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IIUM Engineering Journal, Vol. 2, No. 2, 2001  M. Azram 

 
29

ANTI-PLANE SURFACE DISTURBANCE 

M. Azram 

Department of Science in Engineering, Kulliyyah of Engineering 
International Islamic University Malaysia, Jalan Gombak, 53100 Kuala Lumpur

Abstract:  Solution to a problem of half-space subjected 
to anti-plane surface disturbance via Cagniard-de Hoop 
method has been investigated.  The solution is very 
much dependent on the choice of branch cuts, branches, 
and their behavior along the appropriate selected 
contour.  Investigations show that the contour, which 
simplifies the inversion integral, may or may not exist 
in some quadrants of -plane and is very much 
dependent on the position of the point . Consequently, 
the solution is the Hankel function of first kind 

( )(1)02
i

H k r . 

 
Keywords: Cagniard-de Hoop method. Inversion 
Integral. Hankel function. Anti-plane Shear. Waves 
propagation.  Fourier Transform. 

1. INTRODUCTION 

Surface disturbances give rise to waves.  Waves are a 
ubiquitous and important feature of the physical world, 
and throughout the history it has been a major challenge 
to understand them.  They can propagate on surfaces of 
solids and of fluids.  Chemical waves control the 
heartbeat.  In recent years, elastic waves transmitted 
through the human body have been used for medical 
diagnoses and therapy.  Waves propagation as one of 
basic phenomena in solids is of both theoretical and 
practical importance.  Waves propagation in elastic 
solids is a very broad and attractive research field.  The 
study of waves propagation in elastic solids has a long 
and distinguished history [1]. 
Cauchy and Poisson have studied the propagation of a 
disturbance in an elastic aether as early as middle of the 
19th  century.  Their contribution and work is now, 
generally known as the “theory of elasticity”.  Early 
investigations on the propagation of waves in elastic 
solids have been carried out by Cauchy, Clebsch, 
Christoffel, Green, Lame, Ostrogradsky, Poisson, and 
Stokes.  This work has been briefly discussed in the 
book “The mathematical theory of elasticity” written by  
Love [2]. Another classical book is by Timoshenko [3].   
Lamb, Love, and Rayleigh contributions in this area are 
of significant importance. 
Many professionals in engineering, physics, geology, 
mathematics, and life science encounter problems 
requiring an understanding of elastic waves.  The study 
of wave propagation in elastic solids have stimulating 
and wide-ranging engineering applications, such as 
earthquake phenomena, nuclear explosion, seismology 
[4-6], structure under high rate of loading, high-speed 
machinery, ultrasonics and piezoelectric, material 

science, applied mechanics, in-situ safety and reliability 
control of complex structural components by acoustic 
emission, quantitative non-destructive material testing 
by ultrasonics, dynamic fracture mechanics, fluid 
mechanics, aerodynamics, anistrophy, geophysics, and 
elastodynamics[7], etc.  For other significant 
contributions in the field of engineering, reader is 
encouraged to consult references [1,8-13]. 
Parallel to the study of waves in elastic solids, the 
propagation of waves has also been investigated within 
the context of applied mathematics, electromagnetic 
theory and acoustics[11, 14-19]. 
Exposure to mathematical formulation of governing 
equations of wave motion is essential to understand the 
basic principles of wave motion in solids.  A wide 
range of problems in this area has been studied based 
on mathematical, physical, experimental and numerical 
approaches.  Consequently, large numbers of methods 
have been developed in this domain.  Currently, the 
most commonly used methods are finite differences, 
finite elements[20, 21] , spectral methods and boundary 
elements, soap-film method, photo elastic method, 
integral transform, discrete Fourier transform, 
Maliuzhinets’ method, numerical modeling, Wiener-
hopf technique, and Cagniard-de Hoop method.  
Numerical modeling of elastic waves is important in 
many applications, such as geophysics, non-destructive 
testing, and fluid-structure interaction.  The Wiener-
hopf technique[22] has, since its invention in 1931, 
proved to be an elegant method in engineering and 
mathematical physics.  Advantage of this method is that 
it provides an exact solution of integral equations.  
Geophysics, elastic and electromagnetic waves, 
diffraction of acoustic, fracture mechanics etc. are in 
the domain of its applications.  Parallel to Wiener-hopf 
technique, an immensely important method is the 
Cagniard-de Hoop method.  It is within the context of 
the application of Laplace transform but a considerable 
amount of complex analysis is required.  This method 
was firstly established by[6] and then modified by de 
Hoop[23].   

