t♠ ♠rt♦♥ t

❬❪

Pr ♦t♥r s♦ó♥ ♥♠ér ♠♥t ♠ró♥ s s ♦♥srr

♥ ♠♦♦ q ♦♥sst ♥ r♠♣③r rt♦r ♣♦r ♥ sr ♥ts

rts s q ♥ t = 0 ♠t♥ sñ q s ♣r♦♣ st s♣r

♦♥ s ♥ ♦s r♣t♦rs ❬❪ s s♠ ♥ ♦♥ró♥ ♥ s

♣♦s♦♥s s ♥t♥s ♠s♦r ② r♣t♦r ♦♥♥ ♥rí rstr

♥ ♣♥t♦ s♦r s♣r ♣r♦♥ ♠♥♦s ♠s♦r rt♦r ②

rt♦r r♣t♦r q ♦♥♥ ♥ s s♦ t♠♣♦ t♦t

sñ s ♦s s t♠♣♦ ♥tr ♠s♦r ② rt♦r P♦r

st r③ó♥ ♥ st ♠♦♦ ♦♥srrs ♥ ♦ q♥t q

♦rrs♣♦♥ ♠t ♦ r ♣r♦♣ó♥ ❬❪

♣rtr st ♠♦♦ s ♣ r q ♠t♦♦♦í ♠ró♥

♥rs ♥ t♠♣♦ ♦♥sst ♥ r♣rr ♠♣♦ t = 0 ♣rtr ♦s

t♦s qr♦s ♥ s♣r st ♠♥r s ♣♦s ♦t♥r ♥ ♦r♠

♥rt ♦③ó♥ s ♥ts rts s♦s ♦♥ ♦s rt♦rs

q ②♥ ♣r♦♦ ♦s ♠♣♦s Pr ①tr♣♦r ♦s t♦s trás ♥

t♠♣♦ s ♣ t③r ó♥ ♦♥s sr ♥ ♦s ♠♥s♦♥s

❬❪∂2E

∂x2+

∂z2=

v2(x, z)

♦♥ E s ♠♣♦ sr q s ♣r♦♣ x s ♦♦r♥ s♣

♦r③♦♥t z s ♦♦r♥ s♣ rt v(x, z) s ♦ ♣r♦

♣ó♥ ② t s t♠♣♦ ♥ st ó♥ ♥♦ s ♥ s♣♦s♦♥s s♦r

♣r♦♠ ís♦ q s rs ♣♦r st r③ó♥ s ♣ qr ♦♥

q s ♣r♦♣ q s ♣ ♣r♦①♠r ♣♦r ♥ rs♣st sr ♦♥ ♥

stró♥ ♦ ♦♥ ró♥ s♣ v(x, z)

❯♥ ♣♦s ♣r rs♦r ♥♠ér♠♥t ó♥ s ♣

r ♥ r♣rs♥tó♥ r♥s ♥ts s♥♦ ♦r♥ ♥tr P♦r

♠♣♦ ♣r♠r tér♠♥♦ ó♥ s r♠♣③ ♣♦r

∂2E(x, z, t)

∂x2≈ E(xk+1, zj, ti)− 2E(xk, zj, ti) + E(xk−1, zj, ti)

♦♥ h s ♥r♠♥t♦ s♣ r rs♣st E(xk, zj, ti) ♥

♣♥t♦ r (xk, zj) t♠♣♦ ti stá ♣♦r ❬❪

E(xk, zj, ti) = 2(1− 2A2)E(xk, zj, ti−1)− E(xk, zj, ti−2)+

A2[E(xk+1, zj, ti−1) + E(xk−1, zj, ti−1)+

E(xk, zj+1, ti−1) + E(xk, zj−1, ti−1)]

♦♥ v(x, z)∆t/h ∆t s ♥r♠♥t♦ t♠♣♦r

♥ts ♣r ♠ét♦♦ ♠ró♥ s r③♥ st♥t♦s ♣r♦s♠♥

t♦s ♠♥ ♦♠♦ r♠♦ó♥ ♦♥♦ ♦♠♣♦♥♥t r♥

② ♣ó♥ tr♦s s♣ ② t♠♣♦r st♦s ♣r♦♠♥t♦s st♦r

s♦♥♥ ♠♣t ♦ r♦ sñ ♣r♦ ♣r♠t♥ ♦t♥r rst♦s

♦rs ♣r ♦s ♥♦s ♠ás ♣r♦♥♦s

Pr ♠♣♠♥tr ♠ét♦♦ ♠ró♥ s srr♦ó ♥ ó♦ ♦♠♣

t♦♥ st s ♠♣♠♥tó ♥ ♥ ♠♥r tr s

♦♠♣t ♦♥ ♦tr♦s ♣r♦r♠s tr♦ ♦ s③ó♥ t♠é♥ s

rr♦♦s ♥ s ♥t♦r♥♦

♦♠♦ ♠♣♦ ♣ó♥ ♠ét♦♦ ♠ró♥ s ♦♥sr s♦

♥ ♦t♦ á♠tr♦ 0,05 ♠ q s ♦③ ♥ ♣r♦♥ 0,80 ♠

r ♦t♦ s rtr③ ♣♦r ♥ ♣r♠t rt ǫr = 5,25

② ♥ ♦♥t σ = 2,5 ♠♠ ② ♠♦ r♥♥t ♣♦r ǫr = 3,5 ②

σ = 1 ♠♠ ♠♦ s♦r ♦s s r r♥ ♥tr s ♦♥s

r ♣ó♥ ♠ét♦♦ ♠ró♥ ♠♦♦ ♦♥

♥ rt♦r rrr♠ ♥r♦ só♥ ♠r

♠ts s fc = 500 ③ ♦♥t ♦♥ λ = 0,32 ♠ ♥tr♦ s♦ ♦♥

st♦s ♣rá♠tr♦s s r③ó ♥ s♠ó♥ ♥♠ér rs♣st s♣r

♣r ♥ s♦♥♦ ♦♥ ♦st r♦ Pr ♦ s t③ó ♠ét♦♦ r♥s

♥ts ♥ ♦♠♥♦ t♠♣♦ rs♦r♦♥ ♥♠ér♠♥t s

♦♥s ① ♣r ♦t♥r ♠♣♦ Ey s♦r ♥ r ♥♦r♠

② s r♦♥ s ♦♥♦♥s ♦♥t♦r♥♦ ♠♥r tr r①♦♥s ♥

♦r r ♥ ♣é♥ s ♥ ♠ás ts ♣r♦♠♥t♦

♣r ♦t♥r ♦s t♦s s♠♦s

♥ r s ♠str ♣r ♦rtr s♠♣ ♦t♥♦ ♣r

♠♦♦ r sñ r①ó♥ rt♦r s ♣ ♦srr

♦♠♦ ♥ ♣ér♦ ♦♥ ért ♥ 11 ♥s ♣r♦①♠♠♥t ♦ s ♠♥ó

sñ rt ♦s t♦s ② s ♣ó ♠ét♦♦ ♠ró♥ t③♥♦

♥ ♦ ♦♥st♥t v = 0,16 ♠♥s ♥ r s ♣ r ♦♠♦

rst♦ q ♥rí str ♥♠♥t ♥ ♣ér♦ ró♥

♦♣s♦ ♥ ♣♦só♥ rt♦r

t♦♦♦ís ♦rtr ♠út♣ ♦♠

♠♦♥ ♣♦♥t

♥ ♥♦s s♦s t③♥♦ ♠t♦♦♦í ♦rtr s♠♣ s ♦t♥♥

sñs ♠s♦ és ② q t♥♥ sr ♦♥ss r♥t ♥tr♣rt

ó♥ ♦♠♦ ♦♥s♥ ♦ ♦♥trst q ♣rs♥t♥ s s♦♥t♥s

♥trés ♦♥ rs♣t♦ ♠♦ r♥♥t Pr ♦rr ♥ ♠♦r ♥t

ó♥ ② rtr③ó♥ st t♣♦ sñs s t t③r ♠t♦♦♦ís

♦rtr ♠út♣ ♥ s q ♣r ♣♦só♥ s♦r í♥ s♦♥♦

s qr♥ rs tr③s ♣r st♥ts ♣♦s♦♥s rts ♠s♦r ②

r♣t♦r ♠t♦♦♦í ♠ás ♥ s ♣♥t♦ ♠♦ ♦♠ú♥ ♦♠♠♦♥

♠♣♦♥t P st ♠t♦♦♦í s t③ t♥t♦ ♣r st♠r ♦

♣r♦♣ó♥ ♥ ♥ó♥ ♣r♦♥ ♦♠♦ ♣r ♦t♥r s♦♥s r

ts ♠♦rs ③♦♥ st♦

♥ ♠ét♦♦ P s t③ ♥ ♥t♥ ♠s♦r ② ♥ ♥t♥ r♣t♦r

s ♥t♥s s ♦♦♥ ♠♦s ♦s ♥ ♣♥t♦ s♦r í♥ s♦♥♦ ♦♥

♥ s♣ró♥ tr♠♥ ② s qr ♥ tr③ ♦ ♠s ♥t♥s s

♥ s♠étr♠♥t ♣♥t♦ ♠♦ ♠♥t♥♥♦ ♣♦só♥ st

♣♥t♦ ② s qr ♥ s♥ tr③ ♣r♦♠♥t♦ s r♣t st q

s qr ♥ ♦♥♥t♦ tr③s s♦s ♦♥ ♠s♠♦ ♣♥t♦ ♠♦ r

r st ♣r♦s♦ s r③ ♣r ♥♦ ♦s ♣♥t♦s í♥

s♦♥♦ ♦ qsó♥ ♦s t♦s ♥ sr ♣r♦s♦s ♠♥r

♣r♦♣ ♣r ♦t♥r ♥♠♥t ♥ ♠♦r ♥ s sñs ♠ét♦♦ P

♠♣ ♦s t♣s ♣r♦s♠♥t♦ ♥ ♥ ♣r♠r t♣ s ♣♥ rts

♦rr♦♥s ♥ s tr③s ♣r ♦♠♣♥sr s r♥s t♠♣♦s

♣r♦♣ó♥ ♦ s st♥ts st♥s ♥tr ♥t♥s ② ♥ s♥

t♣ s ♣r♦ rr♣ó♥ ♦ ♣♠♥t♦ s tr③s ♦t♥♥♦

♥ tr③ rst♥t ♣r ♣♥t♦ ♠♦

r♥t rr♣♠♥t♦ ♦ s♠ tr③s s sñs ♥trés s r

st♥ ♣♦r ♥trr♥ ♦♥strt ② s r r♦ ♦♠♦ ♦♥s♥

♥trr♥ strt ♠ ♥ú♠r♦ ♣♠♥t♦ ♥t

s♣r♦♥s ♠s♦r r♣t♦r q s ♦♥sr♥ ♥ s♠

t♠♣♦ ♥ q s r ♥ sñ ♦ rrs ♥ ♥ s♣r

s♦♥t♥ ♣♥ ♦ ♣r♦♣ó♥ ♥ ♠♦ ②

r ❯ó♥ s ♥t♥s ♠s♦r ② r♣t♦r

♥ ♠ét♦♦ P

st♥ r♦rr ♥ ♠♥♦ ♠s♦r rt♦r r♣t♦r Pr s♣r♦♥s

♠②♦rs ♥tr ♠s♦r ② r♣t♦r ♦s ♠♥♦s r♦rr♦s ♣♦r sñ s♦♥

♠②♦rs ② ♥ ♦♥s♥ t♠é♥ s♦♥ ♠②♦rs ♦s t♠♣♦s s

♥sr♦ ♦rrr st r♥ ♥ts s rr♣r s tr③s ♠♥r q

s♠ s ♠s♠s ♣r♦③ ♥ sñ ♠②♦r ♠♣t q q

s ♦sr ♥ ♥ s tr③s ♣♦r s♣r♦ st ♦rró♥ s ♦♥♦

♦♥ ♥♦♠r ♥♦r♠ ♠♦♦t ♥ r s ♣♥ r

♦s t♠♣♦s tv sñ r ♥ ♥ ♥trs ♦r③♦♥t

♥ ♥ó♥ s♣ró♥ ♠s♦r r♣t♦r x ♥ s♦ r①♦♥s

♣r♦s ♣♦r ♥ ♥trs ♣♥ ♦r③♦♥t ♦③ ♦ ♥ ♠♦

♥ ♦ ♣r♦♣ó♥ s ♥♦r♠ ♦s t♠♣♦s s

♣ ①♣rsr ♣♦r ♠♦ s♥t ó♥

t(x) =

t20 −x2

♦♥ t0 r♣rs♥t t♠♣♦ sñ ♥ s♦ ♥ q s ♣♦s♦♥s

s ♥t♥s ♠s♦r ② r♣t♦r ♦♥♥ ♦st r♦ ② vs s ♦

♣♠♥t♦ st ♦ s q ♣r♠t ♦t♥r ♠♦r rst♦ ♣♦s

rr♣r s tr③s ♠s♠ ♣♥ ♦ ♣r♦♣ó♥ ②

s♣s♦r t♦s s ♣s ss♦ q s ♥♥tr♥ ♣♦r ♥♠

♥ tr♠♥ ♥trs r♥ ♥ t♠♣♦ sñ ♥tr

♥ ♦♥ró♥ ♦♥ s♣ró♥ r♦ t0 ② ♥ ♦♥ s♣ró♥ x t(x) stá

♣♦r ∆t ② s ♣ ①♣rsr ♦♠♦ ❬❪

∆t = t(x)− t0

r ♠♣♦s tv sñ r ♥

♥ ♥trs ♦r③♦♥t ♥ ♥ó♥ s♣ró♥ ♠s♦r r

♣t♦r x s♥ ♣r ♦rró♥ ② ♦ ♣r

♦rró♥ ♦♥ ♥ ♦ ♠♥♦r q

② ♠②♦r q

s♦r ♦s t♦s r s r③ ♦rró♥ ó♥

s ♦t♥ rst♦ q s ♠str ♥ r ♦♥ s ♣

r q t♠♣♦ sñ ♦♥ ♣r t♦♦ x ♣r♦ ♣r ♦t♥r

st t♣♦ rst♦ ♦ t♥r ♦r ♦ ♦r

♦ s ♠s♦ ♦ r ♦ ♠s♦ t♦ r ♦s

t♠♣♦s ♥♦ stá♥ ♥♦s ② s♠r s tr③s ♥♦ s ♦r ♠♦rr

sñ ♠♥r st♦ ♥ q s ♥sr♦ r③r ♥ st

♦ ♣r ♦t♥r ♥ rst♦ ♥ ♥ q s sñs ♣rs♥t♥

♠②♦r ♠♦r ♣♦s st♦ ♣r♦♣♦r♦♥ ♥ ♠t♦♦♦í ♣r♦♣ ♣r

♦t♥r ♥ st♠ó♥ s ♦s ♣r♦♣ó♥ ♣r♦♠♦ ♣r

♣♥t♦ ♠♦ ♦♠ú♥ ② ♦ ♣r ♥r♠♥tr s ♥t♥ss s sñs

♥trés ♥ ró♥ r♦ r♥♥t ♥ s tr③s rst♥ts

♦♠♦ ♠♣♦ ♣ó♥ ♠ét♦♦ P s ♦♥sr s♦

♥ rt♦r ♣♥♦ ♦r③♦♥t ♦③♦ ♥ ♣r♦♥ 0,80 ♠ r

strt♦ s♣r s rtr③ ♣♦r ♥ ♣r♠t ǫr = 3,5 ②

♥ ♦♥t σ = 1♠♠ ♠♥trs q strt♦ ♣r♦♥♦ ♣♦r ǫr = 4

② σ = 2 ♠♠ r♥ ♠♣♦ s 500 ③ s ♣ ♥ tó♥

t♦r s ♠trs ♣r♠t ② ♦♥t ♦♥ st♦s

♣rá♠tr♦s s r③ó ♥ s♠ó♥ ♥♠ér rs♣st s♣r ♣r

r Pr ♦st 0,45 ♠ ♣r P n = 4

♦st 0,45 ♠ vst = 0,16 ♠♥s

♥ s♦♥♦ r③♦ ♦rr Pr ♦ s t③ó ♠ét♦♦ r♥s

♥ts q s sr ♥ ♣é♥

♥ r s ♠str ♥ ♣r ♦st 0,45 ♠ ♦t♥♦ ♣r

♠♦♦ r sñ r①ó♥ ♣r ♥trs s ♣

♦srr ♥ t = 11 ♥s ♣r♦①♠♠♥t ♥ r s ♠str

rst♦ ♣r P ♦♥ ♥ ♥ú♠r♦ ② ♦ ♣♠♥t♦ n = 4 ② vst =

0,16 ♠♥s rs♣t♠♥t ♦♠♣rr s rs ② s ♣ r

q ♠ét♦♦ P ♠♦r sñ q s ♦t♥ ♦♥

♣ít♦

♣ó♥ ♦s ♠ét♦♦s ss

♥ ♦rr

♥ ♥♦s s♦s ♣♦r ♠♣♦ ♥♦ ♦s ♥♦s ♣rs♥t♥ ♦ ♦♥trst

♦♥ rs♣t♦ ♠♦ r♥♥t ♦ ♦♠trís ♦♠♣s ♦s st♦s r

③♦s ♦♥ ♠t♦♦♦ís ♦rtr s♠♣ ♣r♦♥ rst♦s ♠♦s

♦ ís ♥tr♣rtr ♥ ss st♦♥s s ♣♥ st♥ts té♥s ②

s ♣r♦s♠♥t♦ s③ó♥ ♦ qsó♥ t♦s ♣r r ♥

♥tr♣rtó♥ ♠ás r ♦s rst♦s ♦♠♦ ♠♣♦ s ♣rs♥t♥ ♥ st

♣ít♦ st♦s q ♠str♥ ♣ó♥ ♦s ♠ét♦♦s ♠ró♥

sñs P ② ♠♦♦ ♣r rtr③ó♥ strtrs ♥ st♦

rq♦ó♦ P♦ ♥♦ ❬❪

t♦ rq♦ó♦ P♦ ♥♦

st♦ rq♦ó♦ P♦ ♥♦ s ♥♥tr ♥ ♠á

♥ ró♥ s♠sért ♥ ♣r♦♥ t♠r r♥t♥ r r

st st♦ stá r♦♥♦ ♦♥ ♥ s ♣r♠rs ♦♠♥s rí♦s

♣st♦rs ró♥ q s srr♦ó ♣r♥♣♠♥t r♥t Prí♦♦

♦r♠t♦ ♦♥ ♣r♦①♠♠♥t ñ♦s ♥tü

♦ s rtrísts ♦rás ③♦♥ P♦ ♥♦

stá ♦♥t♥♠♥t ①♣st ♥ ♥t♥s s♠♥tó♥ ♦♠♦ ♦♥s♥

♦s s♠♥t♦s ♥♦r♠♠♥t ♥trr♥ ♥ t♠♣♦s rt♠♥t ♦rt♦s ♦s

r ❯ó♥ st♦ rq♦ó♦ P♦ ♥♦

♣ ♦♥ ♦③ó♥ s strtrs rq♦ós st

s s♦♥ t♠s ♦♥strs ♦♥ ♣rs

s♦♥ ♥ú♦s t♦♥s ♦♥ ♣rs t♣

♦t♦s q stá♥ ♥ s♣r ♣r♦s♦ ♥tr ♦rtr ♦s ♣r♦t

♥ts ñ♥♦s ♦♠♦ ♥t♦ ♠♦s ♣ró♦s ♥ ♠ ♠♥

t ② t♠é♥ ♥tró♥ tr ♥ ♠r♦ ♦s ♦t♦s rq♦ó♦s

♣♥ qr t♠♣♦r♠♥t ①♣st♦s ♦ ♠♦♠♥t♦ ♥tr s

♠♥t♦s ♦s♦♥♦ ♣♦r ♣r♦♣♦ ♥t♦ ② ♦s ♦s ♥♦ ♣♦s

s tó♥ s sí q ♦s ♣r♠r♦s st♦s rq♦ó♦s ♥ ③♦♥ ❬❪

❬❪ rr♦♥ ♣qñs ♣♦r♦♥s ♣rs t♣ q ♣rí♥ ♥ s

♣r q ♣r♠tr♦♥ ♥tr ♥♦ ♥s rs♥s ♦♥ r♥t

♦♠♣ rqttó♥ r♦ ♦♥ s sr♣♦♥s r③s sts

strtrs ♣rs♥t♥ st♥ts ♦r♠s ② str♦♥s s♣s ♣r♦ ♥

♥r ♦♥sstí♥ ♥ trs ♦ tr♦ t♦♥s ♦r♠ rt♥r ♦♥t

s rt♠♥t ♦ ♣♦r ♠♦ árs ♣r♠♥t rts ♦ ♣t♦s ♥

♣rs♥t s♠♥tó♥ t♦ rr s strtrs q r♦♥ ♦sr

s ♣♦r ♦s rqó♦♦s ♥ ♦s ñ♦s st♦ rst ♥ ♥ s ♠②

♥s s ♦ rt♠♥t ♥ ♥♥♥ s ♥ ♠②♦rí

♦s s♦s

♥tr s strtrs st♦ ①st♥ trs ♦♥str♦♥s ♥rrs ♦♥

♦r♠ ♣r♦①♠♠♥t s♠sér sts r♦♥ srts ♣♦r ♦s

t♥ts ró♥ ♥ ♦s ñ♦s ♥♦s ñ♦s s♣és sr♠♥t♦

sts r♦ t♣s ♥♠♥t ② ár t③ ♦♠♦ ñ♦ ♣♦r s

tr♥t ñ♦s ♥t♠♥t ♥ ♠r♦ ♥ ♥stó♥ rq♦ó

♦s sts t♠s ♥ts ♦♠♦ ② r r r♦♥ ♦

③s ♣rtr rstr♦ rq♦ó♦ ①st♥t ② ①s ❬❪ sts

t♥í♥ á♠tr♦s ♣r♦♠♦ ♥tr 2,7 ♠ ② 3,3 ♠ ss ♣rs s ①t♥í♥ s

0 − 0,2 ♠ ♥ ♣r♦♥ ♠á①♠ 1,2 ♠ ② s ♥♦ st ♥

r♥♦ 0,5− 0,7 ♠ ♥ ♠r♦ trr t♠ ♥♦ ♣♦ sr ♥♦♥tr

♦♠♦ s♦♠♥t s ♦♥♦í s ó♥ ♣r♦①♠ ② ♦ q ♥st

ó♥ rq♦ó rqrí ♦③ó♥ ♣rs strtr s r③ó

♥ ♣r♦s♣ó♥ ♦♥ ♦rr ♥♦s ♦s rst♦s st ♣r♦s♣ó♥

s ♣rs♥t♥ ♥ ♣ró①♠ só♥ ♥ ♠♦strr ♥ q ♠ s ♣♥

rtr③r st t♣♦ strtrs ♣rtr t♦s ♦rtr s♠♣ ②

♣ó♥ ♠t♦♦♦í ♠ró♥

P♦r ♦tr♦ ♦ ♥ st t♣♦ st♦s rq♦ó♦s s s ♣rs♥

ss ♦tr♦s ♠♥t♦s ♥trr♦s ♥ s tó♥ ♥♦ s ♥ ♦t♦

♥tr ♦s st♦s ♦♥ ♦rr ♦ s ♣qñ♦ t♠ñ♦ rst

♠♦ ♥trés tr♠♥r t r♦♥♦r s sñs q ♥r

rí♥ st t♣♦ strtrs P♦r t ♠♦t♦ s r③r♦♥ st♦s ♦♥tr♦♦s

t♥♥ts tr♠♥r ♣♦s tó♥ ♥ ♣rtr s t③r♦♥

♠ét♦♦s ♦rtr s♠♣ ♦♥ rs t ♥s ♣r ♥ s

③ó♥ t rs♦ó♥ ss♦ ♠ás ♦rtr ♠út♣ P ②

♠♦♦ ♣r ♥ ♠♦r ♥tó♥ s sñs ♣r♠rs q ♣r♠t

♦t♥r ♥ ♥tr♣rtó♥ ♦s t♦s ♥ st ♣ít♦ s ♣rs♥

t♥ ♠♣♦s ♦s st♦s ♦s s ♠str♥ ♥s s ♣r♥♣s

rtrísts ② ♥s st♦s ♠ét♦♦s

♦③ó♥ ♥ strtr ♣rs

♦♠♦ s sró ♥ só♥ ♠ét♦♦ ♠ró♥ ♣r♠t ♦t♥r

♠á♥s ♠ás rsts ♦s ♦t♦s ♥trr♦s ② s t ♥ q♦s

s♦s ♥ q s sñs s♦♥ ♦♠♣s ♥ st só♥ s sr ♣ó♥

♠ét♦♦ ♠ró♥ ♥ st♦ rq♦ó♦ P♦ ♥♦ ♦♥ ♦t♦

r♦③r

♠♦♠♥t♦ r③r ♣r♦s♣ó♥ s í st♠♦ q ♣s

t rí♦ ♥ ③♦♥ ♣r♦♠♥t í str♦ stró♥

♥tr ♦s strt♦s ♠ás s♣rs ② ♠③♦ ss ♠trs sí só♦

s s♣r♥ sñs és ② ♣♦♦ ♦♥t♥s ♦s ♥♦s rq♦ó♦s

♦ ♦s P♦r ♦tr♦ ♦ s í♥ ♦sr♦ ♥ s♣r ♥ ♥t

♠♣♦rt♥t r♦s s♣rss ②♦ t♠ñ♦ ♥♦ r s♣r ♥ ró♥ ♦♥

s r♦s s t♠s ② P♦r ♥ ♥♦♥trrs r♦s ♥rsts

♥ s♦ ss sñs ♣♦rí♥ s♦♥r s sñs q s st♥ s♥♦

♥ t s♦ ♥áss rstrí ♠② ♦♠♣♦

♦s sts rtrísts ♥♥ q ♥tó♥ rt s

sñs s ♣rs ♣rs♥trí ts ♠♣♦rt♥ts ♦♠♦ ♥

♠♥♦ ♣♦s ♣r ♦t♥r sñs ♠ás s♥s ② rs s ♣r♦♣s♦ ♣

r ♠ró♥ ♦s t♦s st s ♣r♦s♠♥t♦ ♣rs♥t ♥ sr

♥ts ♦♠♦ s ♠♥♦♥ó ♦s ♣rs ♠r♦s ♦♥stt②♥ ♥ ♠♥ ♠ás

rst ss♦ ♣♦r ♦ q s s♠♣ ♥tr♣rtó♥ s strtrs

♦♠♣s ♥♦ ♣r♦s♦ ♥♦q q stá ♠♣ít♦ ♥ ♠ró♥

s s♣r q ♦♣s s ♣ér♦s ró♥ ② q ♣r♦③ sñs

s ♣rs ♠ás ♦♥trst♥ts ② rs♦s

♥ ♥ ♣r♠r ♠♣ñ r③ ♥ ♠r♦ st ss s ♠tó ♥

ár ♠♣ ♥ q s s♣♦♥í q s ♥♦♥tr ② s qrr♦♥ ♣rs

9m ♦♥t s♦r ♥ r r ♦♥ 1m s♣ró♥ ♥tr í♥s

♥ ♠s r♦♥s ♦s ♣rs qr♦s ♥ r♦♥s ♣r♣♥rs

♠♦r♥ ♣♦s tó♥ ♥tó♥ s ♣rs ② q s

r♦s q s ♦r♠♥ ♣♥ t♥r rt ♥ ♥ ró♥ s♦♥♦

♣r♦ s ♣r♦ q ♠r ró♥ rt ♠♥t

r③ó ♥ ♣r♦s♣ó♥ t③♥♦ ♥ ♦rr ♥♥r st♠

♠♦♥♦stát♦ s ♥t♥s ♠s♦r ② r♣t♦r s ♥ ♥ ♠s♠ ♥

♦st ♦♥ r♥ ♥tr 400 ③

❯♥ ③ qr♦s ♦s t♦s t♦♦s ♦s ♣rs ést♦s s ♣r♦sr♦♥

♠♥r ♠♦rr s③ó♥ s sñs ♥ ♥♦ ♦s ♣rs s

♣ó ♦ ♦rró♥ ♦r♥ t♠♣♦ ♠♥ó♥ sñ rt ②

♥♥ ♥ r s ♠str ♥ rrr♠ rtríst♦ ♦t♥♦ ♥

st st♦r ♥s ③♦♥s rrr♠ rrs ♥ r ♣rs♥t♥

sñs ♦♥ ♠②♦r ♥t♥s ♣r♦ ♥♦ stá♥ r③♦♥♠♥t ♥s ♣♦r ♦

q ♥♦ s ♣♥ ♥tr r♠♥t ♦♠♦ ♣♦ss ♥♦♠ís s s

♣rs s ③♦♥s ♦♣♥ ♦s ♥tr♦s x = (0,0; 1,1) ♠ x = (3,8; 4,6)

♠ x = (5,1; 5,9) ♠ ② x = (7,2; 7,8) ♠ ② st t = 10 ♥s

Pr st♠r ♦ ♠ró♥ s ♥③r♦♥ t♦♦s ♦s ♣rs ♦t

♥♦s ② s s♦♥r♦♥ t♦s s ♣ér♦s ró♥ q s ♥♦♥tr♥

♥tr♦ r♥♦ ♣r♦♥s s♣rs ♣r s ♣rs st 1,2 ♠

sú♥ ♥♦r♠ó♥ rq♦ó ♦ s stó ♥ ♣ér♦ ♥ ♥

s sñs ró♥ ♦♠♦ s ①♣ó ♥ só♥ ♥ ♠♣♦

st r③♦ s ♣ r ♥ r ♥♠♥t s r③ó ♣r♦♠♦

t♦♦s ♦s ♦rs ♦t♥♦s ♦r rst♥t v = (0,18± 0,01) ♠♥s

t③ó ♠s♠ ♦ ♠ró♥ ♣r t♦♦s ♦s ♣rs

♥ r s ♠str só♥ ♠r ♣r ♦s t♦s

r ♦sr ♥ ♠♦r ♠♦r ♥ ♥ó♥ s ♥♦♠ís

♣♦r ♠♦ ♣r♦s♦ ♠ró♥ ♦♥ ♥s ③♦♥s q ♣rs♥t♥ ♥ ♥

t♥s rt ♥t♦r♥♦ ♠②♦r q ♥ ♦s t♦s s♥ ♠rr ♣rtr

st♦s rst♦s ♦s s ③♦♥s ♠♥♦♥s ♥ ró♥ ♦♥ ♦s t♦s s♥

♠rr s ♣♥ str ♠ás r♠♠♥t ♦♠♦ ♥ó♠s r♦ s

r rr♠ ♦t♥♦ ♥ ár t♠

♦rrs♣♦♥ ♣r y = 8 ♠ r ó♥ ♠

r ó♥ ♠r q s ♣ó t♦ ♣r s

♠♣ts s♣r♦rs

r rr♠ ♦t♥♦ ♥ ár t♠

♦rrs♣♦♥ ♣r x = 2 ♠ r ♥ q s

st ♥ ♣ér♦ ♥ ♥ s sñs ró♥ q

s ♦sr♥

♥t♥s ② ♦r♠ sts r♦♥s s♦♥ s q s ①t♥♥ ♦ r♦ ♦s

r♥♦s x = (3,8; 4,6) ♠ ② x = (5,1; 5,9) ♠ ♣r ♣r♦♥s s 0,15 ♠

st 0,9 ♠ ♣r♦①♠♠♥t s s rst♥ ♦♠♣ts ♦♥ ♥♦r

♠ó♥ rq♦ó s♦r s t♠s P♦r ♦♥trr♦ ró♥ ♥ r♥♦

x = (7,2; 7,8) ♠ s ♣ srtr ♦♠♦ sñ ♦r♥ ♥ ♥ ♣r ♦

s ♠♥♦r ♥t♥s rt ② ♦r♠

Pr ♦t♥r ♥ ♠♦r s③ó♥ s ♣♦s ♥♦♠ís s t③ó ♥

t♦ s♣r♦r ♣r s ♠♣ts ♠rs s r s ó r♦ t♦♦s

♦s t♦s ♠r♦s ②♦ ♦r s♦t♦ ♠♥♦r q ♥ ♠r ♣r♥♦

P♦r ♠♣♦ ♥ r s ♠str ♠♥ q rst r

♦ ♣ó♥ t♦ s ③♦♥s ♥ó♠s q s ♠♥♦♥r♦♥ ♥

♣árr♦ ♥tr♦r s ♣♥ ♥tr r♠♥t ♥ st r trs ③♦♥s

q s ①t♥♥ ♦ 1,5 ♠ ♦♥ tr ♠♥♦r q 0,5 ♠ ♦ ♠ás s

q 0,3 ♠ s srtr♦♥ ♦♠♦ ♥♦♠ís ♣♦rq ♦♥tr♥ ♥♦r♠ó♥

rq♦ó

❯♥ r♣♦ r♦♥s ♥ó♠s ♦♥ rtrísts s♠rs s sr♣ts

♥ ♠♣♦ ♥tr♦r s ttr♦♥ ♦ r♦ ③♦♥ st♦ sts s

①t♥r♦♥ ♥ ró♥ rt ♣r♦①♠♠♥t ♥tr z = (0,15; 0,30) ♠

st z = (0,75; 1,00) ♠ ♦♥ ♥♦ ♥tr 0,45 ♠ ② 0,85 ♠ rr♦♥ ♥ ♥

♠♣ ② s ♥tr♣rtr♦♥ ♦♥sr♥♦ s rtrísts s♣rs ♣r

♥ r s ♠str ♥ ♠♣ s ♥♦♠ís ♥ ró♥ ♥ q s

♣r♦ r ♣♥t ♥ ♥tr♣rtó♥ ♦s rst♦s

♥ r s ♠str ár s♣és ①ó♥ ♥

s ♣ ♦srr t♠ r♦③ ♠ó á♠tr♦ ♠♦

r ♣ s ♥♦♠ís ♣r ② s ♥tr♣rtó♥

r ♣♥t

r ♠ ♦ ①ó♥

t♠ ①♣st s ♦t♦ ♥ r♥♦ 2,8 ♠ 3,2 ♠ s ♣rs s

①t♥r♦♥ ♥ ♣r♦♥ s 0,15− 0,30 ♠ st 1,1− 1,2 ♠ ② s ♥♦

0,5− 0,8 ♠ st♦s rst♦s s ♣ ♥♦tr q ♣r♦♥ s

♣rs ♠②♦r q st♠ 0,75 − 1,00 ♠ st ♦ ♥ q s

r①♦♥s ♥ s ♥trss ♥tr s r♦s s ② s♦ ♦ s

♥♦ r♦♥ ♥ts ♦♠♦ ♦♥s♥s s ♥t♥s ♦♥ rs♣t♦

♦trs sñs r♥♥ts P♦r ♦♥trr♦ s ♦t♦ ♥ ♥ st ♥tr

♦s ♦rs ♠♦s ② ①♣r♠♥ts ♣r ♦s ♦tr♦s ♣rá♠tr♦s

♥ s rí r tt♦ ♦s ♣rs ♥ ♥♦ ♦s ♣rs

q r③♥ t♠ só♦ ♥ s tt ♥ ♥♦ ♦s ♣rs

r r ♥ st♦s s♦s ♥ tó♥ ♣r♦♠♥t ♦r

♥ ♥ ♦♠♥ó♥ ♦s s♥ts t♦rs ♦♠trí ② s♣♦só♥

s r♦s s ♣rs rs♣t♦ s♣r ② ♣♦r③ó♥ rr

q ♦♠♥t ♣r♦♥ rt ♦♥trst ♣r♠t ♥ s

♥trss ♥tr s ♣rs ♥♦ tts ② s♦ ♥♦ ♦ s♥t♠♥t

t♦ ② ♠♦♠♥t♦ s♦ ♦ rtr tró ♦r♥

♦s ♥t♦s ♠♥r q s ♣r♦♦ ♥ ♥♦q ♠♥♦s ♥t ♠rr ♥

♦♥só♥ s ♥ ♣ó♥ té♥ ♠ró♥ ♣r♠tó ♦③

ó♥ ♦♥stró♥ s r♦ q ♥ ♠♦s s♦s s ♠♣rs♥

t③ó♥ ♠t♦♦♦ís ♦rtr ♠út♣ q t♥♥ ♥r♠♥tr

s sñs ♥trés ♦♠♦ ♣s♦ ♣r♦ ♠ró♥

tó♥ ss ♥trrs

ét♦♦ ♦rtr s♠♣

♦♥t♥ó♥ s sr ♥ st♦ r③♦ ♥ ró♥ ♦♥ tó♥

ss ♥trrs ♥ ♣rtr s ♠str♥ ♠♣♦s ♥ ♦s s s

♣♥ s ♠t♦♦♦ís ♦rtr s♠♣ ② s③ó♥ rs

t ♥s

r♥t ♥ s♥ ♠♣ñ r③ ♥ st♦ rq♦ó♦ P♦

♥♦ s ó ♦ ♥ ①♣r♠♥t♦ ♦♥tr♦♦ ♥ q ♥ s ♦♥

á♠tr♦ ♠á①♠♦ 0,2 ♠ s ♦♦ó ♥ ♣r♦♥ 1 ♠ ② s ró ♦♥

s♦ ♥tr ♦♠♣st♦ ♣r♥♣♠♥t ♣♦r r♥ ② ♥ ♠♥♦r ♣r♦♣♦ró♥

r ♥ s rs ② s ♣ r ①ó♥ ② t

s rs♣t♠♥t ♥ s rs ② s ♠str♥ ♥ só♥

rt ② ♦tr ♦r③♦♥t rs♣t♠♥t ♦♥ ♦♥ró♥ t③ ②

ó♥ s í♥s s♦♥♦ r③s s ♥t♥s ♠s♦r ② r♣t♦r

s ♦③r♦♥ ♥ ♥trs r s♦ ♦♥ s♣ró♥ 0,25 ♠ ♥tr

s ♦s s s ♥t♥s s r♦♥ ♣r♣♥rs í♥ s♦♥♦

t③ó sst♠ rr ♥s♦rs ♦tr Ps P ② ♥t♥s

500 ③ ♠tó ♥ ár 5 ♠ ① 5 ♠ ② s qró í♥s ♣rs

♦♥ ♥ s♣ró♥ 0,05 ♠ ♥tr y = −0,5 ♠ y = 0,5 ♠ s♣ró♥

♥tr í♥s s s♦♥ó ♠♥r ♦t♥r t♦s ♥s s♥t

♣r ♥rr ♥ s③ó♥ ♦♥ t rs♦ó♥ sñ ♣r♦

♣♦r s

♥ s s♦♥s qrs s ♣r♦sr♦♥ ♣r ♦t♥r ♥ s

③ó♥ s sñs ♥trés ♠♥ó sñ rt ♦

s ♣ó ♦ s ♦rró ♦r♥ t♠♣♦s ② ♥♠♥t s ♣ó

♥♥ ♥ r s ♠str rst♦ ♦t♥♦ ♣r só♥ q

♦rrs♣♦♥ y = 0 ♠ ♦♦r♥ x ♥ rá♦ s rr ♣♥t♦

♠♦ ♥tr s ♣♦s♦♥s ♠s♦r ② r♣t♦r sñ ró♥

♦t♦ ♥trr♦ s ♣ r ♥ r ♦♥ ért ♥ (x; t) = (0 ♠; 10 ♥s)

