Lecture 12: Standing Waves; Wave Interference
Readings: Textbook pages 505-511
Wave Interference
Interference
is one of the most fundamental and important phenomena of the physics of waves.
Intereference is what happens when you have more than one wave. It is how they add up
Two identical waves
Two waves, same frequency but different amplitude
Two identical waves but traveling in opposite direction
Two waves, same frequency but different amplitude, traveling in opposite direction
How to create multiple waves ? One way - reflect the wave of the boundary of the medium. Let us look at again at
very useful applet
in our textbook.
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Principle of superposition: Wave functions simply add up
If I have two waves, 1 and 2, together they are described by simple sum of their wave functions
y( x,t ) = y
_{1}
( x,t ) + y
_{2}
( x,t )
Principle of superposition reflects
linear (in y(x,t) ) nature of the wave equation
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Standing Waves
Especially important example of superposition is the effect of
standing
wave
Standing wave is formed when two waves of the same amplitude and freguency, travelling with the same speed in opposite directions interfere (add up).
Experiment
Theory
Wave function of the standing wave obtained by reflection at
x=0
(inverted amplitude) is
y ( x,t ) = A cos( k x - ω t) - A cos( k x + ω t )
or
y ( x,t ) = 2 A sin( k x ) sin ( ω t)
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Properties of the Standing Waves
Amplitude is twice
A
_{sw}
= 2 A
Different places
x
experience different oscillations
Some have no motion at all,
sin( k x ) = 0
. These are called
nodes
.
Nodes are positioned at
x = ½ n λ
Some experince maximum swing,
sin( k x ) = 1
. The are called
antinodes
.
Antinodes are positioned at
x = ½ n λ + ¼ λ
All points between two nodes move
in phase
There is no transport of energy over large distance