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As regards ty, the potential of the waves diverging in two dimensions, we must use different forms when r is small (compared with X) and when r is large*. When r is small
and when r is large,
By the same argument as for a point resonator we find, as the condition that no work is done at r=0, that the imaginary part- of I/a is — iV/2. For maximum resonance
so that at a distance ^ approximates to
which expresses the width of primary wave-front carrying the same energy as is dispersed by the linear resonator tuned to maximum resonance.
A subject which naturally presents itself for treatment is the effect of a distribution of point resonators over the whole plane of the primary wave-front. Such a distribution may be either regular or haphazard. A regular distribution, e.g. in square order, has the advantage that all the resonators are similarly situated. The whole energy dispersed is then expressed by (29), though the interpretation presents difficulties in general. But even this would not cover all that it is desirable to know. Unless the side of the square (6) is smaller than A,, the waves directly reflected back are accompanied by lateral "spectra" whose directions may be very various. When b < A,, it seems that these are got rid of. For then not only the infinite lines forming sides of the squares which may be drawn through the points, but a fortiori lines drawn obliquely, such as those forming the diagonals, are too close to give spectra. The whole of the effect is then represented by the specular reflexion.
In some respects a haphazard distribution forms a more practical problem, especially in connexion with resonating vapours. But a precise calculation of the averages then involved is probably not easy.
Theory of Sound, § 341.v. p. 60 (1907) ;  Scientific Papers, Vol. v. p. 409.e first we consider the value of ty and its modulus at a great distance r from the resonators. It is evident that \/r is symmetrical with respect to the line R joining the resonators, and if 6 be the angle between r and R, r^ — rz = R cos 0. Thus