Geophys. J. R. ustr.Soe. (1970) 20, 101-126.
Seismic Waves in a Quarter and Three-Quarter Plane
Z. S. Alterman and D. Loewenthd
(Received 1970 February 2)
Summary
The complete motion of an elastic quarter plane and of a three-quarter
plane with free boundaries caused by an explosive point source, is obtained
by finite difference methods.
Varying ratio /?/aof the shear to compressional wave velocity shows
that in the quarter plane the amplitude of motion at the corner increases
with increasing /?/a,in the three-quarter plane it decreases. The motion
in the quarter plane differs from the sum of reflections at perpendicular
half planes. The amplitude of diffracted P waves varies mainly with
distance from the corner. The amplitude of diffracted S waves varies
mainly in angular direction. Comer-generated surface waves and elliptical
particle motion in the waves are analysed. At the corner of a quarter
plane, the amplitude of the Rayleigh wave is three to five times as large
as on a half plane, the particle motion is elliptic and the major axes of the
ellipses are inclined at 45” to the free surface.
1. Introduction
The problem of diffraction of elastic waves at a corner has been discussed extensively in the literature (Lapwood 1961; Ma1 & Knopoff 1966; Gangi 1967)-however,
no complete analytic solution has yet been presented. A complete solution using
finite difference methods has recently been given, in the case when a point source is
located on the diagonal of homogeneous elastic quarter plane with free surfaces
(Alterman & Rotenberg 1969). A pulse of short but h i t e duration is emitted from
the point source. It propagates in the quarter plane. Due to the free boundaries a
large number of reflected and diffracted pulses, as well as surface waves, was found
and analysed. At most points several different pulses arrive almost simultaneously
and the displacement of the point is determined by their combined effect.
We have now added to the previous results an analysis of several finite difference
schemes for the boundary conditions, the effect of variation of elastic parameters
and the motion caused by a source located off the diagonal. In addition, we investigated the motion of a three-quarter plane with free boundary lines, as caused by an
impulsive point source. Diffraction at the corner causes P, S and Rayleigh waves
as in the quarter plane. However, unlike the case of the quarter plane no multiple
reflections occur in the three-quarter plane. The diffracted pulses occur well separated
from reflected pulses and we have been able to study their properties more easily in
this case. The results for the three-quarter and quarter plane are compared with the
well-known motion of the half-plane, known as Lamb’s problem (Lamb 1904).
101
1
102
Z. S. Alterman and D. Lawenthal
The motion at the corner (Section 8) of the quarter plane has previously been
found to be three times the amplitude of the initial pulse. This result applies for
fl/a = 0.55. Here b denotes the shear wave velocity, a the compressional wave
velocity; varying fl/a in the range 0.50 to 0-70changes the amplitude by 22 per cent.
In the quarter plane the amplitude at the corner increases with increasing p/a. In
the threequarter plane it decreases.
The motion in a quarter plane is compared with the sum of reflections at two
perpendicular half planes (Section 9); the two cases differ appreciably due to the
effect of the comer-generated waves.
Diffraction effects (Section 10) at the corner of the three-quarter plane show that
the amplitude of the diffracted P wave depends mainly on the distance from the
corner and not on direction. It decreases linearly with distance from the corner
in the range considered. The amplitude of diffracted S waves varies mainly in
angular direction and is almost independent of distance from the corner in the
range of distances under consideration. By filtering of the lowest frequencies, the
variation of these results with frequency is found.
Comer generated surface waves (Section 11) occur in the quarter plane and in the
three-quarter plane. The particle motion shows separate elliptic paths in the direct
Rayleigh wave in the transmitted or reflected surface wave and in the surface wave
which is caused by diffraction at the corner. The component of motion parallel to
the surface in the three-quarter plane is similar to the displacement in the half plane.
The motion differs in the components which are perpendicular to the surface. A
comparison of the quarter and half planes shows similarity in the perpendicular
components and large differences in the parallel components.
When the source is near to one of the surfaces of the quarter plane, the motion
at the corner shows in addition to the initial pulse also a surface wave. The amplitude of the direct pulse is found to depend on distance from the corner, not on the
deviation of the source from the diagonal. The amplitude of the surface wave is
found to be several times larger than the amplitude of a Rayleigh wave caused by a
source at the same depth in a half plane. In one instance the Rayleigh wave at the
corner is more than four times as large as in a half plane.
Graphs of the elliptic particle motion at the corner are given. The particle ellipses
are inclined at 45" to the free surfaces of the quarter plane.
2. Equations of motion
Let us write the equations of motion for waves in a plane elastic medium in the
vector form (See Ewing Jardezky Press 1957)
U,,= AU,, +BUxy+ CU,,.
Here
u and u are the components of displacement in the x and y directions respectively,
a and B the dilatational and shear velocities.
The boundary conditions are that all stresses on the free edges are zero. Choosing
co-ordinates with the origin at the comer and taking the boundary lines along the
x and y axes to represent either an elastic quarter plane or a three-quarter plane, the
103
Seismic wavm in quart- and threequarter plane
vanishing of stresses is expressed by
a2 u,+ (a2- 28’) uy = O
uy+ux = 0
I
on x = 0 for all t
(3)
on y = 0 for all
(4)
u,+ux = 0
(a2
-282) u, + 012 uy = 0
t.
The equations (I) are solved by the method of finite differences subject to the
boundary conditions (3), (4) and to the initial-or source--conditions which will
be described in the following section. As up to any finite time, the wave has propagated only a finite distance from the source, it suffices to consider a bounded part of
the plane. We solve the problem in quarter or three-quartersof the rectangle bounded
by x = f.L, and y = f L,. For this region we are able to find a solution up to the
time when waves which are reflected from the lines x = & L, and y = i-L2 arrive
at the point of observation.