♥ r ♠é♥ ♣r♥ rs ♣ér♦s ró♥ ♦♥s

② r①♦♥s ♣r♦①♠♠♥t ♦r③♦♥ts ♣r♦♠♥t ♣r♦s ♥ ♦s

í♠ts trs ①ó♥ ♥ r ♥ s s♦♥t♥s ♥

trs s♦ ♥ r ② ♥ í♠t ♥r♦r ①ó♥ ♥

r st♦s ♣tr♦♥s s r♣t♥ ♥ t♦s s í♥s ♦t♥s

s ♣r♦ ♥ sñ t ♥t♥s ♣♦s♠♥t ♦♠♦ ♦♥s

♥ ♣rs♥ r ♥ s ♥tr♦r q ♣rs♥t ♥ t♦ ♦♥trst

♦♥ rs♣t♦ ♠tr q r♦ ❬❪ ❬❪ st sñ s r♠♥t r♦

♥♦ ♦ s ♦r♠ P♦r ♦tr♦ ♦ ♥♦ s r♦ ♦r♥ sñ ♥

(x; t) = (0 ♠; 15 ♥s) ♣r♦①♠♠♥t s ♦♥sr ♦ ♣r♦

♣ó♥ st♠ ♣r ③♦♥ ♥ só♥ v = (0,18 ± 0,01) ♠♥s s

♦t♥ ♥ ♣r♦♥ (1,3 ± 0,1) ♠ ♣r s♦♥t♥ q ♦r♥

sñ st ♦r s ♦♠♣t ♦♥ ♣r♦♥ í♠t ♥r♦r

①ó♥

t ♥s í♥s qrs ♣r♠t ♥tr♣♦r tr♥srs♠♥t

♦s t♦s ♦♥ ♥ ♠② ♥ rs♦ó♥ ② ♦ ♥rr ♥ st sñ

r ①♣r♠♥t♦ ♦♥tr♦♦ ①ó♥ r③ ②

r ♦♥ró♥ ①♣r♠♥t ♠♥s♦♥s

①ó♥ ② ♦③ó♥ s í♥s s♦♥♦

r s♣st ♣r ♣r ♦t♥♦ ♥ y = 0♠ sñ

ró♥ s sñ ♦r♥ ♥ í♠t tr

①ó♥ sñ ♥♦ ♦s strt♦s s ♦r③♦♥ts ②

sñ ♦r♥ ♥ í♠t ♥r♦r ①ó♥

r ❱s③ó♥ ♦s t♦s sñ ró♥

s sñ ♦r♥ ♥ í♠t tr ①ó♥

② sñ ♥♦ ♦s strt♦s s ♦r③♦♥ts

s ♥ r s ♠str ♥ ♠♣♦ st t♣♦ s③ó♥

♥♦♠♥tr ♦♥ q s ♥t♥ s sñs ♠ás r♥ts s ♠s♠

q ♥ r ♥ r s ♣ ♥tr sñ ♦r♥ ♥

s ♥ r sñ ♦r♥ ♥ í♠t tr ①ó♥

② s sñs ♦r♥s ♥ s s♦♥t♥s ♥trs s♦

♦♦

♥ ♦s rst♦s só♥ ♥tr♦r rs ② s ♦sr

sñ ♣r♠r s ♥trr t♠é♥ s ♣♥ r ♥s sñs

s♥rs ♦r♥s ♣♦s♠♥t ♥ ♦s í♠ts trs ♥r♦r

①ó♥ ② ♥ s s♦♥t♥s ♥trs s♦ Pr ♦♥trstr s

ts ♥tr♣rt♦♥s s ♦rr♦♥ ♠♦♦s ② s s♠r♦♥ ♥♠ér♠♥t s

rs♣sts ♦rr

st♦s ♠♦♦s s♦♥ r♣rs♥t♦♥s s♠♣s ♥ st s♦ ss♦

①♣rss ♥ tér♠♥♦s s ♣r♦♣s tr♦♠♥éts ② ♦♠

trí ♦s ♥♦s ♥trr♦s ♥ ♥r st♦s ♠♦♦s sr♥ st♦♥s

♦♠♣s ♥ s q ♣r♦♠ tr♦♠♥ét♦ ♥♦ t♥ s♦ó♥ ♥ít

♣♦r ♦ s s♠ ♥♠ér♠♥t rs♣st sst♠ ♦rr

♥ r s ♠str ♥ ♠♦♦ ♣r♦♣st♦ ♣rtr s rt

rísts ♦♥♦s ③♦♥ st♦ r♦♥ ♦rs ♣r♠t

② ♦♥t tí♣♦s ♣r ♦♠♣♦só♥ s♦ ③♦♥ st♦

♠♦♦ ♦♥t♥ trs strt♦s s♣r♦s ♣♦r ♦s ♥trss s ♦r③♦♥ts

s ♣r♦♥s 0,6 ♠ ② 0,9 ♠ strt♦ s♣r♦r s rtr③

♣♦r ǫr = 3,5 ② σ = 1 ♠♠ strt♦ ♥tr♠♦ s rtr③ ♣♦r ǫr = 4 ②

σ = 2 ♠♠ ② strt♦ ♥r♦r s rtr③ ♣♦r ǫr = 5 ② σ = 2,5 ♠♠

í♠t ♥r♦r ①ó♥ s ♥♥tr ♥ ♣r♦♥ 1,3 ♠

♠tr ♦♥ q s r ③♦♥ ① s rtr③ ♣♦r ǫr = 3

② σ = 1,5 ♠♠ ♥ st s♦ s ó ♥ ♦r ǫr ♠♥♦r ♦

♣r♦♣ó♥ ♠②♦r q ♣r ③♦♥ s♥ ①r ♣♦rq s ♦♥sró q st

♠tr st r♦ ♣♦r r s♦ r♠♦♦ r♥t ①ó♥

s s ♥♥tr ♥ ♣r♦♥ 1 ♠ s♣♦♥ q s s ♥

♥tr ♣r♠♥t ♥ r ǫr = 1 ② σ = 0 ♠♠ ♥ ♠♦♦

♠t ♥r♦r ♠s♠ s ♥♥tr ♦♣ ♣♦r ♠tr ♦♥ q

s r ③♦♥ ① s t♥ 0,2 ♠ á♠tr♦ ♣rs 0,02

♠ s♣s♦r ǫr = 6 ② σ = 2,6 ♠♠ ♥ s ♠trs ♣r♠t ②

♦♥t s ♣ ♥ tó♥ t♦r

♦♥ ♦s ♣rá♠tr♦s ♠♦♦ ♣r♦♣st♦ s r③ó ♥ s♠ó♥ ♥♠é

r rs♣st s♣r ♣r ♥ s♦♥♦ ♦rr ♥ q s ♥t♥s

♠s♦r ② r♣t♦r s♦♥ t♣♦ ♣♦♦ ♠ ♦♥ ② s ♥ ♥ ♥tr

s r s♦ ♦♥ ♦st 0,25 ♠ Pr ♦ s t③ó ♠ét♦♦ r♥s

♥ts ♥ ♦♠♥♦ t♠♣♦ ♥ ♦s ♠♥s♦♥s rs♦r♦♥

♥♠ér♠♥t s ♦♥s ① ♣r ♦t♥r ♠♣♦ Ey s♦r

♥ r ♥♦r♠ ♦♥ ♥r♠♥t♦ 0,01 ♠ ♥ ♠s r♦♥s r

♥ ♠só♥ 500 ③ ♥ ♣é♥ s ♥ ♠ás ts

♣r♦♠♥t♦ r③♦ ♣r ♦t♥r ♦s t♦s s♠♦s

❯♥ ③ st♦s ♦s ♣rá♠tr♦s ♠♦♦ s rs♣st ♣r

♠s♦r ♥ ♥ s ♣♦s♦♥s í♥ s♦♥♦ ② s rstr ♠♣♦

Ey ♥ ♣♦só♥ r♣t♦r tr♠♥ sú♥ ♦st ♦ s r♣t

♣r♦♠♥t♦ ♣r ♥ s ♣♦s♦♥s ♠s♦r s♦r í♥

s♦♥♦ ♥♠♥t s r♣♥ s tr③s ♦t♥♥♦ sí rstr♦s s♠♦s

♥á♦♦s ♦s ♦t♥♦s ①♣r♠♥t♠♥t ♣rtr st ♣♥t♦ s s

♣r♦♠♥t♦ t③♦ t♠♥t ♦s t♦s s ♦r♥♥ ♥ ♥ rrr♠

② s ♣♥ s té♥s ♣r♦s♠♥t♦ ♥srs ♣r ♠♦rr

♠♥ ♦rró♥ ♦r♥ t♠♣♦ ♠♥ó♥ sñ rt

② ♥♥

♥ r s ♠str rrr♠ ♦t♥♦ ♣r ♦s t♦s q

rst♥ ♠♦♦ r ♥ rrr♠ sñ ♦r♥

♥ s s ♣ r ♦♥ ért ♥ (x; t) = (0 ♠; 12 ♥s) ♥ r

ést s ♥t♥s t ♦♠♦ ♦♥s♥ t♦ ♦♥trst ♥tr r ♥

♥tr♦r s ② ♠♦ q r♦ s sñs ♥rs ♥ ♦s

trs ①ó♥ s ♣♥ r ♣♦r ♠♣♦ ♥ (x; t) = (−0,8 ♠; 5 ♥s)

♥ r ♠é♥ ♣rr♦♥ ♥s sñs ♦r♥s ♥ s ♥tr

ss ♥tr ♦s st♥t♦s strt♦s ♣♦r ♠♣♦ ♥ t = 8 ♥s ② ♥ t = 12 ♥s

♥ r ♥♠♥t ♥ (x; t) = (0 ♠; 16 ♥s) ♥ r s ♣

r sñ ♦r♥ ♥ í♠t ♥r♦r ①ó♥ st ♣rs♥tó ♥

s♣t♦ s♠r ♦sr ♥ ♦s rrr♠s ①♣r♠♥ts r

só♥ s sñs ♦r♥s ♥ ♦s í♠ts ①ó♥ ② ♥

s ♥trss q s♣r♥ ♦s st♥t♦s strt♦s t♥♥ ♠♥♦r ♥t♥s q

sñ q s ♦r♥ ♥ s ♦ q s ♦♥s♥ ♠♥♦r ♦♥trst

r ♦♦ s ♥trr strt♦ s♣r♦r

ǫr = 3,5 σ = 1 ♠♠ strt♦ ♥tr♠♦ ǫr = 4 σ = 2 ♠♠

strt♦ ♥r♦r ǫr = 5 σ = 2,5 ♠♠ ♠tr ♦♥ q s

♥ ①ó♥ ǫr = 3 σ = 1,5 ♠♠ s ♣r♦♥

0,9 ♠ á♠tr♦ 0,2 ♠ ♣rs 0,02 ♠ s♣s♦r ǫr = 6

σ = 2,6 ♠♠ s ♣ ♥ tó♥ t♦r st♦s

♣rá♠tr♦s rr♠ s♠♦ ♣r ♥ s♦♥♦ ♦rtr

s♠♣ ♦♥ ♦st 0,25 ♠

q ♣rs♥t♥ s s♦♥t♥s

s ♦♠♣r♥ ♦s rrr♠s ①♣r♠♥t r só♥

② s♠♦ r s ♣♦s ♦srr s♠t q ♣rs♥t♥ ♥

s ♥sr♦ ♥ ó♦ ♣r ♠♦r ♠♥t ♥ts t♣♦ ♣♦♦

② ♥♦s ♦③♦s ♦♠♦ s ♥trr s ♣ r q ó♦

t③♦ ♣r♠t r♣r♦r s ♣r♥♣s rtrísts s sñs

♦t♥s ♥ ♥ s♦♥♦ ♦♥ ♦rr

♥ st s♦ ♠♦♦ ♣r♦♣st♦ ② rrr♠ s♠♦ ♣rtr

♦ ♠♦♦ ♣r♠tr♦♥ str ró♥ ♥tr s sñs ♦srs ②

♦ ♦t♦s ♥trss q s ♥rr♦♥ ② rr♠r ♥tr♣rtó♥ r③

♣rtr ♦s t♦s ①♣r♠♥ts ♥ ♥r ♦s t♦s s♠♦s t♥♥

t♠é♥ ♦trs ♣♦♥s ♥tr s ♠ás ♠♣♦rt♥ts ♥ s t♣s ♣rs

r③r ♥ ♣r♦s♣ó♥ ♣r ♣♥r ♦s ♠ét♦♦s ② strts

qsó♥ ② r♥t ♥áss ♣♦str♦r ♥tr♣rtó♥ ♦s t♦s ♣r

♥tó♥ sñs ♣♦r ♦♠♣ró♥

ét♦♦ ♦rtr ♠út♣ P

Pr ♦♥t♥r st♦ sñ q ♥r ♥ s ♥trr s

♥③r♦♥ s ♠♦rs q ♣r♦ ♠ét♦♦ P ♥ sñ ♣r♠r

s ♠é♥ s str♦♥ s ♠♦rs ♥ s sñs s♥rs

♦r♥s ♥ s s♦♥t♥s ♥trs s♦ ② ♥ ♦s í♠ts

①ó♥ Pr ♦ s qrr♦♥ tr③s ♥ ♦♥ró♥ P ♦♥ ♥r

♠♥t♦ 0,05 ♠ ♥ s♣ró♥ ♥tr s ♣♦s♦♥s ♠s♦r ② r♣t♦r

② ♥ s♣ró♥ ♠á①♠ 1 ♠ st♥ ♥tr ♣♥t♦s ♠♦s Ps

s♦r í♥ s♦♥♦ s ♠str ♥ r 0,05 ♠

♥ ró♥ ♦♥ r só♥ s ♠♥♦♥ó sñ ♥

strt♦ ♣r♦①♠♠♥t ♦r③♦♥t ♦ ♥ t = 11 ♥s ♥tr x = 0,5 ♠ ②

x = 2,0 ♠ st strt♦ s t③♦ qí ♣r str ♦ ♣

♠♥t♦ Pr ♦ s ♣ó ♦rró♥ sñ s♥♦ st♥ts

♦s ♣♠♥t♦ ② s s♦♥ó q q ♣r♦♦ ♠♦r st

♦r rst♥t v = (0,17 ± 0,09) ♠♥s r ♥ r s

♥♥ ♦♥ í♥s tr③♦s s sñs ♥ts ② s♣és ♣r

♥ r s ♠str rst♦ ♣♠♥t♦ ♦♥ n = 4 ♦♠

r ♦③ó♥ í♥ s♦♥♦

♣r♥♦ ♦s rst♦s ♦t♥♦s ♣r ♥ s♦♥♦ ♦rtr s♠♣ r

só♥ ② ♣r P r s ♣♦s ♦srr q

t♥t♦ s sñs ♣r♦s ♣♦r s ♥trr ♦♠♦ ♣♦r rt♦r s

♦r③♦♥t ♣rs♥t♥ ♥ ♠♦r ♣é♥♦s♦s r ♦♥ ♠②♦r r

♠s♠ ♦r♠ sñ ♦r♥ ♥ ♦♥♦ ①ó♥ s ♥ ♦♥ ♥

♥ r ♣rs♥t ♥ ♠♦r s♥t s♥t ♣r sr

♥t ♦♥ r

♥ st ♠♣♦ ♣ó♥ ♠ét♦♦ P ♣r♠tó ♠♦rr s st♥

ts sñs q s ♣rs♥t♥ ♥ rrr♠ ♥ ♠r♦ ♥ ♥♦s s♦s

s ♠♦rs q s ♦t♥♥ ♦♥ ♠ét♦♦ P ♥♦ s♦♥ s♥ts ♦ s r

trísts ♦s ♥♦s ♥trr♦s ♥♦ ♣r♠t♥ t③r ♦ ♠ét♦♦ ♦ q

srr♦r ② ♣r ♠ét♦♦s ♦rtr ♠út♣ tr♥t♦s

r ♠♣♦s tv ♥ ♥ó♥ s♣ró♥

♠s♦r r♣t♦r x s♥ ♣r ♦rró♥ ② ♦

♣r ♦rró♥ ♦♥ ♦ v = (0,17 ± 0,09)

♠♥s

r s♣st ♦t♥ ♦♥ ♠ét♦♦ P

♣ít♦

ét♦♦ rr♦s s♥tét♦s

♠s♦rs ♦rr

♠t♦♦♦í ♦rtr s♠♣ ♣r♠t rs♦r t♦t ♦ ♣r♠♥t

♠② rss st♦♥s ①♣r♠♥ts ♦♠♦ ♣♦r ♠♣♦ s ♥s ♥

♣ít♦ ♥tr♦r ♣rtr s♦ st♥ts té♥s ♣r♦s♠♥t♦

♥tr♣rtó♥ sñs ♥ ♠r♦ ② st♦♥s ♦♥ sñs ♠② és

② ♣♦r ♦ t♥t♦ ♦♥ss ♥ s q s ♥sr♦ t③r ♦trs ♠t♦♦♦ís ♦♥

♦rtr ♠út♣ ♣r ♠♦rr s s ♥ st ♣ít♦ s ♣rs♥t

♥ ♠ét♦♦ ♦rtr ♠út♣ tr♥t♦ ♥♦♠♥♦ ♠ét♦♦ rr♦s

s♥tét♦s ♠s♦rs ♦rr ②♦ ♦t♦ ♥♠♥t s ♠♦rr r

ó♥ ♥t♥s sñ ♣r♠r rs♣t♦ ♦trs sñs s♥rs

sí ♦♠♦ s ♦♥t♥ tr st ♠ét♦♦ ♦♥sr rr♦s ♥t♥s

♣r ♦t♥r ③♦♥s ♠♥s ♠ás strs ♦♥♥tr♥♦ ♠♣♦ s♦r ♦s

♥♦s ♥trés ② s♠♥②é♥♦♦ ♥ s ③♦♥s ♣rérs ♥r♦rs

sñs s♥rs ♥ss

♥ ♣r♠r tér♠♥♦ s ♦♥sr♥ rr♦s ♠s♦rs ♦s ♥ ♠♦s

♥♦r♠s ② s st ró♥ ♠♣♦ ♥♦ ♦♥ ♦t♦ strr

♠♥r ♥ q s ♣♦s ♦♥tr♦r ♣tró♥ ró♥ t♦t rr

♦ ♦♥t♥ó♥ s sr ♠ét♦♦ rr♦s s♥tét♦s ♠s♦rs

♦rr ② s st♥ s rtrísts ♦s ♠♣♦s tr♦♠♥ét♦s

♥r♦s ♣♦r st♦s rr♦s ♥ ró♥ ♠♣♦ r♥♦ ♣r ♠s♦rs

♦s ♥ s♣r s♣ró♥ r trr ♣rtr s ♥③♥

s rs♣sts q s ♦t♥♥ ♣r ♠ét♦♦ ♥ ♦s s♦s ♥♠♥ts

♦t♦ ♣qñ♦ ② rt♦r ①t♥s♦ ♥♠♥t s ♣rs♥t ♥ ♠t♦♦♦í

♣r♦s♠♥t♦ rs♣st ♦t♥ q ♣r♠t ♠♦rr ♦♥t♥

tr s sñs ♣r t♦♦ s r♥♦ ①t♥só♥ t♥♦ sí

♥tr♣rtó♥ s ♠s♠s ❬❪ ❬❪

rr♦s ♠s♦rs

♥ ♥r s ♥t♥s ♠♥ts ♠ás s♠♣s ♦♥strr ♦♠♦ s

t♣♦ ♣♦♦ ♠ ♦♥ ♣rs♥t♥ ♣tr♦♥s ró♥ ♦♥ rt

❯♥ ♣r♦♠♥t♦ q s ♣ t♠♥t ♣r ♦t♥r ♥ ♠

♦r rt ♦♥sst ♥ t③r rs ♥ts ♠♥ts strs ♥

♥ ♦♠trí ♣r♦♣ s q ♦r♠♥ ♥ rr♦ st ♠♥r s ♦r

♦♥♥trr ② rr ♥rí r ♥ ró♥ tr♠♥ ♥

s♣♦

♥ ♥ ♠♦ ♥♦r♠ ② ♥ ró♥ ♠♣♦ ♥♦ s ♣♦s ♦t♥r

①♣rs♦♥s ♥íts q sr♥ stró♥ ♥rí ♥ s♣♦

♣r♦ ♣♦r rr♦ Pr ♦ s ♦♥sr ♥ ♣♦♦ ♦ ♦♥ ♦♥

t l ♦♠♦ s ♠str ♥ r q ♥r ♠♣♦s r♠ó♥♦s

r♥ ω st ♥t s r♣rs♥t ♣♦r ♥ ♥s ♦rr♥t étr

♥♠♥s♦♥ ~Jorig(~r, t) ♦ r♦ ♣♦♦ z

~Jorig(~r, t) =

I0e−iωt sen(π

2− k|z|)δ(x)δ(y)z s |z| ≤ λ

0 s |z| > λ4

♦♥ I0 s ♠♣t ♦rr♥t k = 2πλ

s ♥ú♠r♦ ♦♥ λ s

♦♥t ♦♥ ♥ ♠♦ δ s t r ② sí♥ orig ♥

q ♥t s ♦③ ♥ ♦r♥ ♦♦r♥s ♥♦ s ♦♥sr ♥

♦♥t l = λ2♣r ♥t♥ stó♥ r♣rs♥t ♣r♦①♠♠♥t ♥

♣♦♦ r♥t ♠ ♦♥

♣♦t♥ ♣♦r ♥ á♥♦ só♦ dPdΩ

♦ ♥t♥s ró♥ s

♣ ①♣rsr ♦♠♦

[cos(π2cos(θ))

sen(θ)

♦♥ η =√

ǫs ♠♣♥ ♠♦ ♦♥ µ ♣r♠ ♠♥ét

② ǫ ♣r♠t ♣♦t♥ ♣♦r ♥ á♥♦ só♦ ♥ rr♦

r ♣♦♦ ♠s♦r ② sst♠ ♦♦r♥s

♣♦♦s ♥ ró♥ ♠♣♦ ♥♦ s ♣ ♦t♥r s♣r♣♦♥♥♦ ♦s ♠

♣♦s ♦s ♣♦♦s ♥s r ó♥ ♥ ♣é♥ ♣♦t♥

♣♦r ♥ á♥♦ só♦ ♣r ♥ rr♦ N ♠♥t♦s (dPdΩ)N s ♣

①♣rsr ♦♠♦(dP

N= |ΓN |2

♦♥ ΓN s t♦r rr♦

t♦r rr♦ s ú♥♦ q ♦♥t♥ s ♣♦s♦♥s s ♥ts~dn ② ss ♠♣ts an ② ss ϕn rts ② s q ♠♦ ♣tró♥

♠♥t♦ ♥r♦r ♠s♠♦ s ♦ ♣ ①♣rsr s♥t ♠♥r

ΓN =N−1∑

ane−iϕne−i~k·~dn

s♥♦ t♦r ó♥ s ♥s ♣♦t♥ ♠♥t♦

♥r♦r

♠♥r ♠ás s♠♣ ♥rr ♣tr♦♥s ♦♥ rtrísts s♣ís

s ♦♥sr♥♦ rr♦s rrs t ♦♠♦ r ❯♥ rr♦

rr s ♦♠♣♦♥ ♣♦r rs ♥ts q t♥♥ é♥t ♠♣t ♣rs♥t♥

♥r♠♥t♦s ♦s ♥ ss ss ② ♣♦s♥ stró♥ rr ♥ s♣♦ ♥

s s♦ t♦r rr♦ s ♣ ①♣rsr s♥t ♠♥r

|ΓN |2 =sen2(Nx(αx+kxdx)

sen2( (αx+kxdx)2

sen2(Ny(αy+kydy)

sen2( (αy+kydy)

sen2(Nz(αz+kzdz)2

sen2( (αz+kzdz)2

♦♥ Ni di ② αi ♦♥ i = x, y, z r♣rs♥t♥ ♥ú♠r♦ ♥ts s♣ró♥

② s rt rs♣t♠♥t ♥ s r♦♥s x y ② z r ó♥

r rr♦ rr ♦♥ Nx = 2 Ny = 2 Nz = 4 dx =

3,0dy ② dz = 1,5dx

♥ ♣é♥ ♣♥♥ ♥r stá ♠♣ít ♥ ♦s tér♠♥♦s

ki ó♥ stá ♦♠♣st ♣♦r trs t♦rs r♦♥♦s ♦♥ ♦s

♣rá♠tr♦s r ♥ s r♥ts r♦♥s st♦ s♠♣ ♥áss

s rtrísts ♣tró♥ rst♥t ♦♠♦ ♥ó♥ ♦s ♣rá♠tr♦s

rr♦ ♦♥ ♥ s♠♣ó♥ ♦♥ ♥♦ s ♦♥sr♥ rr♦s ♥

♥ ♦ ♦s ♠♥s♦♥s ♣♦r ♠♣♦ ♣♦♦s ♥♦s ♦ ♥ rr♦ ♣♥♦

rs♣t♠♥t

♥ s♦ rr♦s rrs ♦♥ ♠♥t♦s str♦s ♦ r♦

y ó♥ q

|ΓN |2 =sen2(N(α+kd sen θ senφ)

sen2( (α+kd sen θ senφ)2

♦♥ s s♣r♠r♦♥ ♦s sí♥s ② s r♠♣③ó ky = k sen θ senφ ♣♦

♥♥♦ q t♦♦s ♦s ♠♥t♦s t♥♥ ♠s♠ ♥t♥s ♦s ♣rá♠tr♦s q

tr♠♥♥ s rtrísts ♣tró♥ ró♥ s♦♥ ♥ú♠r♦ ♥ts

N s♣ró♥ d ② s rt α ♥tr ♦s ♠♥t♦s

Pr str ♣♥♥ ♣tró♥ ró♥ ♦♥ ♦s ♣rá♠tr♦s

rr♦ s sñó ♠♣♠♥tó ó♦ ♦♠♣t♦♥ rr② st

ó♦ srr♦♦ ♥ ♥ ♣r♠t r ♣tró♥

ró♥ rr♦s ♣♦♦s ♠ ♦♥ ♦♥ r♦♥s ♣rs ②

s ♦♥sr♥♦ ♥ ♦♥ró♥ rr ♦ rrr ♦♠♦ s ó♦

♠str rs♦s rá♦s ♥tr ♦s t♦r rr♦ Pr tr

t③ó♥ ó♦ rr② s srr♦ó ♥ ♥t♦r♥♦ ♥trt♦ q ♣r♠t

r Ptró♥ ró♥ ♣r rr♦s ♦♥ d = 0,50λ

α = 0 N = 1 N = 2 N = 4 Pr♦②ó♥ ♥ ♣♥♦

x− y ♣tró♥ ró♥ ♦s rr♦s ♥tr♦rs

rr ♦s ♣rá♠tr♦s rr♦ ② ♦srr s♠tá♥♠♥t ♣tró♥

ró♥ rst♥t ♦ ♥t♦r♥♦ ♥t ♠ás ♦♥ rss rr♠♥ts

q t♥ ♥áss ♦s rst♦s r sr♣ó♥ ó♦ ♥

♣é♥

♥ s rtrísts ♣tró♥ ró♥ ♣♥♥ ♦s ♣rá

♠tr♦s rr♦ ♥ s ♦♥♥t♦ ♥ú♠r♦ ♥ts N s♣ró♥ d ② s

rt α ♥tr ♦s ♠♥t♦s s ♣ ♦srr q ♥♦ ♥r ③

r ró♥ ♦♥ ♥ú♠r♦ ♠♥t♦s ♦♠♦ ♠♣♦ s ♦♥sr s♦

♥ ♣♦♦ ♦♥ r♥ ③ ♥ í♦ ǫ = ǫ0 µ = µ0 ② σ = 0

N = 1 r ♣tró♥ s t q ♥ ♣♥♦ ♠♦ ♣♦♦ ♣♥♦

x−y ♥rí r ♣rs♥t ♠s♠ ♥t♥s ♥ t♦s s r♦♥s

♥ s♦ ♥ rr♦ ♦♥ N = 2 d = 0,50λ ② α = 0 r ♣

tró♥ ró♥ ♣rs♥t ♦s ó♦s q ♦♥♥tr♥ t♦ ♥rí ♠t

♥ ♥r ♦♥♥tró♥ s ♠ás ♣r♦♥♥ ♥♦ s ♥r♠♥t

♥ú♠r♦ ♣♦♦s ♥♦s r

r Ptró♥ ró♥ ♣r rr♦s ♦♥ N = 4 α = 0

d = 0,25λ d = 0,50λ d = 0,90λ Pr♦②ó♥ ♥

♣♥♦ x− y ♣tró♥ ró♥ ♦s rr♦s ♥tr♦rs

tr♦ ♥ó♠♥♦ q s ♦sr ♥r♠♥tr ♥ú♠r♦ ♥ts s

♣ró♥ ó♦s s♥r♦s P♦r ♠♣♦ s N = 4 d = 0,50λ ② α = 0 s

♦t♥ ♥ ♣tró♥ ró♥ ♦♥ ♦s ó♦s ♣r♥♣s q ♦♥♥tr♥

♠②♦r ♣rt ♥rí r ② tr♦ ó♦s s♥r♦s ♠②

♥t♥s r ♥ r s ♠str♥ rá♦s ♣r♦②ó♥

♣tró♥ ró♥ ♥ ♣♥♦ x − y ♣r ♦s s♦s s rs

② ♦♥ s ♣ r t♥t♦ ♦♥♥tró♥ ♥rí ♠t

♦♠♦ ♣ró♥ ó♦s s♥r♦s

♥♦ s♣ró♥ ♥tr ♣♦♦s s ♥r♠♥t s ♦sr ♥ ♠♥t♦

♥rí q s ♣r♦♣ ♥ s r♦♥s ♦s ó♦s s♥r♦s st

♥ó♠♥♦ s ♣ ♦srr ♥ s♦ ♥ rr♦ ♦♥ N = 4 α = 0 ② ♦♥

d = 0,25λ r d = 0,50λ r ② d = 0,90λ r

♥♦ rí s rt ♥tr ♦s ♠♥t♦s rr♦ s ♦sr

♥ ♠♦ó♥ ♥ ♦r♥tó♥ ♣tró♥ ró♥ ♦♥t♥♥♦ ♦♥

♠♣♦ ♦♥ N = 4 ② d = 0,50λ s α = 0 r ♦s ó♦s ♣r♥♣s

r Ptró♥ ró♥ ♣r rr♦s ♦♥ N = 4 d =

0,50λ α = 0 α = 0,25π α = 0,50π Pr♦②ó♥ ♥

♣♥♦ x− y ♣tró♥ ró♥ ♦s rr♦s ♥tr♦rs

s ♦r♥t♥ ♥ á♥♦s 0 ② π s rt t♦♠ ♦rs α = 0,25π r

♦ α = 0,50π r s ♣ ♦srr ó♠♦ rí á♥♦

♦r♥tó♥ ♦s ó♦s q ♦r♠ ♥ ♣tró♥ ró♥

♥ rs♠♥ t♦rí rr♦s ♥ts t♣♦ ♣♦♦ ♥ ♥ ♠♦ ♥

♦r♠ ② ♥ ró♥ ♠♣♦ ♥♦ ♠str q ♦♠♥♥♦ rs ♥ts

r♥s ♥tr sí s ♣♦s ♦♥tr♦r rts rtrísts ♣tró♥ r

ó♥ ♦♥♥t♦ ♥ ♣rtr s ♣♦s ♦♥tr♦r ♦r♥tó♥ ② ♦♥

♥tró♥ ♠♣♦ ♠t♦ r♥♦ ♦s ♣rá♠tr♦s rr♦ ♥ú♠r♦

♥ts s♣ró♥ ② s rts ♥tr s

ét♦♦ rr♦s s♥tét♦s ♠s♦rs

♦♠♦ s ♠♦stró ♥ só♥ ♥tr♦r t③r rs ♥t♥s ② str

rs ♦♥ ♥ ♦♠trí s ♣♦s ♦t♥r ♥ rt

♣r ♦♥♥t♦ ② ♦♥tr♦r ró♥ ♥ q s ♦r♥t ♥rí ♠

t st♦ s s♣♠♥t út ♥ s♦s ♥ ♦s q s ♥t♥s ♥s

t♥♥ rt ♦♠♦ ♣♦r ♠♣♦ s ♥t♥s ♦rr ss

❯♥ ♦♥s♥ rt s q ♥rí ♠t s s♣rs

♥ ♥ ♦♠♥ r♥ ♣♦r ♦ q s ♦t♥ ♥t♥s ♣r s sñs

♣r♦♥♥ts ♦s ♥♦s ♥trés ❯♥ ♠♥r s♣rr st ♣r♦♠ s

♦♥♥trr ♥rí s♣♦♥ s♦r ♦s ♥♦s ♣r ♦ s ♣r♦♣♦♥ t③r

rr♦s ♠s♦rs ♦rr

♥ ♠t♦♦♦í ♦rtr s♠♣ s t③ ♥ ú♥ ♥t♥ ♠s♦r ②

♥ ♥t♥ r♣t♦r ♥ ♠ét♦♦ ♣r♦♣st♦ s r♠♣③ ♥t♥ ♠s♦r

♣♦r ♥ rr♦ s ♥t♥s ♦ q ♦s q♣♦s ♦rr t③♦s

t♠♥t ♥t♥ ♦♥ ♥ ú♥ ♥t♥ ♠s♦r s ♣r♦♣♦♥ t③r st

ú♥ ♥t♥ ♦♦á♥♦ ss♠♥t ♥ ♥ s ♣♦s♦♥s ♦s

♠♥t♦s rr♦ ② ♦ s♣r♣♦♥r ♦s rstr♦s ♥s st

♠♥r ♠ás s t ♦♣♠♥t♦ ♥tr ♦s ♠♥t♦s ♠s♦rs st

r♥t ♠ét♦♦ ♦ ♠♠♦s ♠ét♦♦ rr♦s s♥tét♦s ♠s♦rs

s②♥tt ♠ttr rr②

♠t♦♦♦í q s sr ♦♥t♥ó♥ ♦♥sst ♥ ♦s t♣s q

♦♠♣r♥♥ ♦t♥ó♥ t♦s ② ♣r♦s♠♥t♦ ♦s ♠s♠♦s ♦s ♣

rá♠tr♦s s♦♥♦ s♦♥ ♥ú♠r♦ ♣♦s♦♥s ♠s♦r ♥ q s ♥

qrr t♦s N ② s♣ró♥ ♥tr s st♥ts ♣♦s♦♥s ♠s♦r

d P♦r s♠♣ s s r ♠s♠♦ ♦r ♣r s s♣r♦♥s

s ♣♦s♦♥s ♠s♦r ② r♣t♦r ♦ q s s♣♠♥t út ♥♦ s

tr ♥ ♦rt♦r♦ ♦ ♠♣♦

Pr ♦t♥ó♥ s tr③s s r♣t♦r ♥ ♣r♠r ♣♦só♥

í♥ s♦♥♦ ♠s♦r ♠♥♦r st♥ ♣♦s ② s ♦t♥ ♥

tr③ ♦ s s♣③ ♥t♥ ♠s♦r ♥ st♥ d s ♣♦só♥ ②

s qr ♥ s♥ tr③ ♣r♦♠♥t♦ s r♣t st ♥③r

♠á①♠ s♣ró♥ ♣rst ♥tr s ♥t♥s ♠s♦r ② r♣t♦r r r

r sq♠ ♦s ♣s♦s q rqr ♠♣♠♥tó♥

♠ét♦♦

sí s ♦t♥ ♥ ♦♥♥t♦ tr③s s♦s ♣r♠r ♣♦só♥

r♣t♦r r r

♦ s s♣③ r♣t♦r ♥ st♥ d s♥t ♣♦só♥ s♦r

í♥ s♦♥♦ s ♦♦ ♠s♦r ♠♥♦r st♥ ♣♦s ② s r♣t t♦♦

♣r♦♠♥t♦ ♦♥ ♦ s ♦t♥ ♥ s♥♦ ♦♥♥t♦ tr③s s♦s

s♥ ♣♦só♥ r♣t♦r ♦♥t♥ú♥ r③♥♦ ♦s ♠s♠♦s ♣s♦s st

rr t♦ í♥ s♦♥♦ ♦♥ ♦ q s ♥③ t♣ qsó♥

♥ t♣ ♣r♦s♠♥t♦ s t♦♠ ♦♥♥t♦ t♦s s♦♦s

♥ s ♣♦s♦♥s r♣t♦r ② s ①tr♥ s tr③s q ♦rrs♣♦♥♥

♥ú♠r♦ ♥ts s♦♥♦ ♥ ♠♣♦ ♦♥sr♦ s ♣ ♦srr

q t♠♣♦ ♥ q sñ r♣t♦r s ♥r♠♥t ♠♥tr

s♣ró♥ ♠s♦r r♣t♦r r s♦r ♥ s tr③s s

r③ ♥ s♣③♠♥t♦ t♠♣♦r ♣r♦♣♦ s ♣♦s ♦rr q s sñs

r①ó♥ ♥ ♥ s ♥ ♥ s r♣t♦r ♠♥r q

s♠r s tr③s s ♦t♥ ♥ sñ ♠②♦r ♠♣t q s sñs

s ♥ts ♥s r ♥ s♥t ♣s♦ ♣r♦s♠♥t♦

s ♣ st t♣♦ s♣③♠♥t♦ t♠♣♦r ♥ s tr③s ②

♥♠♥t s s s♠ ♣r ♦t♥r ♥ ú♥ tr③ s♦ ♣♦só♥

r♣t♦r r s tr③s ♦t♥s ♣r ♥ s ♣♦s♦♥s

r♣t♦r s rú♥♥ ♣r ♦t♥r ♥ rrr♠ r

♦s s♣③♠♥t♦s t♠♣♦rs s ♥♥ ♠♥r q ♣r ♠és♠

♥t s♣③♠♥t♦ t♦t sté ♦ ♣♦r ∆tm = mdt ♦♥ − (N−1)2

≤ m ≤(N−1)

