Chapter 18: Superposition and Standing Waves
Analysis Model: Waves in interference
o Superposition principle:
If two or more traveling waves are moving through a medium, the resultant
value of the wave function at any point is the algebraic sum of the values of the
wave functions of the individual waves
Linear waves obey this.
o Amplitudes of these waves are much smaller than their
Non-linear waves violate this
o Characterized by large amplitudes
The combination of
separate waves in the
same region of space to
produce a resultant wave
o The superposition principle is the
centerpiece of the analysis model
called waves in interference.
o In many cases, waves combine
according to this principle and
exhibit interesting phenomena
with practical applications
Superposition of Sinusoidal Waves
o If we have two waves, traveling in
the same direction with the same
frequency, wavelength, and
amplitude, differing only in
phase, we can express their wave
function as y = y1 + y2 o The result is this:
The resultant wave function y also is sinusoidal and has the same frequency and
wavelength as the individual waves
The amplitude of the resultant wave is 2Acos(phi/2)
In general, constructive interference occurs when cos(phi/2) = +/- 1
True when Phi = 0, 2π, 4π …
Destructive interference occurs when cos(phi/2) = 0
When the crest of one wave occurs at the same
time the trough of another wave occurs
Interference of Sound waves
o Sound from loudspeaker S is sent into a tube at point P where
there is a T-shaped junction.
o Half the sound energy travels in one direction, half the other way
o The sound waves that reach the receiver R can travel along either
o Distance along any path from the speaker to the receiver is called
path length r. Can create constructive or destructive interference
by sliding the metal covering up or down on the tube
o Suppose we had two speakers, and they faced each other:
o Two identical waves travel in opposite directions in the same medium.
o The waves combine according to the interference model.
o The standing wave function is as follows:
o Standing waves have no sense of motion in the direction of propagation of either
o Describes a special type of Simple harmonic motion.
o Points of zero amplitude are called nodes Located when x = 0,
o Points of greatest amplitude are called antinodes.
Located where x =
The distance between adjacent antinodes is equal to λ/2
The distance between adjacent nodes is the same as above
The distance between a node an