The finite difference scheme then involves the imposition of a grid on the rectangle. The mesh spacing is taken to be h, in the x direction and h2 in the y direction,
and further L, and L , are taken to be integral multiples of h, and h2.
The derivatives in equation (1) are approximated by centred finite differences.
The time increment is denoted by k. The finite difference formulation for equation
(1) is then
U(X,y , r +k) = 2U(x, y, 2)-U(x, Y , r-k)
-2U(X, y, 2 ) + U(X, Y -hi, 01.
(5)
All finite difference approximations appearing in equation ( 5 ) are correct to the
second order in the increments. Equation ( 5 ) is an explicit difference equation.
When values of the displacement components u and t, are known at times t - k and
t , the displacement at the next time level is determined explicitly by equation (5).
In order to find the conditions for stability of equation ( 5 ) let us consider the
propagation of a disturbance in displacement 1
9 = 9 0 e I(mah,+nbh2)
cp.,
( = eeh.
Here qo is a constant vector, qo = (ql, q2) and x = mh,, y = nh,.
equation ( 5 ) , and after rearranging terms, we obtain
(6)
Inserting q in
104
Z. S.Alterman and D. Loewenthal
Here
Dl
= 1-2&’(X+ySY)
D2
= 1- 2 ~ ~ ( 6 X + y Y )
D3
= 4(1-6)2y~4XY(l-X)(1-Y)
and
I
ak
X = sin2-ah 1 ;
2
bh2 .
Y = sin2 2
For non-zero solutions the determinant of coefficients of qo must vanish, so that
(t;2-2D1 t;+ 1)(t;2-2O2 t;+ 1) - D, (2 = 0
which can be written as
(t;2-2A,t;+1)(t;2-2A,t;+1)
where
(9)
=0
A , = +{Dl+ D , + J [ ( D , -D2)2+D,1} = 1-&z(1+6)(x+yY)+&2(1-6)Js
A2
= +{D1+ D2 - J [ ( D l -
and
D2)’+ 0 3 1 )
= 1-&‘(I +6)(x +yY) -&’(I
s = (X-yY)2+4yXY(l-X)(1-
-6)
JS
(10)
Y).
< 1 and according to equation (9) this implies that
!All < 1 and lA21 < 1.
Equations (10) and / A l l < 1 imply that
- 1 < 1-&Z(1+6)(X+yY)+&y1-6)Js < 1
For stability lt;l
which is equivalent to
0 < &2(1+8)(x+yY)-&y1-6)Js
< 2.
(11)
Equations (10) and IA21 < 1 imply
< 1-2(1+8)(X+yY)+&2(1-8)Js < 1
0 < &y1+6)(X+yY)+&2(1-6)JS < 2.
-1
or
From
O<X<l,
O<Y<l,
0<6<1,
(12)
730
it follows that s is positive and that the left-hand-side inequalities in (11) and (12)
are satisfied for any value of 8.
By using the estimate Js
IX -y Y I in the right-hand-side inequality condition
of (10 ,
&2(1+6)(X+yY)-&2(1-6)Js < & 2 ( 1 + 6 ) ( x + y Y ) - & 2 ( 1 - 6 ) ~ x - y Y (
and (11) is satisfied if the right-hand side is less or equal to 2. Hence
1
E2
2
< ( I +S)(X+yY)-(l
X+y6Y
X<yY
(13)
-G)IX-yYI
=[
1
6X+yY
X2yX
Seismic waves in quarter and threequarter plane
105
The strongest condition for e occurs when X = Y = 1. In this case
1
-
Y < l
1
y+6
Y21.
1+y6
l+yS' y+6
An upper bound for Js is given by
Js<
(
<1
X+yY-2yXY
y
X+yY-2XY
y 2 l
and is applied to the expression in (12)
&2(1+6)(x+yY)+e2(1- 6 ) J s
Y<1
G
e2 [(1
+6)(X +y Y) + (1 -6)(X 4- y Y -2 x Y)l]
y 2 1.
The right-hand inequality in (12) is satisfied if the last expressions are less or equal
to 2, or
{yY+X[l-(1-6)yY])-'
y <1
e2 <
I({yY+X[1-(1-6)
Y]}-1
y 3 1.
In (15) the strongest condition for e occurs when X = Y = 1. Inserting these
values of X and Y in (15) we get the same results as in (14) which can be expressed
in the original notation by
For equal grid spacing in the x and y direction, h, = h, = h, condition (16)
reduces to :
3. Impulsive point source
At time t = 0 an impulsive point source within the elastic medium starts to emit
compressional waves. The displacement caused by the source at time t and at a
distance r from the source differs from zero in the r-direction only. Let us denote it
S(r, t). It is defined according to Afterman & Rotenberg (1969) by
t < r/u
-{G2(r,
f)-2G2(r, t-A)+G2(r, t-2A))
f
> r/u.
(18)
106
Z. S. Alterman and D. Lmwenthnl
Where
t < r/a
("
S(r, t ) has a singularity at the source at r = 0, is zero for t
r # 0 has its maximum at
T,,, = ( 4 A + /[4A2+9
r
A' u
=-++A+%-+
a
r
< r/a, and for
given
(+)'I)
... .
We see that the lag of the maximum behind the ray arrival time r/a depends on r.
For A Q r the arrival time of the maximum can be taken as r/a+4A/3, and the time
difference between maximum and first arrival has then the constant value 46/3.
Through the parameter A, S(r, t ) can be adjusted to fit any desired range of pulse
duration. In the present calculations the time-step k is chosen so that A includes
a few grid points in time.
The source function is unbounded near r = 0. The finite difference scheme
cannot be used at r = 0 as well as at a few points in its neighbourhood. This
difficulty was overcome by subtracting S(r, t ) from the total displacement U(x,y, t )
and applying the finite difference scheme to the remaining displacement UR:
UdXY YY t ) = U(XY YY O--S(r(x, Y)Y 2).
