86 ON THE PROPAGATION OF WAVES [!
there attached. In the layer of transition, if so it may be called, the str remains straight during1 the passage of the waves.
If, as in the more ordinary use of the term, we require the transitioi be such that T€- moves always in one direction from the first terminal va to the second, the problem is one already considered. The minim thickness is such that Jo2 has throughout it a constant intermediate va so chosen as to make the reflections equal at the two faces.
It would be of interest to consider a particular case in which /c2 va continuously and always in the one direction. As appears at once from (' d-r)jdx-, as well as d'n/dx, must vanish at both ends of the layer, and tt must also be a third point of inflection between. If the layer be from x to x = /3, we may take
-y— = Ace (x — a) (x — /3).........................(S5
We find that /9 = 2a, and that
77 — 77! I5sc3 (a2 aas a"
772-7?! 4a° [20 43 1 d'T] I5as
---------- _ - a) (# - 2a)...................(87
7j2 - ^ da? 4a5 v 'v ' ^
From these kz would have to be calculated by means of (72), and question would be to find how far a might be reduced without interfei with the prescribed character of k2. But to discuss this in detail would 1 us too far.
If the differences of quality in the variable medium are small, ( simplifies. If 770, k0 be corresponding values, subject to JcQ2 = (74?70~4, may take
where 77' and 8Jc- are small, and (72) becomes approximately
-y-r + 4&0277' = — 770<$/Ca...........................(8£
ax*
Replacing os by t, representing time, we see that the problem is the SE as that of a pendulum upon which displacing forces act; see Theory Sound, § 66. The analogue of the transition from one uniform medi to another is that of the pendulum initially at rest in the position equilibrium, upon which at a certain time a displacing force acts. .r. force may be variable at first, but ultimately assumes a constant value, there is to be no reflection in the original problem, the force must be such a character that when it becomes constant the pendulum is left at in the new position. If fche object be to effect the transition between two states in the shortest possible time, but with forces which are restric never to exceed the final value, it is pretty evident that the force ms 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.