When two (or more) waves of the same kind propagate through the
same region, they produce a composite wave.
This phenomenon is called interference.
It is constructive, when the waves reinforce each other.
It is destructive, when they reduce each other’s amplitude.
Usually the disturbances (displacements) the waves produce are
added algebraically. This is called superposition principle.
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Superposition principle:
adding two square waves
Superposition principle: adding two harmonic waves
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Superposition of pulses:
Constructive Interference
Superposition of pulses: Destructive Interference
For a moment it the string becomes a straight line – no disturbance
is seen. Does the energy of wave motion disappear? Where does it
go? How can the two waves go on after that?
Two kinds of energy in the wave motion: potential – depends on deflection of
the string from the straight line; kinetic – depends on velocity of motion of the
string. In the very moment, when the string becomes straight, it is actually
moving very fast - high K.
http://www.kettering.edu/~drussell/Demos/superposition/superposition.html
When there are two interfering waves with close but different
frequency the result of the interference is the beats.
When there are two interfering waves with close but different
frequency the result of the interference is the beats.
They are perceived as a wave with an average frequency, but
with a slowly oscillating amplitude.
http://www.kettering.edu/~drussell/Demos/superposition/superposition.html
When there are two interfering waves with close but different
frequency the result of the interference is the beats.
They are perceived as a wave with an average frequency, but
with a slowly oscillating amplitude.
y1 (t )  A cos(1t )
y2 (t )  A cos(2t )
The resulting composite wave:
y(t )  y1 (t )  y2 (t )  A cos(1t )  A cos(2t )
1
1
y (t )  2 A cos[ (1  2 )t ]  cos[ (1  2 )t ]
2
2
http://webphysics.ph.msstate.edu/jc/library/15-11/index.html
When there are two interfering waves with close but different
frequency the result of the interference are beats.
They are perceived as a wave with an mean frequency, but with a
slowly oscillating amplitude.
1
1
y (t )  2 A cos[ (1  2 )t ]  cos[ (1  2 )t ]
2
2
slowly oscillating amplitude
mean frequency
What about wave shapes?
Are all periodic waves (not pulses) harmonic?
Harmonic waves are simplest kind of waves.
Other periodic waves (rectangular, triangular etc.) and even wave
pulses can be represented as sums of harmonic waves with
frequencies, which are multiples of the basic one (higher harmonics),
and different amplitudes.
Fourier decomposition.
Fourier decomposition.
A harmonic wave is the simplest kind of wave.
Real waves may have ugly profiles.
It is often convenient to present those complex waves as sums of
simple harmonic waves with frequencies which are multiples of the
same basic frequency. Fourier decomposition.
http://www.kettering.edu/~drussell/Demos/Fourier/Fourier.html
If there are big waves and ripples on the water surface, which are
going to run faster?
The massive December 26, 2004 tsunami traveled 375 miles (600
km) in 75 minutes. That's 300 mph (480 kph).
For some kinds of waves the wave speed of a simple harmonic wave
depends on the wave length.
This phenomenon is called dispersion.
Example – for waves on the surface of deep water the wave speed:
g
v
2
Limits of applicability of the superposition principle.
Superposition principle suggests that disturbance of the composite wave
is an algebraic sum of the disturbances produced by the individual
waves.
This is NOT true, when the amplitudes of the individual waves are so
large that they change the properties of the medium through which
they propagate.
One can tell that the second wave propagates through a different medium
due to the perturbation made by the first wave. The interference does not
result in algebraic addition then.
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v
F

As the amplitude grows, the string gets more stretched,
the tension force increases and so does the wave
speed. A wave with a higher amplitude goes faster!
Waves breaking at a sea shore…
Waves break when their amplitude becomes comparable with the sea depth.
Bigger waves break further away from the shore.
Two waves may interfere constructively and add algebraically far from
the shore 1.
The amplitude of the composite wave with the larger amplitude
becomes comparable with the depth further away from the shore 2-4.
Therefore the composite wave would
break sooner than any of the two
constituent waves.
The breaking wave is not any more a
sum of the two constituent waves – the
superposition principle is violated, when
the amplitude becomes large compared
with the depth.
Interference in two dimensions:
http://www.colorado.edu/physics/2000/schroedinger/big_interference.html