Brooklyn College 1
STANDING WAVES ON A STRING
Purpose
a. To investigate resonance conditions for a vibrating string.
b. To study dependence of wavelength on the tension and linear mass density of the string.
Theory
In any wave motion, wavelength (
) and frequency (f) of the wave is related by
f
v
(1)
where v is the velocity of the propagation of the wave. The velocity, v of a wave on a stretched string
depends on the tension, T, in the string and the mass per unit of the string ,, and is given by
T
v
(2)
If a stretched and vibrating string is clamped at both ends, like a guitar string, the wave reflects
from the fixed ends and waves travel in both directions. The incident and reflected waves will combine
according to superposition principle. When a proper amount of tension is applied along the string for a
given length of the string, the waves travelling in opposite directions resonate and form a standing wave.
Figure 1 shows two of the many possible modes of making standing waves on a string. In the figures, N
indicates the locations the string is stationary, called
nodes, and A indicates the locations the string is
vibrating with maximum amplitude, called antinodes.
Standing waves are discrete phenomena, meaning that
they only occur at specific values of wavelength.
The distance from a node to an adjacent node
(or from an antinode to adjacent antinode) is half of the
wavelength. In order to form a standing wave a
resonance condition has to be satisfied:
2
n
L
(3)
where L is the length of the string and n is an integer.
For the standing wave in Figure 1a, the value of n is 1,
and the wave pattern is called the fundamental or first
harmonic. For Figure 1b, the value of n is 3, and the
wave pattern is called the third harmonic.
In this experiment you will use a vibrator of constant frequency to vibrate the string. Your
investigations involve making resonances of different modes by adjusting the length and tension in two
different strings.
Apparatus
Two strings (about 2 m length), 120-cycle electric vibrator and C-clamp; weight hanger, one 50 gram
slotted weight, twelve 100 gram slotted weights, hand stroboscope, and triple-beam balance.
Figure 1a. A first harmonic standing wave.
harmonic.
Figure 1b. A third harmonic standing wave.
