The Total Internal Reflection of SH Waves
J. A. Hudson
(Received 1961 November 30)
Summary
In the first part of this paper we discuss the reflection of simple
harmonic plane SH waves at angles of incidence greater than the
critical angle at the interface between two semi-infinite elastic media.
The change of phase which occurs on reflection is related to the rate
at which energy crosses the interface. We then calculate the rate at
which energy travels parallel to the interface in the inhomogeneous wave
and the effect that this has on the group velocity of Love waves.
The total reflection of a plane SH pulse is studied in the latter part
of the paper. The expression for the reflected pulse involves the use
of allied functions as defined by Titchmarsh (1937 5 5.1). General
results concerning the allied function are drawn in the appendix from a
few conditions imposed on an otherwise general function.
A plane pulse has physical meaning as an approximation to a bounded
plane pulse or a spherical pulse. We show here that the approximation
is no longer valid for incident angles near 7 ~ / 2or near the critical angle.
The pulse which is usually known as the head wave is not given by this
approximation but we are able to study the displacements which are
forerunners of the reflected pulse and which arrive ahead of the time
predicted for the reflected pulse by ray theory. A few special cases are
given and deductions are made about the reflected wave when the
incident pulse is of symmetric form with a central maximum.
I.
The displacements
We shall consider plane SH waves reflected at incident angles greater than the
critical angle at a plane of discontinuity of elastic properties so that total reflection
occurs.
Let the plane be denoted with reference to Cartesian axes Oxyz by y = o
and let the half-space y < o and y > o contain uniform isotropic media with
density and rigidity p1 and p1 for y < o (medium I), and p2 and p2 for y > o
(medium 2 ) . A. simple harmonic plane wave is incident on the plane of discontinuity from y < 0 ; and /32 > /31. The critical angle of incidence is 8, where
sin 8, = p1//32 and the angle of incidence, 8, of the plane .wave is greater than 8,
(see Figure I).
509
J. A. Hudson
510
The displacements due to the wave are assumed to be in the z-direction only
and may be written as the real parts of
exp(iw( t - x sin9 +y cos9
))+Aexp(iw(t-
B exp(iw
(t
1
xsin9-ycos9))
181
181
x sin9
-w
- -))exp(
in medium
q )
2
181
where
sin29
I
is assumed to be real and positive here and, since 0 > BC, t~ is also real and
positive).
(w
Medium 2
inhomogeneourwave
FIG. to total reflection of a plane SH wave at the interface y = 0.
In order to satisfy the conditions of continuity of stress and displacement
at the interface the constants A and B must be as follows (Jeffreys 1926a, 5 3)
If we write
equations (I .2) become
A
B
exp(2ia)
= 2 cosa exp(ia).
=
The displacements are, therefore,
(
w1 = cosw t w2 =
[(
xsin9+ycosO) +cos w t -
2cosa.cos
181
[(
w t-
.;re)
xsin9-ycos9
- +a]exp(-wq).
P1
(1.4)
The total internal reflection of SH waves
5"
The displacement in the first medium is that of the incident wave and a
reflected wave of the same form and amplitude. In the second medium is the
inhomogeneous wave which travels parallel to the interface with velocity /3l/sin 8
and in which the amplitude decays exponentially with depth.
The phase change in the reflected wave
It can be seen from equation (1.4)for w l that the reflected wave leads the
incident wave by the phase angle 2a and so the wave form of the reflected wave
at the interface appears to be displaced a distance
2.
2aP1
w
sin8
along the interface from the incident wave (see Figure
:
a
Distance dono thrintcrfac.,r
2).
-
FIG.2.-The displacements at the interface due to the incident wave,
the reflected wave, W R , and the inhomogeneous wave, wa; and the
rate of transfer of energy across the interface, Et.
WI,
If the second medium were liquid ( p 2 = 0 ) then a would be zero and the
wave-forms of the reflected and incident waves would be coincident at the
interface.
J. A. Hudson
512
If the second medium were rigid on the other hand (p2 = to) then a would
be 7r12 and the wave reflected 7r out of phase. This means that the nodes of the waveforms of the reflected and incident waves are coincident at the interface but the
displacements are of opposite sign.
Between these two cases (i.e. when o < a < ~ 1 2 )the two waveforms are
displaced from one another at the interface by a distance which depends on the
elastic properties of the two media and on the angle of incidence of the wave.
We define as the magnitude of the displacement of the reflected wave from the
incident wave at the interface, the function A, where
w
A
=
sine
4
2/54
7r
----(“-a)
w
sine
2
for - < u
4
7r
(2.1)
< -.
