Elementary lessons in sound

Contents

P. PAGE

Introduction — Vibratory Motion . . . 229

Reflection of Sound ..... 250

Chapter I Introduction— Vibratory Motion

1. Sound caused by Motion. — If you examine any sounding body you will soon come to the conclusion that the sound is produced by motion of some kind. In many cases the motion is obvious and can be followed by the eye. For example, if you take hold of a stretched string between your finger and thumb, pluck it to one side and then let it go, the string will give out a sound and, at the same time, will be seen to swing from side to side. If the rate of motion is very rapid you may not be able to follow each separate swing, but the fact that the string is in motion is shown by its losing its definite outline and presenting the appearance of an indistinct gauzy spindle. By touching the string with the finger the motion can be stopped, and it is then found that the sound ceases at the same time.

The same indistinct appearance is presented by the prongs of a tuning-fork while it is sounding. When any hard object, such as the point of a knife, is held against the prongs, a loud rattling noise is heard ; and quite a little shower of spray is thrown up when the points of the fork are dipped into water. If you want further proof of the motion you can easily get it by striking the fork and then making it touch your lips or teeth.

2. The kind of Motion. — A tuning-fork consists essentially of two steel springs united together at one end, and each prong of the fork moves in much the same way as a single straight spring clamped at one end. By taking a long thin

CH. I spring (instead of the comparatively short and thick prong of the tuning-fork) we can make the motion slower and examine it at our ease.

Expt. 1. — Clamp the lower end of the spring in a vice. Pull the upper end to one side, as at /, Fig. 1. The spring is elastic : it bends or c gives. ’ In ? / order to bend it you have to exert

force upon it. In virtue of its elasticity the spring tends to recover its original form, and if you wish to keep it bent you must continue the pressure against it.

Now let the spring go. It flies back : but it does not stop when it has got back to its original position (vertical). It overshoots the mark and moves, with gradually diminishing velocity, to about an equal distance on the opposite side (/'). It then begins to return, again overshoots the mark, gradually comes to rest, starts back again, and so goes on swinging from side to side.

Owing to the resistance of the air, etc., the amplitude of the excursions gradually diminishes, and finally the spring comes to rest.

3. Vibration. — When a body or point moves in the manner above described it is said to vibrate or oscillate . Referring again to Fig. 1, a movement from / to /' and back again to / is called a complete vibration. The amplitude of the vibration is measured by the extent of the motion on either side of the mean position or position of rest, i.e. half the distance //'.

If you take a watch and examine the rate of vibration of the spring, you will find that this rate is regular and independent of the amplitude of vibration. Whether the excursions made by the end of the spring are great or small, the number of vibrations executed in a given time is constant. Thus the vibration is regular or periodic. The time required to perform

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VIBRATORY MOTION a complete vibration is called the period of vibration. We shall denote this time (measured in seconds or a fraction of a second) by r. If the number of complete vibrations performed in a second be denoted by n , then n

We may call n the frequency of ’ vibration or the vibration-number.

4. Vibration of Pendulum. — The periodic vibration of an elastic spring is very similar to that of a pendulum swinging under the action of the earth’s attraction. When the bob of the pendulum is displaced to one side and then let go, it swings to a nearly equal distance on the other side, stops, returns, and goes on swinging from side to side regularly, but with the amplitude of the oscillations gradually diminishing, and finally the pendulum comes to rest. The vibrations are regular : the period of vibration is constant and independent of the amplitude. As in the case of the spring, each complete vibration may be divided up into four similar parts.

The pendulum is in its mean position, or position of rest, when the string is vertical and the bob is at its lowest point (N, Fig. 2). Now suppose the bob to be displaced to one side, say to A. The bob not only moves to the left but rises, for it moves in an arc of a circle. In order to raise it through the distance HA you have to exert force and do work in lifting the weight of the bob upwards. You have thus given to the bob the power of doing an equal amount of work in falling through the same distance, i.e . to its position of rest. It possesses energy of position, or potential energy } When the pendulum is released it moves with gradually increasing velocity, and this velocity is greatest when the bob is at its lowest point. The energy has now been changed from the potential into the kinetic form. It is this kinetic

1 The energy of a body is its capacity for doing work.

A body is said to possess potential energy when it is able to do work in virtue of its position. A raised weight (as in a clock) and a head of water are examples of bodies possessing potential energy. So also are coiled springs, compressed air, etc. ; so that ‘ position ’ must here be held to include change of form or volume.

