Transverse Waves: Experiment

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EXPERIMENT 5

TRANSVERSE STANDING WAVES

A string attached to a turning fork is set vibrating at the same frequency as the tuning fork. The length and tension in the string are adjusted until standing waves are observed on the string. By knowing the tension in the string and the wavelength of the standing waves, the frequency of oscillation of the string and thus, the tuning fork is found. This value is then compared to a strobelight determination of the frequency.

THEORY

If transverse waves of constant frequency and amplitude are sent down a string that is fixed at one end, then a reflection of the waves occurs and the oppositely directed waves interfere with each other to produce standing waves. A wave traveling in the positive x direction is given by

[pic] (1)

where A is the amplitude, k = 2[pic]/ [pic] with [pic] being the wavelength, and w= 2[pic]f with f being the frequency. For a similar wave traveling in the negative x direction with the same amplitude

[pic] (2)

If the amplitudes of the waves are small, then the waves obey the law of superposition and add linearly. The resultant wave is

[pic] (3)

which represents a standing wave whose amplitude, 2A cos wt is a function of time. Figure 1 shows the standing wave. The diagram indicates that the wave shape is not moving along the string but is only oscillating vertically on the string.

[pic]
Figure 1. A standing wave fixed at x = 0 and x = L. Five loops are shown. The frequency of a wave is given by
[pic](4)

where V is the speed at which the transverse waves propagate along the string, The speed of the wave, in terms of the tension T, and the mass per unit length of the string [pic] is

[pic](5)

The frequency is, therefore,

[pic](6)

An alternate expression that can be used to calculate the frequency is

[pic](7)

where L is the length of N loops. Because the string is being driven at the same frequency as the tuning fork, the frequency of the tuning fork producing the standing waves is also given by (7).

In the experiment a string is attached to one tine of an electrically-driven tuning fork. The other end of the string passes over a pulley to a weight hanger. (Refer to Figure 2.) The vibrating tuning fork sets up standing waves on the string and the amplitude of these waves can then be maximized by changing the weight on the weight hanger or by moving the entire tuning fork assembly toward or away from the pulley. [pic]

Figure 2. The apparatus set up with standing waves on the string.

APPARATUS

0 electrically-driven tuning fork0weight hanger
0 low voltage power supply03 - 50 gram slotted weights
0 2 leads0string, 2.5 meters long
0 table clamp and upright02 meter long meter stick
0 right angle clamp0strobe light
0 rod-mounted pulley0rubber mallet

PROCEDURE

a) Record the mass per unit length of the string, [pic] and its uncertainty, [pic][pic].

b) Set up the apparatus as shown in Figure 2 with a total of 100 grams hanging from the end of the string. Connect the power supply to the tuning fork terminals with the two leads, and plug in the power supply. Record the total mass hanging from the end of the string.

c) Squeeze the tines of the tuning fork together and release, or strike one tine with the rubber mallet. Then turn the adjustment screw on the side of the tuning fork until a constant vibration results. Lock the screw in place.

d) Move the tuning fork assembly toward or away from the pulley until at least three loops are observed and the amplitude of the oscillation is at maximum.

e) Use the meter stick to measure the distance L. from the first node away from the tuning fork to the first node away from the pulley. (The loops at the two ends of the...
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