2
A is actually the least distance between points of maximum displacement in the
incident and reflected waves at the interface without regard to the sign of the
displacements. Thus when o < a < 7r/4the reflected wave can be said to lead the
incident wave at the interface by a distance A, and when 7r14 < a < a12 the
reflected wave can be said to follow the incident wave at a distance A with displacements of the opposite sign.
We now show that there is a close relationship between A and the work done
by either medium on the interface, or alternatively, the rate at which energy crosses
the interface from one medium to the other.
In the two situations noted above where p2 is zero or infinite no energy crosses
the interface and by the definition (2.1)A is zero in each case.
The expression for the rate at which energy crosses per unit area of the interface
from medium I to medium 2 is given by
or
These two expressions (which represent the work done by medium I on the interface and minus the work done by medium 2 on the interface respectively) are equal
owing to the continuity of stress and displacement at y = 0.
Using either of the equations (1.4) we get
The form of this expression is shown with the form of the displacement at
the interface in Figure 2. It has half the wavelength and half the period of the
displacement at the interface. The total amount of energy which crosses per unit
area at any point of the interface in one period of the wave (2a/u) is zero. The rate
at which energy crosses a width of the interface equal to the wavelength of the
displacement at the interface
The total internal reflection of SH waves
5'3
is also zero. This gives rise to the usual assumption that when total reflection occurs
no energy is transmitted into the reflecting medium.
Using equation(2.1) we may write the amplitude of Et in terms of A and eliminate
a:
wA sin 8
p1w2 cos 8
amp(&) =
sin(
p1
(2.4)
1.
PI
Equation (2.4)shows clearly that when A is zero no energy crosses the interface.
Similarly, if Et is zero for all x and t either
wA sin 8
sin(
p1
)
= o
or p1 cos 8 = 0. Both equations imply A = o and so when there is no energy
interchange across the interface A is zero.
We shall clarify further the relationship between A and Et in the next section.
3. The transfer of energy across the interface
The expression (2.3) for Et shows that over any section of the interface of width
equal to the wavelength of Et, viz.
energy crosses the interface one way in one-half of the section and the other way
in the other half. The rates at which energy crosses the section in each direction
are equal. Therefore the rate at which energy crosses in either direction is given by
the integral of lEtl with respect to x over the interval [o, ~ / 3 1 / 2 wsin e], where x
is taken to be zero at a node of Et
n ~ , 1 sin
2 ~B
'S
lEtl dx
=P1W .sin(
0
tan 8
wA sin 8
pl
).
Equation (3.1) then, gives the rate at which energy crosses the interface in each
direction within a width equal to the wavelength of Et.
The rate at which energy is brought to the interface through the first medium
in the incident wave may be calculated using the first expression of (2.2) and writing
for w l the displacement due to the incident wave alone. This is
EI
p1w2
=
cos 8
81
Energy travels away from the interface into the first medium in the reflected
wave at the rate given by
ER =
plw2
-
cos e
81
(3.3)
calculated in the same way as EI.
Adding (3.2) and (3.3) we get the expression (2.3) for Et which is to be expected.
5'4
J. A. Hudson
To find what proportion of the incident energy on average is transmitted into
the second medium, we integrate EI over one wavelength of Et and divide the result
into (3.1)
1
----
(3.4)
2 tan 8
and so the ratio required is
2
F
=
-sin(
r
oh sin 8
Pl
).
(3.5)
The integral of ER over the wavelength gives the same expression as (3.4) with
a minus sign. Equation (3.1)gives, as we have said, the rate at which energy crosses
one wavelength of the interface in either direction and so F also represents the
proportion of the energy of the reflected wave which returns across the interface
from the inhomogeneous wave. This is to be expected since the total rate at which
energy crosses the interface within one wavelength of Et is zero.
If we write A12 as the wavelength of Et (i.e. X is the wavelength of the displacement at the interface, 271/31/o sin e), (3.5) becomes
F = -sin(+
2
2rA
71
which depends very simply on A.
From (2.1),
Xa
for0
7r
<a<4
A=[
-(
A:-a)
r
for- 71 < a
Therefore, since a lies between o and rl2,
and F lies between o and 217~.
a is the angle between o and 7~12given by
< -.r
(3.7)
4
2
A always lies between o and X/4
Figure 3 shows how a, A and F vary as 8 varies from 8, to a/2, keeping A
constant. The constants used were the ratios of the shear wave velocities and
rigidities used by JetTreys (1926a) :
P1
-=- 9
p2
20:
PI
-=-3
P2
4-
Both a and F are zero when 8 = 8, and increase as 0 increases until a = 7114
and F reaches its maximum value, which is when 8 is given by
This value of 8 is the angle of incidence for which the greatest fraction of the
incident energy crosses into the second medium and for which the displacement A
The total internal reflection of SH waves
515
is greatest. Figure 3 shaws that this value is very near BCwhen the elastic constants
are those chosen.
