Wave Interference with Applications
Math Aside
:
The following trigonometry identity will be useful for this
topic:
sin
α
+ sin
β
=
2cos[(
α

β
)/2] sin[(
α
+
β
)/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 = y
1
+ y
2
.
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
.
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I.
Adding two identical sinusoidal traveling waves
differing only by a phase constant,
δ
A. One dimension
Using the trig. identity and superposition principle above,
y = A sin(kx 
ϖ
t) + A sin(kx 
ϖ
t +
δ
)
y = 2A cos[
δ
/2] sin[kx 
ϖ
t +
δ
/2]
Special cases:
δ
= 0
⇒
constructive interference
Corresponding points line up, i.e. crests line up with crests,
and troughs line up with troughs.
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 Spring '08
 Chen
 Physics, δ, 2L, fbeat, Sinusoidal traveling waves

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