To solve this equation, we shall refer to the process of Article 43. There we must make m=3, which gives

3

2

; and, supposing the wave to diverge, or B to

be a function of v or at-r, the equation becomes

where each succeeding coefficient is deduced by an integral of the preceding coefficient. Then R has the form

dY

dF

Then the velocity in y, or or (Article 37), or

yz dR

dt dy'

yed (above), can be found; observing that

r'dr

47. Interpretation of the expression for radial displacement in this problem; the particles, originally in a circle whose center is the center of divergence, will always be found in a circle of the same diameter whose center oscillates.

Suppose that we measure from the center of the wave (or origin of co-ordinates), in the direction z, a

distance Q, a function of t and r, which will therefore be the same at any given time for all particles in the circle whose original radius was r, but will vary with the time; and let the distance of that point from the quiescent place of a particle under consideration be called r'. Then r'=y" + (≈ — Q)'=y'+z"-2zQ nearly =-2zQ, whence

The displacement of the particle in the direction of r is

Disturbed value of r. = Undisturbed value of r.

That is to say, all the particles, which in the quiescent state were originally in a circle whose center was at the origin, will at the time t be found in a circle of the

same diameter whose center has moved through the

space

[R+r]

dR

dr

in the direction of z. Hence the wave will be of the character of "divergent wave with oscillation of the center of divergence in the direction of z, the amount of oscillation being different for waves of different diameters."

We leave to the student the investigation of the motion of each particle in the circumference of the circle to which it belongs.

From this it will easily be understood that, if motion be begun in the form of a circle with oscillating center, it will be propagated in the form of a circle with oscillating center; because the general formula expressing disturbance must be such as will represent the special disturbance at the place of beginning of the disturbance, and only the formula that we have found will represent that special disturbance.

48. Application of this theory to the vibration of air produced by the vibration of musical strings.

We shall hereafter see that the vibrations of a musical string may always be represented by the com

bination of vibrations of equal periods, one in one plane passing through the string, and the other in another plane passing through the string, at right angles to the former; and that in the simplest case they will be a vibration in one plane. Confining our attention to the simplest case, and supposing x to be in the direction of the string's length, and z in the direction of vibration, and taking a plane yz at right angles to the wire, it is seen that, for determining the vibrations of air in that plane, we have precisely the case contemplated in the last sentence of last Article, namely, a motion begun in the form of a circle with oscillating center: and the theorems of Articles 46 and 47 apply to it. The principal practical results are: that in the plane xy, or in the plane normal to the plane of vibration, the vibrations of air to and from the wire, on which the audible sound mainly depends, and which here have for factor, are small; and that, in all directions, the magnitude of vibrations depends principally on R, whose most important term has for factor r, which diminishes rapidly as the distance increases.

49. Application of the theory to the vibrations of air produced by the vibrations of a tuning-fork.

The tuning-fork is a small instrument in the form represented in Figure 10, constructed of highly elastic metal. In use, one of its branches is struck, and the