A body is said to possess kinetic energy when it is able to do work in virtue of its motion. Thus the wind does (useful) work in turning the sails of a wind-mill ; a cannon-ball does (destructive) work in crashing through the walls of a fort.

CH.

energy possessed by the bob that prevents it from coming to rest at its lowest point and carries it over to the other side (A'), at the same time lifting it up against the action of gravity. At the end of the first swing (or semi-oscillation) the bob comes to rest for an instant and then retraces its path. In the first quarter-period the energy changes from the potential to the kinetic form, and in the second quarter-period from the kinetic to the potential form. In the third and fourth quarters the same changes are gone through with velocities reversed.

Observe that the velocity is least (zero) at the points where the displacement is greatest (A and A') : the velocity is greatest when the displacement is zero (N).

Fig. 3-

5. Simple Straight -line Vibration. — -If the distance through which a pendulum swings is small compared with the length of the string, the path of the bob is very nearly a straight line. The following is a better illustration of such vibration.

Expt. 2. — Coil some thin wire round a pencil or corkborer. Hang the spiral up and to its lower end attach a lead bullet (N, Fig. 3). Pull the bullet down to A and then let it go. It flies up to A' and continues for some time to oscillate" up and down along a vertical line. This vibration along a straight line may be regarded as a type of the kind of motion which accompanies sound-waves in air.

VIBRATORY MOTION

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6. Elasticity. — The force tending to bring the pendulum back to its position of rest is due to the attraction of the earth on the bob. In the case of the spring the force is due to the elasticity of the spring itself. When the spring is bent or pulled out, work is done in overcoming the resistance due to its elasticity* and this work is stored up as potential energy in the spring itself. After being released it moves in the same way and goes through the same changes of energy as the pendulum. Like the latter it is gradually brought to rest by resistances due to the air, etc.

The elasticity of a body may be defined as being that property iti virtue of which it requires force to change its form or volume , and recovers its original form or volume when the force is removed. +

Solids in general offer resistance to any change either of form or volume, and are therefore said to possess elasticity of form as well as elasticity of volume. Fluids exhibit the latter only, for they do not offer resistance to change of form (see p. 27).

In common language we speak of substances like indiarubber as being very elastic, because they 4 give ’ or stretch readily. In scientific language the term is used in quite a different sense. Without entering into any details of the methods of measuring elasticity, we may here state generally that the elasticity of a substance is measured by the force required to produce a given change of form or volume. If the substance yields readily when force is applied, its elasticity is said to be small : but if it offers great resistance, it is said to be highly elastic. Thus the elasticity of steel and glass is great, for both these substances require the application of considerable force to produce even a small change of form. Again, the elasticity of water is very much greater than that of air : if the atmospheric pressure were to be doubled, the volume of any given quantity of air would (according to Boyle’s law) be reduced to one-half ; whereas such a change of pressure would produce scarcely any appreciable effect upon the volume *of a given quantity of water.

7. Graphical Representation of Vibration. — The

following experiment shows how the vibration of a tuning-fork may be studied and represented graphically.

CH. I

Expt. 3. — Smoke one side of a strip of glass by holding it over the flame of burning turpentine or camphor. Make a light pointed style of thin sheet-brass or wire : fix this to the end of one prong of a tuning-fork, as shown in Fig. 4. A bristle attached with wax may be used instead, but is apt to

Fig. 4.— Wave-Line traced by Fork.

bend or break off. Strike the fork, and immediately draw the style lightly but quickly over the smoked glass. After a little practice you will be able to produce a beautifully regular wavy line on the glass. This c wave-line ’ is a record of the motion of the prong in its own handwriting.

8. Musical Sounds and Noises. — By musical sounds are meant such as are produced by the human voice, the violin, the organ, and other instruments used for musical purposes. It is not possible to draw any very sharp distinction between a musical sound and a noise. The sounds produced by certain instruments used in military bands — eg drums and triangles — cannot strictly be called musical, and yet they do not destroy the harmony of the music, but rather strengthen and brighten it.

We may, however, broadly distinguish between the two classes of sounds as follows. A noise is produced by confused and non-periodic movements, — movements which are irregular both in respect of time and strength : whereas a musical sound is produced by regular and periodic vibrations. Our business lies entirely with the latter class of sounds, and we shall in general use the word sound as denoting a musical sound.

