This is my learning object about standing waves on a string. I talk about the harmonics, the equation for calculating the frequency for a wave on a string, and gave an example problem.

1. Standing Waves on Strings
The wave speed of any wave is described by the equation 𝑣 = 𝑓𝜆
The frequency for normal modes (same frequency and phase) of vibration on a string. Where T is the
tension of the string and 𝜇 is the linear mass density.
𝑓 =
𝑚
2𝐿
√
𝑇
𝜇
The wavelength of a fixed string for a standing wave is described by:
𝜆 =
2𝐿
𝑚
𝑚 = 1, 2, 3 …
Where m is a positive integer, it describes the number of half wavelengths or the number of antinodes
found between the fixed ends of the string. L is the length of the string.
The fundamental frequency or first harmonic has
the lowest frequency of the wave and considered
to be the longest wavelength.
The frequencies are known as resonant
frequencies because they can be described by
equation 𝐿 =
𝑚
2
𝜆 which is the rewritten form of
wavelength.
Frequencies higher than the first harmonic are
called overtones.
Question:
The string of a ukulele is 0.66m long and has a linear mass density of 4.00 x 10-4
kg/m. For the first
normal mode, the string has a frequency of 750 Hz.
a) Calculate the wavelength of the third harmonic.
b) What is the speed of the wave?
c) What is the time for the wave to travel to and back from both ends of the string?
d) What is the tension in the string at the third harmonic?
Answer:
a) Using this equation 𝜆 =
2𝐿
𝑚
and plugging in values, where m=3 since at third harmonic we get:
𝜆 =
2𝐿
𝑚
=
2(0.66)
3
= 0.44 𝑚

2. b) We are told the string has a frequency of 750 Hz at the first harmonic. We first need to know the
wavelength of the first harmonic. We can repeat the calculation done in part (a) but instead use
m=1. We solve to get 𝜆 = 1.32m.
We can now use the equation for the speed of a wave.
𝑣 = 𝑓𝜆 = (750)(1.32) = 990 𝑚/𝑠
c) To get the time it takes for wave to travel to both ends of the string we can use:
𝑣 =
𝑑
𝑡
We rearrange the equation to solve for time and get
𝑡 =
𝑑
𝑣
Where our distance is the length of the string and we use the speed found from part (b).
𝑡 =
0.66
990
= 0.00067 𝑠
d) We know that the wavelength at the third harmonic is 0.44m. We also know the speed of the
wave is 990 m/s. From this, we can simply calculate the frequency at the third harmonic to be
2250 Hz. We can now rearrange the equation for frequency on a string and plug in m=3 to solve
for tension.
𝑓 =
𝑚
2𝐿
√
𝑇
𝜇
(
2𝑓𝐿
𝑚
)
2
= (√
𝑇
𝜇
)
2
4𝑓2 𝐿2
𝑚2
=
𝑇
𝜇
𝑇 =
4𝑓2 𝐿2 𝜇
𝑚2
We now plug in our values for our equation:
𝑇 =
4(2250)2(0.44)2 (0.0004)
(3)2
=
1568.16
9
= 174.24 𝑁