Wave Interference with Applications
Math Aside:
The following trigonometry identity will be useful for this
topic: sin + sin = 2cos[(-)/2] sin[(+)/2]
The superposition principle shows how to find the wave
function when more than one wave occupies a string. We
call this wave interference, wave addition or wave
superposition; these all mean simply, add the waves,
y = y1 + y2 .
Applications
One result of this principle is the phenomenon of
constructive and destructive interference. We will also
examine two more applications: standing waves and beats.
I.
Adding two identical sinusoidal traveling waves
differing only by a phase constant, 

A. One dimension
Using the trig. identity and superposition principle above,
y = A sin(kx - t) + A sin(kx - t + )
y = 2A cos[/2] sin[kx - t +/2]
Special cases:
=0 
constructive interference
Corresponding points line up, i.e. crests line up with crests,
and troughs line up with troughs.
 
destructive interference
From the trig. identity below we see this is like a /2
translation of one of the waves.
sin(C + ) = sinC cos + cosC sin
= -sinC
(=was assumed)
Comment: We see that the result of this addition is a
sinusoidal traveling wave with the average phase shift and
a modified amplitude. Each point of the string is in SHM
with the same frequency as before.

B. Two dimensions
When two point sources of waves interfere at a point P the
interference conditions may be written in terms of the path
difference traveled by the two waves, r, using r/
Examples of this are the famous Young’s double slit
interference experiment and also examples are common in
acoustics with speakers. Place below the detailed derivation
given in class.


=n2or r=n 
n=0,1,2. [construc. interf.]
=(n+1/2)2 or r=(n +1/2) n=0,1,2….[destruc. interf.]
II. Adding two identical sinusoidal traveling waves
differing only by their direction of motion:
standing waves
Using the trig. identity and superposition principle above,
y = Asin(kx - t) + Asin(kx + t)
y = 2Asinkx cost
This is an altogether new thing, it is not a traveling wave
but is called a standing wave. Each point of the string is in
SHM with the same angular vibration frequency, , and
amplitude that depends on the place x. These are called
normal modes of vibration.
Points for which sinkx is zero are called nodes, they never
move. Antinodes are places of maximum sinkx and
therefore vibrate with amplitude 2A.
If both ends of the string is a node then kL must equal an
integer times  and therefore, L = n/2 n=1,2,3… and
using v=f gives only the following frequencies will
generate a standing wave pattern,
fn = nf1 , where the fundamental is given by f1=v/2L.
III. Adding two identical sinusoidal traveling waves
differing only by frequency: beats
Using the superposition principle,
y = Asin[kx-t] + Asin[k’x-’t] = A[sint +sin’t]
where I’ve assumed x=0 in the last step. The trig. identity
gives,
y = 2Acos[(’-)t/2] sin[(’+ )t/2].
A graph of this gives an envelope inside of which a rapidly
varying function is enclosed. The beat period is half that of
the envelope.
This leads to a beat frequency of
fbeat = f’ – f .
EXAMPLES [in class]