1) The document discusses standing waves on violin strings and how to calculate the wave speed, tension, and harmonic frequencies. It provides the formulas for wave speed, tension, and harmonic frequencies. 2) As an example, it calculates the wave speed and tension required for a 30cm violin string vibrating at 440Hz. 3) It then uses the same formulas to calculate the second and third harmonic frequencies produced when the string is pressed at different positions, finding the second harmonic is 880Hz and the third is 1320Hz.

1. Learning Object 6: Standing Waves and Music
An example of standing waves on the strings of a violin is used to illustrate how the formulas
presented for standing waves on a string are applied.
*N=nodes in this diagram
1. A tuned string on a violin is 30.0 cm long, and vibrates with a fundamental frequency of 440
Hz (the A4 note). The string has a linear mass density of 0.652 g/m.
a) What is the wave speed?
b) What tension must another 45.0 cm string with the same linear mass density as the violin
string have to have in order to vibrate at 440 Hz? (assume the length of the string does not
change while increasing the tension)
2. For people who have played strings instruments before, we know that when we gently press
our finger at certain positions on the string, harmonic notes are played.
What are the frequencies that are produced when the original violin string is played at its a)
second harmonic? b) third harmonic?

2. Solutions and Explanations
1. a) The speed of the wave v is given by the formula:
v = λ*f
λ of a standing wave is given by 2L/m where m is an integer > 1 and is equal to the number of
antinodes between the fixed ends of the string.
v = (2L/m)*f where L= length of the string=0.300m and m=1 because this is the fundamental
frequency and there is only one antinode.
v = (2*0.300m)*(440Hz) = 264 m/s
b) The frequency of a standing wave is given by the formula:
f = [(m/(2L)]*sqrt(T/µ)
m=1
f*(2L)=sqrt(T/µ)
(2f*L)2=T/µ
(2f*L)2µ = T
T = (2*440Hz*0.45m)2/(0.000652 kg/m)
T = 102 N
2. a)
f = [m/(2L)]*sqrt(T/µ)
f=[m/(2L)]*v
we know the wave speed v from the previous question, and m=2 this time because this is the
second harmonic (2 antinodes)
f = (1/0.300m)(264 m/s)
f = 880 Hz
b) The same process is applied as in part a), where m=3 because it is the third harmonic. We end
up with f=1320
Notice that
f=(m/2L)*v
f=m(1/2L)*v and (1/2L)*v is the fundamental frequency of a string, so by increasing m or
increasing the harmonics,
fm=m*f1
where fm is the frequency of the mth harmonic and f1 is the fundamental frequency.