1912]
THROUGH A STRATIFIED MEDIUM, ETC.
81
positive when the transmission is perpendicular, as, for example, in the case of a stretched string. When the transmission is oblique to the strata, k* may become negative, corresponding to " total reflection," but in most of what follows we shall assume that this does not happen. The continuity of <p and d<$>ldos, even though k" be discontinuous, appears to limit the application of (53) to certain kinds of waves, although, as a matter of analysis, the general differential equation of the second order may always be reduced to this form*.
In the theory of a uniform medium, we may consider stationary waves or progressive waves. The former may be either
<£ = A cos lcQx cospt, or <£ = B sin k0x sinpt ;
and, if B= ± A, the two may be combined, so as to constitute progressive waves
</> = A cos ( pt ± &„#).
Conversely, progressive waves, travelling in opposite directions, may be combined so as to constitute stationary waves. When we pass to variable media, no ambiguity arises respecting stationary waves ; they are such that the phase is the same at all points. But is there such a thing as a progressive wave ? In the full sense of the phrase there is not. In general, if we contemplate the wave forms at two different times, the difference between them cannot be represented by a mere shift of position proportional to the interval of time which has elapsed.
The solution of (53) may be taken to be
where ty(x), %(#) are real oscillatory functions of x; A', B', arbitrary constants as regards so. If we introduce the time-factor, writing p in place of the less familiar c of (1), we may take
6 = A cospt. ty (x) + B sin pt . % (x) ; ................ (55)
arid this may be put into the form
&=Hcx)z(pt-e\ ........................... (56)
where /I cos Q = Aty(cc\ .ZI sin 0 = #%(#), ................ (57)
or H* = A*fr (x}J + B*[x(x)J, ..................... (58)
(59)
But the expression for <£ in (56) cannot be said to represent in general a progressive wave. We may illustrate this even from the case of the uniform medium where ^ (x) = cos lex, % (as) = sin kx. In this case (56) becomes
6 = [A* cos2 kx + B2 sin2 kafi cos \pt - tan"1 f-j tan kxj I. .. .(60)
I \^X / )
* Forsyth's Differential Equations, § 59.
R. VI.iable longitudinal density vibrating transversely.ample, suppose that there are two intermediate layers of equal thickness, of which the first is similar to the final uniform medium, and the second similar to the initial uniform medium. •Of the three partial reflections the first and third are similar, but the secondo guide the lantern-plates into position, and thus to ensure their subsequent exact superposition by simple mechanical means.