(21
By linearity, the finite difference equation for UR is identical with equation (5).
However the boundary conditions are changed. The method of calculation of the
reflected field only has been utilized previously and compared with results of calculations for the complete field (Alterman & Aboudi 1968).
4. Boundary conditions for the quarter plane
To satisfy the boundary conditions for the quarter plane the lines of mesh points
at x = - h and y = - h are added to the array as special lines. On these lines the
displacement components u and u are determined such that the boundary conditions
are satisfiedon the lines x = 0, y > 0 and y = 0, x > 0. Here hl = h2 = h. According to the results of Alterman & Rotenberg (1969) derivatives perpendicular to the
boundary are approximated by uncentred differences, derivatives parallel to the
boundary are approximated by centred differences.
On x = 0 the equations are
They determine U on the special lines y = - h and x =
-It.
107
Seismic wavea in quarter a d three-quarter plane
For a source on the diagonal of the quarter plane there is symmetry and one
condition is sufficient. For a source at an arbitrary location we need also the second
boundary condition.
There is a special situation at the corner (see Fig. 1). If we set x = y = 0,
equation ( 5 ) involves the point (-h, -h) which has not been obtained by the
boundary conditions (22) and (23). An additional equation has to be derived to
satisfy the boundary conditions at the corner. Within the accuracy of the grid
spacing, the location of the corner is at some point in the square (0,O); (0, -h);
(- h, 0)( -h, -h), i.e. the grid determines the location of the corner to an accuracy
given by h. Within this square, the boundary at the corner may be approximated
by a smooth curve, such that the tangent to the boundary at the point x = y = -h/2
Y
y 4 Plane
Ilno
FIG. 1. Quarter and three-quarter planes with superimposed grid for finite
difference scheme.
108
Z.
S. Alterman and I). Laewenthal
is at an angle of 45" to both axes. The normal and tangential stresses at this point
pnnand p,,, are given by
Pnn
=3
Pnt
+ +
(25)
~ x x~ x yt ~ y y
= 3(ppg-~x*)*
(26)
Vanishing of these stresses is expressed by the two equations
(u'
+u,,) + pyu, +u,)
- /3"(u,
=0
(27)
u, -uy = 0.
and
(28)
The finite difference form for (27) and (28) is obtained by substituting the approximation
1
[u(O, -h, t)-u(-h, -h, t)+u(O, 0,t)-u(-h,O, t ) ]
2h
for u, and
1
[u( -h, 0,t ) - u( -h, -h, t ) + u(0, 0,r) - u(o, -h, t ) ]
2h
(30)
for uy. Similar expressions approximate u, and u,,. Thus, u and u at (-A, -h) are
determined by
u(-h, -h,
t ) = u(0, 0,t ) +
U 2 -8 2
[U(O, -h, - u( -h, 0,9l
a'
IJ
-[ ~ ( - h 0,
, t)-u(O, -h,
t)]
(31)
t)].
(32)
U2
u(-h, -h,
t ) = u(O,O, t ) -
BZ [u(O, -h,
U2
t)-u(-h, 0,t ) ]
+-
U 2 - p
U2
[u( -h, 0, t)- ~(0,
-h,
A very simple condition results at the corner if we impose at x = y = 0 the
boundary conditions (3) and (4) simultaneously. From the first equation in (3)
and the second equation in (4) we find
u, = u,, = 0
and determine
U( - h,
-h) = ~(0,
-h)
u(-h, -h) = u(-h, 0).
(33)
(34)
(35)
Calculations for several grid sizes show that the two sets of boundary conditions
lead to the same results. E.g. when the source is on the diagonal at x = y = d,
so that by symmetry u(x, y ) = u(y, x), and in a grid of 21 x 21 divisions in the square
between the source and corner, the maximum value of u at the corner was found to
be 1-937d according to the condition in equation (31) and 1-944d according to
equation (34). When the number of grid points is doubled in each direction the
value of u is 1.938d and 1-935d in the two cases respectively.
Seismic wave8 in quarter and threequarter plane
109
5. Boundary conditions for threequarter plane
For the three-quarter plane the conditions on the free boundary lines are on
x = 0, y < 0 and on y = 0, x < 0. The special (' fictitious ') lines are again added
at x = - h and y = - h. The conditions of vanishing stresses on the free boundary
lines in finite difference form lead to the same equations as before, equations (22)-(23).
All the points on the fictitious lines are determined by these equations and no
additional equation is needed at the corner. In fact, the value of U at x = -h,
y = - h is determined twice: once by equation (22) setting y = - h and once by
equation (23) setting x = -h. The two values of U(-h,-h) are not necessarily
equal. We have solved the finite difference scheme separately with each of the
two values of U. The results throughout the three-quarter plane are essentially
unchanged. The largest difference occurs at the corner. The maximum of the
pulse is once 1466d and once 1*023d,a difference of less than 5 per cent in a coarse
grid.
In order to fix ideas about the corner, and to avoid double definitions, let us
assume that inside the square (0,O); (0, -A); ( - h , 0); (-h, -h) the free surface is
rounded (Fig. 1). At (-h/2; -h/2) it is assumed to have a tangent at 45" to the
x axis. The condition of vanishing of stresses is then applied at this point, leading to
u(h, h, t ) = u(0, 0,t ) +
[1- (32]
"0, h, O--(h,
0 9
01
PZ l'u(0, h, t)-v(h, 0,01
+U2
(36)
[u(O, it, t)-u(h, 0,Ol.
(37)
and
o(h, h, t ) = o(0, 0,2)-
- [1-
($)2]
These are the same expressions as in equations (31), (32) with - h replaced by h.
When the source is on the diagonal at x = y = d, so that u(x, y, r) = u(y, x, t ) ,
then
u(h, h, t ) = o(h, h, t )
[
(:)21
= u(O,O, ?)+ 1-2 -
[u(O, h, t)-u(h, 0,t ) ] .