2s N s ♠♣r m = 0 s rr ♥t ♥tr rr♦ q

s ♣ ♥ s♣③♠♥t♦ t♠♣♦r ∆t0 = 0 ♥ ♥ s♣ró♥

♠s♦r r♣t♦r q♥t ♣r rr♦ ♦♠♦ s♣ró♥ ♥tr ♥tr♦

rr♦ ② r♣t♦r

♦♠♦ ♠t♦♦♦í ♣r s♦♥r dt s ♥r ♥ sr rrr♠s

♣r st♥t♦s ♦rs st ♣rá♠tr♦ ② ♥tr ♦s s s♦♥ q ♥

q sñ s ♦♥ ♠②♦r r

♣ó♥ ♠ét♦♦ ♣♥ só♥ ♦s ♣rá♠tr♦s

rr♦ ♥ s♦ st♦s ♥ sr ♠♥t ♦s ♣r ♦t♥r

♥ ♠♦r t ♥ rs♣st P♦r ♦ t♥t♦ s♥t ♣s♦ ♣r ♦♠

♣r♥r ♥♦♥♠♥t♦ ♠ét♦♦ ♦♥sst ♥ ♥③r s rtrísts

♦s ♠♣♦s tr♦♠♥ét♦s ♣r♦♦s ♣♦r rr♦s s♥tét♦s ♠s♦

rs ♦s s♦r s♣r r trr ② str ó♠♦ s ♠♦♥ ♦s

♠♣♦s rr ♦s ♣rá♠tr♦s rr♦

rtr③ó♥ ♠♣♦ rr♦

♠♣♦ tr♥s♠t♦ ♣♦r ♥ ♥t♥ ♦rr ♣♥ s rt

rísts ② ♦r♥tó♥ ♠s♦r ② s rtrísts ♦♥sttts ② ♦♠é

trs ♥trs r s♦ ② s ♣s ♠ás ♣r♦♥s ♥ s♦

rr♦s ♠s♦rs ♦s ♠♣♦s rst♥ts t♠é♥ ♣♥♥ ♦s ♣rá♠

tr♦s rr♦ ♥t ♠♥t♦s N ♣♦s♦♥s rts d ♠♣ts

da ② s♣③♠♥t♦s t♠♣♦rs dt st♦s ♣rá♠tr♦s s ♥ r

♠♥r ♣r rr r♥ ③ tr♥s♠t♦ ② rr

♦ s ♣♦s♦♥s st♠s ♥♦ ♦♥♥tr♥♦ st ♠♥r

♥rí s♦r é Pr r③r ♥ só♥ ♦s ♣rá♠tr♦s

rr♦ s ♣rs♦ ♦♥♦r ♠♥r ♥ q st♦s ♠♦♥ s rtrísts

♠♣♦

♥ só♥ s ♠♦stró q ♥♦ s ♦♥sr s♦ s♠♣

rr♦s ♦r♠♦s ♣♦r ♣♦♦s ♠ ♦♥ ♠t♥♦ ♥ ♥ ♠♦ ♥♦r♠

s ♣♦s ♦t♥r ①♣rs♦♥s ♥íts ♣r ♦s ♠♣♦s tr♦♠♥ét♦s

♥r♦s ♣♦r rr♦ ♥ ró♥ ♠♣♦ ♥♦

Pr♦ ♥ ♣ó♥ ♠ét♦♦ ♣r♦♠ s ♠ás ♦♠♣♦ ♦s

♠♣♦s s♦♥ ♠t♦s s ♥trs r s♦ ② s ♣r♦♣♥ trés

ss♦ ♦♥ ♥trtú♥ ♦♥ rs♦s strt♦s ♦t♦s ♥ts r

♥♦ s ③ st♦s ♥♦s ♣♥ str ♦s ♣r♦♥s q

♦rrs♣♦♥♥ t♥t♦ s r♦♥s ♠♣♦ r♥♦ ♦♠♦ s ♠♣♦

♥♦ ♦♠♦ ♥♦ ①st ♥ s♦ó♥ ♥ít ♣r ♣r♦♠ ♦♠♣t♦ s

rqr ♥ s♦ó♥ ♥♠ér

Pr ♦t♥r s rtrísts ♣r♥♣s ♠♣♦ rst♥t s ♦♥s

ró ♥ ♠♦♦ s♠♣♦ ♥ q ♦s ♠♣♦s s♦♥ ♥r♦s ♣♦r rr♦s

♣♦♦s ♠ ♦♥ ♦s ♥ ♥trs ♣♥ ♥tr ♦s ♠♦s ♥

♦r♠s q r♣rs♥t ♥trs r s♦ z = 0 ♦♠♦ s ♠str ♥

r ♦s s ♦s ♣♦♦s s♦♥ ♣r♣♥rs í♥ s♦♥

♦ ♦r♥t ♦ r♦ x st♥ ♥tr ♣♦♦s s ♥♦t ♦♥ d

r ♦♠trí rr♦ ♠s♦r st♥ ♥tr

♠♥t♦s s ♥♦t ♣♦r d

♦♥t rr♦ = (N − 1)d

Pr tr st♦ ♦s ♠♣♦s ♠t♦s ♣♦r rr♦ ♥ ♣

ó♥ ♠ét♦♦ s srr♦ó ó♦ ♦♠♣t♦♥ ♥t♦r♥♦ st

♣r♦r♠ ♣r♠t ♦t♥r ♠♣♦ ♥ rr♦ ♥ ♥ tr♠♥♦ ♠♦

t③♥♦ rs♣st ①♣r♠♥t ♦ s♠ ♣r ♥t ú♥ ♠s♠♦

♥t ♦♥ ♥ ♥t♦r♥♦ rá♦ ♠ ② ♦♥ rr♠♥ts ♣r ♥áss

♦s rst♦s r ♣é♥

Pr r③r ♦s á♦s ♥♠ér♦s ♠♣♦ ♥ st só♥ s r♦♥

♦rs ♣r♦♠♦ s♦s r♥♦s♦s ♦♥ ♥ ♣qñ♦ ♣♦r♥t r

s♠rs ♦s ♦s ♠♣♦s ♥ ♣ít♦ ♥tr♦r ♠♦ z < 0 r♣rs♥t

♥ ♣ r ② s rtr③ ♣♦r ♥ ♣r♠t rt ǫr = 1 ② ♥

♦♥t σ = 0 ♠♦ z > 0 s rtr③ó ♣♦r ǫr = 4 σ = 1

♠♠ Pr r♥ ♥tr ♠só♥ s ó fc = 500 ③ ♣rí♦♦

t♠♣♦r τ = 2 ♥s q s ♥ ♦r tí♣♦ ♦s q♣♦s ♦rr ♦♥t

♦♥ λ = 0,3 ♠ Pr ♦t♥r s♦ó♥ ♥♠ér ♣r♦♠ s ♣ó

♠ét♦♦ r♥s ♥ts ② ♠♣♠♥tó♥ s t ♥ ♣é♥

ó♦ t③♦ s ♣tó ♣r ♦t♥r ♣trs t♠♣♦ ♦♥st♥t

♦s t♦s ♠♣♦ étr♦ ♥ ♥ó♥ ♣♦só♥ (x, z)

