Physics of Music PHY103 Worksheet #5
Setup for first sonometer lab
1) Proportionality or scaling
The frequency of mode n for a perfect string that is fixed at both ends is given by the following
formula:
fn  n
v
2L
(Equation 1)
This formula predicts a relationship between the length L of the string and the frequencies
fn at which the string will resonate. Here n is an integer. The lowest or fundamental mode would
have n=1. The velocity of the sound wave, v, is set by the tension and weight of the string. The
velocity of waves along a string is given by the following equation:
v
T

(Equation 2)
In this equation, T is the tension in the wire and  is the linear density of the string. The
linear density is the mass per unit length of the string.
Supposing you measure the frequency of the fundamental mode of string of length L, linear
density  and under tension T.
a) By what factor would you increase the string length to halve the frequency of the
fundamental mode?
b) By what factor would you need to increase the tension to double the frequency of
the fundamental mode?
c) By what factor would you need to increase the string density to halve the
frequency of the fundamental mode?
d) Suppose you have measured the frequency, f, of the fundamental mode of a
string. You have also measured its length, L, and linear density  . Can you
calculate the string tension? Find a formula for the string tension in terms of the
quantities you have measured.
e) If we give linear density  in units of g/cm and velocity v in units of cm/s what
are the units of tension T?