288 SOME PROBLEMS CONCERNING THE MUTUAL INFLUENCE OF [390
If we suppose that the scale (t) of the regular structure is very small compared with A,, we can proceed further in the calculation of the regularly reflected wave. Let Q be one of the resonators, 0 the point in the plane of the resonators opposite to P, at which -\|r is required ; OP ~sc, OQ = y, PQ = r. Then if m be the number of resonators per unit area,
e-ikr
y dy-----,
or since y dy = r dr,
•\/r = 2-Trma. / e~lkr dr. J x
The integral, as written, is not convergent; but as in the theory of diffraction we may omit the integral at the upper limit, if we exclude the case of a nearly circular boundary. Thus
O --rf n 11 /v
-***, ...........................(44)
T., ,„ , , //1r.N
and Moda'f = — y2 — ............................... (45)
The value of A2 is given by (19). We find, with the same limitation as above,
v cos kR f°° T -r, , D A
2, — TT — = ZTTTO cos IsR dR = 0, R h
2 — 73 — = 2-Trm I sin kRdR = 2vrm/&. R Jo
Thus A2 = l/(/e+27rm/&)2
and Mod»^ = 77— ......................... (46)
r (&2 + 27TW)2 '
When the structure is very fine compared with ~K, Ic1 in the denominator may be omitted, and then Mod2-^ = 1, that is the regular reflexion becomes total.
The above calculation is applicable in strictness only to resonators arranged in regular order and very closely distributed. It seems not unlikely that a similar result, viz. a nearly total specular reflexion, would ensue even when there are only a few resonators to the square wave-length, and these are in motion, after the manner of gaseous molecules ; but this requires further examination'.
In the foregoing investigation we have been dealing solely with forced vibrations, executed in synchronism with primary waves incident upon the resonators, and it has not been necessary to enter into details respecting the constitution of the resonators. All that is required is a suitable adjustment to one another of the virtual mass and spring. But it is also of interest to be determined. The wh potential is <£ + ^, and it suffices to consider the state of things at the fi resonator. With sufficient approximation