SURFACE REFLEXION OF EARTHQUAKE WAVES. 
381 
1899). The function we have denoted by f may be any arbitrary function provided 
it represents a distui'bance that can be propagated into an undisturbed region (see 
Love, ‘ Theory of Elasticity,’ p. 283). Thus as long as A 2 and A 3 remain real 
precisely the same functional form represents the incident and reflected disturbances. 
This is the case for longitudinal disturbance impingent at all angles e from 0° to 90°, 
for transversal disturbance (vibration horizontal) at all angles e from 0° to 90°, and 
for transversal disturbance (vibration vertical) at the particular angle e = 45° and 
from e = sec -1 /m (about 56°) to 90°. But when A 2 and A 3 becomes complex, as in 
the case of transversal waves (vibration vertical) for angles e between 0° and sec -1 /j., 
we must start by taking a complex form for f and finally select the real parts from 
the various expressions in the manner familiar in optical theory. 
Let f(d) be the incident transversal disturbance where 
6 = t + (x cos e + z sin e)/V 2 , 
then the Fourier resolution of f(6) gives 
f(d) = 7r -1 [ f{x) d\ f COS a (0 — X) da. 
J — OO Jo 
Since in the reflected transversal disturbance 
where 
We also have 
Ao/A = 
the real part of the corresponding disturbance is 
do p 00 
7i" ‘ i f (x) dX cos {a {6 2 —X) +</> 2 | da, 
» JO 
0 2 = t + (x cos e—z sin e)/V 2 - 
A s /A = 
There is no real angle of reflexion of the longitudinal disturbance, but if we write 
9 3 = t + x cos e/Vj and cosh \Js = n cos e, 
the longitudinal disturbance is expressed by 
is — 7r_1 [ /(M cfo | M-, cosh cos j a (0 3 —\) + 0 3 } da, 
J _oc Jo 
£ 3 = 7 r _1 j f(\)dX I M 3 sinh ^e~ v, ~ Wsinh ' / ' sin {a(0 3 — X)+^ 3 } da. 
The character of the reflected transversal disturbance will depend on the precise 
form of f (x), and, in general, the reflected disturbance will differ from the incident 
disturbance. There will be a “ trail.” 
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VOL. CCXVIII.—A. 
