Standing Waves on a String - Ryerson Department of Physics

W2 - 1
STANDING WAVES ON A STRING
The object of this experiment is to study the resonance modes of a stretched string, and to
test by direct measurement the theoretical formula that relates the tension in the string to its
length, mass, resonance frequency and mode of vibration.
Introduction
A string under tension is one example of the many physical systems that show various
modes of vibration with discrete, characteristic frequencies. These frequencies are sometimes
called eigen-frequencies, a name that is derived from German, meaning the system's own
frequencies. Eigen-frequencies are characteristic of waves confined within fixed boundaries.
The phenomenon of characteristic frequencies in a string can be understood as a superposition of
two waves which travel in opposite directions. When these waves have the same frequency, the
same amplitude, and when integral multiples of half the wavelength fit into the length between
the string supports, then the result is a stationary mode of vibration, called a standing wave. The
two waves continue to reflect at the ends of the string to keep the standing wave pattern
preserved. Nodes are the locations along the string where there is no motion of the string,
though there is a periodic variation of tension at these points. Antinodes are those points where
the motion of the string is the greatest and where the tension remains constant. The fixed ends of
the string are nodes. Resonance occurs when an outside periodic force is applied with a
frequency that matches one of the system's own characteristic frequencies.
Theory
The speed v, of a wave is related to its wavelength λ, and to its frequency f, by the following
equation:
v = fλ
(1)
For a wave travelling along a stretched string it can be shown that the speed is also dependent on
the tension T, and the mass of the string per unit of length µ, according to:
T
v = (2)
μ
The interested student can find a derivation of this equation in most general physics text books.

W2 - 2
Combining equations (1) and (2) results in the following equation that will be convenient for this
experiment:
1 T
f = (3)
λ μ
For standing waves the distance L, between the fixed supports of the string is an integer multiple
of half the wavelength:
λ
L = n
(4)
2
For the basic or fundamental mode of vibration n = 1, so λ = 2L. For the second mode also
called the second harmonic n = 2, λ = L. For the third mode or the third harmonic n = 3, et
cetera. These modes of vibration are shown in the Figure 1 below:
n = 1
n = 2
n = 3
Figure 1. Modes of vibration in a string
Combining equations (1), (2) and (3), we obtain an equation for the resonant frequencies f n :
n T
1
f n
= or f n
= n T
(5)
2L μ
2L μ
This is the equation of a straight line, y = mx + b . If we let y = fn
and x = n T , then the
resulting line should have a slope of :
1
slope = (6)
2L μ

W2 - 3
The parameters of the system in this case are the Tension (T), the Length (L) and the mass per
unit length (μ) of the string. If we fix the values of any two the third can be measured. The
easiest to fix are T and L and use a measurement of f n to determine μ.
Procedure
Pulley
Bridge
L
String
Rigid
Support
Weight
Variable
Frequency
Vibrator
Figure 2. <strong>Standing</strong> waves apparatus
1. Set up the apparatus as shown in the Figure 2 above. Slide the wooden bridge under the
string to adjust the portion of the string that can be set into resonance by the vibrator to be
between 1 and 1.5 metres. Measure the length carefully. Set the tension (T = Mg) by
making the suspended mass M =100g.
2. Adjust the frequency of the vibrator to get a standing wave pattern with n = 1 and record
the value of the frequency. Increase the frequency until a standing wave with 2 loops is
obtained. Repeat to a maximum frequency of about 350Hz. The wavelength for each
standing wave pattern may be found from the equation λ = 2L/n.
3. Increase the mass on the string to 200g and repeat steps 1 and 2 above. Do this again for
300g. In each case the tension in the string will be the mass times g (9.8 m/s 2 ).

W2 - 4
Analysis
1. For each combination of f, n and T a value of μ can be obtained and these could simply be
averaged. Alternatively one could group the observations and for each value of T plot the
values of f vs. n T and then use the slope to find μ, as suggested by equation (6):
n
μ =
1
( 2L ⋅ slope) 2
2. Although this value is small, a measure can be obtained by direct weighing of a sample of
string using the electronic balance in the lab. You will now try to measure μ directly and
compare this value to what you obtained through experiment. Measure a length of string
about 2 – 3 metres long. You may use the same string you did the experiment with, or a
new length of string. Weigh this string on the electronic balance, and calculate the value
for μ.
3. Compare the value for μ obtained by each method. Analyse the sources of uncertainty for
each measurement and comment on the relative accuracy of the values obtained by direct
and indirect measurement.