Experiment 11

**Experiment 11: Standing Waves in a String**

Purpose

To relate standing waves and resonance.

Apparatus

A stretched-string apparatus, with hangers; a set of metric weights; a meter-stick.

Theory

(A) Waves on a stretched string: When a string holds tension, mechanical waves may be set up such that any displacement in at one end automatically transfers that displacement along the length of the string... much the same as a water wave does when a rock is thrown into it.

There is a relationship between the wave's lengththe length of the wave from one peak to another) and its frequency f (how many times the string oscillates in one second). The product of these two properties of the wave yields the waves velocity through the string.

v = f

where v ≡ speed of the wave, ≡ the wavelength, and f ≡ the frequency.

There are two types of waves: transverse and longitudinal. Transverse waves travel in a direction perpendicular to its vibration. If the transverse wave is traveling to the right then the wave vibrates up and down. Waves on a string are transverse waves. Longitudinal waves travel in the direction of its vibration. If a longitudinal wave is traveling to the right, then vibrations of the wave are left and right. (See Fig. 1.)

For transverse waves traveling along a stretched string, the speed of the wave is related to the tension, T, and the linear mass density (mass per unit length) of the string:

v =T / (2)

Vibrating one of its ends will generate waves traveling down a string. Waves that reach the end are reflected back. The actual shape of the string, at an arbitrary time, can be deduced by adding the incident and reflected waves (superposition).

(B) Standing Waves: Under certain conditions, superposition of incident and reflected waves create simple patterns in the stringin several segments. (See Fig.2.) In these patterns the string is not vibrating at certain points along its length (nodes) and is vibrating maximally at other points (antinodes). All points within a segment vibrate in phase. Nodes, the points where the spring is motionless, separate these segments.

Such patterns are called "standing-wave patterns," or natural modes. For a string which is fixed at its ends, formula (3) applies:

= 2 L / n(3)

where L = length of the string, = wavelength, and n = number of segments (1, 2, 3...).

Sketches of such standing-wave patterns are shown in Fig. 2. Notice that the length of a segment is always equal to /2.

(C) Resonance: Natural modesare defined by an integer n = 1, 2, 3.... (See Fig. 2). For a given string length and wave velocity, each mode has its own “frequency” which is determined by the relationship given by Equation 1. Continuously exciting this mode by external oscillations at its frequency will maintain vibrations in that mode. This is the principle of resonance.

However, note that by keeping L constant but changing the value of v (very drastically) we can excite a given natural mode by a frequency of our own choice.

The stretched-string apparatus allows observation of the string-segments in various natural modes as well as measurements of the key parameters. A sketch of this device is shown in Fig. 3.

Note: The effective length of the string is the length between the oscillator and the pulley, as shown in Fig. 3. It is expected to be slightly different for various modes. Your lab instructor will explain to the entire class how to operate the apparatus and how to achieve the most accurate observation of the modes.

Procedure

(a) Place the string over the pulley and attach it to the mass hanger. Plug in the motor for the oscillating strip.

Load your hanger with a mass, M, until you achieve a stable standing-wave pattern in the n=1 mode. Then adjust the load by adding and decreasing M slightly until you achieve themaximum possible amplitude of stable vibrations. I.e. until the string's height is as high as you can achieve it.

On your data-sheet, construct a table with columns for n, M, and L. Record the value of M to 1 g. (Remember to include the mass of the hanger.) Record the value of L to 0.1 cm.

(b) Obtaining n = 2 through n = 5 modes: Unload the hanger. Holding it with your fingers, exert a downward pull, much weaker than that in step (a) above. Change the strength of pull until you glimpse the mode n = 2 (two segments).

Now load the hanger with an equivalent amount of mass to equal the strength of the pull. You should see a glimpse of the n = 2 node. Adjust the mass as in part (a) above. Measure both M and L, and record them on your data sheet.

(c) Repeat part (b) for n = 3, 4, and 5 nodes. As the load M becomes smaller, you may have to use the 5-gm hanger.

(d) Try to achieve patterns with n = 6, 7, 8…. If successful, call your instructor to verify your achievements. Record such data as well.

Lab Report

Part I:

(1) Draw a table as shown for n=1, 2, 3, 4, and 5.

n / M + mhanger / T / v / L / / f / |f - <f>|1

2

3

4

5

Average < f >

(2) Calculate the %-discrepancy between the average <f > of your data and the actual frequency fo = 120Hz of the vibrating metal strip. The value of 10-3 g/cm.

(3) Answer these questions:

Q#1: What is the major reason for limiting your table to n=5, even if higher modes could be observed?

Q#2: Your string is similar to strings in a violin, in a piano, etc., which emit musical sounds. Did you hear some sound, and if you did, in which mode was it? If not, what should have been the main reason? (Hint: assume your hearing is perfect.)

Q#3: Suppose n=1. If a 4%-error is made in measuring M and a simultaneous 1%-error is made in measuring **L, then how big %-error in f this would cause, in the worst possible case**? What if it were ‘the other way around’ (1% for M and 4% for L)?(Note: you may want to try the method of differentials to answer this question. Otherwise some algebra will do.)

1

Stated simply, this means: All points within a segment reach their highest displacements at exactly the same instant of time.

These frequencies are known as overtones. They can be found in all vibrating objects. In fact anything that makes a sound is exhibiting this behavior. In this lab, a string is used to demonstrate the phenomenon. However, musical instruments, from percussion to stringed instruments to the horn section are all typical examples.