frans pretorius university of alberta. 동일 진동수 빛의 중첩 phys 124, section a2, chapter...

Frans PretoriusUniversity of Alberta

동일 진동수 빛의 중첩

PHYS 124, Section A2, Chapter Chapter 17.1-17.6: Principle of Linear Superposition and Interference 2

( , ) sin( ( )) sin( )o oE x t E t kx E t

두 빛의 중첩 ( 더하기 )1

2

( )1 1 1

(

1

2)

2 2 2

sin( )

sin( )

i to o

i to o

E E t E e

E E t E e

1 2

1 2

( ) ( )1 2i t i t

o o

E E E

E e E e

1 2 1 2

1 2 1 2

( ) ( ) ( ) ( )1 2 1 2

( ) ( )2 201 02 01 02

2 201 02 01 02

2 201 02 0 0 11

1

22

22 cos( )

2 cos( ( ))

* i t i t i t i to o o o

i i

I EE E e E e E e E e

E E E E e e

E E E

k xE x

E

E E E


2 201 02 01 02 1 22 cos( ( ))I E E E E k x x 두 빛이 더해진 밝기의 표현 식

계산을 간단히 하기 위해서 E01=E02 라고 가정하자 . 그리고 E012=E022=Io 라 하자0 0 1 2

1 2

01

0

0

2 2

2 cos( ( ))

2 cos(1 )

4

( ( )

(co )s2

I I I k x x

k x x

k x x

I

I

I


Standing waves Standing waves are waves that look stationary, but have an amplitude that

changes with time. Several situations can produce standing waves, including the superposition of left and right moving waves on a string the “natural” modes of vibration of a string fixed at both ends (stringed instruments

work like this) sound waves in a tube open at one or both ends (wind instruments work likes this) sustained 40mph winds set up standing waves in the Tacoma Narrows Bridge in 1940,

causing it to collapse:


Standing waves on a string fixed at both ends

광물리학 빛의 중첩 , 회절 간섭 6

공명 (resonance)

서로 반대 방향으로 진행하는 , 같은 진동수의 빛이 만나면 제자리에서 진동하는 파동이 생긴다 .

마디와 마시 사이 거리는 파장의 절반이다 .

node

antinode

정상파 (Standing


Beats When two tones of similar frequency f1 and f2 are

added together, interference will create what is called a beat frequency at the difference between the two frequencies : fb=f1-f2

Example:

A 200hz tone:

A 200hz + 201hz tone:(beat frequency is 1 hz … this is a 5 second sample, so we should hear ~5 beat cycles)

A 200hz + 210hz tone:(beat frequency is 10 hz …should hear ~50 beat cycles)


Interference of two waves sources vibrating in phase

Two wave sources, S1 and S2, are emitting waves in phase, and of exactly the same frequency and amplitude. Consider a point p that is a distance d1 from source 1, and a distance d2 from source 2.

If

where n is a non-negative integer and is the wavelength, then p will be a point of complete constructive interference

If

then p will be a point of complete destructive interference

d1 d 2

p

S1 S2

ndd || 21

21|| 21 ndd


Example AConsider the

configuration of loudspeakers and listener shown to the right. Assume both loudspeakers are playing the exact same music. What set of frequencies will the listener not be able to hear at all?

Image courtesy John Wiley & Sons, Inc.


Diffractiondiffraction is the bending of a wave

as it moves past edges or obstacles


Single Slit Diffraction With single slit diffraction, we have a

sound wave of wavelength passing through an opening of width D. On the other side of the opening there will be interference between parts of the wave, and at an angle given by

there will be complete destructive interference (the so-called first-minimum)

Note: the above formula only works if D>>

D sin


Standing waves on a string fixed at both ends Since both ends of the string are fixed, the only

possible set of wavelengths are

n=1 gives the first or fundamental harmonic n=2 gives the second harmonic or first overtone, n=3

the third harmonic or second overtone, etc.

Given the relationship f=v, the set of frequencies corresponding to these wavelengths are

nLn /2

Lvnfn 2


Standing waves in a tubeA resonance can be used to set up

standing sound waves in a tube this is a longitudinal standing wave (compared

to the transverse standing wave on a string) If both ends are open, the possible set of

natural frequencies are (as with the string) :

Lvnfn 2


Standing waves in a tubeIf only one end is open, the following

set of resonant frequencies are possible:

Lvnfn 4


Example B: (ch 17, prob. 44)A tube, open at one end, is cut into

two shorter, unequal length pieces. The piece that is open at one end has a fundamental frequency of 675hz, while the piece that is open at both ends has a fundamental frequency of 425hz. What was the fundamental frequency of the original tube?


Example B

Answer: 162Hz

frans pretorius university of alberta. 동일 진동수 빛의 중첩 phys 124, section a2, chapter...

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