Page 16 - Spring 2009

Fig. 1. The Lloyd Mirror geometry.
absence of refraction can be determined directly from the geometry
(2)
When one assumes simple harmonic sources, exp (-iωt), with outward propagating spherical waves, exp (-ikr), the resultant pressure at the receiver R, ρ(rh, zr, t), with a surface reflection coefficient μ is
(3)
(3)
One can define three regions: a nearfield, an interference field and a farfield. The nearfield is when the direct arrival from the source is dominant. The nearfield region is defined, following Urick (1967), as distances less than the range, rhnf, at which the intensity of the image source is 1⁄2 that of the intensity of the direct source arrival. Expanding the source and image radial distances, in Eq. 3, with a Taylor series and neglecting the second order terms, z2s,i /r2, yields the expres- sion for intensity at R:
(4)
When the surface reflection coefficient, μ, is equal to (-1) the resultant intensity is proportional to two times the source intensity times the bracketed cosine term.
(5)
The argument of the cosine determines the maxima and minima in the intensity as a function of r. Maxima occur when the cosine is equal to -1 for
. (6)
The interference peaks at these ranges are four times the free field intensity, Ios.
The above specify the interference field and the farfield expression can be obtained by use of trigonometric relationships
First at a constant distance, r, the quantity
(7)
(8)
Fig. 2. The change in the vertical directionality of the monopole source beneath the pressure release surface, (a) monopole source λ/4 below the pressure release surface, (b) monopole source λ/2 below the pressure release surface, (c) monopole source 3λ/4 below the pressure release surface.
Lloyd’s Mirror 15