Small Corrections in the Theory of Surface Waves
Harold Jeffreys
(Received 1961 April 13)
Summary
Rayleigh’s principle is used to derive expressions for the group
velocities of Love and Rayleigh waves without numerical differentiation.
It is also used to find expressions, accurate to the first order, for the
effects of small changes of the elastic properties on the wave and group
velocities.
I.
Introduction
The standard method of determining group velocities of surface waves is to
assume values of the wave-velocity c and of K , where ~ T / K is the wavelength;
solve the differential equations; and for each K or c interpolate a value of the other
so that the boundary conditions are satisfied. Then the group-velocity C is
determined from
d
C=-(
dK
dc
One disadvantage of this procedure is that for any new suggested structure
the whole process must be carried out ab initio, becoming more complex with each
additional complication suggested. Another disadvantage is that the differentiation
to give C has to be carried out numerically. But the whole range of c is small, and
the intervals used cannot be less than 0.1 of the range of c to give an adequate
determination of the behaviour of C; in practice two figures are lost in the differentiation.
E. Meissner (1926; see also Jeffreys 1959, p. 1 1 0 ) gave a method for Love
waves, based on Rayleigh’s principle, that avoids numerical differentiation. This
method could be adapted to Rayleigh waves, but this does not appear to have been
done. Rayleigh‘s principle can also be used to estimate the effects of small changes
of the distributions of density and elastic moduli. Since it is fairly easy to work out
the solution for a model that can be regarded as a useful first approximation, it
appears that such a method should permit rapid testing of any reasonable suggested
structure.
2.
Love waves
If the displacement is
H*
I
16
Harold Jeffreys
we have, apart from constant factors,
and by equating T and V,
1oy2 =
where
11K2+12
Then Rayleigh's principle says that if we work out (4)using a slightly incorrect
form of v , then for given K (3) will give y2 with an error of the second order.
If we change KO to KO SK but use for K = KO+ 8~ the form of v calculated
for KO, this has a first order error, but
+
10(3/0
+ 6y)'
= Ii(K0
+
8K)'
+1 2 + O(8K)'
(5)
whence to the first order in SK
IoyoSy
But sincep/K
= c,
(6)
= 11KOSK.
dy/dK = C, this is
cc = 11/10.
(7)
This is Meissner's result, which is exact. It replaces numerical differentiation by
the integration of two known functions.
But in the same way, if we also vary p and p, and retain the same values of v ,
we shall have
(10+810)(yo+sy)~=
(11+611)(K0+6~)~+1~+612+
where 610, SI1, 612 are independent of
(10
K.
Hence if 8~
o((8p)2,
(SP)~, ( 8 ~ ) ~ (8)
)
= 0,
+61o)(yo+ 6y)2 = (11+ 611)KO2 +1 2 + 612,
determining c with a second-order error for a fixed
K;
(9)
and from (7)
( ~ O + ~ ~ ) ( C O +=
S C(11+611)/(10+611),
)
(10)
giving C also with an error of the second order.
A method of mine (Jeffreys 1928)used Rayleigh's principle to allow for increase
of the velocity of S with depth in the lower layer, by means of ( 5 ) ; but C was then
determined as usual by numerical differentiation.
3. Rayleigh waves
The displacement is
We find
u1, 242, u3 = u(z) sin(yt-
zT
=
2r/T =
KX), 0,
w(z) cos(yt - K X ) .
fpyZ(u2 + w2) d z
I {h(w'
- w ) ~ + ~ ~ ( K ~ u ~ + w ' ~ ) + ~ ( u ' + K W ) ~ ) ~ Z
Small corrections in the theory of surface waves
1’7
and
r210 =
K211
+ 2 K 1 2 +1 3
(4)
where
1
10 = p(u2 + w2) dz
11 =
J{(A+~p)i12+pw2)dz
I2 =
J ( -AW‘U +~ u ‘ wdz)
13
Varying
K
=
J {(A + 2p)w’Z +pu’2) dz.
as for Love waves we have
IoCC
= 11+1~/K.
4. Conclusions
The use of these results would be that it would be possible to examine the effects
of small changes of p, A, p over non-overlapping intervals of z, and hence to use
observed group-velocities to identify more closely the intervals where such changes
are needed. They do for surface waves what Bullen’s transformation (1960) does
for bodily waves.
Huntingdon Road,
Cambridge,
1961 April 12.
I 60
References
Meissner, E., 1926. Proc. 2nd Congr. Appl. Math., Zurich, pp. 3-11.
Jeffreys, H., 1928. Mon. Not. R. Astr. SOC.Geophys. Suppl. 2, 101-III.
Jeffreys, H., 1959. The Earth (4th edition), (Cambridge: University Press).
Bullen, K. E., 1960. Geophys.J., 3,258-269.