(38)
The maximum of the pulse at the corner has now the value lumax1= 1*037d,in
a coarse grid. It is intermediate between the values obtained by the previous methods.
In a fine grid [u,,,,,~~
= 1-038d,which indicates the accuracy of results.
6. Filtering of zero frequency
Fig. 2 shows the variation in time of the initial pulse emitted by the source.
,, during a time-interval of
After an initial increase from zero to a maximum S
about 46/3 units (see Section 3) the amplitude of the pulse decreases at a similar
rate down to a value S , which is near to
A
lim S(r, Z) = - = s,
2r
t'oD
*
(39)
110
Z.S. Alterman and D. Loewenth1
S max.
x /d
18/21
1.8
'i
16/21
1.2
-
0.6
-
12/21
8/21
0
0.0
1
FIG.2. Amplitude S of initial pulse at x / d = y / d = 18/21; 16/21; 12/21; 8/21; 0
without filtering of zero frequency (upper curves) and with filtering (lower curves),
source at x = y = d.
For small distances from the source S , and S , are almost as large as S,,,. E.g. in
the curve for x/d = y/d = 18/21 in Fig. 2 S,,JS, is near one. A similar situation
occurs in the reflected and diffracted pulses. The slow decrease of the pulse from
S , to S , results from the two-dimensional geometry under consideration as discussed
by Roever, Vining, Strick (1959). In three dimensions it is easy to determine a pulse
which is a solution of the wave equation such that the duration of the variable part
of the pulse is not more than twice the rise time from zero to maximum amplitude.
Also reflected and diffracted pulses in three dimensions vary in a similar fashion.
In order to obtain from the present two-dimensional problem of a comer results
which are closer to a three-dimensional situation we have filtered out the lowest
frequencies in the pulse. The second set of curves in Fig. 2 shows the initial pulse
after filtering. We see that the pulse decreases from its maximum value to near
zero during a relatively short interval.
In an analysis of results, in the following sections, we find that it is far easier to
distinguish between various pulses in the filtered seismograms than in the unfiltered
results (see Fig. 6).
Seismic waves in quarter and three-quarter plane
111
The filtering is performed by the transformation
+
V(x, y, t ) = c[U(X,y, t ) - U(X,y, t - At) V(X, y , t - At)].
The z-transform of this filter is
c(l - z )
F ( z ) = -.
1-cz
(40)
It suppresses zero frequency and reduces low frequencies, depending on the
value of the constant c (Shanks 1967).
The lower set of curves in Fig. 2 shows the initial pulse arriving at the points
x/d = y/d = 18/21; 16/21; 12/21; 8/21 and 0, after transformation by equation (40)
with the constant c = 0.9.
The curves decrease to about 1/5 of S,,, during a time interval which is equal to
the rise time and after that decay rapidly to zero. The maximum of the filtered
pulse is smaller than S,,,, and in a discussion of total amplitudes the unfiltered
results should be analysed. However, in differentiating between various pulses and
their relative contributions to the total displacement at a point, the filtered seismograms are of help.
7. Accuracy of the finite difference calculation
In the subsequent discussion of results several figures show the displacement of a
point in the elastic medium as a function of the dimensionless time-parameter at/d.
For comparison, the arrival-times ut,/d of several waves are indicated by little
arrows. The notation of waves is as defined by Alterman & Rotenberg (1969).
atJd determines the arrival of the maximum of the direct and reflected pulses? and
is obtained from the ray-arrival time plus 4A/3 (see Section 3).
The finite difference calculations were performed for several grid sizes and for
two values of the time constant A fo the pulse. In the region between the corner
and the source, 0 < x < d , 0 < y < d, where the reflection and diffraction phenomena
are of interest, the narrow, peaked pulse of A = 5*2d/21was investigated. Results
were obtained in a coarse, medium and fine grid of Ax = d/21; d/42 and d/84. Fig. 3
gives an example of the accuracy obtained in the three grids. It shows the u component of displacement in a quarter plane at the observation point x/d = 4/21,
y = 0. We see that the results for the medium and fine grids in the dashed and
dotted curves respectively are close. The solid curve for the coarse grid shows
slightly different values at the minimum and maximum of the pulse.
The curves are slightly displaced one with respect to another, so that the peak
of the pulse in the fine grid arrives first, in the medium grid it is second and in the
coarse grid it is last. According to the choice of one-sided boundary conditions in
equations (22)-(23) the boundary should be considered as locattd in the middle
between the lines x = 0, x = - h on one boundary line, and y = 0, y = - h on the
second boundary. With decreasing h the distance between source and boundary
decreases and the pulse is reflected earlier than for large h. At a larger distance from
the source, we considered a pulse of A = l-O4d/a for the investigation of surface
waves. The grid sizes were chosen as Ax = h = d/10 and Ax = h = d/5. The
second graph in Fig. 3 shows the v component of displacement at a distance of 5d
from the source in a three-quarter plane-at the point x = - 4 4 y = d. The results
for the coarser grid (dashed curve) are near enough to the results for the fine grid
(solid curve) to conclude that for an analysis of the motion at thc distant points of
the order of 5d away from the source, a coarse grid of h = d/5 is sufficient. Near
the source, calculations were performed in the grid of h = d/42 with occasional
checking of results obtained with the other grids. In this case A = 0.25.
x 12
Z.S. Alterman and D. Loewenthal
0.4r
I
0
x=4d/21 y'o
I
0.5
I
I .o
-
20
I
I
I
I 6 2 0
1
1.5
at/d
1
1
2.5
1
3.0
I
35
-0.1
1
0
1
4
1
8
at/d
12
FIG.3. The vertical component of displacement, v, in a coarse mesh (solid line),
medium mesh (dashed line) and a h e mesh (dotted line) in a quarter plane (upper
curves) and in a three-quarter plane (lower curves).
