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EP 236 WAVES LABORATORY
7. FORMATION OF STANDING WAVE ON A SPRING
A. STANDING LONGITUDINAL WAVES ON AN HELICAL SPRING
PURPOSE:
There are different objects for the standing waves on a spring experiment. Some of are related
to the standing longitudinal waves on an helical spring such as the investigation longitudinal
waves in helical springs fixed at both ends, generating standing waves as a function of the
excitation frequency f, determining the wave velocity V and investigating the effect of the spring
tension on the phase velocity V and the wavelength λ. The others are related to the circularly
polarized waves such as generating standing, circularly polarized thread waves for various
tension forces F, thread lengths s and thread densities m*, determining the wavelength
of
thread waves as a function of the tension force F, the thread length s and thread density m*.
THEORY:
A wave is formed when two systems capable of coupled oscillation sequnetially execute
oscillatins of the same type. One example of this is the propagation of a longitudinal wave on a
helical spring. The propagation velocity V of the oscillation state is related to the oscillation
frequency f and the wavelength λ through the formula
V=λ.f
(7.1)
This is termed the wave velocity or phase velocity. We can say
V=√(D/m0)
D: spring costant, m0: mass of spring, s: length of spring
(7.2)
When the helical spring is fixed at both ends and excited to oscillation, reflections ocur t both
ends, and the outward and reflected waves are superposed. Standing waves form at certain
excitation frequencies as stationary oscillation patterns. The distance between two oscillation
nodes or two antiodes of a standing wave corresponds to one half the wavelength. For a
standing wave with n oscillation antinodes, we can say
(λ n / 2) = (s/n)
with n = 1,2,3,
For the excitation frequencies, (11.1) and (11.2) give us
fn = V . (n/2s)
(7.3)
(7.4)
respectively
fn = √(D/m0) . (n/2)
(7.5)
Thus, a change in the length s of the extended helical spring does not change the respective
frequency fn required to excite n oscillation antinodes.
In this experiment, two different helical springs are mounted vertically one after another and the
bottom end is caused to oscillate by means of an electric motor with an oscillation lever. The
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Enginering of Physics Department
excitation frequency is continuously adjustable using a function generator. The length s of the
extended spring can be varied easily by adjusting the suspension material.
PROCEDURE:
Set up the experiment as shown in Fig. 1. Connect the stand bases using the short stand rod
and mount the long stand rod in the stand base. Attach the pointer and clamping block to the
stand rod. Also attach the motor (b) in the stand base using the cables and connect it to the
function generator. Measure the length of the unextended helical spring. Using a piece of rubber
string 15-20 cm long, tie a loop which passes through the eyelet of the oscillation lever is
approximately horizontal when the helical spring is tensioned. Hook the end of helical spring 1
into the eyelet of the oscillation lever and attach the other end to the clamping block using
support clip (a). After extending the helical spring to around three times its original length by
moving the claping block, connect the function generator to the 12 V output of the transformer.
On the function generator, set the output voltage U= 3V p, frequency range “x 10 Hz” and signal
form “~” .
First experiment with spring 1:
Starting from the lowest frequency range, slowly increase the frequency f and carefully seek
those frequecies at which standing waves form; read off the frequencies from the scale of
frequency control knob (c).
Length unextended:
Length extended:
n
1
2
3
4
5
f / Hz
λn
Table 1: Frequencies fn required for generating standing waves with n oscillation antinodes.
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Enginering of Physics Department
Second experiment with spring 1:
Once again, generate the standing wave with two oscillation antinodes (three oscillation nodes),
mark the positions of the top and bottom oscillation nodes using the pointers and measure the
wavelength λ as the distance between these oscillation nodes using the tape measure. While
maintaning a constant frequency, extend the helical spring further by moving the clamping block
and observe how the satnding wave persists. Once again, measure the distance between the top
and bottom oscillation nodes.
s / mm
f / Hz
λ / mm
V / m.s-1
Table 2: Frequency f, wavelength λ and phase velocity for two oscillation antinodes as a
function of spring length s.
Experiments with spring 2:
Place helical spring 2 in the experiment setup, extend it to about twice its original length and
repeat the measurements.
Length unextended:
Length extended:
n
1
2
3
4
5
f / Hz
λn
Table 3: Frequencies fn required for generating standing waves with n oscillation antinodes.
s / mm
f / Hz
λ / mm
V / m.s-1
Table 4: Frequency f, wavelength λ and phase velocity for two oscillation antinodes as a function
of spring length s.
Results:
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Enginering of Physics Department
Standing waves can be generated in a helical spring fastened at both ends. The number of
oscillation antinodes increases with the frequency.
When the helical spring is extended further, the stationary oscillation state remains the same
when the excitation frequency f is unchanged. The waveength λ and the phase velocity increase
in proportion to the extension.
CALCULATION:
Calculate the spring constant D, and the wavelength λn for each case. Plot the graph of V-f
accordig to your datas for spring 1 and spring 2.
B. CIRCULARLY POLARIZED WAVES
Two waves with the same frequency, wavelength, and amplitude traveling in opposite directions
will interfere and produce standing waves. Let the harmonic waves be represented by the
equations below
(7.6)
Adding the waves and using a trig identity we find
(7.7)
This is a standing wave -- a stationary vibration pattern. It has nodes - points where the
medium doesn't move, and antinodes - points where the motion is a maximum. A standing wave
on a thread might look like
Figure 1. Standing waves on a thread
Consider a string of length L that is fixed at both ends. The thread has a set of natural patterns
of vibration called normal modes. This can be determined very simply. First remember that the
ends are fixed so they must be nodes. This means a certain number of wavelengths or half
wavelengths can fit on the thread determined by the length of the thread. The first three
possible standing waves are shown below.
