The air wave surrounding an expanding sphere.

When the surface of a sphere vibrates in any assigned manner the spherical sound waves which are propagated outwards can be represented by wellknown formulae provided that the motion is such that only small changes in air density occur. When the motion of the spherical surface is radial the velocity potential of the sound wave is <}> = r~1f{r-at),(1) where a is the velocity of sound and r is the radial co-ordinate. The velocity, u, and the excess, p —p0, of pressure over the atmospheric pressure p 0 are u = r~2f{r — at) — r~xf'{r — at), (2) P~Po = -p a r -xf ‘{r-at).(3) If R is the radius of the sphere which, by its expansion, is producing waves, .R is a function of t and the surface condition is R = R~2f(R — at) — R~xf {R — (4) Equation (4) is an equation for finding the function/. A simple case in which equation (4) can be solved is when R is constant so that the sphere is ex­ panding at a uniform velocity. Taking 0 when = 0 the radius at time t can be exx ressed in the form R = oat, (5) where a is a non-dimensional constant. The limitation that the changes in density are small implies that equations (l)-(3) are true only where is small compared with 1. Writing w — R — at — (a — 1 )atequation (5) becomes /y 1 I n r 1\^ ----/'(w) (------) f{w) + 0. (6) aw \ aw The solution of equation (6) which is valid for negative values of w is