8. The displacement at the comer. Effect of P/a
In Sections 8-11 we study the motion of the quarter or three-quarter plane,
caused by a source located on the diagonal at x = y = d .
The displacement at the comer is of special interest. It shows a single large
maximum at the arrival of the direct pulse, and then decreases to zero. This maximum is larger than the maximum of the initial pulse and includes the effect of reflection at the surface and of diffraction at the corner.
Fig. 4 shows the displacement at the corner of the quarter plane and of the threequarter plane as compared with the motion in a homogeneous medium at the same
distance. The ratio of shear to compressional wave velocity P/a varies from 0.50
to 0.70. We see that for /3/a = 0.55 at the comer of a quarter plane the maximum
of displacement is about three times the maximum of the homogeneous medium (or
of the incident wave). At the corner of a three-quarter plane the maximum is only
about 1.5 times the initial value.
Let us consider the form of the slowly varying part of the curves for t 2 2*5d/a.
In the three-quarter plane it is similar to the form of the initial wave. In the quarter
113
Seismic waves in quarter and three-quarter plane
0.0
7
-0.4
/... .................*.’................_
d0.8
3
::
-1.2
t
-1.6
-2.0
I
0
I
I
1
2
--at/d
1
3
I
4
I
5
FIG.4. The displacement components I(, v at the comer of a quarter plane and
of a three-quarter plane having elastic constants j3la = 0.50; 0.65; 0.70. Source
at x = y = d .
plane the curvature is different-a Rayleigh wave arrives from 3 to 4 a/d time units
after the maximum of the direct pulse. The arrival times of all other waves, including
reflected and diffracted waves, coincide with the arrival time of the direct pulse,
and simply change its amplitude.
Considering the displacement at the corner of the quarter plane for several values
of P/a, we see that the amplitude decreases with decreasing P/a. It is interesting to
note that in a three-quarter plane the dependence on @/ais different. The maximum
of the displacement at the corner increases with decreasing B/a.
9. The motion on the diagonal
For a source on the diagonal of the quarter or three-quarter plane the motion of
points on the diagonal is specially simple. On the diagonal u = tl and the displacement vector U is along the diagonal, so that the component perpendicular to the
diagonal is zero and IUl = uJ2.
The motion in the three-quarter plane is specially simple as no reflected pulses
arrive at points on the diagonal and the effect of any surface waves should be negligible. In addition to the direct P-pulse a P and an S wave return from the corner
through diffraction. They are denoted P(D) and S(D). In plane wave propagation
the S component has no longitudinal component and would be zero on the diagonal.
Due to sphericity, the S(D) has a small non-zero amplitude. The solid lines in
Fig. 5 show the displacement at the points x = y = 10421 and x = y = 6d/21.
They show the direct P-pulse and an increase in amplitude near to the arrival of
P(D) and S ( D ) . However, in order to obtain more specific information about
P(D) and S(D) the results have to be analysed further. Let us subtract from the
total motion the contribution of the direct pulse. The residual displacements U,
are drawn as solid lines in Fig. 6. The dashed line shows U, after filtering of the
lowest frequencies.
114
Z. S. Alterman and D. Loewenthal
.1
x=y= 6d/2l
0.0-
$-0.8
i
-1.2
-1.6
L
I
-0.8
t
- 1.2
I
P
...................................
,
I
pp
\
\\
/
0
'I
\
I
\
\"/
I
-1.6
,/---------
/
,
.
\
I
~RCR
PCS
PbP
FIG. 5. The displacement components u, v in a three-quarter plane (solid line),
a quarter plane (dashed line) and a homogeneous plane (dotted h e ) at a receiver
located at x / d = y / d = 6 (above) and x / d = y / d = 10/21 (below).
At x = y = 6d/21 the arrival times of P ( D ) and S ( D ) are close and the two
pulses combine and occur as a single pulse. At x = y = 10d/21 the separate contribution of S(D) is noticeable. The variation of the maximum amplitude with distance
from the corner of the three-quarter plane is given in Table 1 and is discussed in the
next section. For comparison, the initial pulse, as it propagates in a homogeneous
medium, is added in Fig. 5 as a dotted line.
The motion on the diagonal of the quarter plane is more involved as several
reflected pulses and surface waves arrive. The possible reflected pulses are PP,
PS, PPP and one of PSS or PPS. The dashed curve in Fig. 5 shows the displacement u = v at x = y = 10421 and 6d/21 in the quarter plane. The direct P is the
same as in the three-quarter plane. As expected by ray-theory the PP-pulse has a
small amplitude of opposite sign to P at x = y = 6d/21 and of the same sign as P at
x = y = 10d/21. The ray-path of the PPP pulse coincides with the path of P(D)
and the corresponding pulse has a far larger amplitude than P(D)in the three-quarter
plane. It is the largest pulse that arrives at points on the diagonal. Except for
points near the corner, for which x = y < 2d/21, the amplitude of P P P + P ( D ) is
115
Seismic waves in quarter and three-quarter plane
0.08
Q32
0.40
1
10
1
15
I
I
-
I
2.0 25 3.0
at/d
I
36
I
40
I
4s
0.16.
\
3
t 0*24.
t
0.4
Oe3*
0
FIG. 6. Radial component of the reflected field, uR. in a three-quarter plane
with filtering of zero frequency (dashed tine) and without filtering (solid line)
at the location x = y = 6d/21 (above) and x = y = 10421 (below).
0.7 of the maximum at the corner and is independent of the distance from the corner
in the range 2 < 21 x/d < 10.
In order to find the specific contribution of the quarter plane, let us compare it
with the result of adding reflections at the free surfaces of two halfspaces. Let us
assume that one halfspace has the free surface at y = 0 and extends over y 2 0.
