The Vibrating String
Introduction
For the next two weeks your lab work will involve an exploration of the properties of waves. As the title above suggests, in this first exercise you will be investigating the waves produced in a vibrating string. In the lab that follows next week, the waves will be produced by an ultrasonic source that generates sound vibrations with frequencies above the limit of human hearing.
Equipment
String,
weight hanger and weights, audio oscillator, amplifier,
Statement of the Theory
When both ends of a string are fixed in place so that they cannot move, waves produced by vibrations in the string are limited to only certain fixed wavelengths. These so-called standing waves must have a definite relationship to the length of the string. Since the ends of the string can’t move, only half-integral multiples of wavelengths can fit into the length between the two ends of the string. In other words, if L is the length of the string and λ is the wavelength of the standing wave, then
L = n (λ /2) (1)
where n = 1, 2, 3, 4, …
This relationship can also be expressed in terms of the frequencies of the standing waves. For any wave, the wave velocity is equal to the product of the frequency, f, and the wavelength.
v = f λ (2)
For a string of mass density (mass/length), m, under a stretching force (tension), T, the wave velocity is given by
v = Sqrt (T/m) (3)
Combining this expression for v with the equation for λ in terms of L and the relationship between λ, velocity and frequency, yields the following equation for the frequency of the standing waves in terms of the length of the string:
fn = (n/2L)Sqrt(T/m) (4)
(Verify this for yourself. First, solve equation (1) for λ and substitute this in equation (2). Then, substitute the expression for v from equation (3) into equation (2). Finally, solve the resulting equation for frequency, f. The result is equation (4)).
The frequency corresponding to n = 1 is called the fundamental or first harmonic, fn for n = 2 is the second harmonic, fn for n = 3 is the third harmonic, and so on.
An audio oscillator and amplifier will be used to provide a variable driving frequency to the PASCO mechanical vibrator. This combination is used to drive a string of length L that has been fixed at one end and stretched over a pulley. Tension in the string is supplied by weights on a weight holder attached to the end of the string hanging over the pulley. The mechanical vibrator should be attached near the fixed end of the string at a position that maximizes energy transfer to the string without constraining its motion.
As the frequency of the generator is varied, standing waves should be observed at frequencies corresponding to the values predicted by equation (4). At these resonant frequencies the amplitude of vibration of the string will be large. At other arbitrary frequencies, no stable wave pattern will be produced.
Start by listing the parameter(s), dependent variable(s) and independent variable(s) of the theory. Changing the n, L and T values can each alter the frequency. The standard procedure is to hold two of the three constant, while varying the third, and observing the effect this has on the resonant frequency. The problem is to determine useful values for the quantities that are held constant. They should be chosen to have the standing wave frequencies spread over as large a range as possible. It will take some experimenting on your part to find suitable values.
1. Measure the length of the string and weigh it. Find the mass/unit length, m.
2. Set up the experimental equipment as described above. Adjust the fixed end support so that the portion of the string that is vibrating is about 1 meter long. This is L in equation (4). Now experiment with values of T by changing the hanging masses to find a combination that permits you to observe at least six modes of vibration (n = 1, 2, 3…6) – try for as many as you can get. The sum of the hanging masses plus the hanger, when multiplied by g (9.801 m/s2) is the tension in the string. Record the resonant frequency for each mode and the values of L and T. Watch for stretching of the string, as this will change the mass/length ratio.
3. Choose one mode of vibration (for example n = 2) and for fixed tension, vary L by moving the fixed end support. Record the length and the resonant frequencies for at least six values of L, varying over as wide a range as your experimental set-up will allow (again, the more data points the better).
1. Make two preliminary computer plots of the data from steps 2 and 3 above. For each the data should lie along a smooth curve. (Not necessarily a straight line). Use this as a criterion to identify data points that should be re-examined. This analysis should be carried out as the data are being collected.
2. For the data from steps 2 and 3 make a computer plot of the measured frequencies versus the expected frequencies calculated from equation (4). Determine the slope and intercept of this graph and compare them with the expected result of slope = 1 and intercept = 0.
3. For the data from step 2, plot frequency versus n. Find the slope and intercept and compare them to the expected values predicted by equation (4).
4. For the data from step 3, plot frequency versus (1/L). As in Step 3, find the slope and intercept and compare them to the expected values.
5. Compute the percent difference from their expected value for each of the slopes in Steps 2, 3 and 4 of this Analysis.