lecture2 ch2 waves
DESCRIPTION
Phys Lec 2TRANSCRIPT
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Optics
Rayoptics
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Chapter 2Wave Motion
Lecture 2
Introduction to waves Wave equation Harmonic waves
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One dimensional wave
Classical traveling wave: self-sustaining disturbance of a medium, which moves through space transporting energy and momentum.
Example: sound waves
Longitudinal waves:the medium is displaced in the direction of motion
Transverse waves:the medium is displaced in the direction perpendicular to motion
Note: disturbance advances, not matter
One dimensional waves
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Human wave
A typical human wave has a phase velocity of about 20 seats per second.
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One dimensional wave: math
Disturbance must be a function of position and time:
txftx ,, Shape of disturbance at any instant represents the profile of the wave:
)(0,,0
xfxftxt
at time = 0
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Special case:the shape of wave does not change in time
One dimensional wave: math
Assume:- wave moves at speed v- at time t=0 its profile is f(x)
At time t the disturbance moved distance vt along x axis, but its shape is the same:
txftx v,If we have a snapshot of a wave shape at time zero we can find a full time-dependent equation of the wave.
2exp axxf Gaussian function
2exp txa v
txftx v,What is this?:Regardless of shape, variables x and t must appear as a single unit (xvt)
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Differential wave equationFix time in equation for wave (x,t=const)
- get shape of the wave in spaceFix x in equation for wave (x=const,t)
- get dynamics of disturbance at particular coordinate
Variables x and t are independent, but appear as (xvt) There should be a connection between the shape in space and dynamics in time.
To relate the space and time dependencies of (x,t) we will consider partial derivatives - derivatives of the function in respect to only one coordinate while the other is fixed
This approach was introduced in 18th century (Jean Le Rond dAlembert)
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v
tx'1'
xx
Differential wave equation
)'(, xftx Wave: , where txx v'Partial derivative at fixed time:
xx
xf
xf
x
'
'
'xf
x
Partial derivative at fixed x:
tx
xf
t
''
'xf
t
v
xt
v The rate of change of with t and with x are equal, to within a multiplicative constant
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Differential wave equation
Second partial derivatives:
2
2
2
2
'xf
x
2
22
2
2
'xf
t
v
Differential wave equation
2
2
22
2 1tx
v
homogeneous, linear, second-order differential equation - is in every term- if is solution, then N is also
any integer
This describes undamped system, i.e. it has no sources of damping (energy loss) as the wave propagates - the shape of wave does not change as it propagates.
For damped system: add term /t
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propagationnumber
amplitude
Harmonic waves(sinusoidal waves: described by sin or cos)
Note: any wave can be synthesized as a superposition of harmonic waves
Chose profile at time zero: )()sin(, 0 xfkxAxtx t
Replace x x-vt to get progressive wave traveling right at speed v: )(sin, txkAtx v
This is sinusoidal disturbance both in time and space (the wave is periodic in space and time)
One dimensional waves
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Harmonic waves: wavelength )(sin, txkAtx v
Space period (wavelength): distance between two maxima (minima).If we increase/decrease x by the disturbance should not change:
txtx ,, )(sin txkA v )(sin txkA v ktxkA )(sin v
ktxktxk )(sin)(sin vvThat is only true if 2k
2kPropagation number and wavelength are connected
)sin(, Atx Can rewrite:
phase
)( txk v
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Harmonic waves: temporal shape
)2/(sin2/,2/
tkAtftxx
vv
Example: consider temporal behavior of disturbance at x=/2
t
period
period
v
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Harmonic waves: period, frequencyTemporal period (): time between appearance of two maxima (minima).If we increase/decrease t by the disturbance should not change:
txtx ,,
1
)(sin txkA v )(sin txkA v vv ktxkA sin
vvv ktxktxk )(sin)(sin Therefore: 2vk
2vk2k
22 vv
Frequency is number of oscillations per unit time, since one oscillation occurs in time :
combine vangular temporal frequency: 22
wave number (spatial frequency): 1
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Harmonic waves: summary
1v
)(sin txkA v
2k
Functional shape: Wave parameters:k - propagation number - wavelength - period - frequency - angular temporal frequency - wave number
v
Alternative forms:
txA 2sin
txA 2sin tkxA sin
txA v
2sin
- for wave moving right+ for wave moving left
mostlyused
These eq-ns describe an infinite monochromatic (monoenergetic) wave.Real waves are not infinite and can be described by superposition of harmonic waves. If frequencies of these waves cluster closely to a single frequency (form narrow band) the wave is called quasimonochromatic
single frequency
22 1
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Periodic waves
- wavelength - the length of one profile-element - period - the duration in time for one profile-element - wave number - number of profile-elements per unit length- etc
Waveform produced by saxophone:
profile-elements - when repeated can reproduce the whole waveform
Can use the same parameters to describe:
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Harmonic waves: example
1. Write an equation of a red light wave that propagates along x axis (at speed of light c) and has a wavelength 600 nm.
Solution:)(sin txkA v 2k
)(m106
2sin 7 txA c
2. What is the frequency of this light?
Solution: v
cv Hz105
m10600m/s 103 14
9
8
Hz1/s
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Harmonic wave: Initial phase
tkxAtx sin,Consider wavetkx phase:
When written like that it implies that 0,00 txtx
With a single wave we can always chose x axis so that above is trueBut in general case 0,
00 txtx
x
This is equivalent to the shift of coordinate x by some value a a
taxkAtx sin, katkxAtx sin, tkxAtx sin, - initial phase tkxphase:
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Harmonic wave: Phase
x
tkxA sinCan use cos():
tkxAtx sin, 2cos, tkxAtx
equivalent equations
Special case: = = 180o phase shift
x
tkxAtx sin, kxtAtx sin, 2/cos, kxtAtx
Note: sin(kx-t) and sin(t-kx) both describe wave moving right, but phase-shifted by 180 degrees ().
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Harmonic wave: Phase derivatives
tkx Phase:Partial derivatives:
xtrate of change of phase with time is equal to angular frequency (=2)
kx t
rate of change of phase with distance is
equal to propagation number
tkxAtx sin,
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phase velocity of a wave
Harmonic wave: Phase velocity tkx Phase:
What is the speed of motion of a point with constant phase?
v
kxt
tx
t
x
from the theory of partial derivatives
sign gives direction
In general case, for any wave we can find the phase velocity:
t
x
xt
valways >0by definition Add sign to give direction:
+ in positive x direction- in negative x direction
Phase (red) vs. group (green) velocity(to be discussed later)