On the propagation of waves through a stratified medium, with special reference to the question of reflection

Abstract

The medium is supposed to be such that its properties are everywhere a function of but one co-ordinate x, being of one uniform quality where x is less than a certain value x1, and of another uniform quality (in general, different from the first) where x exceeds a greater value xm-1; and the principal problem is the investigation of the reflection which in general ensues when plane waves in the first medium are incident upon the strati-fications. For the present we suppose the quality to be uniform through strata of finite thickness, the first transition occurring when x = x1, the second at x = x2, and the last at x = xm-1. The expressions for the waves in the various media in order may be taken to be ϕ1 = A1ei[ct + by - a1 (x - x1)] + B1ei[ct + by + a1 (x - x1)], ϕ2 = A2ei[ct + by - a2 (x - x1)] + B2ei[ct + by + a2 (x - x1)], ϕ3 = A3ei[ct + by - a3 (x - x2)] + B3ei[ct + by + a3 (x - x2)], and so on, the A's and B's denoting arbitrary constants. The first terms represent the waves travelling in the positive direction, the second those travelling in the negative direction; and our principal aim is the determina­tion of the ratio B1/A1 imposed by the conditions of the problem, including the requirement that in the final medium there shall be no negative wave.