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Enginering of Physics Department
The wavelengths can be related to the length. The frequency, then, is found from the
wavelength. Standing, circularly polarized waves are generated in a thread with known length s.
The tension force F is varied until waves with the wavelength
(7.8)
The propagation speed of a wave in a medium is calculated using d’Alembert’s wave equation.
For an elastically tensioned thread, we compare e.g. the restoring force acting on a section of
the thread deflected from its resting position with the inertial force of this piece of thread. The
result for the propagation speed is
(7.9)
wherer F is the tension force, A is the thread cross-section and
material.
is the density of the thread
Thus, at a fixed excitation frequency f, the following applies for the wavelength :
(7.10)
PROCEDURE: Set up the experiment as shown in Fig.2
Figure 2. Arrangement for the experiment
(a) Cam
(b) Mounting point for thread length s = 0.35 m
(c) Mounting point for thread length s = 0.48 m
(d) Deflection pulley
(e) Holding arm
(f) Dynamometer
Firstly, Cut up the thread supplied with the apparatus into two pieces of different lengths:
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Cut off a piece 0.65 m long as thread 1 for part a.
Cut off a piece 0.50 m long as thread 2 for part b.
Part a) Wavelength as a function of the tension force F
– Set up the holding arm (e) of the vibrating thread apparatus at position (c).
– Tie one end of thread 1 to cam (a).
– Tie a loop in the other end, hang this on the dynamometer (f).
– Measure the distance between cam (a) and the center of the deflection pulley (d) (= thread
length s) and write this value in the experiment log.
– Switch on the motor of the apparatus.
– With the adjusting screw loosened, vary the force F by changing the height of the holding arm
(e) until a standing wave of maximum amplitude with the wavelength
= 2 s is formed (one
oscillation antinode).
– Read off the corresponding force F1 and write this value in the experiment log.
– By slowly and carefully varying the height of holding arm (e), determine the forces Fn at which
standing waves with n = 2, 3, 4 and 5 antinodes are formed.
– Write down the number n of nodes, the corresponding force Fn and the frequency f in the
experiment log.
– Switch off the motor.
Part b) The influence of thread length s and thread mass m:
– Set up holding arm (e) of the vibrating thread apparatus at position (b).
– Attach thread 2.
– Measure the distance between cam (a) and the center of the deflection pulley (d) (= thread
length s) and write this value in the experiment log.
– Switch on the motor of the apparatus.
– Determine the forces Fn and the frequencies f at which standing waves with n = 1, 2, 3 and 4
antinodes are formed.
– Switch off the motor.
CALCULATIONS:
Calculate the wavelength λn for each number of oscillation nodes n.
Calculate the value of Fn and plot the graph λn versus to Fn
Complete the table 1 and table 2 with datas.
n
Thread 1
Thread 2
F(N)
F(N)
1
2
3
Table 1
n
1
2
3
Thread 1
Fn
(m)
Thread 2
Fn
(m)
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Enginering of Physics Department
Table 2
OUESTIONS:
1. Why is a pulse on a string considered to be transverse? How would you create a
longitudinal wave in a stretched spring? Would it be possible to create a transverse wave
in a spring?
2. By what factor would you have to multiply the tension in a stretched string in order to
double the wave speed?
3. When traveling on a taut string, does a pulse always invert upon reflection? Explain.
4. Does the vertical speed of a segment of a horizontal taut string, through which a wave is
traveling, depend on the wave speed?
5. If you shake one end of a taut rope steadily three times each second, what would be the
period of the sinusoidal wave set up in the rope?
6. A vibrating source generates a sinusoidal wave on a string under constant tension. If the
power delivered to the string is doubled, by what factor does the amplitude change?
Does the wave speed change under these circumstances?
7. Consider a wave traveling on a taut rope. What is the difference, if any, between the
speed of the wave and the speed of a small segment of the rope?
8. If a long rope is hung from a ceiling and waves are sent up the rope from its lower end,
they do not ascend with constant speed. Explain.
9. How do transverse waves differ from longitudinal waves?
10. When all the strings on a guitar are stretched to the same tension, will the speed of a
wave along the most massive bass string be faster, slower, or the same as the speed of a
wave on the lighter strings?
11. If one end of a heavy rope is attached to one end of a light rope, the speed of a wave will
change as the wave goes from the heavy rope to the light one. Will it increase or
decrease? What happens to the frequency? To the wavelength?
12. If you stretch a rubber hose and pluck it, you can observe a pulse traveling up and down
the hose. What happens to the speed of the pulse if you stretch the hose more tightly?
What happens to the speed if you fill the hose with water?
13. In a longitudinal wave in a spring, the coils move back and forth in the direction of wave
motion. Does the speed of the wave depend on the maximum speed of each coil?
14. Both longitudinal and transverse waves can propagate through a solid. A wave on the
surface of a liquid can involve both longitudinal and transverse motion of elements of the
medium. On the other hand, a wave propagating through the volume of a fluid must be
purely longitudinal, not transverse. Why?
15. In an earthquake both S (transverse) and P (longitudinal) waves propagate from the
focus of the earthquake. The focus is in the ground below the epicenter on the surface.
The S waves travel through the Earth more slowly than the P waves (at about 5 km/s
versus 8 km/s). By detecting the time of arrival of the waves, how can one determine the
distance to the focus of the quake? How many detection stations are necessary to locate
the focus unambiguously?
16. In mechanics, massless strings are often assumed. Why is this not a good assumption
when discussing waves on strings?
17. A rope hangs vertically from the ceiling. Do waves on the rope move faster, slower, or at
the same speed as they move from bottom to top? Explain.
18. A rope hangs vertically. You shake the bottom back and forth, creating a sinusoidal wave
train. Is the wavelength at the top the same as, less than, or greater than the
wavelength at the bottom?