9. Pitch. — The long thin spring used in Expt. 1 gave out no sound while vibrating. A certain rate of vibration must be reached before any audible sound is produced. By gradually shortening the spring you can make it vibrate more rapidly, until at last it begins to produce a deep sound or a note of ‘ low pitch.’ By still further shortening the spring (or by using a shorter and thicker one) you can go on increasing the rate

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VIBRATORY MOTION of vibration : this will make the note still higher or 4 raise the pitch.’ Thus the pitch of a note depends upon the rate of vibration. Further proofs of this will be given later on.

10. Intensity of Sound. — The intensity or loudness of a sound depends upon the amplitude of vibration of the sounding body. You can verify this statement sufficiently well for present purposes by watching a vibrating string while it is coming to rest : as the amplitude of vibration diminishes so the sound gradually dies out. In the case of a tuning-fork it can be verified by a slight modification of Expt. 3.

But there are other circumstances that affect the intensity of the sound produced by an instrument. A vibrating string (or a tuning-fork) does not offer a large surface to the air : it cannot directly set any large mass of air into vibration, and consequently it produces only a feeble sound. But if it is made to impart its vibrations to an elastic body presenting a larger surface to the air (generally a thin wooden board called a ‘sounding-board5), the intensity of the sound is greatly increased. This principle is employed in the construction of most stringed instruments, such as violins and pianos. Tuning-forks, also, when used for experimental purposes, are generally mounted on sounding-boxes (Fig. 5), which greatly strengthen the sound.

Expt. 4. — Take hold of a tuning-fork by the stem and set it into vibration. You hear only a feeble sound as long as the stem is held in the hand : but the sound swells out much more loudly when it is lightly pressed against a table or door-panel ; or, better still, against its own sounding-box.

Chapter II

11. Water-Waves, f— You cannot begin the study of wave -motion better than t)y examining carefully the waves which travel over the surface of a pool of water when a stone is dropped into the middle of it. Where the stone drops there is at first produced a depression, which immediately begins to spread outwards from the centre of disturbance in the form of a circular trough. The disturbance which thus travels to the edge of the pool is caused, not by any bodily motion of the water outwards, but by a downward motion of the water-particles, which spreads outwards from point to point. This downward motion is followed by a swing upwards, producing a crest, which follows the trough and travels after it with the same velocity across the surface of the pool. By this succession of moving troughs and crests is produced a series of ripples or water-waves. The distance between one crest and the next, or between one trough and the next, is called a wave-length (Arts. 15 and 16).

That the water itself does not move in the direction in which the waves travel is shown by the behaviour of chips of wood or bits of straw lying on its surface. These simply rise and fall, floating idly on the surface of the water and showing no tendency (unless the disturbance be very violent) to move forward in the direction in which the waves are travelling.

12. Two kinds of Waves. — Observe that in the case of water-waves the motion of the particles is an up-and-down motion ; whereas the waves themselves moY§ horizontally ii-i4 across the surface of the water. Such waves are called transverse waves , because they are produced by motion which is transverse or at right angles to the direction in which the waves are propagated. Strictly speaking, the motion of the water particles is circular, but all that we wish to do here is to point out the broad distinction between the two classes of waves, which are called respectively transverse and longitudmal waves . ^

13. Transverse Waves are such as are produced by a vibratory motion of particles executed in a direction at right angles to that in which the waves are propagated. As an instance of the production of a transverse wave may be mentioned the sudden jerk which a bargeman sends along a rope when he wishes it to clear an obstacle in its path.

Get a rope a few yards in length and lay it straight along the floor. Take hold of one end and, by jerking it rapidly from side to side, send a series of right- and left-handed pulses along it.

If an instantaneous photograph of the wave were taken it would present an appearance like that of the wave-line in Expt. 3. The waves appear to travel along the rope in the direction of its length : but the appearance is simply due to a vibration of each part of the rope which is executed transversely (or at right angles to its length) and which is communicated from each part of the rope to the next.

14. Longitudinal Waves are such as are produced by a vibratory motion of particles executed along the line in which the waves are propagated.

