By this expression, in conjunction with the quarter-period acceleration of phase, the law of the secondary wave is determined.
That the amplitude of the secondary wave should vary as r-1 was to be expected from considerations respecting energy; but the occurrence of the factor λ-1, and the acceleration of phase, have sometimes been regarded as mysterious. It may be well therefore to remember that precisely these laws apply to a secondary wave of sound, which can be investigated upon the strictest mechanical principles.
The recomposition of the secondary waves may also be treated analytically. If the primary wave at O be cos kat, the effect of the secondary wave proceeding from the element dS at Q is
| dS | cos k(at − ρ + ¼λ) = − | dS | sin k(at − ρ). |
| λρ | λρ |
If dS = 2πxdx, we have for the whole effect
| − | 2π | ∫ | ∞ | sin k(at − ρ)x dx | , |
| λ | 0 | ρ |
or, since xdx = ρdρ, k = 2π/λ,
| −k ∫ | ∞ | sin k(at − ρ)dρ = [ −cos k(at − ρ) ] | ∞ | . | |
| r | r |
In order to obtain the effect of the primary wave, as retarded by traversing the distance r, viz. cos k(at − r), it is necessary to suppose that the integrated term vanishes at the upper limit. And it is important to notice that without some further understanding the integral is really ambiguous. According to the assumed law of the secondary wave, the result must actually depend upon the precise radius of the outer boundary of the region of integration, supposed to be exactly circular. This case is, however, at most very special and exceptional. We may usually suppose that a large number of the outer rings are incomplete, so that the integrated term at the upper limit may properly be taken to vanish. If a formal proof be desired, it may be obtained by introducing into the integral a factor such as e-hρ, in which h is ultimately made to diminish without limit.
When the primary wave is plane, the area of the first Fresnel zone is πλr, and, since the secondary waves vary as r-1, the intensity is independent of r, as of course it should be. If, however, the primary wave be spherical, and of radius a at the wave-front of resolution, then we know that at a distance r further on the amplitude of the primary wave will be diminished in the ratio a : (r + a). This may be regarded as a consequence of the altered area of the first Fresnel zone. For, if x be its radius, we have