♥ r s ♠str ♠♣♦ étr♦ ♦♠♣♦♥♥t Ey ♣r♦♦

♣♦r ♥ ú♥♦ ♣♦♦ ♦ ♥ ♦r♥ ♦♦r♥s ♣r ♥ t♠♣♦ ♦

t = 10 ♥s t = 5τ s♣és ♠só♥ ♥ r s ♣ ♦srr

q ♦♠♦ ♦♥s♥ ♣rs♥ ♥trs r s♦ ♦♥

tr♥s♠t ♥ ♠♦ ♣rs♥tó ♠♣t ♣r♦①♠♠♥t ♦♥st♥t ♥

♠②♦r ♣rt r♥♦ ♥r ♦♥ ♦s ♣qñ♦s ♠á①♠♦s ♦s ♣r á♥♦s

♣r♦♣ó♥ ♣r♦①♠♠♥t 48♦ ♦♥ rs♣t♦ rt

Pr ♦t♥r ♠♣♦ t♦t rr♦ s só ♣r♦♠♥t♦ ♣r♦♣st♦

♥ só♥ s r s s♣r♣sr♦♥ ♦s ♠♣♦s ♥ts ♥s

s ♥ ♥ s ♣♦s♦♥s rr♦ ♦s ♠♣♦s ♥r♦s

♣♦r ♥ts ♥ ♣♦s♦♥s st♥ts ♦r♥ ♦♦r♥s s ♦tr♦♥

r③♥♦ ♥ trsó♥ ♥ ró♥ x ♦s ♠♣♦s ♦t♥♦s

♣rtr s♠ó♥ ♥♠ér ♣r ♥t ♥ ♦r♥

♦♦r♥s ♥ r s ♠str ♠♣♦ ♥r♦ ♣♦r ♥ rr♦

♣♦♦s ♠t♥♦ s♠tá♥♠♥t rr♦ stá ♥tr♦ ♥ ♦r♥

st♥ ♥tr ♣♦♦s s d = 0,3λ ♥ st s♦ s ♦♥sró ♥ s♣③♠♥t♦

t♠♣♦r dt = 0τ ② ♠♣ts rts da = 1 ♣r t♦♦s ♦s ♠♥t♦s

rr♦ Pr ♥ ♦♠♣ró♥ s♠♣ í♥ ♣♥t♦s ♥ r ♥

r♥t ♦♥s ♥ ú♥ ♥t

♦♠♣r♥♦ s rs ② s ♣ ♦srr q ♠♣♦

rr♦ s ♠ás ♥t♥s♦ ② ♦♥♥tr♦ r♦r ró♥ ♥♦r♠

♥trs q ♠♣♦ ♥ s♦ ♥t Pr á♥♦s ♥ ③♦♥ ♣rér

♠♦s r♥ts ♦♥s t♥♥ rr ♦tr q t♦ ♦♥♥tró♥

s ♠ás ♥t♥s♦ ♥♦ s ♥r♠♥t N ♣♦r ♠♣♦ ♣r N = 5

r ♠♥trs q ♦s ♦s ♠á①♠♦s ♦s q s ♦sr♥ ♥ r

N = 1 s♣r♥ s ③ r♥t rst♥t rr♦ t♥

trs ♥ s ③♦♥ ♠ás ♥t♥s ♥♦ ♥ú♠r♦ ♥ts s ♥r♠♥t

s s♦♥s ♦♥ rs♣t♦ ♦♠♣♦rt♠♥t♦ ♥ ú♥ ♥t t♠

é♥ s ♥r♠♥t♥ ♥♦ ♣♥t♦ ó♥ s r rr♦ ♠♥

t♥♥♦ ♦s rst♦ ♦s ♣rá♠tr♦s ♦♠♦ ♥ ①♣ó♥ ♣r♦①♠

st ♦♠♣♦rt♠♥t♦ s ♣ ♦♥srr q rr z s ♦♥ts ♦s

♠♥♦s ♥s s ♣♦♦ ♣♥t♦ ó♥ r♥

③ ♠ás ♦♥ rs♣t♦ ♦♥t ♦♥ ♠♣♦ tr♦♠♥ét♦ q s

♣r♦♣ ♣♦rq ♦s ♠♣♦s ♥s ♥trr♥ ♠♥r ♠ás ♦♠♣

② ♦♥ rst♥t s t♦r♥ ♠ás ♦♠♣ st t♦ s ♣ ♦srr

♣♦r ♠♣♦ ♦♠♣r♥♦ s rs ② s s ♦rrs♣♦♥♥

t = 2,5τ t = 5τ ② t = 7,5τ ♦♥ N = 7 d = 0,5λ ② dt = 0τ

♠♥trN ♥♦ ③♦♥ ♥tr ♠♣♦ rr♦ s♠♥②

♥♦ ♠s♠♦ s ♥ st♥ ♥s ♣♦s ♦♥ts rr♦

r ♠♣♦ étr♦ Ey tr♥s♠t♦ ♥ t♠♣♦ ♦

t = 5τ ♦ ♠só♥ ♣r ♥ ú♥♦ ♣♦♦ ♣r ♥

rr♦ ♣♦♦s ② ♣r ♥ rr♦ ♣♦♦s ♦♥ dt = 0τ

② d = 0,3λ ② ♠♣ts rts da = 1 s í♥s ♣♥t♦s

♥♥ ♦r♠ r♥t ♦♥s ♥ ú♥ ♥t

r ♠♣♦ étr♦ Ey tr♥s♠t♦ ♣♦r ♥ rr♦ ♦♥

N = 7 d = 0,5λ ♥ t♠♣♦ ♦ t = 2,5τ t = 5τ ② t = 7,5τ

♦ ♠só♥ ♦♥ dt = 0τ s í♥s ♣♥t♦s ♥♥

♦r♠ r♥t ♦♥s ♥ ú♥ ♥t

♦ ♠ás ♦s ② s ♥r♠♥t ♥♦ s ú r rr♦ ♣r♠r t♦

♦rr ♣♦rq ♠♣♦ tr♥s♠t♦ ♣rs♥t ♣r♦①♠♠♥t ♦r♠ ♥

③ r♥t ♦♥ r♥ q s♠♥② ♠♥tr ♦r N

s♥♦ t♦ s ♦♥s♥ ♥r♠♥t♦ t♠ñ♦ rr♦ ♦ q

♥tr♠♥t ♣r♦ ♥ ♠♣♦ ♠ás ♥♦ r ♦s ♠♥t♦s

♥♦ s ♥③ ♣♥♥ ♦♥ d ♠♥t♥♥♦ ♦s ♦s ♦tr♦s ♣rá

♠tr♦s s ♣ ♦srr q ③ tr♥s♠t♦ s ♠♥♦s r♥t ♣r

♦rs r♥s st ♣rá♠tr♦ st ♠♥r ♠♣♦ rst♥t s

♥♦st ♥ s♣rs ó♥ s rr♦ st t♦ s ♣

♦srr ♦♠♣rr s rs ② s q ♦rrs♣♦♥♥

d = 0,1λ d = 0,3λ ② d = 0,5λ ♦♥ N = 7 ② dt = 0τ ♥ ♥ t♠♣♦ ♦

ó♥ t = 5τ ♣rtr st ♦♠♣ró♥ s ♣ ♥♦tr ♠ás q

♦s ♠♣♦s ♥s s ♥ ♥ts ♥ s ③♦♥s ♣rérs ♦

s r♥s r♥s ♥tr s ♦♥ts ♠♥♦ ♥s ♠♥trs

q ♣rt ♥tr ♠♣♦ rr♦ r ♠ás ♦♥ rs♣t♦ r♥t

♦♥s ♥ ú♥ ♥t

❯♥ s♣③♠♥t♦ t♠♣♦r dt ♥ rr♦ t♠é♥ t♥ ♥ t♦ s♦

r r♥t ♦♥s rst♥t P♦r ♠♣♦ ♥ r s ♠str

♠♣♦ ♥r♦ ♣♦r ♥ rr♦ ♦♥ N = 7 d = 0,3λ ② ♥ s♣③♠♥t♦

t♠♣♦r dt = 0,05τ ♥tr ♦s ♠♥t♦s ♣ ♦srr q t♦ ♠ás

♥t ♣r ♥ s♣③♠♥t♦ t♠♣♦r s q ♠ ró♥ ♥

q s ♣r♦♣ ③ tr♥s♠t♦ á♥♦ tr♥s♠só♥ ♥♦ s

ró♥ ♠á①♠♦ ♠♣♦ st ró♥ ♥♦r♠ ♥trs ♥

st s♦ s −9,6♦ ♠é♥ s ♣ ♥♦tr q r♥t ♦♥s tr♥s♠t♦

♣r♦①♠♠♥t ♠♥t♥ s ♦r♠ ♦r♥ ② q s ③♦♥ ♠á①♠ ♥t♥

s s s♣③ tr♠♥t s♦r r rr♥ ♥ ú♥♦ ♠s♦r

♥ r s ♣ ♦srr ♥ rá♦ á♥♦ ♦r♥tó♥

♠♣♦ ♥ ♥ó♥ s♣③♠♥t♦ t♠♣♦r dt ♥tr s ♥ts

♥ rs♠♥ ♥tr ♦s rst♦s ♦t♥♦s s ♣ str ♣♦r ♥ ♦

q ♥ó♠♥♦ ♦♥♥tró♥ s r♦♥ ♦♥ ♦s ♦rs N ② d ② q

♦r♥tó♥ ♠♣♦ s r♦♥ ♦♥ ♦r dt s ♦♥sr♥ ♦s

♠♣♦s q ♠t ♥ ♦♥♥t♦ ♥ts s ♥ ♥trs ♥tr ♦s

♠♦s ② ♥ ró♥ ♠♣♦ r♥♦ s ♦sr q s ♣♦s ♦♥tr♦r

♦♥♥tró♥ ② ♦r♥tó♥ ♠♣♦ ♠♥r s♠r ♦ q ♦rr ♥

í♦ ♣r♦ ♦♥ rts rtrísts ♣rtrs q s♦♥ ♦♥s♥

r ♠♣♦ étr♦ Ey tr♥s♠t♦ ♣♦r ♥ rr♦

♦♥ N = 7 ♥ t♠♣♦ ♦ t = 5τ ♦ ♠só♥ ♦♥

dt = 0τ ② d = 0,1λ d = 0,3λ d = 0,5λ s í♥s

♣♥t♦s ♥♥ ♦r♠ r♥t ♦♥s ♥ ú♥ ♥t

r ♠♣♦ étr♦ Ey tr♥s♠t♦ ♣r ♥ rr♦

♦♥ N = 7 d = 0,3λ ♦♥ s♣③♠♥t♦ t♠♣♦r dt = 0,05τ

♥tr ♠♥t♦s

r ♥♦ ♦r♥tó♥ ♠♣♦ étr♦ Ey ♥

♥ó♥ s♣③♠♥t♦ t♠♣♦r ♣r ♥ rr♦ ♦♥ N = 7

d = 0,3λ ♥ ♥ t♠♣♦ ♦ t = 5τ

♣rs♥ ♥trs ② r♥í ♥tr ♣♥t♦ ó♥

♠♣♦ ② ♥t ♥ st s♦ rr♦

s♣st ♠ét♦♦ ♥ s♦s ás♦s

♦♥♦s s rtrísts ♠♣♦ ♥ ♥ó♥ ♦s ♣rá♠tr♦s

rr♦ s♥t ♣s♦ ♦♥sst ♥ ♥③r rs♣st q ♣r♦ ♠é

t♦♦ ② qé ♠♥r s ♦t♥♥ sñs ♠♦rs Pr ♦ s st♥

♠♥t s♠ó♥ ♥♠ér ♦s ♠♣♦s ♦s st♦♥s áss

♣r♠r ♥② ♥ ♦t♦ ♣qñ♦ ② s♥ ♥ rt♦r ①t♥s♦ ♦♠♦ s

rá ♠ás ♥t ♦tr♦s s♦s ♠ás ♦♠♣♦s s ♣♥ ♥tr♣rtr ♥ tér♠♥♦s

st♦s

Pr ♦t♥r rstr♦s s♠♦s ♦rr s ♦♥sró ♥ ♠♦♦ s♠

♣♦ ♥ q ♦s ♠♣♦s s♦♥ ♥r♦s ♣♦r ♣♦♦s ♠ ♦♥

♦s ♥ ♥trs ♣♥ ♥tr ♦s ♠♦s q r♣rs♥t ♥trs r

s♦ ② ♦s s ♦s ♣♦♦s s♦♥ ♣r♣♥rs í♥ s♦♥♦

t③ó ♠ét♦♦ r♥s ♥ts ② ♠♣♠♥tó♥ s t ♥

♣é♥ ó♦ s ♣tó ♣r q ♦r♠t♦ s ♦s t♦s

♦rrs♣♦♥ q♦s qr♦s ♦♥ q♣♦s ♦rr st♦ ♣r♠tó

trtr ♦s t♦s s♠♦s ♦♥ ♦s ♣r♦r♠s ♣r♦s♠♥t♦ ② s③ó♥

srr♦♦s ♣r♠♥t ♥tr♦ ♠r♦ ss

❯♥ ③ q s tr♠♥♥ ♦s ♣rá♠tr♦s ♠♦♦ s ♦♦ ♠s♦r

♥ ♥ ♣♦só♥ s♦r s♦ ② s s♠ rs♣st ♦t♥♥♦ tr③s

♣r ♥ ♥ú♠r♦ ♣♦s♦♥s r♣t♦r s ♠♥r t rr

t♦♦ r♥♦ ♦sts t③r ♦ s r♣t ♣r♦♠♥t♦ ♣r ♦trs

♣♦s♦♥s ♠s♦r ♥♠♥t s r♣♥ qs tr③s ♦♥ ♣♦s♦♥s

♦♠♥s r♣t♦r ♦t♥♥♦ sí rstr♦s s♠♦s ♥á♦♦s ♦s ♦t♥

♦s ①♣r♠♥t♠♥t ♣rtr st ♣♥t♦ s s ♣r♦♠♥t♦ ♣r

♦t♥r rs♣st sr♣t♦ ♥ só♥ s t♦♠♥ s tr③s q

♦rrs♣♦♥♥ ♥ s ♥ts rr♦ s ♣ s♣③♠♥t♦

t♠♣♦r s s♠♥ s tr③s ② ♦s rst♦s s ♦r♥♥ ♥ ♥ rrr♠

♣r ♦t♥r rs♣st t♦t

tr sñr rr♦ ♣r♦♣♦ ♣r s♦ ♠♣ str

r♦s ♣rá♠tr♦s ts ♦♠♦ st♥ ♥tr ♦s ♦♠♣♦♥♥ts rr♦

r t♦r ♦♥ 0,1 ♠ á♠tr♦ ♦③♦ ♥

♣r♦♥ 0,75 ♠ ♠♦ r♥♥t s rtr③ ♣♦r

ǫr = 4 ② σ = 1 ♠♠ ♦t♦ s rtr③ ♣♦r ǫr = 5 ②

σ = 2 ♠♠ ♣ ♥ tó♥ t♦r st♦s

♣rá♠tr♦s

② ss s♣③♠♥t♦s t♠♣♦rs rt♦s ❯s♠♥t ♠♣t rt

s t♦♠ ♦♠♦ ♦♥st♥t ♣r s♠♣r ♣r♦s♦ Pr r③r ó♥

♦s ♣rá♠tr♦s s ♥ ♦♥srr s ♦♥♦♥s s♦♥♦ ts ♦♠♦

s ♣r♦♣s ♠♦ ♣r♦♥ ♦s ♥♦s ② r♥

♠só♥ q♣♦ ♦rr Pr tr ♠♣♠♥tó♥ ♠ét♦♦

s srr♦ó ♥ ♠ó♦ ♥♦ ♥ ó♦ ♦♠♣t♦♥ ♥t♦r♥♦

st ♠♥r ó♦ ♣r♠t sñr ♦s rr♦s ♣r ♦♣t♠③r s

rtrísts ♠♣♦ s♦r ♥♦ ② ♦t♥r rs♣st ♠ét♦♦

t③♥♦ t♥t♦ t♦s ①♣r♠♥ts ♦♠♦ s♠♦s ♠ó♦ ♥t ♦♥

♥ ♥t♦r♥♦ rá♦ ♠ ② ♦♥ rr♠♥ts ♣r ♦♥tr♦r s③ó♥

♦s rst♦s q s♠♣♥ ♥♦t♠♥t ♣ó♥ ♠ét♦♦ r

♣é♥

♦♠♦ ♣r♠r s♦ ás♦ s ♥③ r①ó♥ ♥ ♥ ♦t♦ ♣qñ♦

♠♦♦ ♦♥sr♦ s ♠str ♥ r ♦t♦ t♥ 0,1 ♠ á

♠tr♦ ② s ♥♥tr ♥ ♣r♦♥ 0,75 ♠ ♠ s s♣r

st ♣rt s♣r♦r ♦t♦ ♣r♠t rt ② ♦♥t

s♦ s♦♥ ǫr = 4 ② σ = 1 ♠♠ rs♣t♠♥t ♠♥trs q ♣r

♦t♦ s♦♥ ǫr = 5 ② σ = 2 ♠♠ ♣ ♥ tó♥ t♦r

♥ ♦s ♣rá♠tr♦s ♦♥sttt♦s ♠♦ ② ♦t♦ ♦ r♦ t♦

r ♠♣♦ ♠t♦ t♥ r♥ ♥tr 500 ③ ♦♥t ♦♥

♥ ♠♦ s λ = 0,3 ♠

♦♠♦ ♠t♦♦♦í ♥r ♣r r ♠ét♦♦ rrr♠

r rr♠ ♣r ♥ ú♥ ♥t ♦♥ s♣r

ó♥ 0,5 ♠ rr♠ ♣r ♥ rr♦ ♦♥ N = 5

d = 0,1 ♠ ② dt = −0,2 ♥s dt = 0,0 ♥s dt = 0,2 ♥s

r ♠♣♦ étr♦ ♥ ♥ t♠♣♦ t = 7 ♥s ②

rs♣st ♦t♥ ♣r N = 1 ♠♣♦ étr♦ ♥ ♥ t♠♣♦

t = 7 ♥s ② rs♣st ♦t♥ ♣r ♥ rr♦ ♦♥ N = 5

♠♣♦ étr♦ ♥ ♥ t♠♣♦ t = 7 ♥s ② rs♣st ♦t♥

♣r ♥ rr♦ ♦♥ N = 10 ♥ t♦♦s ♦s s♦s d = 0,1 ♠ ②

dt = 0,0 ♥s í♥ ♣♥t♦s ♥ s rs ② r♣rs♥t

♦r♠ r♥t ♦♥s ♣r N = 1 ② ♥ s rs ②

r♣rs♥t ♦r♠ sñ ♦t♥ ♣r N = 1

rst♥t ♠s♠♦ s ♦♠♣r ♦♥ ♦t♥♦ ♣rtr ♥ ♠t♦♦♦í

♦rtr s♠♣ r ♠str ♥ rrr♠ ♣r ♥ ú♥♦ ♠s♦r

♦♥ ♦st ♦ 0,5 ♠ ♣r ♠♦♦ r ♦♦r♥ x ♥

rá♦ s rr ♣♥t♦ ♠♦ ♥tr s ♣♦s♦♥s ♠s♦r ② r♣t♦r

♥ r s ♣ ♦srr ♥ sñ ♦♥ ♦r♠ tí♣ ❱ ♥rt ♥

sñ ró♥ q ♦rrs♣♦♥ r①ó♥ ♥ ♦t♦ ♥trr♦

r ♠str rrr♠ q rst ♥ rr♦ ♦♥ N = 5

d = 0,1♠ ② dt = −0,2 ♥s ♥ st s♦ s♣ró♥ ♥tr r♣t♦r ② ♥t

♥tr rr♦ s 0,5 ♠ s♣③♠♥t♦ t♠♣♦r st♥t♦ r♦

q ♠♣♦ s ♣r♦♣ ♦ r♦ ró♥ θ = 18 ♦♥ rs♣t♦

ró♥ ♥♦r♠ ♥trs r s♦ ♦♦r♥ x s rr ♣♥t♦

♠♦ ♥tr s ♣♦s♦♥s ♥tr♦ rr♦ ② r♣t♦r ♦♠♣r♥♦

s rs ② s ♣ ♦srr q ♥ s♦ rr♦ sñ

r①ó♥ s ♥t♥s ♥ ♥tr♦ x = (−0,8;−0,2) ♠ ② s t♥ú r

é st♦ ♦rr ♣♦rq ③ tr♥s♠t♦ ♠♥ ♥tr♠♥t r♣♦

♥ s ♥tr♦ ② só♦ ♦ ♠♥ ♠r♥♠♥t r é s r q s

♣♦ ♠♦rr ♣rt sñ ♥r♠♥tr r♦♥ ♠♣♦

tr♥s♠t♦

♥ s rs ② s ♠str rst♦ rr ♦r

dt ♠♥t♥♥♦ ♦s ♦s ♦tr♦s ♣rá♠tr♦s rr♦ st♦ s q♥t

♠r á♥♦ tr♥s♠só♥ ♦s ♦rs q s t③r♦♥ dt r♦♥ 0,0

♥s ② 0,2 ♥s ♣ r ó♠♦ s rst♥ ♣♦r♦♥s ♦♥sts sñ

ró♥ ♠r dt

♦♠♦ s ♠♦stró ♥ só♥ r♥t ♦♥s ♠♣♦ ♥r♦

♣♦r rr♦ s ♣r♦rs♠♥t r♥t ♦♥ rs♣t♦ s♦ ♥

ú♥♦ ♠s♦r ♣r ♦rs r♥s N Pr ♥③r t♦ s♦r s

ñ s ♦♠♣r rs♣st ♦t♥ ♣r ♥ rr♦ ♦♥ N = 5 s♦ ♥

q s ♦r♠♦♥s ♥ ♠♣♦ s♦♥ ♣qñs ② ♦♥ N = 10 ♥ q s

♦r♠♦♥s s♦♥ ♣rs Pr ♠♦s s♦s d = 0,1 ♠ ② dt = 0 ♥s

♥②♥ ♦s rst♦s ♣r s♦ N = 1 rs ② ♣r tr

♦♠♣ró♥ ♥ r s ♣ ♦srr q ♣r N = 5 ♠♣♦

♣rs♥t ♦r♠♦♥s ♣qñs ♥ ③♦♥ ♠②♦r ♥t♥s ♥ ♦♥s

♥ ♦r♠ tr♠♦ sñ q s rst ♦♥ ♣r♦①♠♠♥t

♦♥ ♦r♠ sñ ♣r ♥ ú♥ ♥t ♦♠♦ s ♣ r ♥ r

í♥ ♥ ♦r♠ sñ ♦t♥ ♣r N = 1 P♦r ♦tr♦ ♦

r ♦♦ ♦♥ ♥ rt♦r ①t♥s♦ ♣ s♣r

s rtr③ ♣♦r ǫr = 4 ② σ = 1 ♠♠ ② ♣ ♣r♦♥ ♣♦r

ǫr = 5 ② σ = 2 ♠♠ ♣ ♥ tó♥ t♦r

st♦s ♣rá♠tr♦s

s ♦r♠ r♥t ♦♥s rr♦ r s♥t♠♥t r♥t

♦♥s ♥ ú♥ ♥t r ③♦♥ q s rst ♥♦ ♦♥

♦♥ ♦r♠ sñ q s ♦t♥ ♣r ♥ ú♥ ♥t r ♥

♥r s ♥ ♥r♠♥t♦ N ♥t♥s sñ ♥trés ♥ ró♥

♦♥ ♦trs sñs ♣rérs ♠ás á ♥ rt♦ í♠t ♦s rrr♠s r

st♥ts t♥♥ sr ♣♦♦ ♦♠♣rs ♦♥ ♦s ♥t ú♥ ② ♥ ♥r

♣rát♠♥t ♠♣♦ss ♥tr♣rtr

♦ s♠r ♦rr ♦♥ d ②♦ ♦r ♠á①♠♦ stá ♠t♦ ♣♦r ♣ró♥

♥rí q ♥♦ s tr♥s♠t ♥ ró♥ s♣r ♦♠♦ ♥ ♣♦s

s♦ó♥ s ♣♥ t③r rr♦s ♦♥ d ♣qñ♦ ② N rt♠♥t r♥

♦ q ♣r♦ ♦s t♦s ♣r♦♠♦ ♥t♦s tr♥t♠♥t ♥

♣rát N ② d s ♥ t♠é♥ ♠t♦s ♣♦r rr♦r ♥ ♣♦só♥

♦♥t í♥ ♣r♦s♣ó♥ ② t♠♣♦ s♣♦♥ ♣r

♦♠♦ s♥♦ ♠♣♦ s ♥③ s♦ r①♦♥s ♥rs ♣♦r ♥

rt♦r ①t♥s♦ ♥ r s ♠str ♠♦♦ ♣r♦♣st♦ ♣r

st♦ ♣ s♣r♦r s rtr③ ♣♦r ǫr = 4 ② σ = 1 ♠♠ ♠♥trs

q ♠ás ♣r♦♥ ♣♦r ǫr = 5 ② σ = 2 ♠♠ rt♦r stá ♦③♦

♥ ♣r♦♥ ♣r♦♠♦ 0,9 ♠ r♥ ♠t s 500 ③

λ = 0,3 ♠ s ♣ ♥ tó♥ t♦r ♥ t♦ r

♦s ♣rá♠tr♦s ♦♥sttt♦s

r ♠str ♥ rrr♠ ♣r ♥ ú♥♦ ♠s♦r ② ♦st ♦

0,5 ♠ ♣r ♠♦♦ r q ♥ ♠♣♦ ♥tr♦r

r rr♠ ♣r ♥ ú♥♦ ♠s♦r ♦st 0,5 ♠

rr♠ ♣r ♥ rr♦ ♦♥ N = 5 d = 0,1 ♠ ② dt = 0,00

♥s dt = 0,15 ♥s dt = 0,35 ♥s ♥ ♦♥ s s ③♦♥s

rst sñ

♦♦r♥ x ♥ rá♦ s rr ♣♥t♦ ♠♦ ♥tr s ♣♦s♦♥s

♠s♦r ② r♣t♦r r ♠str rrr♠ ♣r ♥ rr♦

♦♥ N = 5 d = 0,1 ♠ ② dt = 0,00 ♥s ♦ st ♦r dt ③

tr♥s♠t♦ s ♣r♦♣ ♦ r♦ ró♥ rt ♦♠♣r♥♦ s

rs ② s ♣ ♦srr q s ♥r♠♥tó ♠♣t

♣rt sñ q stá s♦ s♠♥t♦ ♥♥♦ ③qr♦

♥trs st ③♦♥ s ♥ ♦♥ ♥ ♥ r P♦r ♦tr♦ ♦ s

♣ ♦srr q rs♣st ♣r s♠♥t♦ ♥♥♦ r s

s ♥ st ♠♣♦ ♠ét♦♦ ♥t♥só ♥t♠♥t ♣rt

③qr ♥♥ sñ ② q ③ tr♥s♠t♦ s r s♦r ♣♥♦

rs♣t♦ rt♠♥t r♣t♦r ♦ ♠② r é st ♠♥r s

♦t♥ ♠á①♠ ♠♣t ♥♠♥t ♠ét♦♦ ♥♦ s t♦ ♣r ♠♦rr

s♠♥t♦ ♥♥♦ r ② ♦s s♠♥t♦s ♦r③♦♥ts ② q ♥

st♦s s♦s só♦ ♣rt ♣rér ③ r♦ ♥③ r♣t♦r

s ♣♦s rstr ♦trs ♣♦r♦♥s sñ r r♥♦ s

♣③♠♥t♦ t♠♣♦r dt P♦r ♠♣♦ s s ♣ dt = 0,15 ♥s s ♦t♥

rst♦ q s ♠str ♥ r ♥ q s rst♥ t♦s s

♣♦r♦♥s sñ s♦s ♦♥ ♦s s♠♥t♦s ♦r③♦♥ts rt♦r

s ♣ dt = 0,35 ♥s s ♦t♥ rst♦ q s ♠str ♥ r

♦♥ s ♣ rr ó♠♦ s rst ♣rt sñ s♦

s♠♥t♦ ♥♥♦ r♦ rt♦r ♥ ♥r ♦♠♦ ♥ s rs

② s ♣ ♦srr ♥ ♦♥♥t♦ sñs s♥rs q s ♥t♥s

♥ ♣r ♠ét♦♦ sts sñs ♣r♥ ♦♠♦ ♥s

♥♥ó♥ ♣r dt ♦♥sr♦ ♥ t♦ ♠ét♦♦ ♥♦ só♦ ♥t♥s

s sñs ♦rrs♣♦♥♥ts ♦s rt♦rs ♥trés s♥♦ t♠é♥ s ♣r♦

s ♥ t♦s s ♥trss q trs ③ ♣r♦♣rs ♥ st s♦

s t♦♥s ♦♥srs ♣r ♠tr③ s♦

♥ rs♠♥ ♦s ♣rá♠tr♦s rr♦ ② s ♣r♦♣s ♠♦ tr

♠♥♥ s rtrísts ♠♣♦ ♦♠♦ á♥♦ ♥ q s ♣r♦♣

♥♦ ♥r ② ♦r♠ r♥t ♦♥s ♥ ♦s ♠♣♦s ♥tr♦rs s

♣ r q ♦s ♦s ♣rá♠tr♦s rr♦ ② s ♣r♦♣s ♠♦ s

rst♥ s sñs t♦♦s q♦s ♥♦s ② ♦♠trí s ♣r

rr ♠♣♦ ♠t♦ ♣♦r rr♦ ♣♦só♥ r♣t♦r s r

q ♦s rst♦s q s ♦t♥♥ ♣r ♠ét♦♦ ♣♥♥ t♥t♦ ♦s

♣rá♠tr♦s rr♦ ♦♠♦ ♦♠trí ② ♣r♦♣s ♥♦ ②

♠♦ r♥♥t st♦ s ♦♥srr ♣r ♠ét♦♦ ♣r ♦t♥r

♥♦s rst♦s ♥tr♣rtr♦s ♦rrt♠♥t

Pr♦s♠♥t♦ ♣r ♠♦rr sñs ♦♠

♦s ♠♣♦s ♥tr♦rs ♠str♥ q ♠ét♦♦ s út ♣r ♠♦rr

r♥ts ♣rts sñ ♣r♠r ♥ ♦t♦ ♣qñ♦ ② ♥ rt♦r

①t♥s♦ rr á♥♦ tr♥s♠só♥ Pr♦ ♥ ♥r ♦ q s s

s ♠♦rr sñ ♦♠♣t st♦ s ♠♣♦rt♥t ♥ st♦♥s rs ♦♥

♠♣t sñ ♣r♠r ② ♦①st♥ sñs ♦♥ ♠♦rr

♦s ♥s ♥t♥s ② ♦♥t♥ tr sñ ♥trés ♦r

♥ ♠♦r ♥ó♥ s ♠s♠s ② ♣♦r ♥ ♥ ♥tr♣rtó♥ ♦rrt

♠tá♥♠♥t s s ♠♥t♥r ♦r♠ sñ q s ♦t♥ ♥♦

♥♦ s ♠♥ ♦♥ r♥t ♦♥s ♥ ú♥ ♥t ② q ♥ s

s♦ s s ♦♠♦ ♥tr♣rtr ♦s rst♦s ♠ás q ést♦s s ♣♥

♣r♦sr ♦♥ ♦s ♣r♦♠♥t♦s ② ♣r♦r♠s q s t③♥ t♠♥t

r♦ st♦s rst♦s ♦s ♣s♦s ♣r♦s♠♥t♦ q s ♣r♦♣♦♥♥

♣r ♠♦rr ♥ sñ ♥ t♦ s ①t♥só♥ s♦♥ ♦s s♥ts

♥rr ♥ sr rrr♠s ♣r r♥ts ♦rs dt t③♥

♦ ♣r♦♠♥t♦ sr♣t♦ ♥ só♥

♥ ♥♦ ♦s tr♠♥r ♦ ♦s s♠♥t♦s ♥ ♦s s sñ

stá r♠♥t ♠♦r ② ♦s ♥tr♦s ♣♦s♦♥s rs♣t♦s ♥

♦s q st♦ ♦rr

♣rr s ♣♦r♦♥s rrr♠ q ♦rrs♣♦♥♥ s♦s ♥tr♦s

r♥r s ♣rts rst♥ts ♦s rrr♠s ② ♥rs

♥ r s ♠str ♥ sq♠ ♣r♦♠♥t♦ ♣r♦♣st♦ Pr

♣♦r t③r ♠t♦♦♦í ♣r♦♣st ♦♥ ♥ ② ♥ ♦r♠ ♠ s

srr♦ó ♥ ♠ó♦ s♣ ♥♦ ♥ ó♦ ♦♠♣t♦♥ ♥t♦r♥♦

st ♥t ♦♥ ♥ ♥t♦r♥♦ rá♦ q ♣r♠t ♥rsr ② ♠♦r ♦♥

t♥t♦ ♦s s♠♥t♦s ♥ ♦s q s r♥♦ ♣♦s♦♥s ♦♠♦

r sq♠ ♣r♦♠♥t♦ ♣r ♠♥tr ♦♥

t♥ tr s sñs

r rr♠ ♦t♥♦ ♣r ♥ ♦t♦ ♣qñ♦

♥♥♦ ♣♦r♦♥s ♦s rrr♠s ♦rrs♣♦♥♥ts

N = 5 d = 0,1 ♠ −0,4 ♥s ≤ dt ≤ 0,4 ♥s á♥♦s tr♥s

♠só♥ ♥tr 41♦ ② −41♦ t③r♦♥ ♥tr♦s

r s♣③♠♥t♦s t♠♣♦rs s♦♥♦s ♣r ♦

t♥r rst♦ r

♦s ♣rá♠tr♦s rr♦ ♥ ♥♦ ♦s ♠♥r ♥t ♦♥ q

s ♦t♥ rs♣st ♥t♦ ♦♥ s rr♠♥ts ♣r str s③ó♥

♠s♠ t♥ ♣r♦s♦ st ♦s ♣rá♠tr♦s rr♦ ♣r

♦t♥r ♥ rs♣st ♥ q s ♠♦r sñ ♦♠♣t r ♣é♥

s ♣ ♣r♦♠♥t♦ sr♣t♦ ♦s rst♦s ♣r ♠♦♦

r só♥ s ♦t♥ rst♦ q s ♠str ♥

r t③ó ♥ rr♦ ♦♥ N = 5 d = 0,1 ♠ ② s s♦♥r♦♥

s♣③♠♥t♦s t♠♣♦rs ♥tr −0,4 ♥s ② 0,4 ♥s ♦s q ♦rrs♣♦♥♥

á♥♦s tr♥s♠só♥ ♥tr 41♦ ② −41♦ Pr ♦t♥r rst♦ q s

♠str ♥ r ♥ q s ♠♦r sñ ♦♠♣t ró♥

r rr♠ ♦t♥♦ ♣r ♥ rt♦r ①t♥s♦

♥♥♦ ♣♦r♦♥s ♦s rrr♠s ♦rrs♣♦♥♥ts

N = 5 d = 0,1 ♠ 0,00 ♥s ≤ dt ≤ 0,30 ♥s á♥♦s tr♥s

♠só♥ ♥tr 0♦ ② −30♦ t③r♦♥ ♥tr♦s

r s♣③♠♥t♦s t♠♣♦rs s♦♥♦s ♣r ♦

t♥r rst♦ r

s t③r♦♥ ♥tr♦s ♥ r s ♠str ♥ rá♦ ♦s ♦rs

s♣③♠♥t♦ t♠♣♦r ♥ ♥♦ ♦s ♠é♥ s ♥♦tr q s

s♦♥s ♥s ♦♥st♠♥t ♠♣♠♥ ♣rt♠♥t ♥tr s st♦

♦rró ♥ ♠♣♦ ♣♦r ♥ ♦ ♣♦rq s s♦♥r♦♥ ♠♥t ♦s

♣rá♠tr♦s rr♦ N ② d s r q r♥t ♦♥s tr♥s♠t♦

t♥í ♦r♠ s♠r r♥t ♦♥s ♥ ú♥ ♥t ♥ ♣♦só♥

♥♦ P♦r ♦tr♦ ♦ s s♦♥r♦♥ ♥r♠♥t♦s dt s♥t♠♥t ♦s

♠♥r q s rstr ♥ s ♣rts sñ ② s♠♥t♦s

♦s ♣r r♦♥strr rrr♠

s♥t ♣s♦ ♦♥sst ♥ ♣r ♠ét♦♦ ♣r♦s♠♥t♦ ♣r ♠

♦rr sñ ♦♠♣t ♥ rt♦r ①t♥s♦ ♥ r s ♠str

rst♦ ♣r st ♣r♦♠♥t♦ ♠♦♦ r só♥

♥ st s♦ s t③ó ♥ rr♦ ♦♥ N = 5 ② d = 0,1 ♠ s s♦♥r♦♥

s♣③♠♥t♦s t♠♣♦rs ♥tr 0,00 ♥s ② 0,30 ♥s ♦s q ♦rrs♣♦♥♥

á♥♦s tr♥s♠só♥ ♥tr 0♦ ② −30♦ ② s t③r♦♥ ♥tr♦s ♥ r

s ♠str ♥ rá♦ ♦s ♦rs s♣③♠♥t♦ t♠♣♦r ♥

♥♦ ♦s ♣ ♦srr q sñ r ♣rs♥t ♥ s♣t♦ s♠

r ♦t♥ ♣r ♥ ú♥ ♥t r só♥ ♠♥t♥♥♦

♥ ♥ ♦♥t♥ ♦r♥ ♦ ♠♦r ♣r♦ ♦♥ ♥ sñ

♦♥ ♠②♦r ♥t♥s st rst♦ str ♥ ♠ét♦♦ ♥ s♦

♥ rt♦r ①t♥s♦

♥ st♦♥s ①♣r♠♥ts s s q ♣rts s sñs ♥♦ ♣♥

sr r♠♥t ♥s ♦ ró♥ sñ r♦ ♥ ♥ só♦

s♠♥t♦ ♣ sr s♥t ♣r ♥r ♦s ♣rts s♣rs sñ ♥

♦ rt♦r ♥ ♠♦s s♦s s ♥sr♦ sr ♥ r♥ ♥ú♠r♦ ♦s

♥t ♣♥ ♦♠♣ rt♦r ❯♥ stó♥ ♦♠♣ ♦

rr ♥♦ ♥♦ stá r♦ ♦♥ ♦♥t♥ú ♥ sñ s♣és ♥ ár ♦♥s

s r ♦♥ á ♥t♦ s ♥r ♣♦r ♠♣♦ ♥♦ s s♣r♣♦♥♥

♥t♦s st♥t♦s strt♦s ♦♥ s♠rs ♥♥♦♥s ♥ st♦s s♦s ♥tr

♦s ♠s♦ ♣qñ♦s ♦ ♥r♠♥t♦s r♥s s♣③♠♥t♦ t♠♣♦r

♣♥ r rst♦s ♥♦rrt♦s ♥t♦♥s s ♦♥♥♥t s♦♥r ♥

r♥♦ ♠♣♦ ♣r dt ② ♥ ♥r♠♥t♦ t♥ ♣qñ♦ ♦♠♦ ♣rí♦♦ ♠s

tr♦ ♦ ♠♥♦r q ést ♥ s s♦ s tr③s rqr♥ ♥tr♣♦ó♥ trs

st♦♥s ♦♠♣s s♦♥ qs ♥ s q s s♣r♣♦♥♥ ♦♥ st♥t ♥

♥ó♥ ♣r♥t ♦s ♦ ♠ás sñs ♥tr♦ ♥ ♥tr♦ ♥ s♦s s♦s só♦

s ♣ ♠♦rr ♥ s sñs ♣♦r ③ ♦♥ st ♠t♦♦♦í

♣ít♦

♣ó♥ ♠ét♦♦

s♦s ♦♥ ♥♦s ♠út♣s

♥ ♣ít♦ ♥tr♦r s ♥③ó rs♣st ♦t♥ ♣r ♠ét♦

♦ ♦s s♦s s♠♣s ♥ ♦t♦ ♣qñ♦ ② ♥ rt♦r ①t♥s♦

♥ s♦s ♠♣♦s st ♣rs♥t sñ ♥ ú♥♦ ♥♦ ♦ s ♣r

s♥tó ♥ ♠t♦♦♦í ♦♠♣♠♥tr ♣r♦s♠♥t♦ rs♣st

q ♣r♠t t♠é♥ ♠♦rr ♦♥t♥ tr s sñs ② t

♥tr♣rtó♥ ♦s rst♦s

♥ st ♣ít♦ s st ♣ó♥ ♠ét♦♦ s♦s ♦♠♣♦s

♥ ♦s q stá♥ ♣rs♥ts rs sñs ♥t t♦s s♠♦s s ♠str

ó♠♦ ♣ ♣rs ♠ét♦♦ ♣r rstr ♥ s sñs ♥ ♦r♠

♥♣♥♥t ♦s rst♦s ♦t♥♦s s ♣♥ ♥ ♦s s♦s ①♣r♠♥ts

♥trés rq♦ó♦ ♦♥ t♦s qr♦s r♥t ♥ s♥ ♠♣ñ

r③ ♥ st♦ P♦ ♥♦ ♣r♠r s♦ ♦♥sst ♥ rtr③r ♥

♣r t♣ ♥trr ♣rt♥♥t ♥ s strtrs t♦♥s

tts ♥ st♦ s♥♦ s♦ ♦♠♣r♥ st♦ sñs

r①ó♥ ♣r♦s ♣♦r ss ♥trrs ❬❪ ❬❪

t♦s s♠♦s

♥ ♦s s♦s ①♣r♠♥ts s t q ♥ ③♦♥ st♦ s ♥

♥tr♥ ♣rs♥ts ♠út♣s ♥♦s ♥trés ♠ás ♦tr♦s ♦t♦s s♥

r ♦♦ ♦♥ ♦s rt♦rs á♠tr♦s 0,1 ♠ ǫr =

5 ② σ = 1,5 ♠♠ strt♦ s♣r s rtr③ ♣♦r ǫr = 4

② σ = 1 ♠♠ strt♦ ♣r♦♥♦ ♣♦r ǫr = 4,2 ② σ = 2,5 ♠♠

♣ ♥ tó♥ t♦r st♦s ♣rá♠tr♦s

r♦s Pr trtr s♦s ♥ ♦s q ①st♥ ♠út♣s sñs t

♥tr♣rtó♥ ♦s rst♦s s ♣r♦♣♦♥ ♣r ♠t♦♦♦í sr♣

t ♥ ♣ít♦ ♥tr♦r ♠♦r♥♦ ró♥ sñ ♣r♠r ♥t♦r♥♦ ②

♦♥t♥ s sñs

♠♦♦ q s ♦♥sr s q s ♠str ♥ r ést ♥②

♦s rt♦rs ♣qñ♦s ② ♥ rt♦r ①t♥s♦ ♦s ♣r♦♥s

0,5 ♠ ② 1,0 ♠ rs♣t♠♥t ♠♦s rt♦rs s rtr③♥ ♣♦r ♥

á♠tr♦ 0,1 ♠ ǫr = 5 ② σ = 1,5 ♠♠ strt♦ s♣r s rtr③

♣♦r ǫr = 4 ② σ = 1 ♠♠ ② strt♦ ♥r♦r ♣♦r ǫr = 4,2 ② σ = 2,5 ♠♠

r♥ ♠♣♦ s 500 ③ s ♣ ♥ tó♥ t♦r

s ♠trs ♣r♠t ② ♦♥t

r ♠str rrr♠ ♦t♥♦ ♣r ♥ ú♥♦ ♠s♦r ♦♥

♦st 0,5 ♠ ♥ s ♣♦s ♥tr r♠♥t s sñs ♦s r

t♦rs r①ó♥ ♥trs ♥trr s ♠② é Pr r st sñ

♠ás s s ♣ó ♠ét♦♦ ♣r ♦ s t③ó ♥ rr♦ ♦♥ N = 5

② d = 0,1 ♠ ♥ st s♦ s t③ó ♥ ♦♥♥t♦ s♣③♠♥t♦s t♠♣♦rs

♥tr 0,00 ♥s ② 0,15 ♥s ♦♥ ♥r♠♥t♦ 0,05 ♥s rrr♠ rst♥t s

♠str ♥ r ♦♥r♠ó q ♠ét♦♦ ♠♦r

sñ ♥trés ♠♦r q ♣rs♥tó sñ ♦♥sstó ♥ ♥ ♠♥

t♦ ♦♥trst ♥tr sñ ② ♥t♦r♥♦ ♣rs♥t ♥ rrr♠ ♦tr

q s sñs ♦s rt♦rs s rstr♦♥ só♦ ♥ ♥ ♣qñ ♣♦ró♥

♣r♦ q ♥ ♥r stá♥ ♠ás ss ♥ st s♥t♦ stá r♦ q

♠ét♦♦ t♥ trr s sñs ró♥ ♥♦ ést s ♣ ♣r

r rr♠ ♣r ♥ ú♥♦ ♠s♦r ♦♥ ♦st 0,5

♠ rr♠s ♣r ♥ rr♦ ♦♥ N = 5 ② d = 0,1 ♠ ♦t♥♦s

♣♥♦ ♠t♦♦♦í ♣r ♠♥tr ♦♥t♥ tr

sñ ♥trs ♦r③♦♥t ② sñ rt♦r

r r♥♦ ♦rs x s ó ♥ ♠ts

② ♠t♦♦♦í s ♣ó ♣r ♠♦rr s sñs ró♥

rs♣ts

rstr sñ ♥ rt♦r ①t♥s♦

tr♥t♠♥t s ♣ ♠♦rr sñ ♥♦ ♦s rt♦rs ♣

qñ♦s P♦r ♠♣♦ ♥ r s rst sñ rt♦r

r ♣ ♦srr ♥ ♠♥t♦ ♥ ♦♥trst sñ ② ♥

♦♥t♥ tr ♥ st s♦ sñ rt♦r ①t♥s♦ s

s♣ró st♦ ♦rró s q ♣r rstr sñ ró♥

s s♦♥r♦♥ s♣③♠♥t♦s t♠♣♦rs ♥ ♦s st♥t♦s ♥tr♦s q ♥♦

r♦♥ ♦s ♣r rstr sñs rt♦rs ①t♥s♦s ♦♠♦ ♠♣♦

♥ ♥ r s rstr♦♥ ♠♦s rt♦rs Pr r st♦ s

ó r♥♦ x ♥ ♠ts ♥ ♥ s s ♣ó ♠t♦♦♦í

♣r♦s♠♥t♦ s sñs ró♥ rs♣ts ♦♥ ss♥t

s③ó♥ ♦s ♦s rst♦s s♠tá♥♠♥t

r♦ st♦s rst♦s ♠ét♦♦ ♣r ♦♠♦ ♥ rr♠♥

t ♣r ♥str ♦♠♣♦rt♠♥t♦ sñs és ♦ ♦ss ♥s♦ ♥

s♦s ♦♥ ♠út♣s ♥♦s

rtr③ó♥ ♣rs t♣

♥ st só♥ s ♠str♥ ♦s ♥♦s ♠ét♦♦ ♣♦r ♠♦

s ♣ó♥ ♥ tr♠♥ó♥ ♥♦ ② ♣r♦♥ ♣rs

t♣ ♥ st♦ rq♦ó♦ P♦ ♥♦ ♦♥♦r ♦s ♦rs s♦s

♣rá♠tr♦s s ♠② ♠♣♦rt♥t ♣r ♦s rqó♦♦s ② q ♣rtr ♦s s

r③♥ st♦s ♦♠♣rt♦s ♥áss stíst♦s t ♠ás ♦s ♦rs

♣r♦♥ s♦♥ r♥ts ♣♥r ♥ ①ó♥ ♣♦r ♠♣♦ ♣r

r ♥t s♦ q s r♠♦r

♥ s r③ó ♥ ♣r♦s♣ó♥ ♦ís ①t♥s ♥ st♦ P♦

♥♦ t③♥♦ ♠t♦♦♦í ♦rr ♦rtr s♠♣ ♥á

ss ♦s t♦s ♣r♦♦ ♠♣s s strtrs rq♦ós ♥trrs

♣rtr♠♥t rt ♥t ♣rs t♣ q ♥í♥ ♥ s

tró♥ ♦♠♣ t♦♥s ❬❪ ♥ ♣♦só♥ s ♣rs ♥

♣♥♦ ♦r③♦♥t s ♣♦ str ♦♥ ♣rsó♥ ♣♦r ♠♦ ♠t♦♦♦í

♣ ♥♠♥t só♦ s ♦tr♦♥ ♦rs ♠♣rs♦s ♥♦ s

♣rs ♦♥ ♦rs ♠② ♦s♦s ♣r♦♥ ♥ ♥s ♣qñs s

♦♥s s ♠s♠s sts ♥rt③s r♦♥ ♦♥s♥s ♥ ♣r♠r r

♦♥trst ♣r♠t ♠♦r♦ ♦ ♥ s s♣rs ♦s ♥♦s

st♦ ♦rr ♣r♥♣♠♥t ♣♦rq ♦s t♥ts ♦r♥s ♦♥str②r♦♥ s

♣rs ♦♥ ♦s ♠trs ♠♦ r♥♥t s♠s♠♦ s ss s

♣rs ♥♦r♠♠♥t ♣rs♥t♥ ♥ tr♥só♥ s ♥ ♣r♠t ♦

♦s ♣r♦s♦s ♦♠♣rsó♥ ② s♦ ♣♦r ♣♦ró♥ r♥t s ♦♥stró♥

Pr ♦t♥r ♦s t♦s s s♣s♦ ♥ í♥ s♦♥♦ 3 ♠ ♦♥

t q r③ ♥ s ♣rs tts ♥ st♦ ♣rt♥♥t

♥ú♦ t♦♥ r só♥ st♠ó ♥ ♣r♠t

ǫr = 3 ♣r ③♦♥ st♦ ♣rtr ♦ ♣r♦♣ó♥

v = (0,18±0,01) ♠♥s ♦t♥ ♣r ③♦♥ ♥ só♥ ó♥ ♣r

♠♥r ♦s ♠♣♦s s ♣r♦♥s s♣rs ♣r ♥♦ ♠♥t

♣r♦r♠ ♥t♦r♥♦ ♣é♥ ♥ó q ♣r ♥ s♣ró♥ ♥tr

♥ts d = 0,1 ♠ N ♣♦í sr ♦r♥ ♦ ♦ s qró ♦♥ ♥

s♣ró♥ ♠á①♠ q rí st r♥♦ st♥ ♠á①♠ 0,7 ♠ ♦s t♦s

s qrr♦♥ t③♥♦ sst♠ rr ♥s♦rs ♦tr Ps

P ② ♥t♥s ③ s ♥t♥s ♠s♦rs ② tr♥s♠s♦r s ♦③r♦♥

♥ ♥trs r s♦ ♦♥ ♦s s s ♥t♥s ♣r♣♥rs í♥

s♦♥♦

r ♠str ♥ rrr♠ ♣r ♥ ú♥♦ ♠s♦r ♥ st s♦

♦st s 0,5 ♠ ♣ rr q ♥ ♦s í♠ts trs ♥ s

♣r s ♣♥ str ♦♥ ♣rsó♥ ♣rtr st rrr♠ ó♦

♥ sñ ♣r♦①♠♠♥t 17 ♥s r í♥ ♣♥t♦s ♥ r

♣r ♥r s ♣r ♥q ♠s♠ s ♠s♦ é ♣r

srr♦

♦ s ♣ó ♠ét♦♦ ♣r ♠♦rr sñ r①ó♥ ♥ s

② ♣r ♥r ♠♦r s r♦♥s ♥ ♦s ♦rs ♥ r s ♠str

rrr♠ rst♥t ♣r ♥ rr♦ ♦♥ N = 5 ② d = 0,1 ♠ ♥ q s

♣ó ♠t♦♦♦í ♣r♦s♠♥t♦ ♣r ♠♦rr sñ s

♣r Pr ♦t♥r st rst♦ s t③ó ♣r♦r♠ ♥t♦r♥♦ ③

tr♥s♠t♦ s ró ♦♥♦ ♣r ② s ró s♣③♠♥t♦

t♠♣♦r s −0,30 ♥s st 0,30 ♥s ♦♥ ♥r♠♥t♦s 0,05 ♥s á♥♦s

tr♥s♠só♥ ♥tr 36♦ ②−36♦ ♠♥r ♠♦rr t♦s s ♣rts r♥ts

sñ ② s♦♥r ♦s s♣③♠♥t♦s t♠♣♦rs ♠ás ♦s

r rr♠ ♣r ♥ ú♥♦ ♠s♦r ♦♥ ♦st

0,5 m rr♠s ♣r ♥ rr♦ ♦♥ N = 5 ② d = 0,1 ♠

♦t♥♦s ♣♥♦ ♠t♦♦♦í ♣r ♠♥tr ♦♥t♥

tr sñ s ♣r ② sñ

ró♥ ♦♥ ért ♥ (x; t) = (37 ♠; 8 ♥s) r♥♦ x

s ó ♥ ♠ts ② ♠t♦♦♦í s ♣ó ♣r ♠♦rr

s sñs ró♥ ♦♥ érts ♥ (x; t) = (37 ♠; 8 ♥s) ②

(x; t) = (38,5 ♠; 10 ♥s)