If you look down from a hillside upon a cornfield when a light summer breeze is blowing over it you will see a good example of wave-motion. The only motion of which the ears of corn are capable is a slight swinging motion ; but as the breeze sweeps along the motion is transmitted from one ear to the next and from this again to its neighbour. Thus a certain state of things (ears of corn tightly packed together) is transmitted from point to point, and every gust of wind produces a wave of condensation which skims across the surface of the field. The waves are not wholly longitudinal. There is some little transverse motion : for each cornstalk swings like an inverted

B

CH. II pendulum. But in the main the motion of the ears takes place in the line along which the wave travels. Observe that the characteristic of wave-motion is the transmission of a certain state of things or state of motion without any corresponding transmission of matter.

The best illustration of longitudinal waves is afforded by the behaviour of a spiral coil of wire.

Expt. 5. — Make a spiral of thick copper wire by winding it tightly round a curtain-pole or a thick glass tube. When drawn out the spiral should be about two yards long.1 * * * * * 7 Hang it up by a series of double threads, as shown in Fig. 6, the

Fig 6. — Wave-Machine (Weinhold).

double suspension being for the purpose of preventing any side swing.

By means of such a ‘ wave-machine * you can study the mode of propagation of waves of condensation as well as waves of rarefaction. A wave of rarefaction is produced by taking hold of one end of the spiral, pulling it out in the direction of the axis with a smart jerk, and then letting it go.

1 A common defect of spiral coils used for illustrating wave-motion is that they

are made too small and slight : when this is the case it is difficult to follow a wave

as it quickly passes along the coil. It is really worth while taking trouble to

make a good spiral, for it can be used afterwards to illustrate reflection, interference,

and stationary vibration. I find that the following dimensions (recommended by

Weinhold) are very suitable, viz. — Diameter of wire, 2 mm. ; diameter of spiral.

7 cm. ; number of turns, 72 ; length of completed spiral, 2 metres } length of threads, 60 CIJ1.

A wave of condensation is produced by striking one of the free ends of the spiral inwards. This is rather apt to make the spiral rock : a better plan is to take hold of the outer turn of wire by the end and push it inward or pull it outward, as the case may be.

Try both methods. Watch the waves as they run quickly along the spiral, and examine the way in which the separate coils move. You will thus learn more about wave-motion than you could by reading many pages of description. If you find it difficult to follow the motion of each turn of wire, gum strips of paper or tie bits of twine to a few of them : you will then easily see that each turn simply moves forward and backward as the wave passes it. Notice that when a pulse of condensation is running along the spiral the coils move forward in the direction in which the pulse is travelling : whereas in the case of a pulse of rarefaction the coils move backward , or in a direction opposite to that in which the pulse is travelling.

Place the blade of a knife between the coils near one end of the spiral and rake it quickly across a few turns towards the

ABC

Fig. 7.

other end. You will thus produce a double pulse or complete wave : for each turn of wire as it escapes from the knife swings backward (or outward), thus producing a pulse of rarefaction (C, Fig. 7) which follows the pulse of condensation (B).

Fasten one end of the spiral to an empty box (a cigar-box). Strike the free end. The pulse runs along the spiral and you hear it deal a smart rap against the box. The energy of your blow is transmitted along the spiral to the box ; the wave carries the energy without carrying the matter to which the energy was first imparted.

15. Wave-line and Wave-length : Phase. — The curved line in Fig. 8 represents a portion of a wave-line (or of a rope along which transverse waves are passing). The dotted line is the axis of the waves, and the arrow shows the direction in which they are travelling. The short

CH. II equidistant vertical lines represent the paths along which the particles a, b, c . . . vibrate.

The whole wave-line (of which only a portion is shown in the figure) consists of a repetition of exactly similar parts, and the shortest of such parts into which it can be cut is called a wave-length. Thus, starting from a , the distance am is a wave-length, for the part of the curve following m is an exact copy of that between a and m. Similarly bn, co, dp . . . are wave-lengths.

The various particles af b, c . . . are moving with different velocities, some up and some down, a, b, c are moving downwards ; d is at rest for an instant ; from e to i all are rising : then j is at rest, and so on. Any two particles which are moving in the same direction and with the same velocity are said to be in the same phase . Thus the pairs a and m, b and n, c and o, d and p are in the same phase. We may therefore define the wave-length as being the distance between the tzvo nearest particles in the same phase . Notice particularly that ag is not a wave-length : it is only half a wave-length, a and g are not in the same phase : for although they have the same velocity they are moving in opposite directions. Nor are d and j in the same phase : for though both are at rest d is just starting downwards and j upwards.

