212 FURTHER APPLICATIONS OF BESSEL'S FUNCTIONS OF [380
escape of energy is connected with the curvature of the surface. If V be the velocity of propagation, and 2ir/k the wave-length of plane waves of the given period, the time-factor is eikvt, and the equation for the velocity-potential in two dimensions is
If cjf> be also proportional to cos my, (1) reduces to
of which the solution changes its form when m passes through the value k. For our purpose m is to be supposed greater than k, viz. the wave-length of plane waves is to be greater than the linear period along y. That solution of (1) on the positive side which does not become infinite with x is proportional to Q-x^W-W, so that we may take
cj> = cos Ic Vt . cos my . e~x Vf"**-*8) ...................... (3)
However the vibration may be generated at x — 0, provided only that the linear period along y be that assigned, it is limited to relatively small values of SB and, since no energy can escape, no work is done on the whole at aa = Q. And this is true by however little m may exceed k.
The reason of the difference which ensues when the vibrating surface is curved is now easily seen. Suppose, for example, that in two dimensions <£ is proportional to cos n6, where 0 is a vectorial angle. Near the surface of a cylindrical vibrator the conditions may be such that (3) is approximately applicable, and <f> rapidly diminishes as we go outwards. But when we reach a radius vector r which is sensibly different from the initial one, the conditions may change. In effect the linear dimension of the vibrating compartment increases proportionally to r, and ultimately the equation (2) changes its form and </> oscillates, instead of continuing an exponential decrease. Some energy always escapes, but the amount must be very small if there is a sufficient margin to begin with between m and k.
It may be well before proceeding further to follow a little more closely what happens when there is a transition at a plane surface sc = Q from a mpre to a less refractive medium. The problem is that of total reflexion when the incidence is grazing, in which case the usual formulae* become nugatory. It will be convenient to fix ideas upon the case of sonorous waves, but the results are of wider application. The general differential equation is of the form
See for example Theory of Sound, Vol. n. § 270.d therefore p cannot. In this case clearly