As 8 again increases to 7~12,a also increases to 7r/2 while F decreases to zero once
again. In the two limits of grazing incidence and reflection at the critical angle no
energy is transferred across the interface.
FIG. 3.-The fraction, F, of the energy of the incident wave which
crosses the interface; and the phase difference, a, between the incident
and reflected waves.
4. The inhomogeneous wave
The displacement in the second medium is (equation 1.4)
and
0
is given by
The two limiting values,
7r/2
and BC, of 8 give rise to two special forms of
w2.
When 8 = n/2, a = n/2 and so cos a is zero and the displacement in the
second medium is zero. In the first medium the reflected wave tends to cancel out
the incident wave as 8 tends to 7~12so that when 8 = 7112 the displacement in the
first medium is also zero. This indicates that it is impossible to set up the steady
motion of a plane wave travelling parallel to the interface in the first medium,
though it is theoretically possible to reproduce steady motion of waves reflected at
grazing angles of incidence, however near to 7r/2 (see Brekhovskikh 1960,p. 17,
J. A. Hudson
5'6
for the similar situation with sound waves). It appears that any attempt to set up a
wave travelling parallel to the interface would fail on account of destructive interference of waves reflected from the interface with the incident wave.
When the wave is incident at the critical angle 8, we have sin 8 = sin 8, = /3@
and so ts = 0 . In this case the wave is not attenuated with depth and in fact w2
does not depend on y at all. This means that, though we showed in Section 3 that
no energy crosses the boundary after steady motion is set up at this angle of incidence, there is a wave in the second medium which is no longer confined to the
neighbourhood of the interface. The displacement in the second medium is no
longer an inhomogeneous wave (a plane wave whose amplitude decreases exponentially in a given direction parallel to the wave-front) but an ordinary plane
wave travelling parallel to the interface. It can be understood more clearly as the
limiting case of a refracted wave as the angle of the incident wave increases to 8,
and the angle of the refracted wave increases to 712.
Energy is continually travelling parallel to the interface in the second medium.
For all values of 8 between 8, and 7 / 2the energy crosses any plane perpendicular
to the interface at a rate equal to the rate at which work is being done by one side of
the plane on the other, which is
m
R2
(-~22)($)dy
=
0
= -4T-Q
hU
x3+a].
.cos2a. sin2[w (t - -
(4.3)
The energy contained in the second medium between the half-planes x = xo
and x = xo + Sx, y > 0 , may be calculated as the sum of the kinetic and potential
energies, which is, to the first order in Sx
=-
p2wSx
U
.cos2a((- sin28 +p12
I )
p22
[(
sin2
. w t-- 5 ~ ~ 8 ) + a ]
[(
+u2 cos2 w t - x;;")+a]).
The increase of this energy in time is balanced by the flux of energy across the
half-planes together with the energy crossing the interface. This energy balance is
represented, to the first order in Sx, by the equation
i.e.
The total internal reflection of SH waves
5'7
As a check on the formulae we will now show that this equation holds. From
equation (4.4)
where
From equation (4.3)
Finally
aEz
at
-+-=
aR2
- 4pzwZa C O S ~ Usin x cos x
ax
and by substitution for tan a from equation (3.8) this becomes
e
- -cos
- .0 2 sin 2a sin 2x
B1
which is the expression for Et in equation (2.3).
The average value of Rz over a wavelength in the x-direction is
2rpz cos2u
Xu
Since cos a always decreases and cr always increases as 0 varies from 0, to 7712, the
average value of Rz decreases monotonically as the incident angle increases. The
greater the angle of incidence 'the less energy is transmitted through the lower
medium when the wave is totally reflected.
Again the two values 77/2and 8, of 8 give rise to special cases.
When 0 = 7712, Rz is zero, but we have already shown that a steady wave
travelling parallel to the interface is not a physical possibility.
As 0 tends to O,, cr tends to zero and Rz tends to infinity. This is to be expected
when it is remembered that incidence at the critical angle gives rise to an unattenuated plane wave in the second medium which is infinite in width and
therefore contains infinite energy.
The conclusion that the energy crossing a given plane in the second medium is
infinite is inconceivable physically and reveals that throughout this paper we have
made two important assumptions in order to simplify the analysis. These are:
I.
2.
that the incident wave is of infinite width;
that both media are unbounded.
