This learning objective consists of two questions about standing waves on a string. An explanation for both solutions are provided with the workings to the solution.

1. You play your guitar and are curious what the speed of the wave
travelling down the string is. The guitar string is measure to be 0.41m and vibrates
20 cycles up and down in 3 seconds, vibrating to the third harmonic.
Now that you have found the wave speed you want to change the
fundamental frequency by 11.0Hz. You measured the tension in the guitar spring to
be 69.0N, assume the tension is kept constant. The linear mass density was also read
off the old guitar string package to be 4.20x10^-3 kg/what would you have to
change and by what amount would you need to change it by? HINT:(You would
change the tension)
Solutions:
To find the wave speed of the string the following equation can be used, f=v/ 𝜆, and
rearranged to v= f 𝜆
Now that the formula for the wave speed is derived we will need to find f
(frequency) and 𝜆.
To find frequency take the given values, 20 cycles in 3 seconds and find the
frequency by dividing all the cycles by the total amount of time.
f=20/3s
To find the wavelength us the fact that the string vibrates to the third harmonic,
recalling that the wavelength would be 2L/3 since it’s the third harmonic. The
length of the guitar string was given in the question, being 0.41m, so the wave
length would be found like so: 𝜆 = 2𝐿/3
𝜆 =
2(0.41𝑚)
3
= 0.273𝑚
We have now found both variables needed to calculate the wave speed so simply use
the v= f 𝜆 and sub in the known variables.
v= f 𝜆
v= (20/3)s^-1(0.273m)
v=1.82m/s
The first step to the second question would be to find the fundamental frequency by
using the given values and the equation for the fundamental frequency, this being
f=1/2L√𝑇/𝜇 L=0.41m T=69.0N 𝜇 = 4.20𝑥10−3
𝑘𝑔/𝑚
Plug in the values and solve for f, the fundamental frequency:
f=1/2(0.41m) √
69.0𝑁
4.20𝑥10−3 𝑘𝑔/𝑚
f= 156.3Hz
Now that the fundamental frequency has been found we know we want to change it
by 11 hertz. So subtract 11 hertz from the fundamental frequency to find the new
fundamental frequency after the change. (F= new fundamental frequency)
F= 156.3Hz-11.0hz
F=145.3Hz

2. All right now we need to go back to the equation for the fundamental frequency
used in the first step of this problem and rearrange it with T on one side by itself.
f=1/2L√𝑇/𝜇
T=4L^2 𝜇(f^2)
Using this rearranged equation sub in the known length, 0.41m, the known linear
mass density, 4.20x10^-3 kg/m, and the known NEW fundamental frequency,
145.3Hz (An explanation why the new fundamental frequency is used will be given).
T=4(0.41m)^2(4.20x10^-3 kg/m)(145.3Hz)^2
T= 59.6N
This calculated tension is the tension needed for the changed fundamental
frequency. So to find the change in the tension subtract the original tension, 69.0N
by the calculated tension at the new fundamental frequency, 59.6N, to get the
change in tension.
Change in tension=69.0N-59.6N
=9.4N