1912] THROUGH A STRATIFIED MEDIUM, ETC. 85
The supposition that 77 varies slowly excludes more than a very small reflection.
Equations (79), (80) may be tested on the particular case already referred to where k = njx. We get
so that \adas = {n -^-} log on.
J \ onj
When n~4 is neglected in comparison with unity, n ^n~l maybe identified with V(2 - i)- '
Let us now consider what are the possibilities of avoiding reflection when the transition layer (a?2 #j) between two uniform media is reduced. If ^D k\ ': f}z> &a are the terminal values, (79) requires that *
ic* = 0*77!-*, V = C4%~4.
We will suppose that tjz>f)i- If the transition from 773. to 773 be made too quickly, viz., in too short a space, dPy/da? will become somewhere so large as to render k- negative. The same consideration shows that at the beginning of the layer of transition (oc^, d^jdx must vanish. The quickest admissible rise of 77 will ensue when the curve of rise is such as to make Tc* vanish. When 77 attains the second prescribed value 772, it must suddenly become constant, notwithstanding that this makes 7c2 positively infinite.
From (72) it appears that the curve of rise thus defined satisfies
= ^-°. ................................. <82>
The solution of (82), subject to the conditions that 77 = 77^ when oc CG-i, is
7?ğ-%ğ=C^1-a(ar-a?1)!l = A?1V(a'-ğl)a ................ (83)
Again, when 77 == 773, x ~ ocz, so that
jyfc-^.S^'., ..................... (84)
giving the minimum thickness of the layer of transition.
It will be observed that the minimum thickness of the layer of transition necessary to avoid reflection diminishes without limit with k: Jc2, that is, as the difference between the two media diminishes. However, the arrangement under discussion is very artificial. In the case of the string, for example, it is supposed that the density drops suddenly from the first uniform value to zero, at which it remains constant for a time. At the end of this it becomes momentarily infinite, before assuming the second uniform value. The infinite longitudinal density at xz is equivalent to a finite loadnegative, but this is the only restriction. When ?? is constant, 7c2 is constant ;