I . An incident plane wave of infinite width is an approximation either to a
spherical or cylindrical wave at very large distances or a bounded plane wave
whose width is very large compared with its wavelength.
5'8
J. A. Hudson
In both cases the rate at which energy crosses any plane in the medium is
finite. The infinite value obtained here for this energy flux is due to the fact that
our approximation is invalid at large distances along the wave-front.
2. Any physical medium must be bounded. However, our approximation to
reality will be valid so long as the physical boundaries are far enough away. After a
finite time, though, the boundaries will begin to affect the displacements at every
point and our solution is no longer true.
This means in any case that we cannot integrate to infinity over any plane in
the two media. The expression (4.3) for the rate at which energy is travelling
parallel to the interface will still be a good approximation, however, if the factor
exp(-woy) in the expression (4.1) for w2 is very small at the boundary of the
second medium.
The function R2 therefore still has meaning for all angles of incidence greater
than Be so long as the depth of the second medium is sufficiently great. The
infinite value of R2 at 8 = Be must be rejected.
5. Love waves
We have shown that energy is transported parallel to the interface in the inhomogeneous wave. This has an important effect on the group velocity of Love
waves.
The previous analysis may be applied to the investigation of Love waves since
every plane in the first medium given by
y = -hn,
where hn =
B1
case (a +nn),
n=
0,
I, 2 ...
(5.1)
is free from stress and so we may take one of them as a free surface. The displacements (1.4)correspond now to the alternate reflection of plane waves at the
free surface and the interface. They may be written in the more familiar form
Taking the free surface to be at y = -hn gives the Love wave in the nth
mode.
The displacements ( 5 . 2 ) correspond to Love waves of period 2 ~ / uand in a
layer of depth
hn
=
&(a + n 4
case
having wavelength
A=
and phase velocity
c =
27$1
w
sin8
(5.3)
-. 81
sin 8
The characteristic equation connecting the phase velocity with the period comes
The total internal reflection of SH waves
5 '9
from the substitution of h, and c from (5.3) into the equation for a. This gives
tan(
= tan(h,w
J(+
-
5))
J($ - $)
J(+ - $)
p2
pi
(5.4)
(cf. Ewing, Jardetzky & Press 1957,equation 4-212).
As 6 increases from Be to 7~12,a increases from o to 7~12.For a given depth of the
layer equations (5.3) show that at the same time both the period and the wavelength
decrease from cut-off values (infinity in the lowest mode) to zero and the phase
velocity decreases from &-to PI.
The two extreme cases of reflection at the critical angle and propagation of the
incident wave parallel to the interface do not correspond to real Love waves in
the fundamental mode since the wavelength and periods become infinite and
zero respectively.
We showed in Section 4 that the rate at which energy travels in the inhomogeneous wave is
and that this is valid, if the depth of the second medium is great enough, for
6, < 6 < n/2.
Similarly we find the rate in the layer to be
= p1w tan8[2(a
+m)
+ sinza] sin2
[ (t- w
+a]*
(5.5)
The ratio of the two is
Rl
pl
- -- R2
p2 '
pi2
-P22
who
.tan6
2-H
+
2(a +n?r) sin 2a
2 cos2a
+sin2a
.tana [2(a +nr)
2 cos2a
1
(5.6
Figure 4 shows R2IR1 for the first 2 modes when pl/p2 = 9/20.
The ratio RllRz increases from zero when a = 0,to infinity when a = ~ 1 2 .
Thus for long wavelengths the energy travels mainly in the inhomogeneous wave.
As the wavelength decreases the energy tends to travel more and more in the
layer, and for very short wavelength the energy travels mainly in the layer. This
agrees with the conclusion reached by Stoneley (1958)that in Love waves of the
J. A. Hudson
520
fundamental mode in a double layer over a half-space the energy is contained
mainly in the underlying half-space when the wavelength is long, and mainly
in the upper layer when it is short.
Group velocity may be defined as the velocity with which the energy travels.
The average rate at which energy is transported within a wavelength of the
Love wave is
h
(R1+R2)dx
=
E.
tan8
2
0
zp22
cos2 a
' tana
2(a +nn) + sinza + pi2
-I*
(5.7)
The average energy content per unit length of the Love wave is the sum of
the potential and kinetic energies integrated over all depth and averaged over one
wavelength.
FIG.4.-The ratio, RaIRi, of the rate at which energy travels in the
underlying medium to the rate at which it travels in the layer, for Love
waves (A, is the cut-off value of the wavelength for the node n = I).
