32 WAVES IN STRINGS

32.1 INTRODUCTION

If a transverse wave is caused to travel along a stretched string, the wave is reflected on reaching the ends of the string. The incident and reflected waves have the same speed, frequency and amplitude, and therefore their superposition results in a stationary wave.

If a stretched string is caused to vibrate by being plucked or struck, a number of different stationary waves are produced simultaneously. Only specific modes of vibration are possible, and these are considered in section 32.2.

32.2 THE MODES OF VIBRATION

Fig. 32.1 Modes of vibration of a stretched string

The ends of a stretched string are fixed, and therefore the ends of the string must be displacement nodes. The three simplest modes of vibration which satisfy this condition in the case of a string of length L are shown in Fig. 32.1.

L L L

... .,., -----A,

2

1st harmonic 2nd harmonic 3rd harmonic (fundamental) (1st overtone) (2nd overtone)

(a) (b) (c)

The simplest mode of vibration (a) is called the fundamental, and the frequency at which it vibrates is called the fundamental frequency. The higher frequencies (e.g. (b) and (c)) are called overtones. (Note that the first overtone is the second harmonic, etc.) Representing the wavelengths of the first, second and third harmonics by 21, 22 and 23 respectively, and bearing in mind that the separation of adjacent nodes is equal to half a wavelength (section 31.2), we see from Fig. 32.1 that:

L

489

L 2

L

3

490 SECTION f.' WAVES AND THE WAVE PROPERTIES OF LIGHT

Therefore, if A., is the wavelength of the nth harmonic,

L 2 n

The frequency,fn, of the nth harmonic is given by equation [23.1] as

v fn =-

A., where

[32.1]

[32.2]

v the velocity of either one of the progressive waves that have produced the stationary wave. (Note that the velocity is the same for all wavelengths.)

Therefore, from equations [32.1] and [32.2]

nv fn = 2L

[32.3]

The frequency,/1, of the fundamental (i.e. the first harmonic) is given, by putting n = 1 in equation [32.3], as

v fi = 2L

Therefore, equation [32.3] can be rewritten as

i.e. the frequencies of the various overtones are whole-number multiples ofthe fundamental frequency.

It can be shown that

[32.4]

where

T = the tension in the string (N)

fJ- = the mass per unit length of the string (kg m - I).

Therefore, by equations [32.2] and [32.4]

(n 1, 2, 3 , 0 0 .) [32.5]

1. A wire of length · 400mm and mass 1.20 x 10- 3 kg is under a tension of 120N. What is: (a) the fundamental frequency of vibration, (b) the frequency of the third har­monic?

2. The fundamental frequency of vibration of a particular string is f What would the funda­mental frequency be if the length of the string were to be halved and the tension in it were to be increased by a factor of 4?

WAVES IN STRINGS 491

32.3 STRINGED INSTRUMENTS

When a guitar string is plucked or a piano string is struck, transverse waves travel along the string and are reflected on reaching its ends. The energy of any wave whose wavelength is such that it does not give rise to one of the allowed stationary waves is very quickly dissipated. The waves which remain have frequencies that are given by equation [32.5], and the string vibrates with all these frequencies simultaneously.

The largest amplitude of vibration, and therefore the predominant frequency, is that of the fundamental. The relative amplitudes of the various overtones depend on the particular instrument being played, and it is this that gives an instrument its characteristic sound (see section 34.2).

32.4 MELDE'S EXPERIMENT

Fig. 32.2 Apparatus for Melde's experiment

Fig. 32.3 Modes of vibration in Melde's experiment

If a string is caused to vibrate by being plucked or struck, it vibrates freely at all of its natural frequencies (i.e. the frequencies given by equation [32.5]). On the other hand, if a string is forced to vibrate at some particular frequency, it will vibrate with large amplitude only if the forcing frequency is one of the natural frequencies of the string. This can be very effectively demonstrated by the apparatus shown in Fig. 32.2, and is known as Melde's experiment.