The second halfspace extends over x > 0. Fig. 7 shows the displacement at
x = y = 10d/21 in the quarter plane (solid line) as compared with the sum of reflections in the two halfspaces (dashed curve). The direct P and once reflected PP and
PS coincide, as they are independent of the second surface. However the later
arriving pulses differ in the two cases. We see that one cannot obtain the motion
of the quarter plane by a superposition of half planes. Specifically the amplitude of
P P P and P(D) in the quarter plane is less than the sum of motions in the two half
planes and there are additional corner-generated waves which will be discussed
further on.
116
Z[S. Altennan and D. Loewenthal
X=y=lOd/2L
,
e
o
n
PS
PPS
I I
'
-0.8
- 1.2
P(D)
I
P
I I
- --------___
t
/--
i
,
PCR RCR
- 1.6
0
t
PP
S(D)
PPPt
I
2
-at/d
3
4
5
FIG. 7. Displacement components u, Y of a quarter plane (solid line) compared
with the sum of contributions of two perpendicular half planes (dashed line).
10. Diffraction
The effect of diffraction at the corner is found clearly in the curves which describe
the motion of the three-quarter plane. At points near to the comer the arrival
times of direct and diffracted pulse are close and the two pulses are superposed.
Subtracting the direct pulse, the residual displacement U, shows the diffracted
P ( D ) and S(D) pulses. In order to separate between P(D) and S(D) let us calculate
the components of U, in polar co-ordinates (r,8) with origin r = 0 at the corner
and 8 = 0 on the positive x-axis. Except for spherical-wave effects, P(D) occurs in
the r-component, S ( D ) occurs in the &component.
Considering the maximum values of P ( D ) and S ( D ) with and without filtering
of zero frequency at scveral points (r, O), we find that P ( D ) depends mainly on the
distance r from the comer. It decreases linearly with increasing r in the range
0 < r/d < 3 and the amplitude of P ( D ) at r, A(r), is given in terms of the amplitude
A . = A(0) at the corner, r = 0, by
A(r) = A,( 1-0*88r/d).
(42)
Here A, = 0.45d. The variation in 8 has only secondary effect.
S ( D ) varies mainly with 8. It is zero at 8 = 45" and its amplitude B increases
linearly with 8. In the range 20" < 8 < 100"
B(e) = (e -450) ~(900).
(43)
Here 8 is measured in degrees. B(8) is negative for 8 < 45" and positive for 0 45O.
B(90") = 0.005d.
Fig. 8 shows A(r) and B(8). The dependence of P ( D ) on 8 is indicated by the
spreading of points around the line for A(r). The points for S(D) are close to the
line B(0) for 8 < 100". For 0 > 100" the amplitude of S ( D ) increases more rapidly.
Fig. 8 shows also the variation in amplitude of P(D) and S(D) after filtering of the
lowest frequencies. Denoting the amplitudes of P(D) and S ( D ) after filtering by
A, and B, respectively, we find
=-
A,(r) =
(1 - 1.1
5)
Seismic waves in quarter and threequartsr plane
117
I
I
-
R/d
Fro. 8. Diffraction at the corner of a threaquarter plane: the amplitude of the
diffracted P as function of distance R from the corner, and the amplitude of the
diffracted S as function of 0. (a) without filtering; (b) with filtering.
Here
A,(O) = 0.2d;
B,(9O0) = 0.003d.
A comparison of equations (42) and (44) for A(r) and A,(!) shows that the
diffracted P wave decays with distance from the corner more rapldly for the higher
frequencies than for the lower frequencies. The equations for B(8) and B,(O) differ
only in the reference value B(90").
Table 1 shows the amplitudes of P(D) at several grid-points. Table 2 shows the
amplitudes of S ( D ) .
Table 1
The amplitude of difracted waves P(D) near the corner of a three-quarter plane
rid
0
0.105
0.133
0.171
0.195
0.238
0.257
0.267
0.290
0-300
0.304
0.319
0,371
0.386
0.390
0.400
0.405
0.448
0.486
e
A(r)IA(O)
45"
1*OOo
Al(r)IAAO)
1*OOo
64"
0.860
0.843
0.793
0.744
0.727
0.702
0-711
0-669
0.603
0.678
0-653
0.628
0.612
0.587
0.612
0.596
0.579
0.537
0.792
0.766
0.727
0.675
0.662
0.623
0.623
0.584
0558
0.597
0.571
0.532
0.519
0.545
0.519
0,519
0.480
0.506
45"
34"
76"
54"
22"
45"
61"
108"
39"
64"
81"
83"
104"
45"
70"
58"
101"
2
118
Z. S. Alterman and D. Loewenthal
Table 2
The amplitude of diffracted waves S(D) near the corner of a three-quarter plane
e
rld
22O
34"
39"
45"
45"
45"
45"
51"
54"
58"
64"
0.257
0.171
0.304
0
0.133
0.267
0.400
0.371
0.238
0.448
0.105
0.319
0405
0.195
0.371
0.386
0.486
0.390
0.300
6 4 O
70"
76"
81"
83"
101"
104"
108"
B(0i B(9W
-0.549
-0.275
-0'137
0
0
0
0
0.118
0.196
0.275
0.412
0.431
0.529
0.745
0.863
0.882
1 a43 1
1.588
1.824
B,(WB,(90?
-0'559
-0.279
-0.155
0
0
0
0
0.124
0.217
0.279
0.435
0.435
0.528
0.745
0.839
0.839
1.242
1.398
1.646
11. Surface waves
In addition to the Rayleigh waves which occur in a half-space, there are surface
waves associated with reflection or transmission of Rayleigh waves at the corner,
R , R, and surface waves due to diffraction of the initial pulse at the corner, P, R.
Their occurrence in a quarter plane has been discussed by Alterman & Rotenberg
(1969). Here we compare the surface waves in the quarter and three-quarter plane
with results for the same source in a half plane.