r r ♠ó♥ ♦t♦ ♣r ♦

sr ①♣st

♣r s♠♥t♦ ♣ ♦srr ♥ r q ♣r♦♠♥t♦

♣♦ ♠♦ró t♠♥t ♥t♦ r♦r 17 ♥s ♦♠♣t♥♦

sñ trés ♣r ♣r♦♥ q s ♥♥tr s

♣r s st♠ó ♥ (1,5 ± 0,1) ♠ ♣r ♦ s t③ó ó♥

só♥ ② ♦ ♣r♦♣ó♥ v = (0,18 ± 0,01) ♠♥s st

♠♥r ♣♦s ♠tr s ♣r s ♥ ♦♥ ♥ í♥

♣♥t♦s ♥ r

r s ♥ ♠♣♦ ó♠♦ s ♣♦s ♠♦rr ♥ sñ r

ó♥ ♥ ♥♦ ♦s ♦rs ♣r sñ q s rstó s ♣ ♥tr

r♠♥t ♥ r ♣♦só♥ s ért s (x; t) = (37 ♠; 8 ♥s) ♣r♦

①♠♠♥t s ♥ ♦♥ ♥ ♥ r ♥♠♥t ♥ r

s ♥ ♠♣♦ ♥ q s rstr♦♥ sñs rt♦rs ♥ ♦s ♦s

♦♣st♦s ♣r ♥♦ r♥♦ x ♥ ♠ts ② ♣♥♦ ♠t♦

♦♦í ♣r♦s♠♥t♦ ♥ ♥ s ♦♠♦ s ①♣ó ♣r♠♥t

sñ rst ③qr s ♠s♠ q ♥ r ♠♥trs

q sñ r t♥ ért (x; t) = (38,5 ♠; 10 ♥s) s ♥ ♦♥

♥ ♥ r ♠é♥ s ♣ ♥♦tr q s sñs ró♥

r♥♥ts ♠♦rr♦♥ ♦ q ♠♣♦ tr♥s♠t♦ t♥ ♥ ♥♦ ♥t♦

st♦s rst♦s ♣r♠tr♦♥ st♠r ♥♦ ♣r ♥ 1,5 ♠ ♦ q s

♦♥r♠ó ♦ sr ①♣st ♥ s rs ② s ♠str♥ ♦t♦s

r ♠ó♥ ② ♣r ♦ sr ①♣st rs♣t♠♥t

♣ó♥ sñs ♣r♦s ♣♦r ss

♦♠♦ s ♦ ♥tr♦r♠♥t ♥ ♦s st♦s rq♦ó♦s s ♦♠ú♥ ♥♦♥

trr ss ♦ ♠♥t♦s s♠rs ♥trr♦s ♦rrt rtr③ó♥

st t♣♦ ♥♦s ♣rs♥t ts ♦ q ♥ ♥r ♣r♦♥

sñs és ♦♠♦ ♦♥s♥ ♦ ♦♥trst q ♣rs♥t♥ ♦♥ rs♣

t♦ ♠♦ r♥♥t ♥ só♥ s ♠♦strr♦♥ ♥♦s rst♦s

r♦♥♦s ♦♥ tó♥ ss ♥trrs ♥ ♦s rrr♠s ♦t♥

♦s ♠ás sñ ♣r♠r s s ♣♥ ♦srr rss sñs

♦r♥s ♥ rt♦rs ♥trr♦s ② s♦♥t♥s ♥trs ss♦

♥ st só♥ s st♥ s ♠♦rs ♥ s sñs q s ♦t♥♥ ♠♥t

♣ó♥ ♠ét♦♦

♦s t♦s s qrr♦♥ t③♥♦ ♦♥ró♥ ①♣r♠♥t sr♣t

♥ só♥ ♥ st s♦ s s♣sr♦♥ ♦s í♥s s♦♥♦ 5 ♠

♦♥t ♠♥r q ésts ♣s♥ ♣♦r ♥♠ s r ②

♥ r ♦r ♥ s í♥s s qró ♦♥ s♣ró♥ ♥tr

♥ts 0,05 ♠ ② ♦♥ ♥ s♣ró♥ ♠á①♠ ♥tr r♣t♦r ② ♠s♦r 1

♠ s ♥t♥s ♠s♦r ② r♣t♦r s r♦♥ ♥ ♥trs r s♦ ♦♥

♦s s s ♥t♥s ♣r♣♥rs í♥ s♦♥♦ t③ó sst♠

rr ♥s♦rs ♦tr Ps P ② ♥t♥s 500 ③

♥ ♣r♠r tér♠♥♦ s ♥③r♦♥ ♦s t♦s ♦t♥♦s ♣r í♥ s♦♥♦

r ♥ r s ♠str ♥ ♣r ♦st 0,25 ♠

sñ ró♥ ♦t♦ ♥trr♦ s ♣ r ♦♥ r ♥ r

♦♥ ért ♥ (x; t) = (0 ♠; 10 ♥s) ♦♦r♥ x ♥ rá♦ s rr

♣♥t♦ ♠♦ ♥tr s ♣♦s♦♥s ♠s♦r ② r♣t♦r ❱rs ♣ér♦s

ró♥ ♦♥s ② r①♦♥s ♣r♦①♠♠♥t ♦r③♦♥ts ♣r♦s

♥ s s♦♥t♥s ♥trs s♦ t♠é♥ ♣r♥ ♥ r

r ♦♥ró♥ ①♣r♠♥t t③ ♣r qrr

♦s t♦s

P♦r ♠♣♦ ♥ sñ ♣r♦①♠♠♥t ♦r③♦♥t ♥ strt♦ s ♣

♦srr r♦r 8 ♥s ♥tr x = 0,8 ♠ ② x = 2 ♠ ♣r♦①♠♠♥t

①tr♠♦ ③qr♦ st sñ ♦♥t♥ú ♥ ♥ sñ ró♥ ért ♥

x = 0,8 ♠ ♦r♥ ♥ ♣♥t♦ ♦♥ ①ó♥ ♦rtó strt♦

r ♠str rst♦ ♠ét♦♦ ♣r ró♥ ♥

s ♥trr ♦s ♦rs N ② d s s♦♥r♦♥ ♠♥r q

r♥t ♦♥s rst♥t ♥♦ s st♦rs♦♥ s♥t♠♥t ♥ ♣♦só♥

♥♦ ♣r ♦ s t③ó ♣r♦r♠ ♥t♦r♥♦ ② s ♦♥sró ǫr = 3 ♦s

♣rá♠tr♦s rr♦ r♦♥N = 4 d = 0,1♠ ② ♦st 0,45♠ ♠♥trs q s

s♦♥r♦♥ ♦s s♣③♠♥t♦s t♠♣♦rs ♥tr−0,6 ♥s ② 0,6 ♥s á♥♦s

tr♥s♠só♥ ♥tr 56♦ ② −56♦ ♣r ♠♦rr sñ ♥ ♥tr♦s ♦♥st♦s

t③r♦♥ ♥ t♦t ♥tr♦s ♥ r s s♦♥ó ♥ r♣♦

r♥t s♣③♠♥t♦s t♠♣♦rs ♥tr −0,5 ♥s ② 0,2 ♥s á♥♦s

tr♥s♠só♥ ♥tr 54♦ ② −22♦ ♦♥ ♣r♦♣óst♦ rstr r①ó♥

♣♥♦ ♣r♦①♠♠♥t ♦r③♦♥t ② ♣ér♦ s♦

♥♦ s ♦♠♣r♥ s rs s ♣ ♦srr q s

sñs s♦♥ ♠ás rs ♣r ♠ét♦♦ q ♣r ♠ás ♠ét♦♦

♠♦ró sñ ♦t♦ ♥trr♦ ♦ strt♦ ♦r♠ st

♥♦ s rstó sñ ró♥ ♣r♥♣ r s t♥ó

sñ ♣r♦①♠♠♥t ♦r③♦♥t ② ró♥ s♦ ② rs r

♥♠♥t s ♥③r♦♥ ♦s t♦s ♦t♥♦s ♣r í♥ s♦♥♦

r ♥ r s ♠str ♥ ♣r ♦st 0,25 ♠ ♦♠♦ ♥

r í♥ s♦♥♦ rr♠ ♣r ♥ ú♥♦

♠s♦r ♦st 0,25 ♠ rr♠ ♥ q s ♠♦r

sñ ♦t♦ ♥trr♦ N = 4 d = 0,1 ♠ ♦st 0,45 ♠

−0,6 ≤ dt ≤ 0,6 ♥s rr♠ ♥ q s ♠♦r

sñ rt♦r ♣r♦①♠♠♥t ♦r③♦♥t ♦③♦ ♥

t ≈ 8 ♥s 0,8 ♠ < x < 2 ♠ ② sñ ró♥ s♦ ♦♥

ért ♥ (x; t) = (0,8 ♠; 8 ♥s) N = 4 d = 0,1 ♠ ♦st 0,45 ♠

−0,5 ≤ dt ≤ 0,2 ♥s

s♦ ♥tr♦r sñ r①ó♥ s s ♣ ♦srr ♦♥ ért

♥ (y, t) = (0 ♠; 10 ♥s) ♦♦r♥ y ♥ rá♦ s rr ♣♥t♦

♠♦ ♥tr s ♣♦s♦♥s ♠s♦r ② r♣t♦r ♥ st s♦ t♠é♥ s

♣♦s ♦srr sñ ♣r♦①♠♠♥t ♦r③♦♥t ♥ strt♦ r♦r

8 ♥s ♥tr y = 0,2 ♠ y = 2 ♠ ♣r♦①♠♠♥t sñ s

♦♣ó ♥ st s♦ ♥ r♥♦ ♣♦s♦♥s ♠♥♦r q ♥ ♣r ♣r

♣♦s♠♥t ♦♠♦ ♦♥s♥ ♦r♠ s ② stró♥

s♠♥t♦s ♥ s ♥tr♦r

♦ s ♣ó ♠ét♦♦ ♣r ♠♦rr sñ ♣r♠r s

♥ r s ♠str rrr♠ rst♥t ♣r ♥ rr♦ ♦♥

N = 4 ② d = 0,1 ♠ s s♦♥r♦♥ ♦s s♣③♠♥t♦s t♠♣♦rs s

−0,6 ♥s st 0,6 ♥s ♦♥ ♥r♠♥t♦s 0,05 ♥s ♠♥r ♠♦rr t♦s

s ♣rts r♥ts sñ ♣ ♦srr q ♥ ③♦♥ ♥tr

−0,75 ♠ ≤ y ≤ 0,75 ♠ ♥t♥s sñ ♠♥tó ♦♥ rs♣t♦ s

sñs r♥♥t ② ♠ás q sñ s ③♦ s ♥ t♦♦ r♥♦ y

♥q é♠♥t ♥ ♥♦s tr♠♦s

♠é♥ s ♣ó ♠ét♦♦ ♣r ♠♦rr r①ó♥ ♣♥♦ ♣r♦

①♠♠♥t ♦r③♦♥t ② ♣ér♦ s♦ ♥ r s ♠str

rrr♠ rst♥t ♣r ♥ rr♦ ♦♥ N = 4 ② d = 0,1 ♠ ♥ q

s ♣ó ♠t♦♦♦í ♣r ♠♦rr ♦♥t♥ s sñs r♥♦

s♣③♠♥t♦ t♠♣♦r s −0,5 ♥s st 0,2 ♥s ♦♥ ♥r♠♥t♦s 0,05

♥s ♦♠♦ ♥ s♦ s ♣ ♦srr ♦♥ ♠②♦r r ró♥

♥ ♦r strt♦ ♠♥trs q r①ó♥ ♥ s t♥ s♣

rr ♠ás s ♠♦ró ♦♥t♥ strt♦ ♥ t♦ s ①t♥só♥

r í♥ s♦♥♦ rr♠ ♣r ♥ ú♥♦

♠s♦r ♦st 0,25 ♠ rr♠ ♥ q s ♠♦r

sñ ♦t♦ ♥trr♦ N = 4 d = 0,1 ♠ ♦st 0,45 ♠

−0,6 ≤ dt ≤ 0,6 ♥s rr♠ ♥ q s ♠♦r

sñ rt♦r ♣r♦①♠♠♥t ♦r③♦♥t ♦③♦ ♥

t ≈ 8 ♥s 0,2 ♠ < x < 2 ♠ ② sñ ró♥ s♦ ♦♥

ért ♥ (x; t) = (0,2 ♠, 8 ♥s) N = 4 d = 0,1 ♠ ♦st 0,45 ♠

−0,5 ≤ dt ≤ 0,2 ♥s

♣ít♦

ó♥ ♠ét♦♦

♥ ♦s ♣ít♦s ♥tr♦rs s ♠♦stró q ♠ét♦♦ ♣r♠t ♠♦rr

s sñs t♥t♦ ♥♦s ♦③♦s ♦♠♦ rt♦rs ①t♥s♦s ♥ t♦♦s

♦s s♦s ♣r r s ♠♦rs s ♦♠♣rr♦♥ tt♠♥t s sñs

♦t♥s ♣rtr ♦s ♠ét♦♦s ②

♥ st ♣ít♦ s ♦♠♣r♥ ♥ttt♠♥t s rs♣sts ♦t♥s

♦♥ ♦s ♠ét♦♦s ② P st út♠♦ ♠ét♦♦ ♦rtr ♠út

♣ ♠ás t③♦ s ♠♦rs ♦t♥s s ú♥ ♥t♥♦ ♥s

s rtrísts sñ q ♥②♥ ♥ s s③ó♥ ró♥

♥t♥s rs♣t♦ s sñs r♥♥ts ② s t♦♥s ♥ t♠♣♦

♥t♦ ② ♥ s ♠♣t s s s ♥♥ ♦♥ ♦♥t♥

sñ ❬❪

♥ó♥ ♥♦rs

Pr r ♥ttt♠♥t ♦s rst♦s ♦s ♠ét♦♦s

② P s ♣r♦♣♦♥♥ rts ♥ts ♦♠♦ ♥♦rs

♠♥ ♦t♥ ♦♠♦ rs♣st ♥③♥ s t♦♥s ♥ t♠♣♦ ② ♥

♠♣t sñ ② ró♥ ♥tr s ♥t♥ss sñ ② ♠♦

r♥♥t

Pr ♠r s t♦♥s t♠♣♦ ♥ q ♦rr ♥ r①ó♥

s tr♠♥ t♠♣♦ ♠á①♠♦ ♠♣t ♣r sñ ♥ tr③

tn s ♦ ♦♠♣r ♦♥ t♠♣♦ s tr③s ②♥ts tn−1 ② tn+1 ② s

♥♦r♠③ rst♦

NTD =|tn − (tn−1+tn+1)

st ♠♥t s ♥♦♠♥ r♥ t♠♣♦s ♥♦r♠③ ♥♦r♠③

t♠ r♥

♦♠♦ ♠♣♦ s ♣r s♦ s♠♦ r①ó♥ ♥

♥ ♥trs ♣♥ ♦r③♦♥t ♥ r s ♠str ♠♦♦

♣ s♣r♦r s rtr③ ♣♦r ǫr = 5 ② σ = 1,5 ♠♠ ♠♥trs q ♠ás

♣r♦♥ ♣♦r ǫr = 6 ② σ = 2 ♠♠ rt♦r stá ♦③♦ ♥ ♣r♦♥

1 ♠ r♥ t③ s ③ ♥ s rs ② s

♠str♥ rrr♠s ♣r ♦st ♦ 0,3 ♠ ♥♦ s ♣♥ t♦♥s

t♦rs ② rs♣t♠♥t ♥ t♦ r ♦s ♣rá♠tr♦s

♦♥sttt♦s

♥ r s ♠str♥ ♦s ♦rs ♦s ♣r s sñs

s rs ② ♥ ♥ó♥ x ♥ s rs s ♣ó ♥ ♣r♦♠

♦ s♦r ♦rs ♦♥st♦s ♣r s③rs ♦♠♦ r s♣rr ♦s

♦rs ♦t♥♦s ♣r s♦♥ ♠♥♦rs ♣r ♠♦♦ ♦♥ t

ó♥ s r ♥♦ sñ s ♣ r ♦♥ ♠②♦r r ♠♥trs q

s♠♥r sñ ♦s ♦rs ♠♥t♥ ♠é♥ s

♣ ♦srr ♥ ♠♥t♦ ♥ tó♥ ♦s ♦rs st s

♥ rtríst s♥♦ ♦r♥ q t♠é♥ s s♦ ♦♥ s♠♦r

♥ s③ó♥ s sñs

Pr ♠r s t♦♥s ♠♣t ♦rrs♣♦♥♥ts r①ó♥

♥trés s t♦♠ tr③ ② ss ♣r♠rs ♥s s s♣③ sts út♠s

♥ t♠♣♦ ♠♥r q ♦s ♠á①♠♦s ♠♣t q♥ ♥♦s ② s

s ♦rr♦♥ ♥ ♥ ♥tr♦ t♠♣♦r ♥tr♦ ♥ ♦ ♠á①♠♦ ② ♦♥

①t♥só♥ ♥ ♣rí♦♦ t♠♣♦r Pr rr rst♦ st ♣s♦

rt♠♥t s ♦ ♠ ♦rró♥ ♥ r s ♠str♥ rs

♦rró♥ ♣r ♦s ♠♦♦s r ♣ r q ♦rs ♠②♦rs

② ♠♥♦s t♥ts ♦rró♥ s s♦♥ ♥ ♠♦r s③ó♥ s

♥♠♥t s ♠ ró♥ ♥t♥s ♥tr sñ ♣r♥♣ ② s

r♥♥ts ♠♥t♦srr♦♥♥ s♥ ♥t♥st② rt♦ ♣r ♦ s

r③ ♥ ♦♥t ♥tr s ♥t♥ss ♣r♦♠♦ s sñs ♣r♥♣ ②

r ♦♦ ♥ ♥trs ♣♥ ♦r③♦♥t 1 ♠

♣r♦♥ ♣ s♣r♦r ǫr = 5 ② σ = 1,5 ♠♠ ♣

♣r♦♥ ǫr = 6 ② σ = 2 ♠♠ rr♠s ♦t♥♦s ♣r

♥ ♦♥ró♥ ♦♥ ♦st 0,3 ♠ r♥ ③ ♣r

♦s s♦s ♥ ♦s q s ♣♥ t♦♥s ②

♠tr③ ♦s ♣rá♠tr♦s ♦♥sttt♦s

r ❱♦rs ♦rró♥ ②

♦s ♣r ♥ ♠♦♦ ♦♥ ♥ ♥trs ♣♥ ♦r③♦♥t ♦♥

tó♥ ② ♥ ♠tr③ ♦s ♣rá♠tr♦s ♦♥st

tt♦s ② ♦st ♦ 0,3 ♠

r r③ ♥ q s ♠str♥ ♦s ♥tr♦s ♥ ♦s q

s ♥ s ♥t♥ss sñ ♣r♥♣ ② s sñs

r♥♥ts

r♥♥ts st♦s ♣r♦♠♦s s ♥ ♥ trs ♥tr♦s t♠♣♦ st♥t♦s

r ♣rí♦♦s ①t♥só♥ ♥♦s ♦ r♦ tr③

♣r♠r ♥tr♦ stá ♥tr♦ ♥ ♠á①♠♦ ♥t♦ ♣r♥♣ ♣r r

s ♥t♥s ♠ ♠♥trs q ♦s ♦tr♦s s s♣♦♥♥ s♠étr♠♥t r

♦r ♥t♦ ♣r♥♣ ♣r r ♦r ♠♦ ♥t♥s ♦s

♥t♦s r♥♥ts ♥ r s ♠str rst♦ ♦t♥♦ ♣r

s sñs r①ó♥ ♦s ♠♦♦s ♥tr♦rs ♣ r q ♦rs

♠②♦rs s s♦♥ ♥ ♠♦r s③ó♥ s sñs

♦♠♣ró♥ ♥tr ♦s ♠ét♦♦s ②

♥ st só♥ s ♣rs♥t ♥ ♦♠♣ró♥ ♦s rst♦s ♦t♥♦s

♦♥ ♦s ♠ét♦♦s ② P st♦ s r③ ♣rtr s ♦s

st♦♥s áss q s ♥③r♦♥ st ♠♦♠♥t♦ ró♥ ♥ ♥

♦t♦ ♣qñ♦ ② r①ó♥ ♥ ♥ ♥trs ①t♥s Pr ♠s st♦♥s

s st♥ t♥t♦ t♦s s♠♦s ♦♠♦ ①♣r♠♥ts

♥ ♠ét♦♦ ♥ ③ q s st♥ N ② d s♣③♠♥t♦

t♠♣♦r dt ♥tr s ♦♥s ♠ts s s♦♥ ♥ ♣♦ró♥ s♣r

♠♥r rr ③ tr♥s♠t♦ ♦ r♦ ♠♥♦ ♦

rr♦ rt♦r r♣t♦r s ♣♦r ♦ ♥♠♥t r ♦s t♦s q

r t③ ♠s♠♦ ♦♥♥t♦ t♦s ♣r ♦s ♠é

t♦♦s ② P

s ♣r♦♥ ♥ ♦s rst♦s ♠ét♦♦ r dt

♥ st ♣ít♦ s ♥③ t♠é♥ ♥♥ r♥ts ♥s

tó♥ ♥ ♠tr③ s♦ sts t♦♥s tú♥ ♦♠♦ ♦tr♦s ♦

t♦s ♥ s♦ ♦s q ♣r♦♥ r①♦♥s s♥rs q ♥trr♥ ♦♥

s r①♦♥s ♥trés ♣♦r ♦ s ♠♣♦rt♥t ♥③r á s ♥

♠ét♦♦ ♣r ♠♦rr ♠♥r t ♦s ♥s ♦♥t♥ ②

♠♣t sñ

♦♠♦ s ♦ ♠ás rr ♥ st ♣ít♦ s ♦♠♣r♥ ♦s rst♦s ♦

t♥♦s ♦♥ ♦s ♠ét♦♦s ② P ♥♦tr q st♦s ♠ét♦♦s s

♣♥ ♣r ♠s♠♦ ♦♥♥t♦ t♦s P♦r ♠♣♦ ♥ r s

q ♣rtr ♦s t♦s qr♦s ♣r ♠ét♦♦ s ♣♦s s♦♥r

qs tr③s q ♦rrs♣♦♥♥ ♥ ♠s♠♦ ♣♥t♦ ♠♦ ♦♠ú♥ ♦♥ s♣

r♦♥s r♥ts ♥tr ♠s♦r ② r♣t♦r st ♠♥r ♥ ♦ q s s

♣♥ ♦s ♠ét♦♦s ② P ♠s♠♦ ♦♥♥t♦ t♦s

t♦s s♠♦s

t♦ ♣qñ♦

♥ r s ♠str ♣r♠r ♠♦♦ ♥③♦ ❯♥ ♦t♦ ♦♥ á

♠tr♦ 0,05 ♠ s ♦③ ♥ ♣r♦♥ 0,80 ♠ ♦t♦ s rtr③

♣♦r ♥ ♣r♠t rt ǫr = 5,25 ② ♥ ♦♥t σ = 2,5 ♠♠ ②

♠♦ r♥♥t ♣♦r ǫr = 3,5 ② σ = 1 ♠♠ ♠♦ s♦r ♦s s r

♥tr♦ r s 0,01 ♠ ♥ ♠s r♦♥s ♥ ♦s ♣rá♠tr♦s

s♦ s ♣♥ t♦♥s t♦rs ♦♥ ♥ ♠♣t ♠á①♠

♦♦s ♦s ♣♦♦s s ♦③♥ ♥ ♥trs r s♦ ♥tr♦s ♥ x

ró♥ s♦♥♦ ♦♥ ♦s s ♦s ♣♦♦s ♣r♣♥rs ♠s♠♦

r♥ ♥tr s ♦♥s ♠ts s fc = 500 ③ ♦♥t ♦♥

λ = 0,32 ♠ ♥tr♦ s♦

♥ r s ♠str ♥ ♣r ♦t♥♦ ♣r ♠♦♦ r

♦♥ ♦st ♦ 0,3 ♠ ♦♦r♥ x s rr ♣♥t♦ ♠♦ ♥tr

s ♣♦s♦♥s ♠s♦r ② r♣t♦r ♥ r sñ r①ó♥

rt♦r s ♣ ♦srr ♦♥ t ♦♠♦ ♥ ♣ér♦ ♦♥ ért ♥

t = 11 ♥s ♣r♦①♠♠♥t ♦ q ♥t♥s sñ ♣r♠r

r ♦♦ ♥ ♦t♦ ♣qñ♦ ♦③♦ ♥

♣r♦♥ 0,80 ♠ ♠♦ r♥♥t s rtr③ ♣♦r

ǫr = 3,5 ② σ = 1 ♠♠ ♦t♦ s rtr③ ♣♦r ǫr = 5,25 ②

σ = 2,5 ♠♠

② ♠♦ r♥♥t s♦♥ ♦♠♣rs ② ♥trr♥

r ♠str ♥ rst♦ ♣r ♠ét♦♦ ♦t♥♦ ♣r

♠♦♦ r s st♥s rts ♥tr ♦s ♦♠♣♦♥♥ts

rr♦ s♦♥ d = 0,1 ♠ ♠♥trs q N = 4 ♦♦r♥ x s rr

♣♥t♦ ♠♦ ♥tr ♥tr♦ rr♦ ② ♣♦só♥ r♣t♦r s t③ó ♥

♦st q♥t 0,45 ♠ ♥♦ ♥tr sts ♣♦s♦♥s s♦♥r♦♥

s♣③♠♥t♦s t♠♣♦rs −0,6 ♥s ≤ dt ≤ 0,6 ♥s ♦s q ♦rrs♣♦♥♥

á♥♦s tr♥s♠só♥ ♥tr 50♦ ② −50♦ ♣r ♠♦rr sñ rt♦r ♥

♥tr♦s ♦♥st♦s x t③r♦♥ ♥ t♦t ♥tr♦s ♣r ♦t♥r

rst♦ r

♦♠♣rr s rs ② s ♣ ♦srr q sñ q

rst ♠ét♦♦ s ♠ás s q sñ ♦r♥ ♥ st ♣♥t♦

♥♦ stá r♦ s ♠♦r stá r♦♥ ♣r♥♣♠♥t ♦♥ ♠♥♦rs

t♦♥s sñ ♥ s♣♦ ② ♥ t♠♣♦ ♦ ♦♥ ♥ ♥r♠♥t♦ ♥

ró♥ ♥tr ♥t♥s sñ ♣r♠r ② s sñs r♥♥ts ♦♥

rs♣t♦ st♦ út♠♦ ♥♦ só♦ s rr③♥ s sñs rt♦r ♣r♥♣

♥ r s♥♦ t♠é♥ ♣rts sñs rt♦rs ♠♦ r♥

♥t s t♦♥s ♥ ♠tr③ s♦ s q s ♣♥ ♦srr ♥

r ♦♠♦ r♥s ♥♥ó♥ ♦♥st♥t ♣rs sñ ♣r♠r

♥tr♦ t♦♦s ♦s ♥tr♦s x ♠ás ♣r♥ sñs ♣r♣♥rs

sñ ♣r♥♣ ♥ ♦s ♥tr♦s ♠ás ①tr♠♦s x = (−2;−1) ♠ ② x = (1; 2)

