Physics of Music Experiment 2
Standing Waves on a String
Introduction We need to understand very well the physics of transverse standing waves on a stretched
string, as it is the prototype for all of the stringed instruments. As we saw in class, standing waves in any
medium are described by the equation v = f , where v is the speed of traveling waves on the string, f is
the frequency, and  is the wavelength. The speed v of the waves will depend on the physical
characteristics of the string: its linear mass density and its tension. If we let the mass per unit length of the
string be  (kg/m) and the tension be T Newtons, the speed should be given by
(1)
v  T / .
For a fixed length of string, a variety of standing wave patterns can be observed, having whole numbers of
antinodes of the standing wave. Clearly, as we increase the number of antinodes, the wavelength
characteristic of the standing wave pattern will be getting smaller. Below is an illustration of a particular
standing wave pattern with four antinodes.
For a fixedlength string, the
wavelength of the
lowest-frequency
standing wave will
be twice the
length of the
string. We could
imagine two different procedures for investigating the string: 1) keeping the tension in the string (and
therefore v) constant, and measuring the frequencies of various possible standing waves, or 2) applying a
constant frequency driving force to the string, and varying the tension (and therefore v) in order to find
standing waves. We will use method 2) to investigate the physics of standing waves on a string.
Procedure The strings are attached at one end to a vibrator with a fixed frequency of 120 Hz.
I. Vary the weight stretching the string in order to get 2, 3, 4, …. 8 antinodes in the standing wave pattern.
Record the number of antinodes and the corresponding mass m, remembering to add in the mass of the
weight hanger. Record the length L of the string, from the vibrator to the pulley. Note that you must add
more mass to get fewer antinodes.
II. The number of antinodes n is related to the mass m pulling on the string by
1
(2)
n  (2 L f  / g )
m
Using your data, make a graph of n (y axis) vs. ( 1 / m ) (x axis). Do you get a straight line? What
should the slope of the graph be, as demonstrated by Equation (2)? Using the slope from your graph,
calculate the value of the linear mass density . Use this result to calculate the mass of the entire string of
length L. Is the number you get reasonable?
Report In your lab report give
a) a brief description of the procedure
b) a derivation of Equation 2, including the necessary equations and explanations in English
c) the graph you have plotted, and
d) answers to the questions asked above.