At a point on the surface of the half plane at a horizontal distance of 5d from the
source the maximum of the direct P arrives at at/d = 6-48 and is followed by a
Rayleigh wave. The dashed curves in Fig. 9 show the u and v components of P
and of R in the half space. The solid curve shows the motion in the three-quarter
plane. The Rayleigh wave, R, is preceded by P, R; it has in both u and v components an amplitude which is less than one half of the amplitude of R in a half
plane. The particle motion in Fig. 13 is retrograde elliptic in the half plane, and
similar elliptic motion occurs in the three-quarter plane. It is more involved, does
not complete a full cycle and has a larger eccentricity. In the three-quarter plane
the elliptic particle trajectory consists of two parts. The first part is the P, R wave
alone, the second part includes R and the continuation of P, R. For large time
the u components coincide and converge to the stationary value -0.28d. The v
component tends to - -05d in the half plane, while v in the three-quarter plane stays
positive at 0.15d. Fig. 10 shows a similar comparison when the observer is at the
depth of the source, y = d, and at 5d horizontal distance from the source. The
attenuation of the surface waves with depth is clearly found. The u-component of
the Rayleigh wave is near zero both in the half plane and in the three-quarter plane.
The decrease in the v component amounts to 1/10 of the surface amplitude in the
half space. The decrease of v in the three-quarter plane is more pronouncedbeing one-fifth of the surface amplitude. For comparison, we have calculated the
analytic solution for a source in a half plane (see Garvin 1956; Alterman & Loewenthal 1969). In this analytic solution the time constant of the source is A = 0. The
direct P-pulse and the reflected pulses have infinite amplitude at their respective
arrival times; however, the Rayleigh waves has finite amplitude. The dotted curves
Seismic wave in quarter and threequarter p b e
1
0
I
4
I
a
-ot/d
I
I2
119
I
I
I s 2 0
FIG.9. The horizontal u (upper curvus) and vertical v (lowercurves) components
of displacement in a threequarter plane (solid line) compared with results obtained
for a half plane (analytic solution, dotted line) and by finite differences (dashed
line). for an observer located on the free boundary at a distance of 5d from the
8 0 W .
in Figs 9 and 10 show the motion of the half plane as caused by this source. Except
for the time shift of +A and a small phase shift the Rayleigh waves for both sources
in the half plane have the same form on the surface. The dashed curve in Fig. 13
shows the particle ellipse for A = 0.
Figs 11 and 12 show a comparison between the motion of the quarter plane
and the half plane. Here, the v components of the Rayleigh waves in the two geometries are similar while the largest difference occurs in the u-components. The
u-component of the Rayleigh wave in the quarter plane is more than twice as large
120
Z. S. Alterman and D. Loeweothal
-7
-0.2T
i
2-0.3-
I
-
-0.4
Pj.:
- 0.5 -
i
f
Ps
-0.6
I
I
4
8
-at/d
I
12
I
16
I
20
' 0
-0.1
f
P(D)
FIG.10. The horizontal u (upper curves) and vertical Y (lower curves) components
of displacement in a three-quarter plane (solid line) compared with results obtained
for a half plane (analytic solution, dotted line) and by finite differences (dashed
line), for an observer located at the depth of the source at 5d distance from the
source.
as in the half plane. After the arrival of the various waves, u in the quarter plane
tends to zero while in the half plane it tends to 0.28d. The u component in Fig. 11
shows that P , R and R , R have larger amplitude than the direct P pulse. They
arrive at a?/d > 12. A comparison with Fig. 12 shows the attenuation of these
surface waves with depths.
In conclusion, the u-component of motion in the three-quarter plane is similar
to the u-displacement in the half plane. The motion differs in the u-components.
A comparison of the quarter and half planes shows the opposite-here the u-components (which are perpendicular to the free surface) are similar, and tht main
difference is in the u-components-i.e. parallel to the surface near which the observer
is located.
Seismic waves in quarter and threequarter plane
121
0.5
0.4
0.2
Pa
I
0.2
0.I
1
0
p
RCR
-0.1
I
0
1
4
1
8
-at/d
I
l2
1
16
1
20
,f-!
O
0.1. 1
0
3 0
f
-0.1 -
-0.2FIG.11. The horizontal u (upper curves) and vertical v (lower curves) components
of displacement in a quarter plane (solid line) and in a half plane (dashed line) at an
observer on the free boundary at a distance of 5d from the source.
122
Z.S. AltamPn and D. Loewenthal
1
0
'
0.2
-
0.1
-
I
4
I
0
-at/d
I
12
1
16
I
20
0
3
0-
-0.1
-
FIG.12. The horizontal u (upper curves) and vertical v (lower curves) components
of displacement in a quarter plane (solid line) and in a half plane (dashed line)
at an observer located at the depth of the source and at 5d distance from it.
The surface waves are largest in the quarter plane, less in the half plane and have
the smallest amplitude in the three-quarter plane. The particle motion in the surface
waves in the quarter plane shows the separate elliptic paths of R, P, R and R, R.
See Fig. 13. A comparison of surface waves at several distances from the source confirmsthe known result that in a half-plane the amplitude of the Rayleigh wave does not
change with distance. A phase shift with distance is found. Fig. 14 shows the
u-components of displacement at 3d, 4d and 5d distance from the source. The
surface wave in the quarter plane is almost unchanged, except for interference with
reflected waves and a slight increase in amplitude with distance. The surface wave
in the three-quarter plane shows some decrease in amplitude and a phase shift. The
corner-generated P, R and R, R are indicated in Fig. 14.
12. Source off-diagonal
For simplicity we have considered up to now a source located on the diagonal
axis of symmetry. However the same analysis applies also when the source is located
at an arbitrary point in the elastic medium.
