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Interference
principle of superposition –displacements of two (or more) waves add
together to give net displacement
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Interference–superposition
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Interference–superposition
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Clicker question
Two identical pulses of opposite amplitude travel along a stretched
string and interfere destructively. Which of the following is/are true?
A. There is an instant at which the string is completely straight.
B. When the two pulses interfere, the energy of the puses is momentarily
zero.
C. There is a point on the string that does not move up or down.
D. There are several points onthe string that do not move up or down.
E. More than one of the above is true.
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Superposition–Fourier analysis
The principle of superposition makes it possible to construct complex shapes
by adding waves of different amplitudes, wavelengths & phases
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position (m)
2
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0
-1
-2
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Superposition–beats
Consider the superposition of two waves with slightly different frequencies.
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position (m)
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Superposition–beats
Superposing two waves with slightly different frequencies gives constructive
interference in some places and destructive in others, giving rise to beats
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position (m)
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-1
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Superposition–beats
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position (m)
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Clicker Question
CT page 121
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Interference in 2D
Examples
water waves
2-slit interference
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Dispersion
non−dispersive
dispersive
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Dispersion
Recall ω=angular frequency and k=wavenumber.
These quantities are related via a dispersion relation:
ω = ω(k)
The group velocity gives the speed of wave packets (envelops), and is
defined as
vgroup =
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dω
dk
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Dispersion
The wave is non-dispersive if
vgroup
dω
=
= something that does not depend on k
dk
and dispersive otherwise.
√
Example: Deep ocean waves ω(k) = 2 gk
Example: Sound waves ω(k) = ck
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Sound waves
sound waves are longitudinal waves that propagate through gases, liquids
& solids. Air molecules are displaced back and forth as pressure and
density is changed.
sound intensity level , β, quantifies sound levels. β = 10 log
I
I0
, and is
measured in decibels. I=intensity, I0 = 10−12W/m2 is a reference level.
Speed of sound in gas: v =
q
γP
ρ
P =pressure, ρ=density, γ=constant that characterizes gas
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Reflection–end is fixed
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Reflection–end is free
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Partial Reflection
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Refraction
To be covered in PHYS 122
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Standing waves
A wave travels along a string,
reflects and travels back.
Its ends are fixed, so the
reflected wave is inverted.
The reflected waves interfere,
giving standing waves.
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Standing waves
modes– allowed waves, also called harmonics, given by L = mλ/2
(L=length of string, λ=wavelength, m=integer)
mode number– m, integer above
fundamental mode– corresponds to m = 1
node– Points where string doesn’t move at all
anti-node– points where wave displacement is maximum
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Doppler effect
The Doppler effect or Doppler shift refers to the phenomena in which an
observer measures a higher frequency from a source moving toward him/her
than an observer moving with the source (frequency is lower if the source is
moving away).
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Doppler effect-stationary source
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Doppler effect-moving source
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