♠ s s s♦♥ ♦♥s♥ ♣ró♥ ♥rí q ♥♦ s tr♥s♠t

♥ ró♥ s♣r ♣r |dt| > 0,4 ♥s ♦ q t♠é♥ ♣ tr

r Pr ♦st 0,3 ♠ Pr N = 4

d = 0,1 ♠ ♦st 0,45 ♠ −0,6 ≤ dt ≤ 0,6 ♥s Pr P

♦♥t♥ ♥t♥s ♦s rst♦s

♥ r s ♠str rst♦ ♠ét♦♦ P ♣r ♠♦♦

r ♥ú♠r♦ ♣♠♥t♦ ♥ r s n = 4 ♦♠♦ ♦

♣♠♥t♦ s ♦♥sr ♦ ♣r♦♣ó♥ ♥ ♠♦ v = 0,16

♠♥s P♦r ♦tr♦ ♦ ♦♠♣r♥♦ s rs ② s q sñ

s ♠ás r ♣r ♠ét♦♦s q ♣r ♠ét♦♦ P st rst♦

s t♥ó♥ s sñs ♣r♦s ♣♦r rt♦rs s♥r♦s

r ③ tr♥s♠t♦ ♥ ♠ét♦♦

Pr r ♥ttt♠♥t ♦s rst♦s ♦s ♠ét♦♦s ②

P s ♥③r♦♥ ♦s ♥♦rs ♥♦s ♥ só♥ s t♦♥s

t♠♣♦ ② ♠♣t ♦ r♦ r①ó♥ ♣r♥♣ ② s ♥t♥s

♦♥ rs♣t♦ ♥t♥s ♦♥♦

♥ r s ♠str r♥ t♠♣♦s ♥♦r♠③ ♣r s

sñs ró♥ ♥ s rs ♦♠♦ ♥ó♥ x s rs

♥ r s♦♥ rst♦ ♣r♦♠r ♦s ♦rs tr③s ♦♥sts

♥ r s ♣ ♦srr q ♠ét♦♦ r ♦ r♦

t♦♦ r♥♦ x ♠ás r ♣rs♥t ♠♥♦s r♦♥s q s

♦trs ② s ①t♥ ♥ ♥ r♥♦ x ♠②♦r ♦♠♦ s ♠♥♦♥ó sts r

trísts stá♥ r♦♥s t♠é♥ ♦♥ ♥ ♠♦r s③ó♥ sñ

ró♥ ♦♠♥t ♦r ♣r♦♠♦ ♦ ♦ r♦

t♦♦ r♥♦ x s 0,60 10−2 1,61 10−2 ② 1,64 10−2 ♣r ♦s ♠ét♦♦s

② P rs♣t♠♥t ♦ q ♥ ♥ ♠♦r s♥t ♦r

♦r♥ ♦♠♦ ♦♥s♥ ♣ó♥ ♠ét♦♦ ♦tr

q ♥ st s♦ rt♦r ♣qñ♦ ♠ét♦♦ P ♥♦ ♦ ♥ rst♦

t♥ stst♦r♦ st♦ s s♣rr ♦ q ♠s♠♦ s♦ ♦r♥♠♥t

sñ♦ ♣r st♦ strt♦s ♣r♦♥♦s ♦♥ ♣qñ ♥♥ó♥ ❬❪

s rs q rstr♦♥ r ♦rró♥ ♣r sñ r

ó♥ r s ♠str♥ ♥ r Pr s③r s rs

s r③ó ♥ ♣r♦♠♦ ♦rs ♦♥st♦s ♦ r♦ s ♠s♠s

♣ ♦srr q ♠ét♦♦ ♥r♠♥tó ♦rró♥ ♥ ♣r♦①

♠♠♥t r♥♦ x ♥♦ ♥ r ♠♥♦s t♥t P♦r

♦♥trr♦ ♥ ♥tr♦ x = (0,25; 0,45) ♠ r stá ♣♦r ♦

r ♣r st s ♦♠♣♦rt♠♥t♦ ♥ó♠♦ ♦rr ♦s♦♥♠♥t

♥ r♦♥s ♥ s q ♥ sñ s♥r ♥t♥s ♠♣♦rt♥t r③

r ♦rró♥ ② ♣r sñ

ró♥

r Pr♦♠♦ ♦rró♥ ②

♥ ♥ó♥ s♣③♠♥t♦ t♠♣♦r dt ♥ ♥tr♦ x =

(−1,5; − 1,1) ♠ ①tr♠♦ ③qr♦ sñ ró♥

Pr♦♠♦ ♦rró♥ ② ♥ ♥ó♥

s♣③♠♥t♦ t♠♣♦r dt ♥ ♥tr♦ x = (−0,2; 0,2) ♠ ♥

♥tr♦ sñ

10−2 10−2 10−2

♦rró♥

♦rró♥ ② ♣r♦♠♦s ♦ r♦

t♦♦ r♥♦ x ♣r ♠♣♦ s♠♦ rt♦r

♥♥ ♦s ♥s tó♥ ♥ ♦s ♣rá♠tr♦s s♦

s út♠s ♦s ♦♠♥s s rr♥ ♦rs rt♦s sts

sñ st ♦ ♦r♠ó♥ st út♠ ♦♥ rs♣t♦

♠ét♦♦ P s ♦sr q ♦rró♥ sñ ♦r♥ ♠♦ró ♥ s

t♥t♦s ♥tr♦s x ♣r♦ ♣♦r ♦ ♦s rst♦s ♠ét♦♦

♦rró♥ ♦ ♣r ♦s trs ♠ét♦♦s ♦ r♦ t♦♦ r♥♦

x s ♥ ♥ t

♥ r s ♠str♥ ♦♥ts ♥tr s ♥t♥ss ♣r♦♠♦

sñ ♣r♥♣ ② r♥♥ts ♣r ♦s ♠ét♦♦s ② P

♦♠♣rr s rs r s ♣ ♦srr q ♠ét♦♦

♣r♦♦ ♥ ♦r ♠②♦r ♥ ♠②♦r ♣rt r

st ♣r♠r ♠♣♦ ♠♦stró q ♠♦r ♥ sñ ♣r♦ ♣♦r

♠ét♦♦ s r♦♥ t♥t♦ ♦♥ r♠♥t♦ s t♦♥s ♦♠♦

♦♥ ♥r♠♥t♦ ♥ ró♥ ♥tr s ♥t♥ss sñ ♣r♠r ② s

r♥♥ts s ♠♥ts ♠♦r ♥ s rs ♦rró♥

② rí♥ ♦♥ ♣♦só♥ ♠é♥ s ♣s♦ rt♦ ♦♥ ♥ ♦rró♥

♥♦ ♣r ♥tr s ♣♦s♦♥s ♦s ♠í♥♠♦s ② ♠á①♠♦s ss rs ♥

♥r ♥♦ s t♦♥s ♥ ♠tr③ s♦ s♠♥②♥ s rs

s t♦r♥♥ ♠ás ss ② s ♣r♦①♠♥ ♦r ♦♥st♥t r♦ ♠♥trs

q ♠♦r rt ♥r ♣r♦ ♣♦r ♠ét♦♦ s♠♥② P♦r

♠♣♦ ♣r ♥ tó♥ ♥ ♦s ♣rá♠tr♦s s♦ ♦r

♥r ♣r♦♠♦ ♦ r♦ t♦♦ r♥♦ x ♣r ♦ ♣♦r

♦r ♥r ♣r s 0,37 stá ♦ ♦♥t rs♣t♦ ♣r

s♦ 0,46 r t ♠r♠♥t ♦r ♥r rt♦

♣r ♠ét♦♦ P ♣s 1,02 1,04 ♠♥t♥é♥♦s ♣rát♠♥t

♦♥st♥t P♦r ♦tr♦ ♦ s rs ♦rró♥ s s③♥ ② s ♣r♦①♠♥

♦r ♦♥st♥t 1 ♦♥ ♥ ♠♦r rt sñ ♣r♦ ♥♦ s

t♦♥s s♦ s r♥ Pr ♥ tó♥ ♦rró♥

♥r rt ♣r ♦s ♠ét♦♦s ② P s♦♥ 1,26 ② 1,07 ♣♦r ♥♠

♦s ♦rs rs♣t♦s ♣r s♦ 1,09 ② 0,98 P♦r ♦♥trr♦ ♦s

♦rs s♦t♦s ② rt♦s s ♥r♠♥t♥ ♥♦ ♥

tó♥ s♠♥② ♠♥trs q s rs ♣r ♦s trs ♠ét♦♦s s ♥

♠♥♦s t♥ts ② ♥ s♣r♣♦♥rs ♥tr sí Pr ♥ tó♥

♦s ♣rá♠tr♦s s♦ ♦r ♥r rt♦

♣r ♦s ♠ét♦♦s ② P s♦♥ 1,14 ② 0,96 ♦s q stá♥ ♦ ♦s

♦rs ♣r 1,26 ② 1,09 rs♣t♠♥t

Pr ♥③r ♦s t♦s só♥ ró♥ ③ tr♥s♠t♦

s♦r ♦s rst♦s ♠ét♦♦ s ♥ ♥ ♥tr♦ x ② s ♣r♦♠♥

s ♠♥ts ♦rró♥ ② ♥tr♦ é ♣r dt s♣

③♠♥t♦ t♠♣♦r r ♠str ♣r♦♠♦ ♣r

② P ♥ st s♦ ♥tr♦ ♥ q s ♣r♦♠ x = (−1,5;−1,1) ♠

s ♦③ ♥ ①tr♠♦ ③qr♦ sñ ró♥ ♣ ♦srr

q s rs ♣r ♣rs♥t♥ ♥ ♣♦ ♦♥ ♠í♥♠♦ ♥ dt = −0,5 ♥s ♦s

♦rs dt ♣♦ ♦rrs♣♦♥♥ á♥♦ tr♥s♠só♥ q ♠♦r r

♥rí ♦ r♦ ♠♥♦ ♠s♦r rt♦r r♣t♦r s rs ♦

rró♥ ② rs ② t♠é♥ ♣rs♥t♥ ♣♦s ♦♥ ♠á①♠♦s

♥ dt = −0,5 ♥s ♦tr q sts rs s ♣♥ sr ♣r s♦♥r ♥

s♣③♠♥t♦ t♠♣♦r q ♦♣t♠ ♦s rst♦s ② q st só♥

♥♦ s rít ♦ ♥♦ ♦♥sr ♦s ♣♦s P♦r ♠♣♦ ♠ét♦

♦ ♠♦r s♠tá♥♠♥t ♦s ♦rs ♦r♥s ♦rró♥

② s dt = −0,6 ♥s dt = −0,3 ♥s ♦ q ♦♥stt② ♥ ♥tr♦

st♥t ♠♣♦

s rs ② s♦♥ ♥á♦s s rs ②

rs♣t♠♥t ♣r♦ ♣r ♥ ♥tr♦ ♣r♦♠♦ ♥tr♦ ♥ ♦r♥

x = (−0,2; 0,2) ♠ s rs ♣r ♣rs♥t♥ rtrísts s♠rs

s ♥tr♦rs ①♣t♦ ♣♦r ♣♦só♥ ss ①tr♠♦s ♦s q stá♥ ♦③♦s

♥ dt = −0,1 ♥s ♥ st s♦ ♠ét♦♦ t♠é♥ ♠♦r s♠tá♥♠♥t

s ♠♥ts ♦s trs ♥♦rs ♥ ♥ ♥tr♦ ①t♥s♦ s dt = 0,0

♥s dt = 0,2 ♥s

t♦r ①t♥s♦

♥ r s ♠str s♥♦ ♠♦♦ ♥③♦ st ♦rrs♣♦♥

♥ rt♦r ♣♥♦ ♦r③♦♥t ♦③♦ ♥ ♣r♦♥ 0,80♠ strt♦

s♣r s rtr③ ♣♦r ♥ ♣r♠t ǫr = 3,5 ② ♥ ♦♥t

σ = 1 ♠♠ ♠♥trs q strt♦ ♣r♦♥♦ ♣♦r ǫr = 4 ② σ = 2 ♠♠

♥♠♥t ♥ tó♥ ♥ ♠tr③ s♦ s ♦s

♦tr♦s ♣rá♠tr♦s ♦♠♦ ♦♥ró♥ ♦s ♣♦♦s ② s rtrísts

sñ tr♥s♠t s♦♥ s ♠s♠s q ♥ ♠♦♦ ♥tr♦r

r ♦♦ ♥ rt♦r ♣♥♦ ♦r③♦♥t ♦③♦

♥ ♣r♦♥ 0,80 ♠ strt♦ s♣r s rtr③

♣♦r ǫr = 3,5 ② σ = 1 ♠♠ strt♦ ♣r♦♥♦ ♣♦r ǫr = 4 ②

σ = 2 ♠♠

♥ r s ♠str ♥ ♣r ♦st 0,45 ♠ ♦t♥♦ ♣r

♠♦♦ r sñ r①ó♥ ♣r ♥trs s ♣

♦srr r♠♥t t = 11 ♥s ♣r♦①♠♠♥t ♥ r s

♠str rst♦ rs♣t♦ ♣r ♦♠♦ ♣r ♠♦♦ ♥tr♦r ♦s

♣rá♠tr♦s rr♦ s♦♥ N = 4 ② d = 0,1 ♠ ♥ st s♦ só♦ s ♥sr♦

♥ s♣③♠♥t♦ t♠♣♦r ♦♥st♥t dt = 0,1 ♥s á♥♦ tr♥s♠só♥

−10♦ ♥ t♦♦ r♥♦ x ♦♠♣rr s rs ② s ♣

r q ♠ét♦♦ ♠♦r sñ q s ♦t♥ ♦♥ ♥ r

s ♠str rst♦ ♠ét♦♦ P ♣r ♠♦♦ r

♥ú♠r♦ ♣♠♥t♦ s n = 4 ♦♠♦ ♦ ♣♠♥t♦ s

♦♥sr ♦ ♣r♦♣ó♥ ♥ strt♦ s♣r v = 0,16 ♠♥s

♠é♥ rst♦ P ♠♦r sñ ♦r♥ ♣r♦ ♥ ♠♥♦r ♠

r Pr ♦st 0,45 ♠ Pr N = 4

d = 0,1 ♠ ♦st 0,45 ♠ dt = 0,1 ♥s Pr P n = 4

10−2 10−2 10−2

♦rró♥

♦rró♥ ② ♣r♦♠♦s ♦ r♦

t♦♦ r♥♦ x ♣r ♠♣♦ s♠♦ rt♦r

♥♥ ♦s ♥s tó♥ ♥ ♦s ♣rá♠tr♦s s♦

s út♠s ♦s ♦♠♥s s rr♥ ♦rs rt♦s sts

q ♠ét♦♦

Pr ♦♥r♠r sts ♦sr♦♥s ♥ s rs s ♠str♥

s rs ♦rró♥ ② rs♣t♠♥t ♣r s sñs

② P s rs s ♦t♥♥ ♣♦r ♠♦ ♠s♠♦ ♣r♦♠♥t♦

t③♦ ♥ s♦ rt♦r ♣ ♦srr q ♦s ♠ét♦♦s ②

P ♠♦rr♦♥ rst♦ ♣r s trs ♠♥ts ♦♥ ♠♦rs rst♦s

♣r ♠ét♦♦ ♦s ♦rs ♥rs ♣r♦♠♦s ♥ t♦♦ r♥♦

x ♦rró♥ ② s♦♥ 0,29 10−2 1,25 10−2 ② 0,83 10−2 ♣r

♦s rst♦s ② P rs♣t♠♥t

♦♠♦ ♥ s♦ rt♦r ♠♦r ♥ s rs ② ♦rró♥

q s ♦t♥ ♦♥ ♦s ♠ét♦♦s ② P ♣r rt♦r ♣♥♦ ♦r③♦♥t

s ♥r♠♥t ♥♦ ♥ tó♥ ♥ ♠tr③ s♦ ♠♥t

r t ♣r ♦s ♦rs rt♦s sts ♠♥ts P♦r ♦♥tr

r♦ ♠♦r ♥ s ♥r♠♥t ♥♦ s t♦♥s ♥ s♦

s♠♥②♥

♥ s rs ② s ♠str♥ ♦s ♦rs ♣r♦♠♦s

♦rró♥ ② ♦♠♦ ♥ó♥ dt ♣r rt♦r ♣♥♦ ♦

r③♦♥t ♥ st s♦ s t③ r♥♦ ♦♠♣t♦ x ♥ ♣r♦♠♦

♣ ♦srr q s rs s♦♥ tt♠♥t s♠rs s rs

♦t♥s ♣r♠♥t ♣r rt♦r rs ♣rtr♠♥t

qs s ♣r♦♠♥♦ ♥ ♥ ♥tr♦ x ♥tr♦ ♥ ♣♦só♥

r ♦rró♥ ② ♣r ♥ r

t♦r ♣♥♦ ♦r③♦♥t

♦♠♦ ♥ó♥ dt

♦t♦ rs ② q ♣r sts ♣♦s♦♥s ♠s st♦♥s

s♦♥ ♣r♦①♠♠♥t s♠rs

♥ st só♥ s sr♥ ♦s ♠♣♦s q ♠str♥ qé ♠♥r

♠ét♦♦ ♠♦r t♦s ①♣r♠♥ts ♦♥sr♥ ♦s t♦s ♦t♥♦s

♥ st♦ ss ♥trrs ♥③♦s ♥ só♥ qr♦s

s♦r í♥ s♦♥♦ t③♥♦ ♦♥ró♥ ①♣r♠♥t sr♣t

♥ só♥ ♥ s rs s ♠str♥ ♦s rst♦s

♥ ♣r ♦♥ ♦st 0,25 ♠ r sñ ró♥ ♦t♦

♥trr♦ s ♣ r ♦♥ r ♦♥ ért ♥ t = 10 ♥s ❱rs ♣ér♦s

ró♥ ♦♥s ② r①♦♥s ♣r♦①♠♠♥t ♦r③♦♥ts ♣r♦

s ♥ s s♦♥t♥s ♥trs s♦ t♠é♥ ♣r♥ ♥ r

P♦r ♠♣♦ ♥ sñ ♣r♦①♠♠♥t ♦r③♦♥t ♥ strt♦ s ♣ ♦

srr r♦r t = 8 ♥s ♥tr x = 0,8 ♠ ② x = 2,4 ♠ ♣r♦①♠♠♥t

①tr♠♦ ③qr♦ st sñ ♦♥t♥ú ♥ ♥ sñ ró♥ ért

♥ x = 0,80 ♠ ♦r♥ ♥ ♣♥t♦ ♦♥ ①ó♥ ♦rtó strt♦

r ♠str rst♦ ♠ét♦♦ ♣r ró♥ ♥

♦t♦ ♥trr♦ N = 4 d = 0,1 ♠ ② ♦st 0,45 ♠ s s♦♥r♦♥ ♦s

s♣③♠♥t♦s t♠♣♦rs −0,6 ♥s ≤ dt ≤ 0,6 ♥s ♣r ♠♦rr sñ ♥

♥tr♦s ♦♥st♦s ♥ r s s♦♥ó ♥ r♣♦ r♥t

s♣③♠♥t♦s t♠♣♦rs−0,5 ♥s ≤ dt ≤ 0,2 ♥s ♦♥ ♣r♦♣óst♦ rstr

r①ó♥ ♣♥♦ ♣r♦①♠♠♥t ♦r③♦♥t ② ♣ér♦ s♦ P♦r

♦tr♦ ♦ r ♠str rst♦ ♣r P ♦♥ ♥ ♥ú♠r♦

♣♠♥t♦ n = 4 t③ó ♦ ♣♠♥t♦ ♦t♥ ♥ só♥

♣♥♦ ♦rró♥ sñ ♣♥♦ ♣r♦①♠♠♥t

♦r③♦♥t v = (0,17± 0,09) ♠♥s

♥♦ s ♦♠♣r♥ s rs s ♣ ♦srr q s

sñs s♦♥ ♠ás rs ♣r ♦s ♠ét♦♦s ② P q ♣r ♦♥ rs

t♦s ♠♦rs ♣r s sñs ♠ás ♦♠♦ s sró ♥ só♥

♠ét♦♦ ♠♦r sñ ♦t♦ ♥trr♦ ♦ strt♦

♦r♠ st ♥♦ s rst sñ ró♥ ♣r♥♣ r

s t♥ú sñ ♣r♦①♠♠♥t ♦r③♦♥t ② ró♥ s♦

② rs r

r Pr ♦st 0,25 ♠ Pr ♥ q

s ♠♦r sñ ♦t♦ ♥trr♦ N = 4 d = 0,1 ♠ ♦st

0,45 ♠ −0,6 ≤ dt ≤ 0,6 ♥s Pr ♥ q s ♠♦r

sñ rt♦r ♣r♦①♠♠♥t ♦r③♦♥t ♦③♦ ♥

t = 8 ♥s 0,8 ♠ < x < 2,4 ♠ ② sñ ró♥ s♦ ♦♥

ért ♥ (x, t) = (0,8 ♠; 8 ♥s) N = 4 d = 0,1 ♠ ♦st 0,45 ♠

−0,5 ≤ dt ≤ 0,2 ♥s Pr P n = 4

r ♦rró♥ ② ♣r r

ó♥ ♦t♦ ♥trr♦

♥ ♥ó♥ dt ♥ ♥tr♦ x = (−1,2;−0,8) ♠ ♥ ①tr♠♦

③qr♦ sñ Pr♦♠♦ ♦rró♥ ②

♥ ♥ó♥ dt ♥ ♥tr♦ x = (−0,2; 0,2) ♠ ♥

♥tr♦ sñ

r ♦rró♥ ② ♥ ♥ó♥

x ♣r rt♦r ♣r♦①♠♠♥t ♦r③♦♥t

♦♠♦ ♥ó♥ dt

rt♦r 10−2 10−2 10−2

t♦r 10−2 10−2 10−2

♦rró♥rt♦r

rt♦r

♦rró♥ ② ♣r♦♠♦s ♦ r♦

t♦♦ r♥♦ x ♣r ♠♣♦ ①♣r♠♥t s út♠s

♦s ♦♠♥s s rr♥ ♦rs rt♦s sts ♥ts

Pr ♥③r s t♦♥s ② ♠♣ts rts s sñs s

r♦♥ s ♠s♠s ♠♥ts q s t③r♦♥ ♥ s s♦♥s ♥tr♦rs

s rs ② ♠str♥ s rs ♦rró♥ ②

rs♣t♠♥t ♣r ró♥ ♥ ♦t♦ ♥trr♦ ♣

♦srr q ♠ét♦♦ ♠♦ró rst♦ ♥ ♠②♦r ♣rt

r♥♦ x ② r♥♦ ♣r s ♠♥ts rs♣ts

♠ás ♠ét♦♦ P ♠♦ró s rs sñ ♦r♥ ♣r♦ ♠♥♦s

q ♠ét♦♦ ♦s ♦rs ♦s s ♠♦rs ♦rró♥

② ♦s ♦ r♦ r♥♦ ♦♠♣t♦ x s ♥♥ ♥ t

♦♠♣r♥♦ s rs ② s ♣ ♦srr

♠ás q ♦s ♠♣♦s ①♣r♠♥ts ② ♥♠ér♦s ♣rs♥t♥ ♦♠♣♦rt♠♥t♦s

s♠rs ♦ r♦ s rs

♥ s rs ② s ♠str♥ rs ♦s ♣r♦♠♦s

♦rró♥ ② ♦♠♦ ♥♦♥s dt ♣r ♥ ♥tr♦

♣r♦♠♦ x = (−1,2;−0,8) ♠ ♦③♦ ♥ ①tr♠♦ ③qr♦ s

sñs ró♥ s rs ② ♠str♥ rs s♠rs

♣r ♥tr♦ ♣r♦♠♦ x = (−0,2; 0,2) ♠ ♦③♦ ♥ ♥tr♦

s sñs ♦♠♦ ♥ ♦s ♠♣♦s s♠♦s s rs ♣rs♥t♥ ♣♦s

♣r q♦s ♦rs dt q ♠♦r r♥ ♥rí ♦ r♦ ♠♥♦

♠s♦r rt♦r r♣t♦r ♥tr♦ dt ♥ ♠ét♦♦ ♠♦r

rst♦ s ♠②♦r q 0,2 ♥s s♥t♠♥t ♥♦ ♣r r ♣♦s

♥ á só♥ st ♣rá♠tr♦

♦♠♦ út♠♦ ♠♣♦ s ♥③ r①ó♥ ♥ strt♦ ♣r♦①♠♠♥t

♦r③♦♥t q s rstó ♣♦r ♠♦ ♠ét♦♦ ♥ r s

rs ♠str♥ rs ♦rró♥ ② ♣r s

sñs ♦s ♠ét♦♦s ② P ♣ ♦srr q ♦s ♠ét♦♦s

② P ♠♦r♥ r ♥ ♠②♦r ♣rt r♥♦ x ♥

②♥♦ ♥tr♦ r♦♥♦ ró♥ s♦ x < 1 ♠ ♦

tó♥ ② ♥ ♠♣t sñ ♦r♥ ♦s ♥♦s

♠ét♦♦ s♦♥ ♠ás ♠♦r♦s ♥ st ♠♣♦ q ♥ ♦s ♥tr♦rs

♦♥ rst♦s s♠rs ♣r ② P ♥♠♥t ♥ s rs

s ♠str♥ rs ♣r♦♠♦ ♦rró♥ ② ♦♠♦

♥ó♥ dt ♣r ♥ ♥tr♦ ♣r♦♠♦ x = (1,0; 2,4) ♠ ♦③♦ ♥

♣rt ♣r♦①♠♠♥t ♦r③♦♥t sñ sts rs ♦♥r♠r♦♥ s

rtrísts ♦srs ♥ ♦s ♠♣♦s ♣r♦s

♣ít♦

♦♥s♦♥s

s♠♥ rst♦s ② ♦♥s♦♥s

♥ st ss s stó ♠ét♦♦ rr♦s s♥tét♦s ♠s♦rs ♦

rr ② s srr♦r♦♥ s té♥s ② rr♠♥ts áss ♣r r t

s ♠♣♠♥tó♥ ♠é♥ s r♦♥ tt ② ♥ttt♠♥t s

♠♦rs ♦t♥s ♥ s sñs ♠♥t ♣ó♥ ♠ét♦♦

♦♠♦ ♥tr♦ó♥ s ♣rs♥tr♦♥ ♦s ♥♠♥t♦s ♠ét♦♦ ♦rr

sí ♦♠♦ s té♥s ♦rtr s♠♣ ② ♠út♣ q s t③♥ t

♠♥t ♥ s ♣ó♥ ♦ s sró r♠♥t ♥ sr tr♦s

r③♦s ♥ st♦ rq♦ó♦ P♦ ♥♦ ♦s q ♥♦r♥ t

ó♥ ♣rs ♥trrs ② st♦ ♣♦s tó♥ ss

♥trrs ♥ ♠♦s s♦s s ♣rs♥t♥ sñs ♦♥ss q ♥♦ ♣♥ sr

♥tr♣rts ♠♥t ♣rtr ♦s ♣rs t♦s ♦rtr s♠♣

♥t ♣ó♥ ♠ét♦♦ ♠ró♥ ♥ s♦ ♣r ②

P ② ♠♦♦ ♥ s♦ s ss ♣♦s ♦rr ♥ ♥tr♣rt

ó♥ ♠②♦rí ss sñs ♦s tr♦s ♥ r♥ ♠ ♠♦tr♦♥

srr♦♦ sst♠át♦ ♠ét♦♦

Pr ♦♠♥③r ♦s st♦s s♦r ♠ét♦♦ s ♥③r♦♥ ♦s ♠♣♦s

tr♦♠♥ét♦s ♣r♦♦s ♣♦r rr♦s ♠s♦rs t♦rí rr♦s

♥ts t♣♦ ♣♦♦ ♥ í♦ ② ♥ ró♥ ♠♣♦ ♥♦ ♠str q

♦♠♥♥♦ rs ♥ts r♥s ♥tr sí s ♣♦s ♦♥tr♦r rts r

trísts ♣tró♥ ró♥ ♦♥♥t♦ ♥ ♣rtr r♥ ②

á♥♦ ♣r♦♣ó♥ ♦ q s ♣♦t♥♠♥t út ♣r ♦rr ♥r♠♥

t♦ ♥rí s♦r ♦s ♥♦s ♥trés ② ró♥ ♥♥

♥t♦r♥♦ s ♥ s s s s ♠ét♦♦ Pr tr ♦s st♦s

r③♦s s srr♦ó ó♦ ♦♠♣t♦♥ rr② st ó♦ ♣r♠t

r ♦s ♣tr♦♥s ró♥ qr stró♥ ♣♦♦s ② r♣r♦

r ♦s r♠s ♦rrs♣♦♥♥ts ♥t♥s rs ♠♥r st♠r s

♣♥♥ ♥r

♦ s ♣r♦♥ s ♠s♠s s ♣r ♦t♥r ♦s ♠♣♦s tr♦♠

♥ét♦s ♥r♦s ♣♦r ♥ts s ♥ ♥trs r s♦ ② s ♦♥

srr♦♥ ♦s ♠♣♦s q ♠t ♦♥♥t♦ ♥ ró♥ ♠♣♦ r♥♦

♦sró q t♠é♥ ♥ s s♦ s ♣♦s ♦♥tr♦r ♦♥♥tró♥ ②

♦r♥tó♥ ♠♣♦ ♠♥r s♠r ♦ q ♦rr ♥ í♦ ♣r♦ ♦♥

rts rtrísts ♣rtrs q s♦♥ ♦♥s♥ ♣rs♥

♥trs ♥tr ♦s ♠♦s ② r♥í ♥tr ♣♥t♦ ó♥

♠♣♦ ② ♥t ♥ st s♦ rr♦ ♥ ♣rtr s ♥③ó ó♠♦ s

♥ s♦♥r ♦s ♣rá♠tr♦s rr♦ ♣r ♦t♥r ♥ r♥t ♦♥s

q rst ♦ ♣r ♣r♦r ♥ ♠♦r t s sñs

①♠♥ó ♠ét♦♦ ♦♠♦ ♥ ♠♥r ♠♦rr s sñs

♦rr ①♣ó ó♠♦ ♥♦♥ ♠ét♦♦ rr♦s s♥tét♦s ♦♥ s

t♥t♦s t♣♦s ♥♦s ② qé ♠♥r ss rst♦s ♠♦r♥ s sñs

♦♥srr♦♥ ♦s st♦♥s ♥♠♥ts ♣rtr s s s ♣♦s

♥tr♣rtr ♦s s♦s ♠ás ♦♠♣♦s ♥ ♦♥t♥♥♦ ♥ ♦t♦ ♣qñ♦ ②

♦tr ♥②♥♦ ♥ ♥trs ①t♥s r♥ts ♣rts sñ r

♣r♦♥ sr rsts r♥♦ á♥♦ tr♥s♠só♥ t♥♥♦ s ③

♦trs ♣rts sñ ♥♦ s ♣rts ♥t♥ss ♦rrs♣♦♥r♦♥

qs ♣♦r♦♥s s ♥trss ♥trrs q rr♦♥ ♠ás ♥t♠♥

t ♥rí r♣t♦r s ♦trs ♣rts ♦rrs♣♦♥r♦♥ s♠♥t♦s

q ♥♦ r♦♥ ♠♥♦s ♦ q rr♦♥ ③ r ♣♦só♥ r

♣t♦r s s ♥r♠♥tó r♠♥t ♣r s ♣rts rsts

sñ ♥ ♥r ♣r♦♠♦ tr③s ♠♣ít♦ ♥ ♠ét♦♦ t♥ó

♥r r♦ t♦r♦ ② t r♥ s♣ ② r♦r③r r①ó♥

♣r♠r

♣♦ st ♠♥r ♠ét♦♦ ♠♦stró sr t♦ ♣r ♠♦rr

st♠♥t ♣rts sñ ♦s ♥♦s ♥q ♥♦ ♣r r③r ♥t♦s

♦♠♣t♦s ♥ ♦♥s♥ s ♠♣♠♥tó ♥ ♠t♦♦♦í q ♠♦ró

s ♦ r♦ t♦ sñ st ♦♥sst ♥ s♦♥r ♥ sr

s♣③♠♥t♦s t♠♣♦rs ♦ q♥t♠♥t á♥♦s tr♥s♠só♥

♣r ♦s s t♦s s ♣♦r♦♥s r♥ts sñ s♦♥ ♠♦rs ♦

s ♣♦r♦♥s ♠♦rs s ♦rt♥ ② s ♦♦♥ ♥ts ♥ rrr♠ ♥

st ♠t♦♦♦í s ♣ó ♥ ♠♦♦ q ♥② ♦s rt♦rs ♣qñ♦s

② ♥ ♥trs ①t♥s ♠♣♦ ♣r♠tó ♦srr ♣ ♠ét♦♦

♣r ♠♦rr ♠♥r st s sñs ♦t♦ ② trr s ♦trs

sñs ♥♠♥t s ♣ó ♥ r♥t st té♥ ♣r rstr

♠♥r s♠tá♥ ♠s sñs ró♥ ♥ rrr♠ rst♥t

♣♦rt ♠ét♦♦ ♣r str rt♦rs q ♥r♥ s♦♠♥t

sñs t♥s ♦ ♦ss s ♠♦stró ♣á♥♦♦ ♥ st♦ s♦s rs

♣r♠r♦ ♦s ♦♥sstó ♥ ♠ó♥ s♣s♦r ② ♣r♦♥

♣rs t♣ ♥ st♦ rq♦ó♦ P♦ ♥♦ ♣r s s s♦♥♦s

♣r♦s ♦♥ ♦rtr s♠♣ í♥ ♦ rst♦s ♥t♦s ♠ét♦♦

rr♦s ♣r♦ó sr ♥t ♣r s③r sñ s ♣r ②

s r♦♥s ♥ ss ♦rs st ♠♥r s♣s♦r ② ♣r♦♥

♣r ♣r♦♥ sr ♠♦s ♥ s♥♦ s♦ r♦♥♦ ♦♥ tó♥

ss ♥trrs ♠ét♦♦ ♠♦stró ♠♣♦rt♥ts ♥♦s ♥ ♥t♦

♣ trr s r ♠♦rr ♥ ♦r♠ st sñs ♦r♥s

♥ st♥t♦s ♦t♦s

srr♦ó ♣r♦r♠ ♦♠♣t♦♥ ♥t♦r♥♦ q t sñ♦

♦s rr♦s ♣r ♦♣t♠③r s rtrísts ♠♣♦ s♦r ♥♦

♦ s ①t♥ó ♣r♦r♠ ♠♥r q ♦♥stt② ♥ rr♠♥t

♣r s st♥ts t♣s ♠♣♠♥tó♥ ♠ét♦♦ st ♣r♦r♠

♣r♠t s♥tt③r ♠♣♦ ♥ rr♦ ♥ ♥ ♠♦ ♥♦r♠ ♦t♥r

rs♣st rr♦ ♣r t♦s ①♣r♠♥ts ♦ s♠♦s ② ♣r st

rs♣st ♠t♦♦♦í q ♣r♠t ♠♦rr ♦♥t♥ tr s

sñs ♥♠♥t s srr♦ó ♥ ♥t♦r♥♦ rá♦ ♠ ♦♥ rr♠♥ts

♣r ♥áss ♦s rst♦s q s♠♣♥ ♣ó♥ ♠ét♦♦

♦ s ♣rs♥tó ♥ st♦ ♥ttt♦ s ♠♦rs ♣r♦s ♥

s sñs ♣♦r ♠ét♦♦ r♦♥ s r♦♥s t♠♣♦ ♥

♦rr r①ó♥ ♦r♠ ♣s♦ r♦ ② s ♥t♥s

♦♥ rs♣t♦ r♦ r♥♥t t♦♦ ♦ ♦♠♦ ♥ó♥ ♣♦só♥

♦♠♣rr♦♥ ♦s rst♦s ♠ét♦♦ ♦♥ ♦s ♦s ♠ét♦♦s ② P

♥③r♦♥ ♦s t♦s r♥ts ♥s tó♥ ♥ ♦s ♣rá♠tr♦s

s♦ ② ó♥ s rt ♥tr ♦s ♦♠♣♦♥♥ts rr♦

r♦♥ t♥t♦ t♦s s♠♦s ♦♠♦ ①♣r♠♥ts ♣r ♦s s♦s

♦t♦ ♣qñ♦ ② rt♦r ①t♥s♦

s sñs ♦t♥s ♦♥ ♠ét♦♦ rstr♦♥ ♠ás ss q s

♦t♥s ♦♥ ♦s ♠ét♦♦s ② P ♦♠♦ ♦♥s♥ t♥ó♥

♥t s sñs s♥rs rt♦rs r ③ q s ♣r♦♣

♠♦stró q ♠♦r ♦rr s ♦♥s♥ s♥s♦s ♥ s

t♦♥s t♠♣♦ r①ó♥ ② ♠♣t ♣s♦ r♦ ② ♥

♥r♠♥t♦ ♥ ró♥ ♥t♥s sñ ♣r♥♣ rs♣t♦ s

r♥♥ts ♠♣♦rt♥ rt st♦s ♦rs ♠ ♦♥ ♣♦só♥

rr♦ ♦ r♦ í♥ s♦♥♦ s♥ ♥ ró♥ ♥t ♥tr s

rs♣ts rs ♥ ♥r s rs ♦t♥s ♣♦r ♠♦ ♠ét♦♦

♣rs♥t♥ ♠♥♦s r♦♥s q s ♠ás ♦tr rtríst r♦♥ ♦♥

♥ ♠♦r s③ó♥ s sñs

♥ ♦s st♦s r③♦s s ♦♠♣rr♦♥ rst♦s ♦t♥♦s ♥♦

♥ú♠r♦ ♥ts t③s ♥ ♠ét♦♦ ♦♥ ♦♥ ♥ú♠r♦

♣♠♥t♦ ♠ét♦♦ P s r③♥ s♦♥♦s ♥ ♦s q s ♥t♥s

s ♣♦s♦♥♥ ♠♥♠♥t s♦r s♣r s ♥sr♦ qrr ♣r♦①

♠♠♥t ♠s♠ ♥t t♦s ♥ ♠♦s ♠ét♦♦s s r q ♦s

♠ét♦♦s ② P rqr♥ s♠rs t♠♣♦s qsó♥ ♥ ♠

r♦ ♠ét♦♦ ♥ ♥r ♠♦str♦ rst♦s s♣r♦rs ♦♠♦ s

st♦ ♣rtr ♦s st♦s r③♦s ♥ st ss

♠♦strr♦♥ ♦s t♦s só♥ ró♥ ③ tr♥s♠t♦

♥♦ ♦♥ s♣③♠♥t♦ t♠♣♦r s♦r ♦s rst♦s ♠ét♦♦

♦s rst♦s ♣r ♠ét♦♦ ♣rs♥t♥ ♠♦rs q s ♠①♠③♥

♣r q♦s s♣③♠♥t♦s t♠♣♦rs q r♥ ♠ás ♥t♠♥t

♥rí ♦ r♦ ♠♥♦ ♠s♦r rt♦r r♣t♦r Pr st♦s s♣③

♠♥t♦s t♠♣♦rs s ♠♦rs ♥ ♦s rst♦s ♣r ♠ét♦♦ ①♥

s ♦rrs♣♦♥♥ts ♠ét♦♦ ♥ ♥tr♦s s♣③♠♥t♦ t♠♣♦r

♦♥ ♥ ①t♥só♥ ♥ ♦r♥ ♦ ♣♦r ♥♠ ♥ ♣rí♦♦ ♠str♦ tí

♣♦ st♦ ♠str q ♥♦ s rít ó♥ st ♣rá♠tr♦ r♥t

♠♣♠♥tó♥ ♠ét♦♦ ② q ♠s♠♦ ♣ sr ♦ ♦♥

rt ① s♠s♠♦ ♦s rst♦s ♦t♥♦s ♣r t♦s s♠♦s ②

①♣r♠♥ts rstr♦♥ tt♠♥t s♠rs ♥tr sí

♥ rs♠♥ ♥ st ss s ♦ró ♥③r sst♥♠♥t ♥ ♦♠♣r♥

só♥ ♠ét♦♦ ② ♥ s ♠♣♠♥tó♥ srr♦r♦♥ s té♥s ②

s rr♠♥ts áss ♥srs ♥ s ♣ó♥ ② s r♦♥ tt

② ♥ttt♠♥t s ♠♦rs ♥ ♥tr♣rtó♥ ♦s t♦s ♦t♥s

♦♥ t③ó♥ ♠ét♦♦ ♦♠♦ rst♦ t♦♦ ♦ ♠ét♦♦

s ♣ ♦♥srr ♥ ♥ ♦♠♣♠♥t♦ ♣r ♦s s♦♥♦s ts

♠ás ♠ét♦♦ s ♣ ♣r ♥ ♣r♦ ♦ s♣és ♦tr♦s ♠ét♦♦s

♦rtr ♠út♣ ♥♦r♠♠♥t s♥ qrr t♦s ♦♥s ♦s rst♦s

♦t♥♦s ♥ ♣rs♥t tr♦ ♠♦strr♦♥ ♥♦s s♥t♦s ♥ ♥

t♦ ♣ó♥ st ♥♦ ♠ét♦♦ ♥ ♦rr ♦ q ♦♥stt② ♥ ♣s♦

♥♠♥t ♣r ♦♥t♥r ♦♥ srr♦♦ ♠ét♦♦

trs í♥s tr♦

♦s rst♦s ♦t♥♦s ♥ st ss ♠str♥ r♥ ♣♦t♥ q ♣r

s♥t ♠ét♦♦ ♥ ♦♥s♥ s ♣r♦♣♦♥♥ s s♥ts í♥s

tr♦ ♣r ♦♥t♥r srr♦♦ ♠s♠♦

③r ♠♣♠♥tó♥ rr♦s ♥♦ ♥♦r♠s ♥ ♥t♦ s

s♣r♦♥s ss ② ♠♣ts rts ♦s st♥t♦s ♠♥t♦s

♠♥r ♦♣t♠③r ♦s ♠♣♦s ♠t♦s s♦r ♦s ♥♦s ♥trés

str t♦ s♦r rs♣st ♠ét♦♦ s st♥ts

♦r♥t♦♥s s ♥t♥s ♠s♦r ② r♣t♦r ♥tr sí ② ♦♥ rs♣t♦

í♥ s♦♥♦ ♠s♠♦ ♠♦♦ str ♦s ♥♦s ♠

♣♠♥tr rr♦s ②♦s ♠♥t♦s t♥♥ st♥t ♦r♥tó♥ s r

st♥t ♣♦r③ó♥

❱♥♦ ♦♥ st♦ ♦♠♣rr ♠ét♦♦ ♦♥ ♠t♦♦♦ís q ♣

♥ r♦tó♥ ♦r ❬❪ ♥ st ♠ét♦♦ s qr♥ rs tr③s

♥ ♣♥t♦ s♣r ♣r st♥ts ♦r♥t♦♥s rts ♥

tr s ♥t♥s ♠s♦r ② r♣t♦r ♦ q ♥ts ♥ ♥t♦

♦③ó♥ ♦t♦s r♦s ② st♠ó♥ s ♦r♥tó♥ ❬❪

♥q ♣rs♥t ts ♥♦ sñ ♣♦s ♠♣t

♥ r♦ ♥ ♠♦ s ♦ ♦ s ♣r♦♥ rr♦rs ♥ ♣♦s

♦♥♠♥t♦ s ♥t♥s ♥ ts st♦♥s ♠ét♦♦ ♣♦rí

♣r♦r ♠♦rs r♥

①t♥r ♣ó♥ ♠ét♦♦ rr♦s ♥ ♦s q ♦s

♠♥t♦s s ♥♥tr♥ str♦s t♥t♦ ♦ r♦ í♥ s♦♥♦

♦♠♦ ♥ ró♥ ♣r♣♥r ♠♥r ♦rr ♥♦str

③ ② ♦♥tr♦r s ♦r♥tó♥ ♥ ró♥ ♣r♣♥r í♥

s♦♥♦

③r st♦s ♠♣♦ ♥r♦ ♣♦r ♦s st♥t♦s rr♦s tr

és s♠♦♥s ♥ ♠♥r ♥③r ♥ ♦♠♣r♥só♥ ♠ás

♣r♦♥ ♥♦♥♠♥t♦ ♠ét♦♦ ② ♦rr ♦♥tr♦r ♥

t♠♥t r♥t ♦♥s rr♦ ♣r ♦♣t♠③r rs♣st

srr♦r ♥ sst♠ qsó♥ t♦♠át q ♣r♠t ♣r

♠ét♦♦ ♥ t♠♣♦s ♦rt♦s ♠♥r r ♣♦s s t③ó♥

♣♦r ♠♣♦ ♥ rs t ♥s ♦ ♣r r③r st♦s ♣rs

①t♥s♦s

③r st♦s s♠♦s ② ①♣r♠♥ts ♥ ró♥ ♦♥ ♦s t♦s

♦s st♥t♦s t♣♦s ♥ts sñs s♥rs q s ♣♥

♦srr ♥ ♦s t♦s ①♣r♠♥ts ♣r♦♣♦♥ str ♦s t♦s

r♦ ♦r♥t ♣r♦♦ ♣♦r ♥ts ♥ ss♦ ♦ r ② ♥♦

♦r♥t ♣♦r ♠♣♦ r♦ tró♥♦

③r st♦s sst♠át♦s t♥t♦ ①♣r♠♥ts ♦♠♦ s♠♦s ♣

r ♥③r s ♠♦rs q ♣r♦ ♠ét♦♦ ♥ rs♦ó♥

sñs ♣ró①♠s ♥t sí ② ♥ ♣r♦♥ ♣♥tró♥

♣r t♦r♠ r♣r♦ q ♥ q ♦s ♣tr♦♥s r

ó♥ ♥ ♥t♥ ♥ ♠só♥ ② r♣ó♥ s♦♥ ♦s ♠s♠♦s ❬❪ ♣r

t③r rr♦s r♣t♦rs ♥ ♦r♠ s♠r ♦ r③♦ ♥ st

ss ♦♥ ♠s♦rs

♣é♥

♠ó♥ ♥♠ér

rrr♠s

♥ s ♣♦♥s ♦♥srs ♠ét♦♦ ♦rr ♦s ♠♣♦s s♦♥

♠t♦s s ♥trs r s♦ ② s ♣r♦♣♥ trés ss♦ ♦♥

♥trtú♥ ♦♥ rs♦s ♦t♦s ♦s ♥♦s ♥trés ♣♥ str ♦s

♣r♦♥s q ♦rrs♣♦♥♥ t♥t♦ s r♦♥s ♠♣♦ r♥♦ ♦♠♦

♠♣♦ ♥♦ ♦ q s trt ♥ ♣r♦♠ ♦♠♣♦ q ♥♦ s

♣♦s ♦t♥r ♥ s♦ó♥ ♥ít s t s♠r ♥♠ér♠♥t

rs♣st sst♠

♦s rrr♠s s♠♦s ♥♠ér♠♥t ♣r♠t♥ str ró♥

♥tr s sñs ♦srs ② ♦s ♥♦s q s ♥r♥ ♦s t♦s s♠♦s

t♥♥ ♦trs ♣♦♥s ♦♠♦ ♣♦r ♠♣♦ ♥tr sñs ♦♥ ♦r♠s

rtrísts Pr ♦ s ♥r ♥ rs♣st s♠ ② ♦ s ♦♠♣r

♦♥ s sñs tts st ♠♥r s st ás s ♣♦rí♥

♦rrs♣♦♥r ♦s ♦t♦s ♥trr♦s ❬❪ tr s ♣♦♥s ♦♥sst ♥

rr ♥tr♣rtó♥ r③ ♦s t♦s ♦ ♥ ♣r♠r ♥tr♣r

tó♥ ♦s t♦s s ♣r♦♣♦♥ ♥ ♠♦♦ s r s ♣r♦♣s étrs ②

♦♠trí t♥t♦ ♠♦ ♦♠♦ ♦s ♥♦s ② s s♠ rs♣st ♦

rr ♣r ♠♦♦ ♣r♦♣st♦ ♦ s ♦♠♣r♥ ♦s rst♦s s♠♦s

♣r ♦ ♠♦♦ ♦♥ s sñs qrs s t q ♦♠♦ rst♦

♥ ♣r♠r ♥áss s ♣r♦♣♦♥ ♥ ♠♦ó♥ ♠♦♦ ♥ ♦ q

♥ ♣r♦s♦ st ♠♦♦ st r ♥tr♣rtó♥ ♥

♦s t♦s

ét♦♦ s♠ó♥

♦♦s ♦s t♦s s♠♦s ♥ st ss s ♦t♥♥ ♦♥ ♠ét♦♦

r♥s ♥ts ♥ ♦♠♥♦ t♠♣♦ ♥tr♥ t♠♦♠♥

❬❪ ♥ st ♠ét♦♦ s rs ♥♠ér♠♥t ♥ rsó♥ srt s

♦♥s ① ♦♥sr♥ s ♦♥s ❬❪

~∇× ~E = −µ∂ ~H

~∇× ~H = σ ~E + ǫ∂ ~E

♦♥ ǫ µ ② σ s♦♥ ♣r♠t ♣r♠ ♠♥ét ② ♦♥t

rs♣t♠♥t ~E s t♦r ♠♣♦ étr♦ ② ~H s t♦r ♠♣♦

♠♥ét♦ ♦ ♣r r③r s♠♦♥s s♦♥♦s r①ó♥ ♥ ♦s

q s ♥t♥s s ♥ ♥ s♣r r s♦ ♦♥ ss s ♣r♣♥

rs í♥ s♦♥♦ q s ①t♥ ♦ r♦ x s s♠

q s ♣r♦♣s íss ♠♦ s♦♥ ♥r♥ts ♥ ró♥ y

♥ s s♦ s ♦♥s ② s ♦t♥♥ ♦♥s s♦

♣s ♣r s ♦♠♣♦♥♥ts ♦s ♠♣♦s tr♦♠♥ét♦s ④Ey Hx Hz⑥

♦♠♣♦♥♥t tr♥srs♦ étr ② ④Ex Ez Hy⑥ ♦♠♣♦♥♥t tr♥srs♦

♠♥ét Pr ♦♠♣♦♥♥t s ♦♥s s ♣♥ ①♣rsr

s♥t ♠♥r∂Ey

∂z= µ

∂x= −µ

σEy − ǫ∂Ey

∂x− ∂Hx

Pr ♦t♥r ♥ s♦ó♥ ♥♠ér s ♦♥s ② s

r ♥ ♦♥♥t♦ ♦♥s ♥ r♥s ♥ts s♦r ♥ r srt

② ♥♦r♠ ♦s ♣♥t♦s r s ♥♦t♥ ♣♦r ❬❪

(x, z) = (j∆x, k∆z)

♦♥ ∆x ② ∆z s♦♥ ♦s ♥r♠♥t♦s s♣s r s ③

♥ó♥ s♣♦ ② t♠♣♦ s sr s♥t ♠♥r

F (x, z, t) = F (j∆x, k∆z, n∆t) = F n(j, k)

♦♥ ∆t s ♥r♠♥t♦ t♠♣♦r ♥♠♥t s rs s♣s ②

t♠♣♦rs ♥ s ♦♥s s ♣r♦①♠♥ t③♥♦ ♠ét♦♦ r♥s

♠♣♠♥tó♥ ♦♠♣t♦♥

t③ ♠♣♠♥tó♥ ♠ét♦♦ r③ ♣♦r r♥ ②

♥t ❬❪ trt ♥ ó♦ ♥ ♥ q ♣r♠t s♠

r ♥♠ér♠♥t ♦s ♠♣♦s ♥r♦s ♣♦r ♥ sst♠ ♦rr ♥ ♦s

♠♥s♦♥s ó♦ t♥ s s♥ts rtrísts

Pr s rs s♣s s t③ ♥ ♣r♦①♠ó♥ r♥s

♥ts rt♦ ♦r♥ ♠♥trs q ♣r s rs t♠♣♦rs s

t③♥ ①♣rs♦♥s s♥♦ ♦r♥

t③♥ ♦♥♦♥s ♦♥t♦r♥♦ t♣♦ ♣rt② ♠t ②r P

♠♥r tr r①♦♥s ♦s ♦rs r Pr ♦ s

r ♥ ♦♥t♦r♥♦ r rt ♥t s ♦♥ ♣r♦♣

s tr♦♠♥éts ts q ♦s ♠♣♦s q ♥ s ró♥ s♥

s♦r♦s ♣♦r ♦♠♣t♦ ♠♥r tr r①♦♥s ♦s ♠s♠♦s

♥tr♦r r

Pr ♥tr♦r ♥ ♥t ♠♣♦ étr♦ ♥ r s ♦♥sr

♣s♦ r ♣r ♠♣♦ étr♦ Ey s s♠ ♥

♣s♦ t♠♣♦r ♥ ♣♦só♥ r ♥ q s ♥♥tr

♥t st♦ s q♥t s♠r ♣s♦ ♦rrs♣♦♥♥t ♥t

♦♠♣♦♥♥t y tr♠♥♦ ♥s ♦rr♥t ♥ s ♦♥s

Pr s♠r ♦s r♣t♦rs s rstr ♠♣♦ étr♦ ♥ ♥ó♥

t♠♣♦ ♥ ♦s ♣♥t♦s r ♥ ♦s q s ♥♥tr♥ ♦s r♣t♦rs

♦ q s trt ♥ ó♦ s ♥ts s♦♥ ♠♥t♦s ♥s

q s ①t♥♥ ♥♥t♠♥t ♥ ró♥ ♣r♣♥r ♣♥♦

s♦♥♦ ♥ ♦♥s♥ ó♦ ♥♦ ♠♦ ♠♥t s♣rsó♥

② ♣tró♥ ró♥ ♥ts t♣♦ ♣♦♦ ♣r ♦ s ♥sr♦

♥ ó♦ ♥ ♠r♦ ó♦ ♣r♠t r♣r♦r s ♣r♥♣s

rtrísts s sñs ♦t♥s ♥ ♥ s♦♥♦ ♦♥ ♦rr

♣r♦r♠ rqr ♦♠♦ ♥tr s ♠trs ♦♥ s ♣r♦♣s

tr♦♠♥éts ♠♦♦ ǫ σ ② µ s ♣♦s♦♥s s ♥ts ②

♦s r♣t♦rs ♠ás r♥ ♥tr ♥t ♦s ♣s♦s

s♣s ② t♠♣♦r ② t♠♣♦ ♥ s♠ó♥

Pr ♥ s ♣♦s♦♥s s ♥ts ó♦ r③ ♥

sr ♦♠♣t tr♦♥s s t = 0 st t♠♣♦ ♥

s♠ó♥ st ♠♥r s ♦t♥ ♥ só♥ t♦s Ey(xr, t)

