V - DWC

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3. If some, but not all, of the casines are imaginary, the coefficients of equation (1)
are partly real partly imaginary. In this case too a solution is impossible. If however
all cosines are imaginary then the terms are partly negative partly positive, and the
equation can, therefore, be solved. Consequently the sines of all occurring angles must
be greater than 1, which is certainly true if the sinus of the smallest of these angles is
greater than 1.
sin il VI sin iz Vz
As --:- -- = \- and -.~- = - and VI > Q31' Vz> SBz, Cl and 'z wil! always be smaller
SIn Cl SB I SIn 1'2 ~lh
than il and iz.
Equation (1) being symmetricaI with respect to the suffixes 1 and 2, we can henceforth
assume SB 2
> ~nl without restricting the problem; th en Cl is the smallest angle.
So wh en all cosines are imaginary sin Cl > 1.
Summarizing these above remarks, we can assert that a solution of equation (1) is
only possible if sin '1 is imaginary or is greater than 1; or putting it otherwise:
SIn 2 Cl < 0 or sin 2 '1> 1.
We can condense these two inequalities into the following one: < 1.
si"z 'I
1
Therefore it is advisable to use -:-2--- as a new variable, which we shal1 call C.
Sl/l' 'I
If
. _ 1 h " nl ,. mI. mz
SUl Cl - --cc, t en sIn II = ~-_', sIn 12 = ----=, SII1 C2 =--=
V( V( V( V(
SBz
V 2
where mI = ill~ and m2 = ill~'
Hence
W
[1t111g
" th
e e
quation derived by STONELEY in this notation we get:
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V;' 1 (el-{?2)2_(el xrl- e2 XI) (el Yz + ez YI)! +
+4 V; lel X z yz-e2 XI YI-(er-ez)! + 4 (,uI-,u2)Z(XI YI-l)(xzyz-l) = 0
where
-------
V----z V-------Z
V; V r _ V r
Xz = l/l-a- z ' YJ = 1-- m 2 'Yz-
V being the phase velocity of the STONELEY waves.
r
l-w~,
V mI ;Ol I
identical with equation (2).
This equation reduces to a very simple form if we take
1°. e2 = 0: (2- 1-~~I~J z = 4 V(T·=;;;1) -Cl--() ;
3°. VI = V z
and mI = m
z
1
V 2
Putting -~ = r; this equation is
~IZ
with ['2 = 0 this is the RA YLEIGH equation.
(WIECHERTS' medium):
(2--()-2 V(f=-vI ()n=EW -== (}~t~ ~~j~; ( Y
. V(T-=-(f(f-=--;;-I 1;).
in which
iVl~E
COS Cl == -------=--.
V(
. i mI Vl-a(
cos 12 = -~----=---- ,
V(
im2 VT-~-(
COS C2 :--'--= --------=
V(
Equation (1) then becomes
(J'l and )'2 being the constants of incompressibility)