This is
$1
cos 8
z(a+n.rr)
cos2a
zP22
+ sin28 sin 2a + sin28 P12
tan a
The total internal reflection of SH waves
521
The ratio (RIE)gives the average velocity of the energy within the wave, which
is the group velocity;
C
=
R
- = /31sin8
E
[
(a +nn) sin a
( a +nn) sin a
+ cosa(sin2a +p2 cos2a)
+ cos a sinW(sin2a +p2 cos2a
wherep = 4 p 1 .
By the use of equations (5.3) and (5.4)this may be shown to be
c = -B12
X
C
(5.10)
which i s the formula derived by differentiating equation (5.4)to get
A2
dw
(-2x)
which is C.
The factor outside thebrackets of(5.9)and(5.10)isj31sin 8 (orP12/c) which would
be the group velocity of the Love wave if it consisted only of a multiply reflected
plane wave, reflected at both surfaces without transfer of energy into the lower
medium. For instance, if p 2 = o (i.e. the lower boundary is free from stress) (5.10)
gives C = /312/c straight away. If p2 = co (i.e. the lower boundary is rigid) the
result is more difficult to obtain. However, equation (3.8) shows that p cos a
remains finite as p -+ GO and a + nl2 and so equation (5.9) again becomes
C = sin 8 = ,912/c.
6. The plane pulse and allied pulse
We will now deal with an incident plane pulse. Any such pulse may be
written as a function of
x sin 8
+y cos 8
B1
The reflected pulse will be a function of
x sine-y cos8
B1
Using Fourier's integral theorem (Titchmarsh 1937, 5 1.9) we may express
an incident pulse of arbitrary shape as the double integral
WI(T+)
=
f fw ~ ( fcos)
f.
?r
0
H
dw
-02
w((-T+)
df
J. A. Hudson
522
m
m
=
[f ( w )cos w + d w + I g(w) sin w+dw
0
0
where
m
-m
and
Equation (6.2)shows that wz may be written as an integral over simple harmonic
functions. We have been discussing the incident wave given by COSWT+ in the
previous sections. The function sin COT+is derived by taking the imaginary parts
of the expressions in (I. I). Following this procedure we obtain instead of (I .4)the
displacements
w1 = sin WT+ sin(wT- +2a)
(6.3)
w2 = 2 cos a. sin(oT +a) exp( - woy)
where
7
=
x sin0
t--*
B1
The results of the previous sections would have been essentially the same
had we used sinon+ instead of COSUT+.
By the principle of superposition we may write down the integral form of the
reflected wave :
m
WR(T-)
= cos 2a{
m
I
f ( w ) COS WT- dw
+
0
m
- sin 2a{I f(w ) sin w- dw 0
g(w) sin WT- do}
m
g(w ) cos COT- dw}
0
s s
I
I
0
= - cos 2a d o w#) cos w(47r
o
-m
I
7r
T-)
@
w~(f)
sin w(4 -T - ) df.
o
(6.4)
-m
The first double integral is the expression for incident pulse WI(T-). The
second integral is the allied function to WI (Titchmarsh 1937 5 5.1 and Jeffreys
& Jeffreys 1956 5 14-11'),
s s
c o o 3
WI'(T-) = f d o w~(f)
sin w(f - T-) df.
?r
o
-m
The total internal reflection of SH waves
523
This function arises in connection with other physical systems. One, in particular,
is the response of a recording instrument (Jeffreys 1940).
Two particularly simple forms of incident pulse and allied function are noted
by both Titchmarsh and Lamb (1904, p. 26). These are
I
where
5 is a positive constant and
w(T+) =
I
E
9.
(T+)2 + E 2
w'(T-) =
-Q
.
(T-)2 +#I2
where E is a positive constant.
Friedlander (1948) derived the expressions for the displacements when an
incident pulse of the form of the Dirichlet function (6.6) is totally reflected.
Brekhovskikh (1960 $ 9 ) investigated the case when the incident pulse has
the formaof(6.7), though he simplified the mathematics in the later stages by letting
E tend to zero, in which case the expressions in (6.7) become the Dirac delta function
and its allied function,
w(T+)= TQS(T+)
w'(T-) = -Q/T-.
I
Arons and Yennie (1950)investigated the total reflection of a plane acoustic
pulse with the potential function
(a being a positive constant) whose allied function is
@'(T-)
=
Q
- -Ei(a.r-)
n
exp( - U F )
where Ei is the exponential integral function.
Once the original function is given for all T+ the allied function is defined for all
values of its argument by equation (6.5). It is not easy, however, to deduce the
form of the allied function when only the general shape of the original function is
known. A few results of this nature when the original function is symmetric
about T+ = o are given in the Appendix.