♥ ♦♥ró♥ ♠s♦r ♦♠ú♥ xr r♣rs♥t s ♣♦s♦♥s r♣t♦r

♦♠♦ s ó♦ ♥r ♥ ♦ t♦s s r ♥ ♠tr③

♦♥ trs ♥s q ♦♥t♥ t♦s s s♦♥s t♦s Ey(xr, t) ♥

♦♥ró♥ ♠s♦r ♦♠ú♥ ♥ ♣r ♣♦só♥ ♠s♦r

r Ps♦ q ♠♥t ♠♣♦ étr♦ Ey ♣r ♥

r♥ ③

♦♦♥s r③s s♦r ó♦

♣r♦r♠ s ♠♦ó ♠♥r tr s ♣ó♥ ♦♥ ♦s ♣r♦♣ó

st♦s s♣í♦s st ss ♥ ♣r♠r s ♠♦♦♥s r③s s

♣tó ♦r♠t♦ s ♦s t♦s ♠♥r q ♦rrs♣♦♥ ♦r♠t♦

q♦s t♦s qr♦s ♦♥ q♣♦s ♦rr st♦ ♣r♠tó trtr ♦s

t♦s s♠♦s ♦♥ ♦s ♣r♦r♠s ♣r♦s♠♥t♦ ② s③ó♥ srr♦

♦s ♣r♠♥t ♥tr♦ ♠r♦ st ss Pr ♦ s ♠♣♠♥tó

♥ r♠♥t♦ ó♦ q ♣r♠t rr ♥ r♦s s♣r♦s ♦s t♦s

♦rrs♣♦♥♥ts ♥ s s♦♥s ♥ ♦♥ró♥ ♠s♦r ♦♠ú♥

♣r ♠♣♦ Ey(xr, t) ♥ st♦s r♦s s ♥②♥ ♠ás ♦s t♠♣♦s ♥

♦s q s rstr ♠♣♦ t ② s ♣♦s♦♥s ♠s♦r ② ♦s r♣t♦rs

xr ♦♥tr ♦♥ st ♥♦r♠ó♥ t ♣♦str♦r trt♠♥t♦ ♦s

t♦s ♦♠♦ ♣♦r ♠♣♦ ♥♦ s ①tr♥ ♦s t♦s ♥sr♦s ♣r s♠r

♦s rstr♦s ♦t♥♦s ♦♥ ♥ sst♠ ♦rr ♥ ♦s st♥t♦s t♣♦s

s♦♥♦s

♦ q ♣r ♥ s ♣♦s♦♥s ♥t ♦s rst♦s s

rr ♥ r♦s s♣r♦s s ♠♦♦♥s r③s ♣rs♥t♥ ♥

t ♦♥ ♣r♠tr t♥r s♠ó♥ ♦ q s ♣rtr♠♥t út

♥ ♦s s♦s ♦♥ rs ①t♥ss ♣r ♦s q s rqr♥ t♠♣♦s r♦s

Pr ♦s st♦s r③♦s ♥ st ss rstó ♥trés ♥③r

♣r♦♣ó♥ s sñs ② ♥tró♥ s ♠s♠s ♦♥ ♦s ♥♦s P♦r

st r③ó♥ ♦tr ♠♦ó♥ r③ s♦r ó♦ ♦r♥ ♦♥sstó ♥

srr♦r ♥ rsó♥ tr♥t ♦ ó♦ ♥ q s ♥♦r♣♦r ♥

r♠♥t♦ ♣r♦r♠ q ♣r♠t ①trr ♣trs ♠♣♦ ♥ ♣s♦

t♠♣♦r s♠ó♥ ♥♦ s ♦rr ♥ rsó♥ ó♦ s

♦t♥ ♦♠♦ s ♥ sr r♦s q ♦♥t♥♥ s ♠trs ♦♥ ♦s

♦rs ♠♣♦ étr♦ ♥ ♣♦só♥ Ey(x, z) ♣r ♣s♦ t♠♣♦r

s♠ó♥

♣ó♥ ó♦ s s♠♦♥s r③s

♦s ♣rá♠tr♦s t③♦s ♥ s s♠♦♥s r③s ♥ st ss

r♦♥

r♥ ♥tr ♠só♥ s ③

s♣♠♥t♦ r s 0,01 ♠

Ps♦ t♠♣♦r dt = 0,02 ♥s

Pr s♠r ♠s♦rs ② r♣t♦rs ♦s ♥ ♥trs r s♦

st♦s s ♥ ♥ ♣r♠r í♥ r ♦♥ ♦s ♣rá♠tr♦s

♠♦

♣é♥

rr♦s ♥t♥s ♥ ♥ ♠♦

♥♦r♠

♥ st só♥ s ♣rs♥t♥ s ♦♥s q sr♥ stró♥

♥rí r ♣♦r ♥ ♣♦♦ ♦s♥t ♠ ♦♥ ② s ♦rr♦♥s

♥srs ♥♦ ♣♦♦ stá s♣③♦ ♦r♥ ♦ s ♣

♣r♥♣♦ s♣r♣♦só♥ ♣r r t♦r rr♦ s♠♥♦ ♣♦r

s♠♣ q ♦s ♣♦♦s s♦♥ ♣r♦s ❬❪ ❬❪

♦♥sr ♥ ♥t♥ ♦♥ ♦♥t l ♦♠♦ s ♠str ♥

r q ♥r ♠♣♦s r♠ó♥♦s r♥ ω s ♥ts

rr♦ s r♣rs♥t♥ ♣♦r ♥ ♥s ♦rr♥t étr ♥♠♥s♦♥~Jorig(~r, t) ♦ r♦ ♥t♥ z

r ♣♦♦ ♠s♦r ② sst♠ ♦♦r♥s

~Jorig(~r, t) =

I0e−iωt sen(π

2− k|z|)δ(x)δ(y)z s |z| ≤ λ

0 s |z| > λ4

♦♥ I0 s ♠♣t ♦rr♥t k = 2πλ

s ♥ú♠r♦ ♦♥

λ s ♦♥t ♦♥ ♥ ♠♦ δ s ♥ó♥ t r ②

sí♥ orig ♥ q ♥t s ♦③ ♥ ♦r♥ ♦♦r♥s

♦♥sr ♥ ♦♥t l = λ2♣r ♥t♥ st stó♥ r♣rs♥t

♣r♦①♠♠♥t ♥ ♣♦♦ r♥t ♠ ♦♥ ó♥ s ♥

♥ ♣r♦①♠ó♥ ♥♦ á♠tr♦ ♥t♥ s ♣qñ♦ ♦♥ rs♣t♦

s ♦♥t ♣r♦ ♥♦ s á ♣r ♥t♥s rss ♥ s s♦ rí q

♦♥srr ♣♥♥ r ~Jorig

♠♣♦ étr♦ rst♥t r♦ ♣♦r ♣♦♦ ①♣rs♦ ♥ ♦♦r

♥s sérs r stá ♦ ♣♦r

~Eorig(~r, t) = i2Iηei(kr−ωt)

cos(π2cos(θ))

sen(θ)θ

♦♥ η =√

ǫs ♠♣♥ ♠♦ ♦♥ µ ♣r♠ ♠♥ét

② ǫ ♣r♠t ♥s ♣♦t♥ dPds

♥t♥ ♣r

srr s ♣tró♥ ró♥

ds= |〈~S〉| =

2Re( ~E × ~H∗)

~S ♥♦t t♦r P♦②♥t♥ ② ♦s ♦rts ♥♥ ♣r♦♠♦ t♠♣♦r ♥

♥ ♣rí♦♦ Pr ♥t ♦♥sr rst

8π2r2

[cos(π2cos(θ))

sen(θ)

♥t♦♥s ♣♦t♥ ♣♦r ♥ á♥♦ só♦ dPdΩ

♦ ♥t♥s r

ó♥ s ♣ ①♣rsr ♦♠♦

orig= r2

[cos(π2cos(θ))

sen(θ)

♥♦ s ♦♥sr ♥ ♣♦♦ s♣③♦ ♦r♥ ♦ r♦ r

ó♥ ~d ♥s ♦rr♥t rs♣t ~Jdesp s ♣ srr s♥t

♠♥r~Jdesp(~r, t) = ae−i(ωt+ϕ) ~Jorig(~r − ~d)

♦♥ ~Jorig ♥ ♥s ♦rr♥t ♥♦ ♥t s ♥♥tr ♥

♦r♥ ó♥ r ♥ tér♠♥♦ s ϕ ② ♥ t♦r ♠♣t

a ♦♥ ♣r♦♣óst♦ ♦♥t③r ♣♦ss ♠♦s ♥ s♦s ♣rá♠tr♦s

♠♣♦ étr♦ rst♥t s

~Edesp(~r, t) = ae−iϕe−i~k·~d ~Eorig(~r, t)

♥♠♥t ♠♣♦ étr♦ ♥ rr♦ ♣♦♦s ♥ ró♥

♠♣♦ ♥♦ s ♣ ♦t♥r s♣r♣♦♥♥♦ ♦s ♠♣♦s ♦s ♣♦♦s ♥

s Pr ♦s ♣♦♦s ♠ ♦♥ ♦♥sr♦s ♥ rr♦ N

♠♥t♦s s sr ♣♦r s♥t ①♣rsó♥

~EN(~r, t) = ~Eorig(~r, t)(

N−1∑

tér♠♥♦ ♥tr ♣ré♥tss s ú♥♦ q ♦♥t♥ s ♣♦s♦♥s s ♥ts~dn ② ss ♠♣ts an ② ss ϕn rts ② s ♦ ♥ ♦♠♦ t♦r rr♦

ΓN =N−1∑

♣♦t♥ ♣♦r ♥ ár r ♣♦r rr♦(

Ns ♣

①♣rsr ♦♠♦(dP

N= |ΓN |2

♦♥(

origs ♥s ♣♦t♥ r ♣♦r ♥ ú♥♦ ♣♦♦ ♦

③♦ ♥ ♦r♥ ó♥ ♥á♦♠♥t ♣♦t♥ ♣♦r ♥

á♥♦ só♦ s ♣ ①♣rsr ♦♠♦

N= |ΓN |2

♣r♠r t♦r st ó♥ t♦r rr♦ t♥ ♥ ♥t

stró♥ s♣ ♦s ♠♥t♦s ss ss ② ♠♣ts rts ♠♥trs

q s♥♦ s ♥s ♣♦t♥ ♠♥t♦ ♥r♦r ♥t♦♥s s

♣ ♣♥sr q t♦r rr♦ ♠♦ ♣tró♥ ♠♥t♦ ♥r♦r

♠♥r ♠ás s♠♣ ♥rr ♣tr♦♥s ♦♥ rtrísts s♣ís

s ♦♥sr♥♦ rr♦s rrs ❯♥ rr♦ rr s ♦♠♣♦♥ ♣♦r rs

♥ts q t♥ é♥t ♠♣t ② ♣rs♥t♥ ♥r♠♥t♦s ♦s ♥ ss ss

r rr♦ rr ♦♥ N1 = 2 N2 = 2 N3 = 4

dx = 3,0 dy ② dz = 1,5 dx

② ♦♥ stró♥ rr ♥ s♣♦ ♥ st s♦ s út ①♣rsr t♦r

rr♦ ♣♦r ♠♦ ♥♦tó♥ rst♦rá ♣♦só♥ ♠♥t♦

rr♦ ~dn s sr ♣♦r ♠♦ t♦r ❬❪

~dn = n1~b1 + n2

~b2 + n3~b3

♦♥ ~b1 ~b2 ② ~b3 s♦♥ ♥r♦rs ♥♦ ♦♣♥rs ② ni = 0, 1, . . . , (Ni − 1)

i = 1, 2, 3 ♦♥ st ♥♦tó♥ ♥ú♠r♦ t♦t ♣♦♦s N s r♦♥ ♦♥

Ni ♣♦r ♠♦ N = N1N2N3 ♠♥trs q ♦♥t t♦t r ♥

ró♥ ~bi s (Ni − 1)|~bi| P♦r s♠♣ ♥ ♦ q s s s♠ q ~b1~b2 ② ~b3 s♦♥ ♣r♦s x y ② z rs♣t♠♥t s r q s ♥

~b1 = dxx,~b2 = dyy,~b3 = dz z

② ♥♠♥t ó♥ s ♣ ①♣rsr ♦♠♦ s

~dn = n1dxx+ n2dyy + n3dz z

♦♥ dx dy ② dz ♥♦t♥ s ♦♥st♥ts r ♦ r♦ ♦s trs s

rts♥♦s

r ♠str ♥ ♠♣♦ ♦♥ N1 = 2 N2 = 2 N3 = 4 dx = 3,0dy ②

dz = 1,5dx s ♠♥t♦ ϕn s ♣ ①♣rsr ♠♥r s♠r

♦♠♦

ϕn = n1α1 + n2α2 + n3α3

♦♥ α1 α2 ② α3 ♦s ♥r♠♥t♦s ♥ s r♦♥s~b1 ~b2 ②~b3 rs♣t♠♥t

♦♥sr♥♦ s ♦♥s ② ② s♠♥♦ q s ♥ts

t♥♥ é♥t ♠♣t ai = 1, ∀i t♦r rr♦ ó♥ s

♣ srr ♦♠♦

ΓN =(

N1−1∑

e−in1(α1+kxdz))(

N2−1∑

e−in2(α2+kydy))(

N3−1∑

e−in3(α3+kzdz))

s ♦♥sr q

N−1∑

e−imγ =sen(Nγ

sen(γ2)e−i(N−1) γ

♥t♦♥s s ♦t♥ s♥t ①♣rsó♥

|ΓN |2 =sen2(N1(α1+kxdx)

sen2( (α1+kxdx)2

sen2(N2(α2+kydy)

sen2( (α2+kydy)

sen2(N3(α3+kzdz)2

sen2( (α3+kzdz)2

♣♥♥ ♥r stá ♠♣ít ♥ ♦s tér♠♥♦s ki ó♥

stá ♦♠♣st ♣♦r trs t♦rs r♦♥♦s ♦♥ ♦s ♣rá♠tr♦s r ♥

r♥ts r♦♥s st♦ s♠♣ ♥áss s rtrísts ♣tró♥

rst♥t ♦♠♦ ♥ó♥ ♦s ♣rá♠tr♦s r ♦♥ ♥ s♠♣ó♥

♦♥ ♥♦ s ♦♥sr♥ rr♦s ♥ ♥ ♦ ♦s r♦♥s ♣♦r ♠♣♦

♣♦♦s ♥♦s ♦ ♥ rr♦ ♣♥♦ rs♣t♠♥t

♣é♥

Pr♦r♠ rr②

ó♦ rr② srr♦♦ ♥ ♥ ♣r♠t r

♣tró♥ ró♥ rr♦s ♣♦♦s ♠ ♦♥ ♦♥ r♦♥s

♣rs ② s ♦♥sr♥♦ ♥ ♦♥ró♥ rr ♦ rrr ♥ ♣r♠r

tér♠♥♦ ó♦ ♣tró♥ ró♥ ♥ ♣♦♦ ♠ ♦♥

♦③♦ ♥ ♦r♥ ó♥ ② q ést s ♠♥t♦ ♣r♠r♦

q ♦♥stt② rr♦ ♦ s s rtrísts rr♦ ♥ú♠r♦

♣♦♦s ♣♦só♥ t s tr♠♥ t♦r rr♦ ó♥

♣r rr♦s rrs ♦ ó♥ ♣r rr♦s rrrs ♥♠♥t

♣tró♥ ró♥ rr♦ s ♦t♥♥ ó♥

♣r♦r♠ ♠str ♦s ♦rs ♥s ♥ts q ♣r♠t♥ s

rr s rtrísts ♣r♥♣s ♣tró♥ ró♥ ♥t♥ ♥

st s♦ rr♦ ❬❪ ❯♥ ♥t ♠♣♦rt♥t s P♦t♥

♦t PT ♠t ♣♦r ♥t♥ q s ♥ ♦♠♦

∫ π

∫ 2π

dΩdΩ

♣ ♥t♥ ♣r rr st ♣♦t♥ ♥ ♥ ró♥

(θ, φ) s♠♥t s ♠ ♣♦r ♠♦ rt D(θ, φ) q s ♥

♦♠♦ ♣♦t♥ ♣♦r ♥ á♥♦ só♦ ♥♦r♠③ ♣♦r t♦r PT/4π

D(θ, φ) =4π

t♦r ♥♦r♠③ó♥ PT/4π r♣rs♥t ♣♦t♥ r ♣♦r ♥

á♥♦ só♦ ♥♦ s ♦♥sr ♥ ♥t♥ ♥♦ r♦♥ s♥ ♣♥♥

♥r st s ♣ ♥tr♣rtr ♠ás ♦♠♦ ♥s ♣♦t♥

♣r♦♠♦ ♣♦r ♥ á♥♦ só♦ ♥t♦♥s ♣r ♣tr♦♥s ♥♦r♠s s

D = 1 ♥ t♦s r♦♥s P♦r ♦♥trr♦ ♣r ♣tr♦♥s r♦♥s s

D > 1 ♣r q♦s á♥♦s ♥ ♦s s s ♣rs♥t♥ ó♦s ♠♥trs q

s D < 1 ♥ ♦s s q ♣rs♥t ♣tró♥ ❯♥ ♥t r♦♥ s

♥♥ ♥t♥ G(θ, φ) ♥ st s♦ ♣♦t♥ t♦t ♣r♦st ♣♦r

♥t PS t♠♥t ♠ P♦t♥ ♥tr r♠♣③ ♣♦t♥

♥t PT ♥ ó♥

G(θ, φ) =4π

s ♣érs q s ♣r♦♥ r♥t ♣r♦s♦ ♠só♥ s♠♥t

s r♣rs♥t♥ ♣♦r t♦r ♥ e q s ♥ ♦♠♦ ♦♥t

PT/PS t♦r e t♦♠ ♦rs ♥tr ② ♦rrs♣♦♥ ♥ s♣♦st♦

s♥ ♣érs ♠♥trs q ♦s ♦rs ♠ás ♦s ♦rrs♣♦♥♥ sst♠s

♠② ♥♥ts t♦r e r♦♥ ♠ás s ♥♦♥s G ② D

G(θ, φ) = eD(θ, φ)

út♠ ♥t q s ♦♥sr s á♥♦ só♦ ③

∆Ω =

∫ π

∫ 2π

D(θ, φ)

♦♥ Dmax s ♠á①♠♦ ♦r D(θ, φ) Pr ♥t♥s t♠♥t rts

s q ♦♥♥tr♥ ♠②♦r ♣rt s ♣♦t♥ r ♥ ♥ ♣qñ♦ á♥♦

só♦ ∆Ω s r♥♦ s ♠í♥♠♦ ♦r P♦r ♦♥trr♦ ♣r ♥t♥s

sótr♦♣s ∆Ω t♦♠ s ♠á①♠♦ ♦r 4π ♥t ∆Ω r♣rs♥t á♥♦

só♦ trés t♦ ♣♦t♥ r s ♥srí s ♣♦t♥

♣♦r ♥ á♥♦ só♦ dP/dΩ r ♦♥st♥t s♦r st ró♥ ♥r

♣r♦r♠ t♥ ♥ ♥trs rá q ♣r♠t ♥ s♦ ♠ ♣♦r

♠♦ ♥ rr ♠♥ú ② r♥ts ♦♥tr♦s rá♦s ❯t③♥♦ st ♥

trs s ♣♥ rr á♠♥t s rtrísts rr♦ ♥ r

s ♠str st ♥trs ♣r rr♦s ♦♥ ♦♥ró♥ rr ♦♣ó♥

r ♥ ♠♥ú rr② ♥ st♦r s♣r♦r ③qr♦ st ♥t♥

♣r♥ ♦s ♣rá♠tr♦s rr♦ rr② ♣r♠trs Nx Ny ② Nz ♥ú♠r♦

♣♦♦s ♥ ♦s s x y ② z rs♣t♠♥t di/λ i = x, y, z ss ♦rrs

♣♦♥♥ts s♣r♦♥s di ♥ ♥s ♦♥t ♦♥ λ ② ♥♠♥t

φi/π s ss φi ♥♦r♠③s π ❯♥ ③ q s ♥ st♦s ♣rá♠tr♦s

s ♥ ♦s ♦rs P♦t♥ ♦t ♦t rt ♣♦r

ó♥ ①♣rs ♥ ❲ts ② ♥♦ ó♦ ③ ♠ s♦

♥ ó♥ ♥ r♥s st♦s ♦rs s ♠str♥ ♥ st♦r

Pttr♥ rtrsts r ♥ ♣r♦ ♣r♥ tr♦ rá♦s

♥ r♥ts ♥t♥s ♦♠♦ s ♠str ♥ r ♥t♥

♣♦ rt♦♥ ♣ttr♥ q ♦rrs♣♦♥ ♣tró♥ ró♥ ♥

♣♦♦ ♠♥t ♥ ♦r♥ (dPdΩ)orig ♥t♥ rr② q ♠str

stró♥ ♥ s♣♦ ♦s ♣♦♦s t♦r rr♦ rr② t♦r

|ΓN |2 ② ♣tró♥ ró♥ rr♦ rr② rt♦♥ ♣ttr♥ (dPdΩ)N

♥♦ s rí♥ ♦s ♣rá♠tr♦s ♥tr s ♥ts ♥ st♦r Pttr♥

rtrsts ② ♦s rá♦s s t③♥ t♦♠át♠♥t st ♠♥r

s á r ó♠♦ r♦♥s ♥ rr♦ ♠♦♥ ♣tró♥ ró♥

♥t♥

♥ sst♦r t♦♥ ♥ ♣♦♥t st♦r Pttr♥ rtrsts s

♠str♥ ♥s rtrísts ♣tró♥ ró♥ q s ♥ ♥

ró♥ ♥ ♣♦r ♦s á♥♦s sér♦s θ ② φ ♥trsó♥ ♥tr sts

r♦♥s ② ♣tró♥ ró♥ s ♠str ♦♠♦ ♥ ♣♥t♦ ♥ ♣tró♥

ró♥ rr♦ ♥t♥ rr② rt♦♥ ♣ttr♥ r r s

♥ts s ♥ st♦r t♦♥ ♥ ♣♦♥t s♦♥ ♥t♥s

ró♥ ó♥ ♥s ♣♦t♥ ó♥

♥♥ ó♥ ② rt ó♥ ♦tr q ♣r

r rt s ♥sr♦ s♥r ♥ ♦r t♦r ♥ e

st ♦r ♥t♦ ♦♥ ♥t♥s I0 ♣♦♦ ♠♥t s ♣♥ str

s♦♥♥♦ ♣t♦♥s ♥ rr ♠♥ú

♠é♥ s ♣♦s ♠r t♣♦ rá♦ s♦♥♥♦ rá♦s t♣♦

s♣r r ♦ r r ♥ st♦r P♦t st ♦r♠ s ♣

♦t♥r ♥ ♦ ♣tró♥ t③♥♦ ♥ rá♦ s♣r ② ♦

s ♣♥ ♦srr ♦s ts t③♥♦ r♥ts ♦rts ♥♦ sr♦

r stá ♠r♦ s ♠str♥ trs rs ♥ ♣r θ ♦♥st♥t ② s ♦trs

♣r ♦s ♦rs r♥ts φ st♦s ♦rs s ♣♥ rr t③♥♦ ♦s

♦♥tr♦s s③s

rr ♠♥ú t♥ ♦♣♦♥s ♣r rr ♥ s♦ ♦♥ró♥

rr♦ ② ♦s rst♦s ② ♣r rr♦s ♥♠♥t P♦r ♦tr♦ ♦

♣r♦r♠ ♥② ♦s ♦♥r♦♥s ♣rtr♠♥s ♥ s ♣r

♥t♥ ♥ s t♦♥í r ② ♦tr ♣r ♥ ♥t♥ ♣♦♥s

r ♥trs rá ♣r♦r♠ rr② ♦♥r♦

♣r rr♦s rrs ♥ st♦r s♣r♦r ③qr♦ s ♣♥

♥tr♦r ♦s ♣rá♠tr♦s rr♦ st♦r Pttr♥ r

trsts ♠str ♥s ♥ts q rtr③♥ ♣tró♥

ró♥ st♦r P♦t ♣r♠t s♦♥r ♦r♠t♦

s③ó♥ s♣r ②♦ rs ♥ tr♠♥s ♣♦s♦♥s ♥

rs ♥♠♥t ♥ rt♦ st♦r s t③ ♥ ♣r♦s♦

r st♦s rá♦s ♠str♥ s③ó♥

♣tró♥ ró♥ ♥ ♣♦♦ ♠ ♦♥ stró♥

♦♠étr ♣♦♦s ♥ rr♦ t♦r rr♦ ②

♣tró♥ ró♥ rr♦

♦♠♦ ♥ ♠♣♦ ♥♦♥♠♥t♦ ♣r♦r♠ rr② s ♦♥sr

♥ rr♦ rr ♦r♠♦ ♣♦r tr♦ ♣♦♦s é♥t♦s ♣r♦s ♥♦s ♦♥

y r ♦s ♣rá♠tr♦s ♦♠♣t♦s st ♠♣♦ s♦ s♦♥

♦s s♥ts(Nx, Ny, Nz) = (1; 4; 1)

(dx, dy, dz)/λ = (0; 0,50; 0)

(φx, φy, φz)/π = (0; 0; 0)

♥ r s ♠str ♣tró♥ ró♥ rst♥t st ♦♥

ró♥ ♣r♦ ♥ ♣tró♥ t♦ ♦♥ ♦s ó♦s ♣r♥♣s q ♦♥♥tr♥

♠②♦r ♣rt ♥rí ♠t ② tr♦ ó♦s s♥r♦s ♠② ♣q

ñ♦s ♣♦só♥ ♦s tr♦ ♣♦♦s sr♣t♦s ♥ ♠♣♦ ♠ ♥

stró♥ rt♥r s♦

(Nx, Ny, Nz) = (2; 2; 1)

(dx, dy, dz)/λ = (0,25; 0,50; 0)

(φx, φy, φz)/π = (0; 0; 0)

♥ ♦♥ró♥ t♠é♥ ♣r♦ ♦s ó♦s ♣r♥♣s r

♣r♦ ♦♥ ♥ ♣♦t♥ r 162 W ♠②♦r q ♦r 115 W ♣r♦

♥ s♦ tr♦ ♣rá♠tr♦ q s ♣ rr ♣♦r ♠♦ ♥trs

rá s s rt ♥tr ♦s ♣♦♦s P♦r ♠♣♦ s s rt φi/π

♠ 0 0,5 ♣tró♥ ró♥ rst♥t r s ♠②

r♥t ♥ ó♦ s♣r ② ♥rí r s ♦♥♥tr ♥ ♥ ú♥

ró♥

♥♦ s trt rr♦s ♥♦ rrs s ♥t ♦♥ ♠②♦r rt ♥

ó♥ ♦s ♣rá♠tr♦s ♥♦ s ♥③ó s♦ r s ♠♦stró

q ♥t♦ ♦♥ ♦s ó♦s ♣r♥♣s ♣r♥ tr♦ ó♦s ♠ás ♣qñ♦s

st♦ s♥ q ♣rt ♥rí s♣♦♥ s ♣r♦♣ ♥ ♥ ró♥

st♥t s ♣r♥♣s ♥t♦♥s s s ♥t♥tr ♠♥♠③r ♦s ó♦s s

♥r♦s ❯♥ ♠♥r r③r st♦ s ♠r ♥t♥s rt ♦s

♣♦♦s Pr♠r♦ s ♥sr♦ ♠r ♥t♥ rrr rr② r

s♦♥♥♦ ♦♣ó♥ rr② ♥ rr ♠♥ú ② ♦ s♦♥r

rrr ♥ st ♥t♥ s ♣ r ♥ú♠r♦ t♦t ♣♦♦s N ②

♠♥r ♥♣♥♥t rr s rtrísts ♠♥t♦ s ♠♣t

s a s ♦♦r♥s ♥ s q s ♥♦ ♦s ♣♦♦s x/λ y/λ

r rr♦ rr tr♦ ♣♦♦s ♦s ♥

② ♦♥ dy = 0,5λ s♦ ② ♣tró♥ ró♥

♦rrs♣♦♥♥t

r rr♦ rr ♦r♠♦ ♣♦r ♦s s ♦s

♣♦♦s ♦♥ dx = 0,25 λ ② dy = 0,5 λ s♦ ② ♣tró♥

ró♥ ♦rrs♣♦♥♥t Ptró♥ ró♥ ♠s♠ s

tró♥ ♣r♦ ♥tr♦♥♦ ♥ ♦rr♠♥t♦ s φx = 0,5 π

♥tr ♦s ♣♦♦s

② z/λ ② s ss ♦rrs♣♦♥♥ts φ/π ♦s ♦rs q s ♠str♥ ♥

r ♦rrs♣♦♥♥ s♦ ①♣t♦ ♣♦r ♥ ♠♦ ♥ s ♥t♥ss

♦s ♣♦♦s ♥ ♣tró♥ ró♥ rst♥t r s ♣ r

q ♦s ó♦s s♥r♦s s♣r♥

♥ ♠♦s s♦s ♣rát♦s sr♣ó♥ té♥ ♥ ♥t♥ ♥②

♣tró♥ ró♥ ♣r♦ ♥♦ s ♦♠trí ♥ s stró♥ ♦rr♥t ♥

♦♥s♥ s t♥ ♥ r♥ ♠tó♥ r③r ①♣r♠♥t♦s ♥♠ér♦s

♣r ♦s s s ♥sr♦ ♦♥♦r ♦s ♠♣♦s tr♦♠♥ét♦s ♥r♦s

♣♦r ♥t♥ P♦r ♠♣♦ ♥♦ s trt r ♣♥tró♥ ♦s

♠♣♦s tr♦♠♥ét♦s ♥ s♦ ♦ sñ ♣r♦ ♣♦r ♥ ♦t♦ ♥

♣rtr ❯♥ ♠♥r s♣rr st ♠tó♥ s ♦t♥r ♥ ♥t q

♥t ♦ ♥ sst♠ ♥ts q ♣r♦③ ♥ ♣tró♥ ró♥ s♠r

♣tró♥ ♦♥♦♦ ♣r ♥ tr♠♥ ♥t♥ ♦s ♠♣♦s tr♦♠♥ét♦s

s ♣♥ r ♦ ♣r ♥t q♥t

♦♠♦ ♠♣♦ s ♦♥sr ♣tró♥ ró♥ ♥ ♥t♥ ♦

rr q s ♠str ♥ r ❬❪ ② s s ♥ rr♦ ♣♦♦s

②♦ ♣tró♥ s st ♦r♥ Pr ♦ s t③ ♦♣ó♥ tt♥ ♠

♥ú ♣r♥♣ ♣r♦r♠ rr② r ♥♠♥t s ♣r♦♣♦♥ ♥

rr♦ ♦♠♣st♦ ♣♦r ♦s ♣♦♦s ♦s ♦ r♦ ① ② s rí s

♣♦só♥ ② ss rts t③♥♦ ♠ét♦♦ ♣r ② rr♦r ♦♠♦ rr♦r

st ♣r♦r♠ só♥ stá♥r Pr ♦♥ró♥

♦s ♣♦♦s s ♦t♥ ♥ ♥ st ♣r dx/λ = 0,17 ② φx/π = 0,5 r

rr♦r st ♣r st ♦♥ró♥ s Pr ♠♦rr st

s r ♦tr♦ ♣♦♦ t♠é♥ ♦ ♥ x ♦ s ♦ s♣③ ♥

♥ ♣♦só♥ ♥ x ♠♥r q rst ♥ rr♦ ♥♦ rr

♠♦r st s ♦t♥ ♣r ♣♦♦ ♥ st♥ 0,31 λ ♦r♥ ② s

rt r♦ rr♦r r ♠str ♣tró♥ ró♥ r

st♥t r♠♥t st rst♦ s ♠♦r ② ♥♦ s ♦ ♣♦rí r ♦t♥♦

t③♥♦ ♥ rr♦ rr

♦s ♠♣♦s ♥tr♦rs ♠str♥ r♠♥t ó♠♦ s ♣♦s str s

rtrísts ♣tró♥ ró♥ ② s ♣♥♥ ♦s ♣rá♠tr♦s

rr♦ ♥ú♠r♦ ♣♦♦s ♣♦s♦♥s ♠♣ts ② ss ♦♥♦♠♥t♦

r♦ ①♣r♠♥tr r♥ts ss rr♦s s ♠♣♦rt♥t ♣r sñr

rr♦s ♦♥ str♦♥s s♣ís ♥rí r

r ♥trs rá ♣r♦r♠ rr② ♦♥

r♦ ♣r rr♦s ♥♦ rrs ♥ st♦r rr② Pr♠trs N

s ♥ú♠r♦ t♦t ♣♦♦s a s ♠♣ts ♦rrs♣♦♥♥ts

x/λ y/λ ② z/λ s ♦♥s ♣♦♦ ② phi/π ss

♦rrs♣♦♥♥ts Ptró♥ ró♥ ♦rrs♣♦♥♥t ♦s ♣

rá♠tr♦s N = 4 a = (1 3 3 1) x/λ = (0,0 0,0 0,0 0,0) y/λ =

(0,0 0,5 1,0 1,5) z/λ = (0,0 0,0 0,0 0,0) ② φ/π = (0,0 0,0 0,0 0,0)

