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 x 1 , and of another uniform quality (in general, different from the first) where x exceeds a greater value x m -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 = x 1 , the second at x = x 2 , and the last at x = x m -1. The expressions for the waves in the various media in order may be taken to be ϕ 1 = A 1 e i [ ct + by - a 1 ( x - x 1 )] + B 1 e i [ ct + by + a 1 ( x - x 1 )] , ϕ 2 = A 2 e i [ ct + by - a 2 ( x - x 1 )] + B 2 e i [ ct + by + a 2 ( x - x 1 )] , ϕ 3 = A 3 e i [ ct + by - a 3 ( x - x 2 )] + B 3 e i [ ct + by + a 3 ( x - x 2 )] , 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 B 1 /A 1 imposed by the conditions of the problem, including the requirement that in the final medium there shall be no negative wave.