Pulley String under tension

Mechanical oscillator

Signal gene(ator

The frequency of the signal generator is slowly increased and, at first, very little happens. Eventually though, a frequency / 1 (say) is reached at which the string vibrates with large amplitude in the form of a single loop (Fig. 32.3(a)). If the frequency is increased beyond this value, the amplitude of the vibrations dies away. When the forcing frequency reaches 2/1, the string again vibrates with large amplitude, but this time it vibrates as two loops (Fig. 32.3(b)). At 3/1 it vibrates as three loops, etc. Substituting the relevant values of L, T and J1 in equation [32. 5] confirms that the forcing frequencies, / 1 , 2/1 and 3/1 , are respectively equal to the frequencies of the first, second and third harmonics of the string. This, then, is an example of resonance - the string responds well only to those forcing frequencies which are equal to its natural frequencies of vibration.

T.---J'"" - ----------(a) (b)

492

Notes (i)

SECTION E: WAVES AND THE WAVE PROPERTIES OF LIGHT

The amplitude of vibration of the oscillator is small in comparison with that of the string, and therefore the string behaves (almost) as if it is fixed at its point of attachment to the oscillator. --

(ii) The reflected waves are not quite as 'strong' as the incident waves, and this prevents the displacements at the nodes being exactly zero.

(iii) The motion of the string can be 'frozen' if stroboscopic illumination is available. This demonstrates very convincingly that each section of the string is in anti-phase with that in an adjacent loop.

32.5 EXPERIMENTAL VERIFICATION OF f, = 21L ~

Fig. 32.4 Sonometer

The frequency, f 1 , of the fundamental mode of vibration of a stretched string is given, by putting n = 1 in equation [32.5], as

f, = -1 fT 2Ly --;; It follows that:

(i) j 1 ex 1I L if T and J1. are constant

(ii) f 1 ex ,jT if L and J1. are constant

(iii) .h. ex 1 I JJi if L and Tare constant.

These relationships are sometimes referred to as the laws of vibration of stretched strings. They may be verified experimentally by using a sonometer (Fig. 32.4), as described below.

Movable Wire under bridge tension

Fixed bridge

Wire f\.--LL-----=--~-------11"'-4-- anchored

Known -- mass(M)

To verify f1 ex: 1/ L

Hollow sounding box

here

Having selected suitable values of T and JL, the position of the movable bridge is altered so that the vibrating length, L, of the wire is such that when the wire is plucked it produces the same note as a tuning fork of known frequency. If the experimenter is not sufficiently 'musical' to detect whether the two notes have the same pitch, he can make use of a resonance technique. A small piece of paper in the form of an inverted vee is placed on the centre of the wire, and the stern of a vibrating tuning fork is held against one of the bridges. This forces the wire to vibrate, and if its length is such that its fundamental frequency of vibration is equal to the frequency of the ~ning fork, the wire vibrates with large amplitude and throws the paper_ off the w1re. The procedure is repeated using tuning forks of other ~own frequencies, and without altering either Tor Jl.. A graph ofj1 against 11 Lis hnear and passes through the origin, thus verifying the relationship.

WAVES IN STRINGS 493

To Verify f1 ex VT With L kept constant at some suitable value, the mass, M, and thererore the tension T ( = Mg), is altered so that when the wire is plucked it produces the same note as a tuning fork of known frequency. The procedure is repeated using tuning forks of other known frequencies, and without changing either Lor J.L. A graph of !I against n is linear and passes through the origin, thus verifying the relationship.

To Verify f, ex 1/ ..Jii This relationship cannot be verified directly if tuning forks are used, because neither the frequencies of a set of tuning forks nor the masses per unit length of a set of wires are continuously variable. However, once it has been verified that f 1 ex 1 I L , it is sufficient to show that L ex 1 I JJi at constant Tand constantf1. First, the mass per unit length, J.L, of a wire is determined by weighing. The length, L, of the wire is then adjusted so that when the wire is plucked it produces the same note as one of the tuning forks . The procedure is repeated using wires of different masses per unit length. Each wire must be under the same tension as the first wire, and in each case the length is adjusted until the wire vibrates at the same frequency as the tuning fork that was used with the first wire. A graph of L against II fo is linear and passes through the origin, thus verifyingj1 ex 1 I Vfi.