could be increased if it is desired to obtain a greater number of points in the curve. Other equidistant points are taken on a straight line

OA passing through the centre of the circle. Drawing perpendiculars to OA through each point on that line, and lines parallel to OA through the corresponding points of the circle, the intersections of the two sets of lines mark the points on the curve. The wave-length is the distance between the two nearest points which have the same phase. If λ be the wave-length, so that the phase is the same for x and x+λ, it follows that pλ must be equal to 2π, or p = 2π/λ.

The difference of phase between two particles at distances 1 and ≈2 from the origin, as obtained from this equation, is

In the further consideration of wave motion, we shall consider principally waves the displacement of which can be represented by the equation (3).

10. Application of Fourier's Theorem. By an important theorem due to Fourier, any function f(x) may between fixed limits x=-c and x = + c be represented as the sum of a series, in such a way that writing a = x/c

f(x) = α + α1 cos a + α2 COS 2a + α3 cos 3a +
+ b1 sin a + b2 sin 2a + b2 sin 3a +

......

·(4).

The constants a。, ɑ1, b1, b2, etc. may be determined from the function ƒ, and we may for our present purpose fix for ƒ (x) outside the specified limits the values calculated from the series on the right-hand side. If waves of all lengths are propagated with the same velocity v, we may obtain the shape at any subsequent time for waves travelling in the positive direction by writing in all terms on the right-hand side x – vt for x, and having done so we may add the series again, when it is seen that the sum now becomes f(x - vt). Hence the condition that normal waves of all lengths travel with the same velocity carries with it the consequence that waves of any shape may be propagated

without change of type. On the other hand, if as in the case of lightwaves travelling through a dispersive medium, the velocity of propagation depends on the wave-length, there must always be a change of type when waves which are not of the simple cosine or sine shape are propagated.

11. Waves travelling along a stretched string. Let us now consider the kinetics of wave propagation.

T

Consider a small portion AB of a curved string and

B

acted on by equal tangential forces at the ends. The resultant force passes by symmetry through C the centre of the circle of curvature of AB and bisects AB. If 20 be the angle subtended by the portion AB of the string at C, the intensity of the resultant is 27'sin, which is nearly equal to 270 if be sufficiently small. As 2r0 is the length of the arc AB, where r is the radius of curvature, the "resultant force per unit length" is Tr, i.e. equal to the product of the tension and the curvature.

Fig. 12.

Let now a string be only slightly curved, so that every part of it is near a straight line which shall be the axis of x. Its inclination to that axis dy/da may be supposed to be sufficiently small to allow its square to be neglected. The force acting on an element of length ds has been proved to be Tds/r, and neglecting the square of dy/dx, we may take the same expression to represent that component of the force which lies in the y direction.

If p be the density per unit length, and hence pds the mass of the length ds, the equation of motion is:

Comparing this equation with (2) it is seen that T/p is the velocity of the waves which are propagated along the string.

B

A' A

Fig. 13.

Let Fig. 13 represent a portion of a string stretched to a constant tension by a weight P. Let it be displaced by outside forces until it occupies a position such as that shown in the figure. If the constraint

is suddenly removed, the tension of the string will, by what precedes,

act in such a way that there is at each point a resultant force towards the centre of curvature. Hence the point B will begin to move downwards while A and C move upwards. If AH has been previously straight, this portion of the string is in equilibrium, but as soon as A is lifted up, the point at which the straight and curved portions join, has been moved to the left. If A' is that point, AA' which was previously in equilibrium, has ceased to be so. It follows that a disturbance will set out from A and travel from right to left, with a velocity which has already been found to be √T/p. A similar reasoning shows that the displaced region AC will also send out a disturbance from C towards K. Two waves travelling in opposite directions will therefore start from ABC.

Now we know from observation that it is possible for a disturbance to travel in one direction only, and it is a matter of interest to examine the conditions under which a displacement such as ABC may be propagated forward only or backward only. In order that it shall travel only forward, it is clearly necessary that the point A should remain in its position in spite of the force acting upwards, and this is only possible if at the time to which the figure applies, A has a velocity downwards, of such magnitude that the force acting at A just destroys the velocity. The force is of the nature of an "impulse" because if there is a discontinuity of slope at A, the curvature is infinite, and hence the force is infinite, and capable of suddenly destroying a finite velocity. Similarly all along ABC a certain relation between velocity and displacement must hold, and this relation must be of such a nature that each portion will have zero velocity as soon as the wave has passed over it. The mathematical relation which must connect the displacement and the velocity at each point when waves are propagated in one direction only, is obtained from (1) substituting the value of v:

where the upper sign holds for waves propagated in the positive direction.

