Date :
Transverse and Longitudinal Waves
8. Transverse and Longitudinal Waves
Background
Traveling Wave
Stationary Wave
Wave equation
Group Velocity
Phase Velocity
Aim of the experiment
1. To determine the phase velocity of the stationary wave produced in an
ordinary string (1) by measuring the wavelength () and the frequency (and (2) by
measuring the tension T in the cord and mass per unit length () of the cord.
2. To determine the velocity of sound in air by resonating closed and open air
columns.
Apparatus required
Square cross-section rubber rope/ordinary string
Supports
Wave generator
Measuring tape
Open and close ended pipes of different lengths
Small speaker
Theory
1. Phase velocity
A cotton cord fixed at one end is inserted through the eye of the plunger of an
electromagnetic vibrator (wave generator). The other end goes over a pulley and attached
to a pan which can carry weights. The frequency of vibration recorded from the vibrator
is so adjusted that stationary wave is produced in the cord held under certain tension. The
wavelength of the wave produced is calculated by measuring the distance between the
pulley and the vibrator head. Nodes are formed at the points where the support and
vibrator are attached.
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Transverse and Longitudinal Waves
Fig. 1. Experimental set up to measure phase velocity of a rope wave
The stationary wave produced in a string or rope as a result of two oppositely
traveling waves
and
is given by,
y1 = A Sin(kx-t)
y2 = A Sin(-kx-t+)
y = y1 + y2 = A [Sin (kx-t) + Sin(-kx-t+)]




= 2 ASin   t  Cos kx   ,
2
2


where  is the phase difference between two waves.
…(1)
For the stationary wave produced in a rope fixed at both ends,
and
y = 0 at
y = 0 at
x=0
x=L
…(2)
…(3)
where L = length of the rope
Boundary condition (2) implies =(2n+1) or
y = 2A Sin kx Cos t
Boundary condition (3) implies Sin kL = 0 or kL = n
where n is an integer.
or
2

L
L
n

n
2
…(4)
Therefore, if the length of the string is an integral multiple of /2, then only stationary
waves are produced.

Fundamental or first harmonic L  0
2
First overtone or second harmonic L  1

Second overtone or third harmonic L  3 2
2
Third overtone or fourth harmonic L  2 3
For nth overtone, there are n nodes and
(n+1) antinodes forms between the ends
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Transverse and Longitudinal Waves
Phase velocity (c) of the rope wave is calculated by multiplying the wavelength () of
the stationary wave produced with the frequency of vibration (), recorded by the
wave generator
i.e. c=

Assuming that the wave speed (c) depends only on tension (T) and mass per unit
length (μ) of the string, we can use dimensional analysis to find how ‘c’ depends on
these quantities. So the result is; phase velocity
T
c

The proportionality constant cannot be fixed by the dimensional analysis. The value
of the constant can be obtained from a mechanical analysis of the problem or from
experiment. These methods show that the constant is equal to unity and the final result
is:
T
c=

So if we measure T and μ we can again calculate c, and this should match
with the earlier obtained result by measuring λ and ν.
2. Velocity of sound in air
We know that in open and closed organ pipes resonates when stationary waves are
formed in side the pipes. If l is the length of the air column then for stationary waves
the following conditions are satisfied.
Open ended:
l = (2n) /4,
(n=1,2,3………)
…………………..(5)
One end closed:
l = (2n-1) /4,
(n=1,2,3………)
…………………..(6)
Procedure
Phase velocity
1.
Pass the cotton cord through the eye of the vibrator plunger.
2.
Fix the one side of the end to a vertical supports and the other side is attached
to a pan and is passed over a pulley.
3.
Put some weight on the pan to produce a reasonable tension.
4.
Measure the length of the cord between the vibrator head and the pulley with
the help of a meter scale. Count the number of loops formed for the resonance.
5.
With different range settings, slowly increase the frequency of vibration of
the wave generator until the rope begins to vibrate with blurry antinodes and
nodes in between.
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Transverse and Longitudinal Waves
6.
Note down the frequency at which the rope is set into vibration.
7.
Repeat for different lengths of the strings and different weights on the pan .
8.
Plot a graph of ν vs 1/Slope of the straight line is the phase velocity of the
wave for fixed length and fixed tension.
9.
Calculate tension in the string and use given linear mass density of the rope
and hence calculate the phase velocity again to compare the result with the one
obtained in step 8.
Note : To excite higher modes use low tension(50 – 100 gms) & long string lengths
(more than 1.5 mts).
Velocity of sound
1. Remove vibrator cords from the oscillator and attach a speaker cords to that
and adjust the frequency multiplier knob to 1000.
2. Put the speaker near one end of the air column. Switch on the frequency
generator. A mild monotonous sound will be heard. Slowly vary the frequency
to observe loud sound. At this moment air column is resonating with the
speaker frequency. Note down the frequency. Increase the frequency slowly
the sound will become milder and again at certain frequency it will become
loud. This is the second overtone. Note the frequency. Similarly obtain other
overtones.
3. Repeat 2 for various open and closed columns.
4. Identify the mode and hence find  for each resonating frequency. Plot a
graph of ν vs 1/for a fixed column lengthSlope of the straight line is the
velocity of the sound wave in air.
The first two modes of open and closed columns
Open
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Transverse and Longitudinal Waves
Observations
Mass per unit length of the string () = ……. Kg/m,
Tension, T, in the rope = (M+m) gm, where ‘m’ is the mass of the pan and ‘M’ is
the mass placed on the pan.
Mass of the pan(m) = 50gm.
Repeat experiment for M= 50gm, 100gm, 150gm
Table I: Phase velocity of rope waves
Sl.
No.
T=(M+m)g
(In Newton)
Length of the
string, L (in
m)
Frequency
ν (in Hz)
No. of loops
(n)
λ=2L/n
Velocity v
(in m/s)
νλ
T

(50+50)g
L1=
(100+50)g
L2 =
(150 +50)g
L3 =
90
Transverse and Longitudinal Waves
Table II: Velocity of sound in air
Sl.
No.
Type of
column,
open or
closed
Length of air
column, L (in
m)
Resonating
Frequency
ν (in Hz)
91
Mode
number
n
(fundamental
or overtone)
λ=2L/n
(open)
λ=4L/(2n-1)
(closed)
Velocity v
(in m/s)
νλ
Transverse and Longitudinal Waves
Error calculation
c= 
Find
c
c

 



Precautions
1. Ensure that the amplitude of vibration is maximum or else, error may come in
correlating nodal distances with the rope lengths
2. Always set the vibration to maximum by turning the frequency knob in one
direction or else backlash error may creep in
3. Ensure that the rope passes freely through the eye of the vibrator plunger
4. Do not place more than 200g on the pan as it may damage or break the string.
Questions
1.
2.
3.
4.
5.
Do we get first, second, third…. (i.e., all the harmonics) harmonics in all cases
of stationary waves produced?
What is the difference between group velocity and phase velocity? Which of the
two in greater?
How can you improve upon the experimental set up to find the velocity of rope
wave without the use of a wave generator?
What are the possible sources of error in the experiment?
With what frequency or frequencies the string would vibrate if it is plucked.
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Transverse and Longitudinal Waves
Graph : Phase velocity of rope wave and velocity of sound
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