16. Wave-length and Velocity. — The waves in Fig. 8 are supposed to be travelling towards the right. The curve indicates the positions of the particles at a particular instant. Their positions at any future instant can be found by shifting the wave-line forward through a certain distance depending on the interval of time.

Consider the particle m. When m has completed a whole vibration — down, up to the top, and back again to its mean position, — the disturbance will have moved forward through the distance am, i.e. through a whole wave-length . Thus we get a third definition of wave-length : it is the distance through which a wave travels in the time required for a complete vibration.

We know that

T

when n denotes the vibration-number and r the period of vibration (Art. 3). Let \ denote the wave-length and v the velocity with which the waves we have just seen that they move through the distance X

, . distance

velocity = — ,

time

T T

This last result is of the greatest importance. The relation between velocity, vibration-number, and wave-length which it expresses holds good not only for transverse waves but for all kinds of waves.

17. Comparison between Longitudinal and Transverse Waves. — The bottom line (R) in the accompanying figure represents a series of particles at rest. The line above (L) represents the positions of the particles when traversed by a series of longitudinal waves. The top line (T) represents the positions of the particles at the same instant when traversed by corresponding transverse waves. An upward displacement in the transverse wave corresponds to a displacement to the right in the longitudinal wave : a downward displacement in the transverse wave corresponds r ^

A !

~ Compression ~ liaref

1 1

1 :

action. Comp 7

essioiv ;

Fig. 9. — Transverse and Longitudinal Waves.

to a displacement to the left in the longitudinal wave. The position of any particle in L is found by displacing the corresponding particle in R through a distance proportional to the displacement in T. The two sets of waves correspond in the following respects : —

A crest in T corresponds to portions of L in which particles are displaced to the right. A trough corresponds to displacements to the left.

The points of no displacement (N) occur alternately on falling and rising parts of the curve T : these correspond to alternate centres of compression and rarefaction in L.

In T the distance NN represents half a wave-length : so in L the distance between a pulse of compression and a pulse of rarefaction is half a wave-length. A complete wave-length in L is represented by the distance between two successive condensations or two successive rarefactions.

travel forward : in the time r.

Thus since

it follows that or

Chapter III

18. Transmission of Sound through Air. — We have seen that sound is caused by a vibratory motion of the sounding body. We have next to consider how this motion travels through the air and reaches our ears.

It is important to understand that when sound is transmitted through the air, the air itself does not move as a whole, but a certain state of motion is transmitted through the air from point to point and from particle to particle. Thus the report caused by the explosion of a gun can be heard for miles around ; but the smoke expelled from the muzzle of the gun only travels forward for a few yards.

Let us return to our spring (Fig. i) and see liow its vibrations affect the air in its neighbourhood. As the spring swings from left to right (/ to /') it compresses the air in front of it. The air, being elastic, resists the compression and tends to expand. In so doing it compresses the air lying in front of it. This again reacts upon the next layer of air, and so the motion of the spring causes a pulse of compression , which travels forward with uniform speed through the air.

Meanwhile the spring has swung backwards towards the left hand (from /' to /). The particles of air in contact with it share its motion, and thus the layer of air lying towards the right of the spring expands or is rarefied. The pulse of rarefaction thus produced travels forward through the air with the same uniform speed as the pulse of compression.

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As the spring begins its next vibration another pulse of compression is produced ; and this again is followed by a pulse of rarefaction. Thus a series of alternate pulses of compression and rarefaction are produced, and follow each other in regular succession through the air (as indicated in Fig. 10). The student must refer to text-books on Physiology for an explanation of the manner in which these sound-waves affect the ear when they fall upon it and produce the sensation of sound.

The wave-length of a given note in air may be defined in any of the ways given in Arts. 1 6 and 1 7 : e.g. it is the

Fig. 10. — Sound-Waves in Air.

distance between two consecutive pulses of compression or two consecutive pulses of rarefaction ; or it is twice the distance between a pulse of compression and the next pulse of rarefaction.

19. Transmission through a Cylindrical Tube. —

Sound-waves are propagated through air contained in a cylindrical tube much like the waves in Expt. 5. The waves are longitudinal : the air-particles vibrate backward and forward in the line along which the waves advance.

Imagine the spiral replaced by a cylindrical column of air, along which pulses are sent by a vibrating tuning-fork or spring at one end. Each forward swing of the fork sends a pulse of condensation along the column : the ensuing backward swing sends a pulse of rarefaction after it at the same speed. A regular succession of such pulses constitutes a train of sound-