SURFACE REFLEXION OF EARTHQUAKE WAVES. 
375 
Let a ray EO be incident at the point 0, making an angle e with OX. This angle 
is usually called the angle of emergence. The term is misleading. It is really the 
co-angle of incidence in the optical analogy, and when occasion arises I propose to 
call it the angle of impingence. 
(l) Let the incident wave be longitudinal. The component displacements & and 
Q are 
(&, £ 1 ) = — A (cos e, sin e)f{t+ (x cos e+z sin 
where f is any arbitrary function. 
This wave gives rise to a reflected longitudinal disturbance making an equal angle 
e on the other side of OZ, expressed by 
(0> Q — — A 2 (cos e, — sin e) f {t + (x cos e — z sin e)/V\}, 
and a reflected transversal disturbance making an angle e! on the other side of OZ, 
expressed by 
( 5 , Q = A 3 (sin e', cos e r ) f {t+(x cos e' — z sin e')/V 2 }. 
At the surface of separation OX the stresses must vanish, and this requires that 
3 -£ + ^ = o 
0z Sx 
and 
( Vl “-2V/) (|| + |i) + 2V/1| = °. 
when 2 — 0 . 
Thus we get the relations 
where 
It may be noted that 
A — A 2 = fj. A 3 cos 2eysin 2e, 
A + A 2 = fj .~ 1 A-o sin 2e'/cos 2e', 
fx = YJV 2 and n cos e! = cos e. 
sin 2e(Ar — A 2 2 ) — sin 2e'A 3 2 , 
which satisfies the energy condition that the rate of arrival of energy by the incident 
waves is equal to the rate at which energy passes away by the reflected waves. 
Solving the above equations, we get 
^ ^ _ (sin 2 e sin 2e’ — /u. 2 cos 2 2e'\ 
\ sin 2 e sin 2e'+ a cos 2 2 e /l 
AJA = -3 
2/x sin 2 e cos 2e! 
(sin 2 e sin 2e! + ^ cos 2 2e'\ 
3 d 2 
