This presentation explains the principals and applications of standing waves.

1. Standing Waves
This presentation will explicate the key
principals behind standing waves.

2. Mathematical Definition of a Standing Wave
● Consider two waves moving in the opposite directions with the same wavelength, amplitude, and
frequency.
o This is the conceptual foundation for a stand wave.
o Now for rough mathematical bits:
Wave 1: D1(x,t) = A sin(kx - ωt) moving in direction of increasing x
Wave 2: D2(x,t) = A sin(kx + ωt) moving in direction of decreasing x
Now we add the waves together.
D(x,t) = D1(x,t) + D2(x,t)
= A sin(kx - ωt) + A sin(kx + ωt)
= A [sin(kx - wt) + sin(kx - ωt)] --Factoring out A
= 2A sin(kx) cos (ωt) --Using trigonometric identities

3. Interpretation of the Math
● When two waves of opposite direction but with equal amplitude, frequency, and wavelength
combine along a string, the result is the string oscillating in Simple Harmonic Motion.Visually, the
result is something as such:
● The result is something where the string is moving only up and down. Moreover, the above
image also shows the points where the amplitude are zero, the nodes, and where it’s at a
maximum at 2A, the anti-node. All other points have amplitude between zero and 2A.
● If we consider wavelength,λ, to be distance between two nodes, we can therefore assume that
the position of node is 0, λ/2, λ, and so forth. Similarly, the anti-nodes occur at λ/4, 3λ/4, 5λ/4,
and so forth.
● Since the nodes are stationary, no energy flows through a standing wave. Similarly to SHM, the
energy continuously transforms between potential and kinetic energy.

4. Application One: Standing Waves on A
String
• Consider a string of length L with fixed ends. The wavelength will
then be defined as λ = 2𝐿/𝑚 where 𝐿 is the length of the string, and
𝑚 = 1,2,3,4 … or the number of anti-nodes on a string. A standing
wave, thus, only occurs at specific wavelengths corresponding to the
length of the string.
• The frequency of a standing wave is defined as 𝑓 =
𝑚
2𝐿
∗ 𝑇/μ
where 𝑇 is the tension of the string, and μ is the linear mass density.
The permitted frequencies are called resonant frequencies, or
harmonics. They depend on the tension, the linear mass density, and
the number of anti-nodes on the string.

5. Application One: Standing Waves on
A String Cont.
• The image to the right adeptly explains these concepts on the last
slide.
• As we can see, as we go from the first harmonic upwards, the
wavelength decreases. Using the previously stated equation for
wavelength, we can determine the corresponding wavelengths for
n=1,2,3… So, λ = 2𝐿 for n=1, λ = 𝐿 for n=2, λ = 2𝐿/3 for n=3, and so on.
• The first harmonic frequency would be 𝑓 =
1
2𝐿
∗ 𝑇/μ for n=1, 2𝑓 for n=2, 3𝑓 for
n=3, and so on.

6. Application Two: Stringed
Instruments
• Previously, we learned that standing waves can occur on a string with two fixed
ends. This model seamlessly models stringed instruments such as guitars and
violins.
• A guitar for instance, will have a fixed string of some length L, some Tension T, and
some linear mass density µ. Thus, the same properties elucidated before apply
here. For instance, the string will form a standing wave at some frequency 𝑓,
where𝑓 =
𝑚
2𝐿
∗
𝑇
𝜇
, and will similarly will have a wavelength of as λ =
2𝐿
𝑚
.