I have discussed this question at length, because it shows clearly the important fact that if waves are sent out from any disturbed region, the displacements in that region are not by themselves sufficient to determine the subsequent motion, the velocities being just as important as the displacements. In the above case, with the same displacements, the velocities might be chosen so as to give a wave wholly moving forward in one direction, or wholly moving back in the opposite direction, while generally there are two portions of the wave, one moving towards the positive, and one towards the negative side.

We confine

12. Transverse Waves in an Elastic Medium. our attention for the present to bodies, the elastic properties of which are independent of direction. Such bodies are said to be "isotropic."*

Consider a medium in which the displacements are the same in magnitude and direction for all points lying in the same plane drawn normally to a given line. In Fig. 14 OX represents this line, and A1B1, A,B2, A3B... are the intersections of a number of planes perpendicular to OX with the

B

15

B3

B2

ย

B4 B5

C1 C2 C3

C4

c's

Cs

A A2 A3 A4 LAS

Fig. 14.

surface of the paper. At each point of these planes the displacements are supposed to be identical, but they may differ in different planes. If the displacements are all normal to OX and in the plane of the paper, each plane may be imagined to slide along itself through distances equal respectively to CC, CC2 etc. We confine the investigation to the case of elastic forces which are such that for the linear displacements contemplated, the restitu

tional force acts backwards in the direction of the displacement. The strain set up in the medium by the displacement is one involving change of shape only, and not any change of volume.

P

If PM and M'P' are the positions in the strained condition of two lines originally parallel to OX; the parallelogram PQNM was originally a rectangle, and the elementary theory connecting strains and stresses shows that the plane A,B, to be maintained in its displaced position must be acted on by an upward force which per unit surface is equal to n tan a, where n is the resistance to distortion and a the angle between PM and OX. Similarly the plane A, B, to keep its position must be acted on by a downward force which per

LA

Fig. 15.

2

* Thomson and Tait, Vol. 1., Art. 676, give the following definition of isotropy:

"The substance of a homogeneous solid is called isotropic when a spherical portion of it, tested by any physical agency, exhibits no difference in quality however it is turned. Or, which amounts to the same, a cubical portion cut from any position in an isotropic body exhibits the same qualities relatively to each pair of parallel faces. Or two equal and similar portions cut from any positions in the body, not subject to the condition of parallelism, are undistinguishable from one another. A substance which is not isotropic, but exhibits differences of quality in different directions, is called eolotropic."

unit surface is n tan a', where a' is the angle between P'M' and OX. The resultant force acting on a small rectangular volume of unit height, thickness NN' and length MN is

If the displacements are denoted by y, we have

so that the resultant force may now be written MN ×t × n

MNxt is the volume considered, and if p is the density, n

day

dx2

ρ

will

denote the resultant force divided by the mass. We have considered the force necessary to maintain the medium in its strained condition, but if that force is removed, the acceleration may be obtained by the third law of motion:

This equation is of the form (2) and shows that the medium is capable of transmitting waves in a direction OX with a velocity √n/p. As the velocity is independent of the wave-length, waves of any shape are propagated without change of type.

If we imagine a second disturbance superposed on the one which has been discussed, and at right angles to it, we arrive at a wave propagation in which each particle describes a plane curve. We may for convenience limit the discussion to waves of the normal type, in which the displacements are therefore represented by

Superposing a similar wave, the displacements being in the z direction, z = b cos (wt − px + 8).

The paths of the particles in each plane are seen to be similar and elliptic, circular or rectilinear, according to the value of 8 and the relations holding between a and b, Art. 6.

One important observation remains to be made. Imagine the medium to consist of a number of detached particles, not acting on each other, but attracted to their position of equilibrium by a force varying as the distance.

Let the particles at the time t = 0 be put into such a position and have such velocities that their displacements may be represented by

y = a cos px