Physics 211 Experiment #13 Standing Waves on a String
Objective: The objective of this experiment is to gain knowledge and understanding of
standing waves driven by an external force. Resonant conditions for standing waves
on a string will be investigated.
Apparatus: Pasco Variable Frequency Wave Driver with string
Pasco Student Function Generator
50 g weight hanger and slotted weight set
2 table clamps
short rod
pendulum clamp
pulley
Digital multimeter
1 or 2 meter stick
Pre-Lab Exercise:
A string is oscillating in a standing wave pattern. The third harmonic is
exhibited.
a. How many nodes are present? How many antinodes are present?
b. If the string is 2.2 m long and has a mass of 9.5 g, what is its mass per unit
length, ?
c. Suppose only 1.2 m of the string is oscillating in the third harmonic, and the
string has a tension of 2.0 N. What is the frequency of oscillation of the
string?
d. Write the equation for the standing wave pattern on the string in the form of
equation 1 (below) with appropriate numbers substituted and simplified as
much as possible.
Theory:
The Standing Wave Pattern:
A transverse wave traveling along a string is reflected from the fixed (pulley) end
and returns interfering with the wave continuously generated by the wave driver. At
resonance, a standing wave pattern will be established. There are antinodes at
intervals along the pattern, nodes at the ends of the string, and there may be nodes
periodically occurring within the pattern as well.
The equation for a transverse wave traveling in the positive x direction along a string is
y1  A sin  kx  t  ,
where k 
2

,   2 f ,  is the wavelength, and f the frequency of the oscillation.
The wave reflected from the fixed end is described by 
y2  A sin  kx  t  ;
the reflected wave travels in the negative x direction.
The combined wave will be the sum of the original wave and the reflected wave
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y  y1  y2  A sin  kx   t   sin  kx   t   .
Expanding the sine functions as the sum or difference of two angles yields:
y  Asin kx cos t  cos kx sin t  sin kx cos t  cos kx sin t  ,
which simplifies to
y   2 A sin kx  cos t .
(1)
The factors in parentheses are time independent and represent the distribution of
displacement along the string.
We now analyze the “amplitude,” 2 Asin kx , for nodes. The nodes will occur when
2 Asin kx  0 ,
which will occur when the angle kx is equal to 0, or a multiple of .
kx  n , n  0,1, 2,
2
x  n

The end points of the string are nodes, thus x = 0, n = 0 and at x = L where n will equal
the number of segments or loops in the standing wave.
Substituting x = L, and solving for  provides the condition for the wavelength for each
pattern or harmonic of the vibrating string.
2

n 
Ln
2L
n
(2)
The nodes divide the string into half wavelength segments (/2), so if the frequency (f)
is known, the speed of a wave can be determined by the fundamental wave velocity
equation
v f

(3)
The frequency of the wave can be found by substituting (2) into (3) and solving for f:
fn  v
If we also note that f1  v
n
2L
1
, then
2L
f n  nf1
(4)
Velocity of a transverse wave in a string:
The velocity of a wave traveling on a string is given by
v
T

(5)
where T is the tension in the string and  is the mass per unit length (m/) of the
string.
Overview: In this lab a string is attached to a mechanism that drives waves at a
particular frequency. The wave speed is determined by the mass per unit length of the
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string, and the tension of the string. The wavelength of the wave can be determined
by the length of the string that participates in the oscillation, and the number of nodes
present. The frequency will be adjusted to set up the fundamental mode of oscillation
(also known as the first harmonic), the second harmonic, third, etc. The properties of
each wave, such as the number of nodes present, the wavelength of the wave, the
frequency of oscillation, and the wave speed, will be studied.


Figure 1. A picture of the apparatus for generating standing waves on a string. Note that the function
generator requires a digital volt meter to determine the frequency output of the function
generator.

Procedure:
Examine the wave driver, and make sure that the oscillating pin is in the locked
position during set up.
Set up:
1. Measure the mass and length of the string, and record these values into an excel
spreadsheet.
2. Tie two loops in the string: one loop at an end, and the other about 1.5 m from the
first.
3. Assemble the two table clamps to the table approximately 1.1 m apart with the
pulley inserted in one clamp and the pendulum clamp extending at a right angle
over the table at the other end.
4. Assemble the string between the two supports with 200 g total mass applied to the
end of the string placed over the pulley. This mass supplies the tension (T) in the
string. (For a 200g mass, what is the tension in the string?)
5. Approximately 10 cm from the pendulum clamp, attach the string to the wave
driver.
6. Attach two wires to the sine wave output of the function generator (check to be
sure the switch on the function generator is set for a sine wave) and attach them to
the two inputs of the wave driver. DO NOT PLUG IN OR TURN ON THE
FUNCTION GENERATOR UNTIL TOLD TO DO SO BY YOUR INSTRUCTOR!
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7. Attach two more wires from the DC volts output of the function generator to the V and COM inputs on the digital voltmeter and select the 20 V DC (20 DCV) setting
on the meter. Turn the meter on if there is a separate on-off switch.
NOTE: When reading the frequency (voltage) displayed on the digital meter you
must use the same multiplier as is used on the function generator to obtain the
correct order of magnitude of frequency.
8. Unlock the wave driver, select the X1 multiplier range on the function generator,
set the dial for 1 Hertz, select an output in the middle of the output range. AFTER
THE SET UP IS INSPECTED BY YOUR INSTRUCTOR, turn on the function
generator.
Data collection:
9. Adjust the frequency of the function generator until the string oscillates in its
fundamental mode. Record the oscillation frequency, the number of nodes, and the
total length of the string that participates in the oscillation. Determine the
distance between the nodes, and the wavelength of the oscillation. Record all
information into an excel spreadsheet.
10. Repeat step 9 for the second, third, etc. harmonic. Measure at least the first 10
harmonics, recording all of the information listed in 9. (This set of runs is called
case 1.)
11. Reduce the tension to one fourth its value for the new length and repeat steps 9
and 10 for the first 6 harmonics. (This set of runs is called case 2.)
12. Repeat steps 9 and 10 for this new tension.
Analysis of the Data
1. For cases 1, and 2, and for each frequency find the wavelength  of the wave, and
the value of n that describes the wave (see equation 2).
 For case 1, plot the frequency of the wave versus 1/; the slope of the line should
be equal to the wave speed. Compare the wave speed determined from the slope of
the line with that obtained using equation 5. 
 Repeat step 2 for case 2, and find the slope of the line. 
 What is the ratios of wave speeds for case 1 compared to 2 (from the graphs). Is
this ratios equal to the ratio of the predicted wave speeds (Apply equation 5 here)?
5. Are the measured frequencies for case 1 equal to nf1, where n is the number of the
harmonic?
6. What is the ratio of the frequency of the second harmonic for case 1 compared to
case 2? Is this ratio the same for the third harmonic? the fourth? the fifth? Is there
a pattern here?
Make sure that you discuss experimental sources of error in your laboratory report.
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