123
Seismic waves in quarter and threequarter plane
ELEee
J/2 Plane
.25-
t
.I5 -
.05
-
\
R
.05
i-.05 -
-
i-.05 -
,
-.I5
-
u
0.3--u
0.5 -.25
02
I
-.05-u
1
0.4
-.I5
I
.05
1
.I5
.25
-0.2
-0.3
-U
-0.4
-0.5
FIG.13. Elliptic particle motion at the corner of a quarter plane (middle) and a
three-quarter plane (right) as compared with the particle motion in a half plane
(left) dashed line-analytic solution, solid line-finite difference solution.
In
n
0.26
lo
0.0
5d
4d
-0.13
3d
V
1
0
1
4
I
8
-aVd
I
I
R
0
I
d0
16
12
I
4
0
I
I
aVd
12
I
4
I
16
1
8
-Wd
I
12
I
16
I
20
FIG.14. Vertical component of displacement at a horizontal distance of 3d; 4d,
5d; from the source, on the surface of a quarter plane (upper left), half plane (upper
right) and a threequarter plane (lower).
124
2.6. Altemao and D. Loewentbl
13/21
-0.5t
w t
3 -1.0
t
-'.5t
- L
2.0
Y\I/I I I/t R
YIv Rt h
RCR
FIG.15. Vertical (upper) and horizontal (lower) components of displacement at
the comer of a quarter plane. The source is at a distance d from the free surface
x = 0 and at distances d, 17d/21; 13d/21 from the second free surface y = 0.
Let us consider the motion at the corner of the quarter plane when the source is
moved away from the diagonal. Fig. 15 shows the horizontal and vertical displacements u and v at the corner, when the source is at x = d and at y / d = 17/21
and y/d = 13/21 as compared with y = d.
Let us consider first the amplitude of the direct P-pulse. As expected, the maximum of the u component A@) increases with decreasing distance from the surface
y = 0, while the maximum A(v) of v decreases. However, on comparing the total
amplitude A = (A(u)'+A(v)')* with the amplitude of the incident wave A , we find
that A/A, is constant-independent of the location of the source. Several additional
source-locations in the range 0 < y < d were checked and in all cases A / A , = 2.9.
We see that the displacement at the corner has about three times the amplitude of
the incident pulse (for elastic constants fi/a = 0.55) and that this result is independent
of the location of the source.
125
j:/
t
0
-6
- - 0
5-
u/d
-4
4
-3
l-LA-A-.l
-5
--u/d4
-3
L
$4
__.c--J
--3U/d
-2
FIG.16. Particle motion at the corner x = y = 0 of a quarter plane for sources
at horizontal distance d from the free surface x = 0 and at vertical distances
5d/21; 8d/21; 9d/21 from the free surface x = 0 (below), compared with the
particle motion at x = y = 0 for the same sources in a half plane (above).
Table 3
The major axis of the particle ellipse: A in quarter plane; B in half plane
Location
of source
Yld
13/21
11/21
9/21
8/21
5/21
Ald
B/d
AIB
0.5
0.1
0.2
5.2
3.3
2.9
0.8
1.2
1.6
3.4
0.4
0.6
1.2
2.7
2.8
In the curve for y/d = 13/21, when the source is nearest to the surface we find
clearly the arrival of a Rayleigh wave. Its amplitude A is five times as large as the
amplitude B of a Rayleigh wave at the same point in a half space. When the source
is even nearer to the surface both A and B increase. Table 3 shows the variation of
A and B with depth of source. We see that A / B decreases when the source nears
the surface, i.e. the attenuation of the corner generated surface waves is smaller
than the attenuation of the Rayleigh wave in a half space. Fig. 16 shows the particle
motion. The upper curves are for the half plane and show the elliptic particle paths.
The major axis of the ellipses is in the u direction or near to it. The lower curves
are for the quarter plane. They show the larger amplitude of the elliptic particle
motion and an inclination of 45" between the major axes and the u and u directions.
126
Z. S. Altermen and D.Laweatha1
Acknowledgment
The computations connected with this paper were performed at the Computations
Centre of Tel-Aviv University.
Department of Environmental Sciences,
Tel-Aviv University.
References
Alterman, Z. & Rotenberg, A., 1969. Seismic waves in a quarter plane, Bull. seism.
SOC.Am., 59,347-368.
Alterman, Z . & Aboudi, J., 1968. Pulse propagation in a laterally heterogeneous
fluid sphere by finite difference methods, J. Phys. Earth., 16, 173-193.
Alterman, Z. & Karal, F., 1968. Propagation of elastic waves in layered media by
finite difference methods, Bull. seism. SOC.Am., 58, 367-398.
Alterman, Z. & Loewenthal, D., 1969. Algebraic expressions for the impulsive
motion of an elastic half space, Is. J. Tech., 7,495-504.
Ewing, W. M., Jardetzky, W.S. & Press, F., 1957. Elastic waves in layered media,
McGraw-Hill, New York.
Gangi, A. F., 1967. Experimental determination of P wave/Rayleigh wave conversion coefficients at a stress-free wedge, J. geophys. Res., 72, 5685-5692.
Garvin, W. W., 1956. Exact transient solution of the buried line source problem,
Proc. R. SOC.A , 234, 528-541.
Lamb, H., 1904. On the propagation of tremors over the surface of an elastic solid,
Phil. Trans. R. SOC.(Lond.) A , 203, 1-42.
Lapwood, E. R., 1961. The transmission of a Rayleigh pulse round a corner,
Geophys. J. R. mtr. SOC.,4, 174-196.
Ma], A. K. & Knopoff, L., 1966. Transmission of Rayleigh waves at a corner,
Bull. seism. SOC.Am., 56, 455466.
Roever, W. L., Vining, T. F. & Strick, E., 1959. Propagation of elastic wave motion
from an impulsive source along a fluid solid interface 11, Phil. Trans. R . Soc.
(Lortd.), A , 251,455-423.
Shanks, S. L., 1967. Recursion filters for digital processing, Geophysics, 32, 33-51.