524
J. A. Hudson
Using (I .4) and (6.3) we obtain an expression for the displacement in the second
medium due to the incident pulse WZ(T+):
m
exp( - wuy)lf(w)
w ~ ( Ty)
, = 2 cosa
COS(WT
+ a)+ g ( w ) sin(wT +a)] dw
0
1
c o r n
+ sin a
dw
wz(g) exp( - ooy)sin w(5-
7)
dt).
(6.10)
--m
If the pulses are of the form given by either (6.6) or (6.7) the displacements represented by w2 die away with depth as ~ / instead
y
of dying away exponentially as
they do when the waves are of simple harmonic form (see Friedlander 1948, and
Brekhovskikh, 1960 5 9).
7. The reflected pulse
Equation (6.4) shows that the incident pulse is distorted on reflection. We saw
in Section 2 that an incident simple harmonic wave train can be understood to
have been displaced on reflection by a distance A given by (2.1). When the angle
of incidence, 8, is given A depends only on the period of the waves. In fact A
is proportional to I/W. Since we integrated over the simple harmonic waves with
respect to w to get an incident pulse (equation (6.2)) the reflected wave is not simply
displaced but distorted. We showed in Sections 2 and 3 that A is closely related
to the fraction of the incident energy which crosses the interface into the second
medium. Thus there will be a similar relation, though more complicated, between
the distortion of the incident pulse and the energy crossing into the second medium.
In the study of the reflection of a spherical pulse at the interface between
two media there appears a wave which is refracted at the critical angle, travels
along the interface, and is refracted back into the first medium (see for instance
Jeffreys 1926b). This is known as the head wave. We are dealing with a plane
pulse here as an approximation to a spherical pulse at very large distances. Since
the head pulse is a diffraction phenomenon which arises because of the curvature
of the spherical wave-front it does not in fact appear in this analysis. However, we
find displacements, arising from the energy which travels along the interface in
the lower medium, which occur before the arrival time predicted for the reflected
wave by ray theory. These correspond to the forerunners mentioned by Spencer
(1958). Spencer describes such displacements in a study on the reflection of a
cylindrical pulse at an interface. They are the displacements which occur after
the initial head pulse and lead into the reflected pulse.
This investigation therefore cannot throw any light on the head pulse but it
gives an approximation, valid at large distances, to the forerunners of the reflected
pulse.
The forerunner is easily picked out when the incident pulse has a finite arrival
time. Suppose for instance that WI(T+) is non-zero for T+ > T O and is zero for T + < TO.
The total internal reflection of SH waves
525
The forerunner may be defined as the part of the reflected pulse which occurs for
T- < TO. Using equation (6.4)’ this is
w~(T-)sinza
for
T-
< TO.
(7.1)
Obviously a forerunner does not appear when a = o or 7712, in which case
the incident wave is reflected in the same form (inverted in the second case).
So the forerunner dies out as the angle of incidence approaches 8, or 4 2 .
When the incident pulse is of the form of the Dirichlet function (6.6) the
forerunner is given by
It is a gradually increasing displacement rising to infinity (the infinity being due to
the idealized nature of the pulse).
Using an incident pulse of the form of a delta function (6.8) gives a forerunner
of the form
Q sinza for
--
T-
<o
T-
(7.3)
Arons & Yennie (1950) show in a diagram the shapes of the reflected waves
with their forerunners for various values of a when the incident pulse is given
by (6.9).
In each of the above cases the forerunner is a steady increase in displacement
leading into the main reflected pulse. The functions used are of idealized form
and give rise to infinities in the expressions for the displacements. If an incident
pulse of the form of (6.7) is used (following Lamb 1904) all the displacements
remain finite. An interesting feature which arises is that a maximum of displacement appears in the forerunner.
Figure 5 shows the shape of the reflected pulse for an incident Lamb’s pulse
for various values of a. The reflected pulse is (using (6.4) and (6.7))
E
WR(T‘-)
=
Q.
cos 2a -T- sin 2a
E2
+(T-)2
(7.4)
This has a stationary value when
[ ~ 2 - ( ~ - ) 2 ] s i n 2 a + z ~ ~ -=~0.
0~2a
(7.5)
The two values of T- given by (7.5) both correspond to maxima of the absolute
value of the displacement. The values are
71 =
- E tana
72 = E
cota.
I
At these two points the displacements are
cos2a
WR(T1) = E
(7.7)
526
J. A. Hudson
When 8 = OC, a = o (see equation (3.8)) and the pulse is reflected in the same
form. 7- = 71 gives the central maximum and 7- = 7 2 refers to a minimum at
infinity.