r Ptró♥ ró♥ ♦rrs♣♦♥♥t ♥ ♥

t♥ ♦rr Ptró♥ ró♥ ♦t♥♦ st♥♦ ♦s

♣rá♠tr♦s ♥ rr♦ rr ♦♠♣st♦ ♦s ♣♦♦s rá

♦ s♣r ② ♣tró♥ ♦r♥ í♥ ♣♥t♦s Ptró♥

ró♥ ♦t♥♦ st♥♦ ♦s ♣rá♠tr♦s ♥ rr♦ ♥♦

rr ♦♠♣st♦ ♣♦r trs ♣♦♦s

♣é♥

Pr♦r♠ ♥t♦r♥♦

♠ét♦♦ rr♦s s♥tét♦s ♠s♦rs ♦rr ♣r♠t ♠♥

tr rt ♦s ♠♣♦s tr♥s♠t♦s ♣♦r ♦s q♣♦s ② ♦♥♥trr

♥rí s♣♦♥ s♦r ♦s ♥♦s ♥trés tr sñr rr♦

♣r♦♣♦ ♣r s♦ ♠♣ str r♦s ♣rá♠tr♦s ts ♦♠♦ s

t♥ ♥tr ♦s ♦♠♣♦♥♥ts rr♦ ② ss ss rts Pr r③r

ó♥ ♦s ♣rá♠tr♦s ♦s s ♥ ♦♥srr s ♣r♦♣s

♠♦ ♣r♦♥ ♦s ♥♦s ② r♥ ♠só♥ q♣♦

♥ ♠r♦ st ss s srr♦ó ♥ ó♦ ♦♠♣t♦♥ ♣r

♠♣♠♥tó♥ s st♥ts t♣s ♠ét♦♦ ♣r♦r♠ s

rr♦♦ ♥ ♥ rsó♥ stá ♦ ♥ trs ♠ó♦s

rr② t♦ rr♠ ♠ Pr♦ss♥ st♦s ♣r♠t♥ ♦

t♥r rá♦ ♠♣♦ ♥ rr♦ ♥ ♥ ♠♦ ♥♦r♠ ♦t♥r

rs♣st rr♦ ♣r t♦s ①♣r♠♥ts ♦ s♥tét♦s ② ♣r ♥ ♠

t♦♦♦í ♣r♦s♠♥t♦ ♣r ♠♥tr ♦♥t♥ tr s sñs

ó♦ t sñ♦ ♦s rr♦s ♣r ♦♣t♠③r s rtrísts

♠♣♦ s♦r ♥♦ srr♦ó ♠ás ♥ ♥t♦r♥♦ rá♦ ♠ ♦♥

rr♠♥ts ♣r ♥áss ♦s rst♦s ♣r s♠♣r ♣ó♥

♠s♠♦

♥t♦r♥♦ ♦♥sst ♥ ♦s ♥t♥s ♥ s s ♦♥t♥ ♥trs

rá ② ♦tr ♦♥t♥ r ♦s rst♦s ♥trs rá ♣r♠t

♥ s♦ s♥♦ ♣r♦r♠ trés ♥ rr ♠♥ú ② r♥ts

♦♥tr♦s stá ♥ trs ♣♥s q ♣r♠t♥ rr ♦s t♦s ♥rr

r ❱♥t♥ ♣rs♥tó♥ ♣r♦r♠ ♥t♦r♥♦

rs♣st rr♦ ② ♠♦r ♠♥ ♦t♥

♥r ♣r♦r♠ s ♠str ♥ ♥t♥ ♣rs♥tó♥ r

♦♥ ♥♦♠r ♣r♦r♠ ② ♦t♦♥s q ♣r♠t♥ ♥rsr ♥♦ ♦s

♠ó♦s rr② t♦ rr♠ ♠ Pr♦ss♥ ♠é♥

s ♠str♥ ♦t♦♥s q ♣r♠t♥ rr ♥t♥ ② ♣ ♦ sr

♣r♦r♠ ①t

♠ó♦ rr② ♣r♠t s♠r ♦♥stró♥ rr♦s ♠

s♦rs ♦rr ♣rtr ♥ts t♣♦ ♣♦♦ ♣rs ♥tr sí ② ♥ ♥

♦♥ró♥ rr s♥tt③r ♠♣♦ rr♦ ② ♠♦strr ♥ rá♦

♠s♠♦ ♠ó♦ t♦ rr♠ ♣r♠t ♦t♥r rrr♠ ♦

rrs♣♦♥♥t ♥ rr♦ ② ♠ó♦ ♠ Pr♦ss♥ ♣r♠t ♦t♥r ♥

rrr♠ ♦♠♣st♦ ♣♦r s♠♥t♦s rrr♠s ♥r♦s ♦♥ st♥t♦

s♣③♠♥t♦ t♠♣♦r

♦♦s ♦s ♠ó♦s stá♥ ♦s ♥ trs ♣♥s ♥♣t t rr② ♣r

♠trs ♠ r ❯♥ ③ q s ♥rs qr ♦s trs

♠ó♦s s ♣ r ♥ ♣rt s♣r♦r ♥t♥ ♥ rr ♠♥ú ♦♥

♦♣♦♥s ♣r ♦♣rr ♦♥ r♦s ♣r ♥rsr ♦s st♥t♦s ♠ó♦s

♣r♦r♠ rr② t♦ rr♠ ♠ Pr♦ss♥ ♦

♥t♥ ② ♣

♦♣ó♥ s♣ ♥ ♠♥ú ♦♥ s ♦♣♦♥s ♣r♠trs r

♥ ♥ r♦ ♦s ♦rs t♦♦s ♦s ♦♥tr♦s q s ♠str♥ ♥ ♥t♥

♦ ♣r♠trs r s ♥ r♦ ♦s ♦rs t♦♦s ♦s ♦♥tr♦s

q s ♠str♥ ♥ ♥t♥ ①♣♦rt t r ♥ ♥ r♦ s ♦s

♦rs rá♦ q s ♠str ♥ r ①♣♦rt r r ♥ ♥

r♦ t ♣ rá♦ q s ♠str ♥ r st ♠♣ ♦s

♦rs q s ♠str♥ ♥ ♦s ♦♥tr♦s ② ①t ♣r♠t sr ♣r♦r♠

ó♦ rr②

♠ó♦ rr② r stá ♦ ♥ trs ♣♥s ♥♣t t

rr② ♣r♠trs ♠ ♣♥ ♥♣t t ♣r♠t ♥rsr ó♥

♦s r♦ q ♦♥t♥♥ ♦s t♦s Pt ♦s r♦s ♥tr ♠ó♦

rr② ♥ ♦♥t♥r ♦s t♦s ♦rrs♣♦♥♥ts ♦s ♦rs ♠♣♦

♥ ♥ó♥ ♣♦só♥ ♣r ♥ ♠♦ ♥♦r♠ ♦ ♣s ♥ ♥r

♦♥ s♠trí trsó♥ ♥ ró♥ ♦r③♦♥t ♦s t♦s ♥tr ♥

♥♦♥trrs ♥ ♥ ♠tr③ rt♥r ♥ ♦r♠t♦ s ② ♦♥ s♣♦ ♦♠♦

s♣r♦r ♥tr♦ ♠tr③ rt♥r ♦s t♦s ♥ str ♦r♥♦s

s♥t ♠♥r ♥ ♣r♠r ♠♥t♦ ♣r♠r ② ♣r♠r ♦♠♥

t♠♣♦ q ♦rrs♣♦♥♥ ♦s t♦s ♦♥t♥♦s ♥ ♠tr③ ♣r♠r

♣♦s♦♥s ♦r③♦♥ts x ♣r♠r ♦♠♥ ♣♦s♦♥s rts z ♥

rst♦ ♦s rs ♦s ♦rs ♠♣♦ ♥ ♦s ♣♥t♦s (x, z) r r

♦♦s ♦s r♦s t♥♥ q t♥r ♠s♠ ♥t s ② ♦♠♥s

♣r♦r♠ r t♦♦s ♦s r♦s q s ♥♥tr♥ ♥ ó♥

♥ ② ♠str ♦s ♦rs ♥ ♥ ② ♣s♦ ♣r t♠♣♦ ② s

♣♦s♦♥s x ② z ♦s t♦s ♦s ♣rá♠tr♦s rr♦ r sr ♦♥sst♥ts

♦♥ st♦s ♦rs

♣♥ rr② ♣r♠trs r ♣r♠t ♥tr♦r ♦s ♣rá♠tr♦s

rr♦ ♥ú♠r♦ ♥ts N s♣ró♥ d ② s♣③♠♥t♦ t♠♣♦r

r ó♦ rr②

rt♦ dt ♥tr ♥ts ② t♠♣♦ ♥ q s qr ♦t♥r ♠♣♦

♣♥ ♠ r ♣r♠t ♠♦strr r♥ts ♠♥t♦s s♦r

♠♥ q s ♣♥ t③r ♥ ♥áss ♠s♠ P♦r ♠♦

♦♣ó♥ ♦♥t♦r ♦ t r♦♥t ♦ s♥ tr♥s♠ttr s ♣♦s s♣r♣♦♥r

r ♥ r ♣r sr t③ ♦♠♦ rr♥ ♣r ♦♠♣rr ♦r♠

♦s r♥ts ♦♥s ♥ ú♥ ♥t ② rr♦ r s ♥ r♦

♣s ♣r♦r♠ ♣r♠t ♥rsr ♦s ♦rs ♦s s♠s rt ②

♦r③♦♥t ♣r♦♠♥t♦ ♦♥sst ♥ ♥rr rá♦ ♠♣♦ ♥

♥t ♣r ♥ tr♠♥♦ t♠♣♦ str r r♥t ♦♥s ② ♦

♥rr rá♦ ♠♣♦ rr♦ ♣r ♠s♠♦ t♠♣♦ st ♦r♠ s

♣♦s ♦srr ♣♦r ♠♣♦ á s r♥♦ ♦s ♣rá♠tr♦s ♣r ♦s q

r♥t ♦♥s ♥♦ s ♦r♠ ♦♥ rs♣t♦ r♥t ♥ ♥t ♦♣ó♥

♥ ♦ t ♠①♠♥ ♠str s♦r r ♥ rt ♥tr ♦r♥

♦♦r♥s ② ♣♦só♥ ♠á①♠♦ ♥t♥s ♠♣♦ ♠str ♥

x1 x2 x3

z1 E11 E12

z2 E21

r ♦r♠t♦ ♦s t♦s ♥tr ♣r ♠ó♦

♥t♥ ♦r ♥♠ér♦ á♥♦ ♦♣ó♥ ♥r ♣rtr ♠str

s♦r r ♦s í♥s q ♥♥ ♦s á♥♦s ①tr♠♦s ♥ ♦s q ♠♣♦

t♥ ♥t♥s ♥ ♠ó♦ s♣r♦r ♥ ♥t♥s ♠r ♥t♥s

♠r IT s ♥ ♥ ó♥ ♦♥ IMax s ♥t♥s ♠á①♠

♠♣♦ ② δ s ♥ ♦r ♠♥♦r ♦ 1

IT = IMax · δ

s ♥t♥ rá s ♣ rr ♦r δ ♠♣t rt♦♥

♣r ♦srr s ③♦♥s st♥t ♥t♥s r♥t ♦♥s t♠é♥

s ♠str ♦r ♥♠ér♦ ♣rtr ♥r ♥♠♥t ♣♦só♥

♦ rr② ♠str s ♣♦s♦♥s rr♦ s♦r r ② s ♣♦s♦♥s

trt♦♥ ② ①s ♣r♠t♥ ♦♥tr♦r s ♦♦rs ② s s♣

rá♦

♦♠♦ ♠♣♦ ♣ó♥ ♣r♦r♠ s ♦♥sr ♥áss ♠♦

♦ q s ♠str ♥ r Pr ♥rr ♦s t♦s s♠♦s s t③

♥ r♥ ♥tr ♠só♥ fc = 500 ③ ♠♦ ♣r z < 0 r

♣rs♥t ♥ ♣ r ② s rtr③ ♣♦r ♥ ♣r♠t rt ǫr = 1

② ♥ ♦♥t σ = 0 s♦ ♣r z > 0 s rtr③ ♣♦r ǫr = 3,5

σ = 1 ♠♠ Pr r③r s♠ó♥ ♥♠ér s t③ ♠ét♦♦

♣é♥ ♥tr♦ r s 0,01 ♠ ♥ ♠s r♦♥s

rá♦ q ♥r ♦♠♦ rs♣st ♣r♦r♠ s ♠str ♥ r

♦♥ ♦s ♣rá♠tr♦s s♦♥♦s N = 5 d = 0,1 ♠ dt = 0,2 ♥s ② t = 8

♥s s ♦t♥ ♥ ♠♣♦ ♦♥♥tr♦ ② ♦r♥t♦ ♥ r t♠é♥ s

♠str♥ ♦s st♥t♦s ♠♥t♦s q ♦♦rr ♥ ♥áss rst♦

r rr♥ q ♣r♠t ♦♠♣rr ♦♥ ♦r♠ r♥t ♦♥s

♥ ú♥ ♥t ♦r♥tó♥ ② ♣rtr ♥r ③ ② s ♣♦s♦♥s

s ♥ts rr♦

r ♦♦ ♥ s♠s♣♦ ♥♦r♠ ♦♥ ǫr = 3,5 ②

σ = 1 ♠♠

ó♦ t♦ rr♠

♠ó♦ t♦ rr♠ r stá ♦ ♥ trs ♣♥

s ♥♣t t rr② ♣r♠trs ♠ ♣♥ ♥♣t t ♣r♠t ♥rsr

ó♥ ♦s r♦ q ♦♥t♥♥ s tr③s Pt

♣r♦r♠ ♠t ♦♠♦ ♥tr t♦s qr♦s ♥ ♦♥ró♥

r♣t♦r ♦♠ú♥ s r q r♣t♦r s ♦♦ ♥ ♥ ♣♥t♦ í♥

s♦♥♦ ② ♦ s qr♥ tr③s ♣r st♥ts ♣♦s♦♥s ♠s♦r s♦r

♠s♠ í♥ Pr ♥ s ♣♦s♦♥s r♣t♦r s♦r í♥

s♦♥♦ s ♦t♥ ♥ ♦♥♥t♦ t♦s q t♠♥t s r ♥ ♥

r♦ s♣r♦ ♥ ♥ ♦♥tr♦ sst♠ ♦rr ♣r♦r♠

r t♦♦s ♦s r♦s q s ♥♥tr♥ ♥ ó♥ ♦s r♦s

♥ ♦♥t♥r ♦s ♦rs ♠♣♦ ♥ ♥ó♥ t♠♣♦ ♣r ♥

s ♣♦s♦♥s s ♣♦s♦♥s ♥ str s♦r ♥ ♠s♠ í♥ ② ♥

s tr③s ♦rrs♣♦♥r ♣♦s♦♥s st♥ts s ♥ts rr♦

♦s t♦s ♥ ♥♦♥trrs ♥ ♥ ♠tr③ rt♥r ♥ ♦r♠t♦ s ② ♦♥

s♣♦ ♦♠♦ s♣r♦r r ♦♦s ♦s r♦s t♥♥ q t♥r

♠s♠ ♥t s ② ♦♠♥s ♥tr♦ ♠tr③ rt♥r ♦s

t♦s ♥ str ♦r♥♦s s♥t ♦r♠ ♥ ♣r♠r ♠♥t♦

♣r♠r ② ♣r♠r ♦♠♥ ♣♦só♥ r♣t♦r xR ♥

♣r♠r s ♣♦s♦♥s ♠s♦r xE ♥ ♣r♠r ♦♠♥ ♦s t♠♣♦s

② ♥ rst♦ s ♣♦s♦♥s ♠tr③ ♦s ♦rs ♠♣♦ ♥ ♥ó♥

♣♦só♥ ② t♠♣♦

♣♥ ♥♣t t ♣r♠t ♣r rt♦s ♣r♦s♦s ♦s t♦s ♦

♠♥ ♦♠♣♦♥♥t r♥ ♦r♠s ♠♣t ♥♦r♠③

r rá♦ ♠♣♦ ♣r N = 5 d = 0,1 ♠ dt = 0,2

♥s ② t = 8 ♥s ♦t♥♦ ♦♥ ♠ó♦ rr②

♠♣t ♥ s tr③s ♣♦r s♣r♦ ② ♦rrt t♠ ♦r♥ ♦rr

♦r♥ t♠♣♦ ♣♥♦ ♦♣ó♥ ♠ rs♠♣♥ s ♣ ♥tr♣♦r

♥ t♠♣♦ ♥ s tr③s ♣♦r s♣r♦ ♥♦tr q ♣s♦ t♠♣♦r

♦s t♦s s ♦rrs♣♦♥ ♦♥ ♠í♥♠ ró♥ q s ♣ ♥tr♦r

♥ s♣③♠♥t♦ t♠♣♦r ♥tr s ♥ts rr♦ ② ♥ ♦♥s♥

♥ á♥♦ ♦r♥tó♥

♣♥ rr② ♣r♠trs ♣r♠t ♥tr♦r ♦s ♣rá♠tr♦s rr♦

♥ú♠r♦ ♥ts N st♥ d ② s♣③♠♥t♦ t♠♣♦r dt rt♦

♥tr ♥ts ② ♦r♥ ♦s ♠♥t♦s r♣t♦r r ♦ ♠s♦r

♣♥ ♠ ♣r♠t ♠♦r s♣t♦ ♠♥ ♥t ♦♥ s

♦♣♦♥s ♥ ♣ ♥♥ r P♦st♦♥ rs♠♣♥ ♥tr♣♦

♥ s ♣♦s♦♥s ♠♦ rt rst sñ rt r

trt♦♥ ♣r♠t rr stró♥ s ♦♦rs rá♦ ②

①s ♣r♠t rr ♦s s r

♦♠♦ ♠♣♦ s ♦♥sr♥ ♦s t♦s ♥♠ér♦s ♦t♥♦s ♣r ♠♦♦

r ❯♥ ♦t♦ ♦♥ á♠tr♦ 0,05 ♠ s ♦③ ♥ ♣r♦♥

r ó♦ t♦ rr♠

xR xE1 xE2 xE3

t1 S11 S12

t2 S21

r ♦r♠t♦ ♦s t♦s ♥tr ♣r ♦s ♠ó♦s

t♦ rr♠ ♠ Pr♦ss♥

r ♦♦ ♥ rt♦r ♣qñ♦ ♦♥ á♠tr♦ 0,05♠

♥ ♣r♦♥ 0,80 ♠ ♦♥ ♣r♠t rt ǫr = 5,25

② ♦♥t σ = 2,5 ♠♠ ♥ ♥ ♠♦ ♦♥ ǫr = 3,5 ② σ = 1

♠♠

0,80 ♠ ♦t♦ s rtr③ ♣♦r ♥ ♣r♠t rt ǫr = 5,25 ②

♥ ♦♥t σ = 2,5 ♠♠ ② ♠♦ r♥♥t ♣♦r ǫr = 3,5 ② σ = 1

♠♠ ♠♦ s♦r ♦s s r ♥tr♦ r s 0,01 ♠ ♥ ♠s

r♦♥s ♥ ♦s ♣rá♠tr♦s s♦ s ♣♥ t♦♥s t♦rs

r♥ ♥tr s ♦♥s ♠ts s fc = 500 ③ ♥ r

s ♠str rs♣st ♦t♥ ♣r N = 5 d = 0,1 ♠ ② dt = 0,2 ♥s

P♦r ♠♦ ♣♥ ♠ s st ♠♦♦ s③ó♥ ♣r ♦srr

♥ ♦r♠ rst♦

ó♦ ♠ Pr♦ss♥

♠ó♦ ♠ Pr♦ss♥ stá ♦ ♥ trs ♣♥s ♥♣t t rr②

♣r♠trs ♠ r ♦s ♣♥s ♥♣t t ♠ ♦♥♥

♦♥ ♦s q ♣r♥ ♥ ♠ó♦ t♦ rr♠

♣♥ rr② ♣r♠trs ♣r♠t ♥tr♦r ♦s ♣rá♠tr♦s rr♦

♥ú♠r♦ ♥ts N st♥ rt ♥tr ♥ts d ② ♦r♥ ♦s

♠♥t♦s r♣t♦r r ♦ ♠s♦r r ♥ st ♣♥ s ♠s

tr♥ ♠ás trs ♦♠♥s ♠♥t s s s ♣♦s ♥rsr ♦s í♠ts

♦s ♥tr♦s ♥t ♣♦st♦♥ ② ♥ ♣♦st♦♥ ② s♣③♠♥t♦ t♠♣♦r

♥ ♥♦ ♦s t t♠ st t♥ s♦rs ♥ r

s ♠str rs♣st ♦t♥ ♣r ♠♦♦ r ♦♥ N = 5

d = 0,1 ♠ ② dt s♦♥♦ ♠♥r rstr sñ rt♦r ♥

♥tr♦

r rá♦ ♠♣♦ ♣r N = 5 d = 0,1 ♠ ② dt = 0,2

♥s ♦t♥♦ ♦♥ ♠ó♦ t♦ rr♠

♣♥ ♠ ♥t ♦♥ ♦♣ó♥ ♦ t ♠ts ♦ t ♥trs q

♠str ♦s í♠ts ♦s ♥tr♦s ♥ ♦s q s ♣♥ ♦s st♥t♦s ♦rs

rtrísts ♥rs ♣r♦r♠

♥ t♦s s ♠♦s ♣r♦r♠ ♥♦ s ♠♦ ♥♦

♦s ♣rá♠tr♦s ♥tr ♦♥tr♦ ♠ ♦♦r ♥♥♦ q ♦r q

s ♠str ♥♦ ♦rrs♣♦♥ rá♦ ❯♥ ③ q s t③ rá♦

♣♦r ♠♦ ♦s ♦t♦♥s ♦ ♦ ♣♣② ♦s ♦♥tr♦s rrs♥ s ♦♦r ♥

♣r♦r♠ ♥t ♦♥ ♥♦s ♠♥t♦s q t♥ s s♦ ♣♦r ♠♣♦ s s

trt ♥rr r s♥ r r♦ ♦s t♦s ♣r♠♥t s ♠str

♥ s♦ ♦ ♠s♠♦ ♦rr s s ♥t♥t ♣r ú♥ t♣♦ ♣r♦s♠♥t♦

♥ts r ♥r♦ r

♣r♦r♠ t♥ rss ♣♦♥s ♣♦r ♠♣♦ s ♦ ♣ t③r

♣r sñr rr♦ ♥ts r③r ♥ st♦ ♠♣♦ ♣r st♠r

♥t ♥ts ② s♣ró♥ ♥tr s q ♣r♠t♥ ♦t♥r

t♦s ♦s ♣r ♣r ♠ét♦♦ ♠é♥ ♣r♠t tr♠♥r ♦s

♣rá♠tr♦s rr♦ ♣r rstr sñs ♥ tr♠♥s ③♦♥s rr

r ó♦ ♠ Pr♦ss♥

r♠ ♣♦r ♠♣♦ st♥ts ♣r♦♥s á♥♦ s t③ ♠ó♦

t♦ rr♠ ② s ♦♥ t♦s s♠♦s ♦ ①♣r♠♥ts ♣r♦

r♠ ♣r♠t str s st♥ts sñs ② rr ♦s ♣rá♠tr♦s rr♦

st r③r ♥ st ♥♦ ♦s ♣rá♠tr♦s st♠♦s ♦♥ ♠ó♦ rr②

♥♠♥t ♠ó♦ ♠ Pr♦ss♥ s ♣ t③r ♥ s♦ ♥

q s sñs ♣rs♥t♥ r♦♥s ♥♥ó♥ r♥s ② q ♥ ú♥

s ♥♦ ♣r♠t rstrs ♥ s t♦t ♥ s♦ ♦srr q sñ

♥trés ♥♦ s rst ♥ t♦ s ①t♥só♥ ♥♦ s ♦sr rs♣st

♠ét♦♦ ♠ó♦ t♦ rr♠ ♠t♦♦♦í ♦♥sst

♥ ♣sr ♠ó♦ ♠ Pr♦ss♥ ② s♦♥r s ♣r♦♣ ♣r ♦s

st♥t♦s ♥tr♦s ♣♦só♥ ♣r ♠♥tr sí ♦♥t♥ tr s

sñs ♥trés

r rá♦ ♠♣♦ ♣r N = 5 d = 0,1 ♠ ② dt

s♦♥♦ ♠♥r rstr sñ rt♦r ♥

♥tr♦ ♦t♥♦ ♦♥ ♠ó♦ ♠ Pr♦ss♥

r♠♥t♦s

r♦ rr ♥ ② ést♦r ♣♦r r♠ ♦ r♦ st ss ♦♥

t♥t ♣♥ ② ó♥

r♦ rr t♦s s ♣rs♦♥s q ♦r♠♥ r♣♦ ♦ís

♣ ② ♠♥t ♣♦r s t♦ ♥ t♦♦s ♦s ♠♦♠♥t♦s ♦♠♣rt♦s

r♦ rr ♠ ♠ ♣♦r ♣♦②r♠ s♠♣r

♦rí

❬❪ ♦s Prs st ♥ ❱ ♥ ♥

♠♦t s♥s♥ ♥ P sts ♦ t♦ t ts ❲str♥ ♥t

P♥s ♥ ♦r♥ ♦ ♣♣ ♦♣②ss

❬❪ ♦♠③ ♥ ♠♦♦ s♣♥ss ♥ ❲ss♠r P

♦♥s ♦ ♣♦st♦♥ ♦♥♣t ♠♦ r♦♠ P

sr② ♦♥ ♣②r♦st♦ ♣♦sts t r♣ ❱♦♥♦ ♥♦♥s ♦

♠♦r♣♦♦②

❬❪ Ptt♥ ♥ ♦♠r ♥ ♥♥♥ P P

♥ ♦♠str② ♠sr♠♥ts ♦ ♥ t s ♥t t♦ st②

♥rsr s♠rt♦♥ ♣t②s ♦♣②ss

❬❪ ♦♥♦♠♦ r♥ s ♥ tt♦ P ♣r♦s♣t♥

♥ ♣rs♣♥ ❲ r♥t♥ ♦r♥ ♦ ♣♣ ♦♣②ss

❬❪ r rr r t♥ss ♦

r♦♥♣♥trt♥ rr sr②s ♥ t ♦t♦♥ ♦ ♥♠r r sts

♥ ♠♦r♥ ♠trs ♦r♥ ♦ ♣♣ ♦♣②ss

❬❪ Pér③r ❱ ss ♣s s♦r♦ rtí♥③ ♥

♥s ♥trt ♥rsr ♦♣②s sr② ♦ t tr

♦ ♦r ♦r♥ ♦ r♦♦ ♥

❬❪ r♥♦ ♦ ❯s♥ ♠t♦♠♣♦♥♥t P t♦ ♠♦♥t♦r rs

♥ st♦r ♥ ♦r♥ ♦ ♣♣ ♦♣②ss

❬❪ P♦rs♥ ♥ ❲ r♦♥♣♥trt♥ rr ♣r♦s ♦r

♠t♣ st t♥s rtt r♠♦ tr♦ t t ♣r♦ss♥

♦♣②ss

❬❪ ♥s♠t ♥ ♦r♦♣♦♦s ♥s♣t♦♥ ♦ rt

♥♥ s s♥ P ♦r♥ ♦ ♣♣ ♦♣②ss

❬❪ r♦♦ Prs♦ ♦♦r ♥ ss② r②st

♥ ♣♣s P♠♦♥t♦r♥ ♠r♦ t♦♠♦r♣ ♥rs♦♥ ♦r

♥ ♦ ♣♣ ♦♣②ss

❬❪ s♠ ❨ ❩♦ t♦ st♠t♦♥ ♦ r♦♥tr

② P ♥ ♥ r t ♠t♣ ♠♦s rt♦♥s ♦r♥ ♦ ♣♣

♦♣②ss

❬❪ ♦♦r Prs♦ ♥ Prs♦ ♣♣t♦♥ ♦ ♠r♦

t♦♠♦r♣② ♥ ②r♦♦♣②ss ♦♠ ①♠♣s ❱♦s ❩♦♥ ♦r♥

❬❪ ♦♥ rr② ♦r③ ♥ ❩♥ ♥trt

♦♣②s ♥ ♦♦ ♥stt♦♥ ♦ tr♦♥♦s qr

♥ ♦♠s ssss♣♣ ♦r♥ ♦ ♣♣ ♦♣②ss

❬❪ s ♦♦ ③③♦ ♥ ♣♥♥ ❱ ♦♣②s

rtrst♦♥ ♦ r♦s ❱ st r♦t♦♥ t② ♦r♥ ♦ ♣♣

♦♣②ss

❬❪ ♥ r ❯s ♦ r♦♥ ♣♥trt♥ rr t♦ ♠♣

ssr r♦♦ trs ♥ ♥ r♥ r ♦r♥ ♦ r♦

♦ ♥

❬❪ r♥s ♥③ ♥ ♦r rs♦t♦♥ ♠t

♥♥ rr ♥stt♦♥ ♦ ♦♠♥ ♥ ♦rtr♥ t② ♦r♥

♦ ♣♣ ♦♣②ss

❬❪ rs♠ ❲r ♥ ♦rst♠②r rs♦t♦♥

P ♠♥ ♦♣②ss

❬❪ ♦ ❨ ♥ ♥ Ptt♥ ♥ st ♥

♠sr♠♥ts ♦ rt♦♥ ♣ttr♥s ♦ ♣♦ P ♥t♥♥s

♦r♥ ♦ ♣♣ ♦♣②ss

❬❪ ③s ♥ Ptrs r ♥s r

♣♦ rt♦♥ ②♥♠s tr♦ ♠♦♥ ♦r♥ ♦ ♣♣

♦♣②ss

❬❪ rr ♥ ♦ t♦st r♦♥ ♣♥trt♥ rr

t ♦r ♠♣r♦ ♠♥ ♥ rs ♦ tr ♦♠♣①t② ♣♣t♦♥ t

t ♠r♥ st ♦r♥ ♦ ♣♣ ♦♣②ss

❬❪ P♣♥ r♦ ♦rt Pr③③♦♥ ♥ ♥tt

♥ ♣r♦ss♥ ♥ ♥tr♣rtt♦♥ ♦ ♠t♦ r♦♥ ♣♥trt♥

rr t s st♦r② r♦♠ ♥ r♦♦ st ♦r♥ ♦ ♣♣

♦♣②ss

❬❪ r Pr♥ ♠♥ ♦♥ rt♦ P ♥ ♥rs

sts ♦ t P r♦♥ ♠t♦ ♦r st♠t♥ s♦

tr ♦♥t♥t r♥ rrt♦♥ ♥ r♥ ♦r♥ ♦ ②r♦♦②

❬❪ r ♣♣②♥ ❱ ♥②ss t♦ P t ♦♣②s

sr ttrs

❬❪ ♦r♥ r ♥♥ ssr ❯s♥ ♠♣t

rt♦♥ t ♦st ♥ ♥♦r♠③ rs ♣♦r③t♦♥ ♥②ss ♦ r♦♥

♣♥trt♥ rr t t♦ r♥tt ♥ P rs r♦♠ strtr♣

♥s ♦r♥ ♦ ♣♣ ♦♣②ss

❬❪ r♦♥ ♦t♦ s ♥ ❱

rs♥ rt♦♥ ♦♥ts ♦r P❱ ♥②ss ♥ tt♦♥ ♦ s

tr ♥ P ♦♥t♠♥♥ts r r ♦♣②ss

❬❪ r♥ r r♥s ts P s♦♥t ♠♥

rs ♥s♥ ♥ rst♦ ♥t rs ♥

rr♦♥ tt② ♠♣♣♥ s♥ ♥ ♠t♥♥ r♦♥♣♥trt♥

♠♥ s②st♠ Pr♦♥s ♦ ♥ ♥♥ ♦♥r♥ ♦♥ t ♣♣

t♦♥ ♦ ♦♣②s ♥ t♦♦♦s t♦ r♥s♣♦rtt♦♥ ts

♥ ♥rstrtr ♦s ♥s ❯ P❨

❬❪ t♦ ♠ ❨ ♥ ❳ ♦♥ ❩♥ ❩ ♥ ♥

P s♥ ♥ rr② ♥t♥♥ ♦r ♥♠♥ tt♦♥ r r

♦♣②ss

❬❪ ♥ ♥ ♦♥ P♥ ♠rt♦♥ ♥ tt ♦♦r♥ts

♦♣②ss

❬❪ t♦ P ♥ ♦ Pst♥ ♥ ♦♠

P♥ ♣t ♠rt♦♥ ♦♣②ss

❬❪ r♦♥r ♥ ♦♥♦♥r ♥stt♥ tt♥

t♦♥ sttr♥ ♣s ♥tr ♥ t♦t t s♥ s♠t ♥trr♦

♠tr ♠s ♦ ♦rst rs r♥st♦♥s ♦♥ ♦s♥s

♥ ♠♦t ♥s♥

❬❪ ❲♥ ❲ ♣♣r♦ ♦ ♣t s②♥r♦♥③t♦♥ ♦r stt

rt♠ ♠♥ r♥st♦♥s ♦♥ ♦s♥s ♥ ♠♦t ♥

❬❪ s♥ r♥ ♥ rr P ♦♣♠♥t ♦ ♦♠

♣t P rr s ♦♥ ♠ts♥s♦rs t♥♦♦② Pr♦♥s ♦

t ♥ ♥tr♥t♦♥ ❲♦rs♦♣ ♦♥ ♥ r♦♥ P♥trt♥

r tr♥s

❬❪ s ❯ ♦r ♥ ♥ r♥♥ Ps rr② t♥♦♦②

♦r P ♥t♥♥ s♥ ♦r ♥r ssr ①♣♦rt♦♥ Pr♦♥s ♦

t ♥ ♥tr♥t♦♥ ❲♦rs♦♣ ♦♥ ♥ r♦♥ P♥trt♥ r

tr♥s

❬❪ ♥ ❩ ③♥ ♥ ♥ ♦①♥ ❳ sr ♦♥ t

♦♠♥t♦♥ ♦r Psrr② r♦♥ P♥trt♥ r ❲♥ ❯♥r

st② ♦r♥ ♦ tr ♥s

❬❪ ② ❨r♦♦② ♥ trt ❲t ♥r ♦

s♥ ♥ ♥ rr②s P ♦ ♥

❬❪ t③ P ♥ Prr♦ Psrr② tr♥s♠ttrs ♦r P sr

②s ♦♣②s ♥

❬❪ s♦♥ ss tr♦②♥♠s ❲② ❨♦r

❬❪ trtt♦♥ tr♦♠♥t ♦r② r ❨♦r

♥ ♦♥♦♥

❬❪ ♥ s ② ♦sé s ♥t ♦♠♣♦rs rq♦♦ís

❯♥ ♠t♦♦♦í ♥trs♣♥r ♣r ①♣♦rr ♣s♦ ♥ó♥

st♦r tr ❯♥rs ♠ó♥s s s

❬❪ t ♣ts s♦♥ ❲s② ♥ r♥s♦

❬❪ ♦ t♦r r♦♥ P♥trt♥ r ♦r② ♥ ♣♣

t♦♥s sr tr♥s

❬❪ ❨♠③ s♠ t ♥②ss ♦t② ♦ ①♣♦rt♦♥ ♦♣②

❬❪ sr ♥ ♥♥♥ P ♦s② ①♠♣s ♦

rrst♠ ♠rt♦♥ ♦ s♥♥♥ r♦♥♣♥trt♥ rr ♣r♦

s ♦♣②ss

❬❪ ♠♣é ♦♥tró♥ rq♦♦í á♥

ss ♦t♦r ❯♥rs ♦♥ Pt ♥♦s rs

r♥t♥

❬❪ ♠♣é s trs r♦rrs ♣rs♣á♥s

á♥ ♥♦stt♠r ♦♥s ♦ r♥t♥

♥tr♦♣♦♦í

❬❪ tt♦ ② ♥ s ♦①st♥ sñ♦s t♥♦

stíst♦s ♥ Prí♦♦ rí♦ ♣r♥♦ s♦ ♥trr♦ ♥ r♥

é r♦② ♥♦st t♠r r♥t♥ st ♥tr

s♦♥s ♥ ♥tr♦♣♦♦í

❬❪ r♦♥ r♦♥ rr s♠t♦♥ ♦r r♦♦ ♣♣

t♦♥s ♦♣②s Pr♦s♣t♥

❬❪ r♦♥ r♦♥rr ♥♠r ♠♦♥ ♣♣ t♦ ♥

♥r♥ ♣r♦♠s r♦♣♥ ♦r♥ ♦ ♥r♦♥♠♥t ♥ ♥♥r♥

♦♣②ss

❬❪ r♥ ♦♥♦♠♦ s ♥ ♣♣t♦♥ ♦ t s②♥t

t ♠ttrrr② ♠t♦ t♦ ♠♣r♦ P s♥s ♦r♥ ♦ ♣♣

♦♣②ss

❬❪ r♥ ♦♥♦♠♦ s Ps♥ ♠♣r♦♠♥t tr♦

t s②♥tt ♠ttr rr② ♠t♦ st② ♦ ts rtrsts ♦r♥

♦ ♣♣ ♦♣②ss ♥í♦ ♥♦♠r ♥s

r♣t ♥♠r PP

❬❪ rt♥♦ ♦♥♦♠♦ s♥♦ s tt♦ tr

♥ P ♣r♦s♣t♥ t P♦ ♥♦ r♦ st ♥♦rtstr♥

r♥t♥ ♦♣②ss

❬❪ ❨ ♠r s♦t♦♥ ♦ ♥t ♦♥r② ♣r♦♠s ♥

♦♥ ①s qt♦♥s ♥ s♦tr♦♣ ♠ r♥st♦♥s ♦♥

♥t♥♥s ♥ Pr♦♣t♦♥ P

❬❪ ❱♥ st ♥ t♦ P ♣♣t♦♥ ♦ ♦r r♦tt♦♥ t♦

r♦♥♣♥trt♥ rr t ♦♣②ss

❬❪ ♥s ♥t♥♥ ♦r② ♥②ss ♥ s♥ ♦♥ ❲② ♥

♦♥s ♥ ❨♦r

❬❪ r♥ ♥ ♥t ♠r ♠♦♥ ♦ r♦♥♣♥trt♥

rr ♥ s♥ ♦♠♣trs ♦s♥s

❬❪ ♥s ♥t♥♥ ♦r② Pr♦♥s ♦ t

❬❪ sr♦t ❲ ♥ r♠♥ ♦ tt P②ss ♥rs

♦ ♦r ❲♦rt

❬❪ r♦♥ t♦♥ Pttr♥s ♦r P ♦rr ♦♥

♦♣②ss

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