Note 22 Standing Waves
Standing waves are the result of the interference between an out-going harmonic wave and its
reflection within a confined medium.
Standing Wave on a String
On a string, the ends must have an amplitude of zero. Here is what the out-going wave look like.
wave source
fixed barrier
When it reaches the other boundary, it reflect with a phase shift of 180° or π.
direct reflection
phase-shifted
reflection
This wave is the sum of the right-traveling wave and its reflection. Note that the right end always
has a displacement of zero since it is fixed.
The above situation is the transient behavior that occur in the beginning moment of the wave
generation. This disappears in a fraction of a second. Eventually, the out-going and the reflected
waves completely overlap.
If the wavelength is a particular value such that the displacements always add to each other, then
we have what is called resonance. The resulting wave pattern are called standing waves.
page 1
For a string fixed at both ends, the patterns look like this. The first highest wavelength or lowest
frequency resonance pattern is called the fundamental or the 1st harmonic. It will oscillate
between the maximum amplitude.
The second pattern is the 2nd harmonic.
The third pattern is the 3rd harmonic.
The fourth pattern is the 4rd harmonic.
The points on the standing wave where the displacement is always zero are called nodes. The
points on the standing wave where the displacement is always maximum are called anti-nodes.
This is the pattern for standing waves with fixed ends shown above.
L=n
λ
2
where n = 1, 2, 3, ...
The value of n is the harmonic number of the resonance. Notice that the value of n is also the
number of anti-nodes or half wavelengths.
page 2
Standing Wave with One Free End
Different systems have different boundary and resonance conditions. If the string is free to travel
on one end, then we have the following standing wave patterns.
The resonance condition for this system is
L = ( 2n − 1 )
λ
4
where n = 1, 2, 3, ...
Here, the value of n always start with 1 to keep the nomenclature in sync with the harmonic
number.
page 3
Standing Wave with Two Free Ends
Here are the standing wave patterns.
The resonance conditions are the same as in a two-fixed-end system.
L=n
λ
2
where n = 1, 2, 3, ...
page 4
Standing Wave Superposition
Here is the standing wave mathematically. For the ends being fixed, we will use the sine function.
There is the original wave starting at the origin the travels to the right.
y+(x,t) = A sin ( kx − ωt )
The reflected wave has the same wavelength and frequency as the original wave but it travels in
opposite direction. It also starts at the other end of the medium at L so its origin is shifted. In the
diagram below, the black line is the original right-traveling wave. The red is the reflected wave
without a phase shift of π. It is a mirror image of the black wave about the reflection point at L.
The blue is the reflected wave with a phase shift of π. Its equation is this.
y−(x,t) = A sin ( k ( x − L ) + ωt ) = A sin ( kx − kL + ωt )
original wave >
< reflected wave
medium, L
< reflected wave
with phase shift
The superposition of these two waves forms the standing wave.
ytotal (x,t) = A sin ( kx − ωt ) + A sin ( kx − kL + ωt )
⎡ sin ( kx ) cos ( −ωt ) + sin ( −ωt ) cos ( kx ) +
⎤
⎢
⎥
ytotal (x,t) = A ⎢
⎥
sin
kx
−
kL
cos
ωt
+
sin
ωt
cos
kx
−
kL
(
)
(
)
(
)
(
)
⎢⎣
⎥⎦
⎡ sin ( kx ) cos ( ωt ) − sin ( ωt ) cos ( kx ) +
⎤
⎢
⎥
ytotal (x,t) = A ⎢⎢ sin ( kx ) cos ( −kL ) cos ( ωt ) + sin ( −kL ) cos ( kx ) cos ( ωt ) + ⎥⎥
⎢
⎥
⎢⎣ sin ( ωt ) cos ( kx ) cos ( −kL ) − sin ( ωt ) sin ( kx ) sin ( −kL ) ⎥⎦
I will now group these in terns of sine and cosine of kx.
ytotal (x,t) = A sin ( kx ) ⎡⎣ cos ( ωt ) + cos ( kL ) cos ( ωt ) + sin ( ωt ) sin ( kL ) ⎤⎦ +
A cos ( kx ) ⎡⎣ −sin ( ωt ) − sin ( kL ) cos ( ωt ) + sin ( ωt ) cos ( kL ) ⎤⎦
Since the ends are fixed, the boundary conditions require that the coefficient of the cosine(kx)
term be zero. This means
−sin ( ωt ) − sin ( kL ) cos ( ωt ) + sin ( ωt ) cos ( kL ) = 0
⎡ cos ( kL ) − 1 ⎤ sin ( ωt ) + ⎡ −sin ( kL ) ⎤ cos ( ωt ) = 0
⎣
⎦
⎣
⎦
Since sine(𝜔t) and cos(𝜔t) are independent functions, their coefficients must be zero
independently. Thus,
cos ( kL ) − 1 = 0 and sin ( kL ) = 0
page 5
The first condition says this.
cos ( kL ) = 1 ⇒ kL = n ( 2π )
⇒
2π
L = n ( 2π )
λ
⇒ L = nλ
The second condition says this.
sin ( kL ) = 0
⇒ kL = n ( π )
⇒
2π
L = n(π)
λ
⇒ L=n
λ
2
This second condition includes the first condition so the overall condition for L is this.
L=n
λ
2
With this condition, the superposition is now
ytotal (x,t) = A sin ( kx ) ⎡⎣ cos ( ωt ) + cos ( ωt ) ⎤⎦ = 2A sin ( kx ) cos ( ωt )
This is a sine-shaped envelope with twice the amplitude of the original wave and the displacement
oscillates within this envelope.
page 6
Example
A pipe that is 1.5 meters long is open on one end and is closed on the other end. What is the
lowest set of three frequencies at which the the pipe will resonate with a standing wave?
Solution
One free end and one fixed end means the following condition. There is an odd number of quarter
wavelengths that can fit inside the pipe.
L = ( 2n − 1 )
λ
4
where n = 1, 2, 3, ...
The lowest frequency is the same as the longest wavelength and that means the smallest n
values. The frequencies are
1
L = ( 2n − 1 ) λ
4
⇒ λ=
4L
2n − 1
⇒ f =
v
2n − 1
=
v
λ
4L
The lowest frequencies are
1 ⋅ 343 m/s
= 57 Hz
4 ⋅ 1.5 m
3 ⋅ 343 m/s
f2 =
= 172 Hz
4 ⋅ 1.5 m
5 ⋅ 343 m/s
f3 =
= 286 Hz
4 ⋅ 1.5 m
f1 =
page 7