FIG.5.-The reflected pulse for an incident Lamb's pulse. (a) a = 0 ,
(b) a = IS", (c) a = 30°, (d) a = 45" (the allied pulse).
As 8 increases a increases and the values of 71 and 7 2 decrease. The main peak
of the displacement arrives earlier than before (i.e. for 7- < 0 ) and its amplitude
decreases. A very shallow minimum of displacement (corresponding to 7- = 7 2 )
occurs very late in time (i.e. for very large T - ) and it increases in magnitude and
arrives earlier for larger values of 8. The forerunner here is a gradual increase in
displacement leading into the main pulse.
When 8 is given by equation (3.9), a = v/4 and the maximum and minimum of
displacement are equal in magnitude and occur on either side of, and equidistant
The total internal reflection of SH waves
527
from T- = 0 . WR has the form of the allied function of the Lamb pulse (see
equation (6.7)).
As 8 increases further, a increases from m / 4 and the minimum of displacement
(at T- = 72) takes the place of the main pulse while the maximum (at T- = 71)
becomes part of the forerunner. When 8 is near m / 2 the magnitude of the maximum
is very small and it arrives very early in time.
When 8 is near Oc or m / 2 the minimum or maximum respectively of WR is very
much displaced from the main reflected pulse. As an approximation to a spherical
pulse or a bounded plane pulse this analysis will not be of much use in these two
cases since at large distances from the main pulse the curvature or boundedness of
the pulse will have an important effect.
In the Appendix we have deduced certain properties of the allied function when
the incident function is symmetric about T+ = o and has a single central maximum.
Lamb's form of pulse is one such function. We will now show qualitatively that
any pulse with these general properties is distorted on reflection in much the same
was as the Lamb pulse.
If the incident pulse is WZ(T+)
then the reflected pulse is (equation (6.4))
WR(T-) = WI(T-) cos 2a +wz'(T-) sin 24.
(7.8)
When a = o ( 0 = 0,) the pulse is reflected in the same form.
When a = m / 4 (8is given by equation (3.9)) the reflected pulse has the form of
the allied function. It is anti-symmetric about T- = o and is negative for positive
T- and positive for negative T- (see Appendix).
When a = m / 2 ( 0 = 4 2 ) the pulse is reflected in the same form but with the
opposite sign.
Both wz and wf are positive when T- is negative and so equation (7.8) shows that
WR is positive for all T- < o when o < u < m/4. The initial displacement in time
is positive and, in general, rises to a maximum before decreasing to zero at a time
later than T- = 0.
If T- = TO refers to this zero of WR, WR(T~)
(i.e. [aWR/aT-] T- = TO) is negative,
and
WZ(TO) cos 2u +WZ'(TO)
sin 2a = 0.
(7.9)
TO varies with a in order to satisfy equation (7.9). Differentiating this equation
we get
I a70
WZ(TO)
sin 2a - WZ'(TO)
cos 2a
--=
2 aa
WI(TO)cos 2a +WZ'(TO)sin 2a
wz and sin2a are always positive and WR(TO)< 0. Therefore TO always
decreases as a increases.
When a = m/4, TO = o and so as u decreases from m / 4 to zero TO is positive
and increases. The point at which WR becomes negative therefore occurs for
larger and larger values of T- (increasingly late in time) until, at a = 0 , the value
as a --f 0).
of TO becomes infinite (aTo/aa -+
Similarly, when 4 4 < a < m/2, WR is negative for all T- >/ 0. In general
the displacement will begin with a forerunner of one or more oscillations. It
passes through zero at T- = TO < o where it becomes negative and decreases to
a minimum and finally tends to zero at infinity.
J. A. Hudson
528
negative again and so equation (7.10) shows that the zero of WR at
occurs for increasingly large negative T O (increasingly early in time) as
a increases to n/2.
If T- = T m is a maximum of WR,
WR(TO) is
T-
=
TO
WZ(Tm) cos 2a +WZ‘(Tm)
Tm
sin 2a
= 0.
(7.4
varies with a according to the equation
I
87,
2
aa
--=
W(Tm) sin 2~
?&(Tm)
- WZ‘(Tm) cos 2a
cos 2a + %Z’(Tm) sin 2a
When a = n/4, WR takes its maximum value when T- = T m < 0. wR(Tm) is
negative since T m refers to a maximum of WR, s i n ~ is
a positive for o < a < n/2
and WZ(Tm) is positive for negative Tm. Therefore, as a increases to n/2, the
position of this maximum moves in the direction of negative T- (earlier in time)
and as a decreases to zero T m increases but cannot go beyond T- = o since ar,/aa
becomes positive when T m > 0. This maximum evidently becomes the central
maximum of the reflected pulse when a = o and the pulse is reflected in the same
form.
Similarly the minimum value of WE occurs for positive T- and moves towards
T- = o as a increases to n/2. This becomes the central minimum of WR when
a = n/2.
We can find out how the magnitudes of the maximum and minimum vary
with a from the equation
1 a
- -[WR(Tm)]
2
am
=
- WZ(Tm) sin 2a + w z ’ ( T ~cos
) 24
(7.13)
(using (7.8) and (7.1I)).
The maximum which occurs for negative T- when a = 7214 evidently decreases
in magnitude as a increases to ““12 since both terms on the right of (7.13) are then
negative. We have shown that this maximum becomes part of the forerunner
for values of a in this range, and the main pulse is the minimum of WR which
occurs for T- > o (late in time).
The minimum which occurs for positive T- when a = n/4 decreases in absolute
magnitude as a decreases to zero since both terms on the right of (7.13) are again
negative. The main pulse for a in this range is the maximum occurring for T- < o
(early in time).
Thus we see how the reflected pulses are similar for all incident pulses of the
same general shape as the Lamb pulse. The details of the form of the reflection
must depend on the exact shape of the incident pulse.
8. Conclusion
We have shown how the continual interchange of energy across the interface
is related to the phase change in plane simple harmonic SH waves and to the
distortion of a plane Dulse when there is total reflection at the interface.
The total internal reffection of SH waves
529
The proportion of energy which travels parallel to the interface in the inhomogeneous wave was found to have considerable effect on the value of the
group velocity of Love waves, particularly when the wavelength is large.
In the study of the reflection of a plane pulse the allied function arises, and in
the Appendix we have deduced a few of its properties when the original function is
symmetric and has a single central maximum. Displacements were found in the
study of the reflected plane pulse which arrive earlier than expected on the basis
of simple ray theory, and which arise from the energy travelling in the reflecting
medium. We have investigated these forerunners in a few special cases and made
deductions about their form when the incident pulse is of the type studied in the
Appendix.
In three of the special cases the incident pulse is of idealized form and the
forerunners consist of a steady increase in displacement. In the fourth case and
in the more general case the displacements are more realistic in that they are
everywhere bounded, and a maximum of displacement appears in the forerunner
for certain angles of incidence.
Throughout this investigation we have shown how the results relate to physical
situations as an approximation to the displacements occurring in a spherical
pulse or a bounded plane pulse. When the incident angle is near 7r/2 or the
critical angle the approximation fails to give a realistic picture.
Appendix
Any function + ( T ) which is of bounded variation and is continuous in the
range [ - 00, 001 and which satisfies
-m
may be written as
a
m
(Titchmarsh 1937, 5 1.9). The integral on the right of (A.2) converges uniformly
in any finite interval.
The function
m
0
a
-m
is the allied function to +(T) (Titchmarsh 1937, 5 5.1).
Titchmarsh (1937, $35.15, 5.16) shows that if + ( T ) satisfies the Lipschitz
condition
I+(.
+4- +(4I< KlW
(o<a<~,K>o)
(A.4)
530
as h +0 , for all x, and if
J. A. Hudson
m
J Id(4IPdT <
43
-co
for somep >
I, then
$(T)
exists and may be written as
+ also satisfies both (A.4) and (A.5).
The reciprocal relation is
Equation (A.6) may be rewritten to give
where P denotes the Cauchy principal value.
If we now assume that $( T) is always positive, symmetric about T = 0 , and has
a single maximum at T = 0 , equation (A.8) shows that +'(T) is antisymmetric about
T = o and is negative for positive T and positive for negative T .
Since +'(T) satisfies (A.4) and (A.5) it must tend to zero as1.1 --f 00 and must be
bounded for all T . It takes its maximum for negative T and its minimum for a value
of T with the same magnitude but with the opposite sign. Subsidiary maxima and
minima may occur for other values of T .
We will now look for an expression for the derivative of +'(T).
Therefore
If &$) is differentiable
+(f +h) -+(O = ME)+ E
where E = o(h).
(A.1 I)
The total internal reflection of SH waves
53'
If we assume that d(6) satisfies equations (A.4) and (A.5), the integral
(A.12)
exists and equals the derivative of $ ( T ) .
With these conditions therefore the derivative of
allied function of the derivative of #(T).
$(T)
exists and equals the
Acknowledgment
The author wishes to thank Dr E. R. Lapwood for his helpful advice during
the writing of this paper.
Jesus College,
Cambridge.
I 96I November.
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