2. BACKGROUND 

Achenbach [1], has used the Cagniard-de Hoop Method 
to solve a problem of half-space subjected to anti-plane 
surface disturbance i.e., he considered the half-space y 
 0 subjected to the surface disturbance of the form  (at 
y = 0 ) 

( )
i t

yz T x e
w

t m=  (1) 



IIUM Engineering Journal, Vol. 2, No. 2, 2001  M Azram 

 30

Where yzt  is the Shear Stress,  is elastic constant, 
T(x) is the strain,  is the circular velocity, and t is the 
time. The anti-plane shear disturbance generates a 
deformation which is governed by  

2 2 2

2 2 2 2
1
T

w w w
x y c t

¶ ¶ ¶
+ =

¶ ¶ ¶
 (2) 

Where w(x,y,t) is the displacement in z-direction and cT 
is the transverse velocity.  If the surface disturbance has 
been operating for a long time, it is reasonable to 
assume that a steady state has been reached, and we 
may consider the displacement in the form 

( , , ) ( , ) i tw x y t w x y e w=  (3) 

The governing equation for w(x,y) follows from (2) as a 
boundary value problem  

( )

2 2
2

2 2

0

0T

y

w w
k w

x y
w

T x
y =

¶ ¶
+ + =

¶ ¶
¶

=
¶

 (4) 

where T
T

k
c
w

= is the wave number, exponential 

Fourier transform along with inverse transform are 
defined as; 

*( ) ( )
i x

f e f x dx
x

x
¥

- ¥
= ò  (5) 

*1( ) ( )
2

i xf x e f dx x x
p

¥ -

- ¥
= ò  (6) 

Exponential Fourier transform (5) applied to (4) yields; 

( )
2 2*

*
2 2

( )
, T

k y

T

T
w y e

k
xx

x
x

- -
= -

-
   (7) 

where *( )T x  is the Fourier transform of T(x).  Since the 
solution w(x,y) is required to be bounded as y, 
hence, w*(,y) in (7) is made bounded as y by 
making the multiple-valued function 2 2Tkx -  as 
single-valued by introducing the branch cuts emanating 
from the branch points  =  kT.  Branches are chosen 
such that 2 2Re( )Tkx -  is non-negative on the real -

axis.  In addition, we choose 2 2Tkx -  positive for 
Re() > kT, Im() = 0.  The exact evaluation is carried 
out by a specific choice of T(x) = (x), where (x) is the 
Dirac delta function, which yields T*() = 1. 
Consequently, the inverse Fourier transform of (7) 
yields the inversion integral  

2 2

2 2

1 1
( , )

2
Ti x k y

c T

w x y e d
k

x x
x

p x
- - -

= -
-ò

 (8) 

Where c is parallel but above and below to the negative 
and positive real axis respectively, as shown in Fig. 1.  
 

 

 

Fig. 1: Path in the x -plane. 

With the introduction of x = r cos  and y = r sin , the 
integral equation (8) reduces as 

{ }2 2cos sin
2 2

1 1
( , )

2
Ti r i k

c T

w x y e d
k

x q x q
x

p x

- - --
=

-ò
 (9) 

This inversion integral is equivalent to the most often 
used inversion integral in elastodynamics given by  

{ }2 2cos sin( , ) ( , , ) T
i r i k

c

w x y f r e d
x q x q

q x x
- -

= ò  

where the contour c is in the complex  - plane. 
Application of the Cagniard-de Hoop method was the 
standard approach to find the solution, in which focus 
was to search a contour c1 such that whenever  is on 
c1,  

2 2cos sinTS i kx q x q= - - .  
This equation can be solved for x as: 

2 2cos sinTS i S kx q q= ± - , which yields the 

hyperbola 
( ) ( )

2 2
Re Im

1
cos sinT Tk k

x x
q q

é ù é ù
ê ú ê ú- =ê ú ê úë û ë û

 provided 

2 2 0TS k- ³ with S to be real. 

The required contour was branch of this hyperbola that 
lies in the right- half of the  - plane as shown in Fig. 2.  
Consequently, the multiple valued function 2 2Tkx -  
was made single-valued function by introducing branch 
cuts emanating from the branch points Tk±  and going 
to the left and right along the real axis.  The branch was 
chosen which has positive real part for  on c1.  The 
ultimate solution was the Hankel function of second 

kind (2)0( , ) ( )2 T
i

w x y H k r= . 

Noble[22] considered the same inversion integral with 
the assumption that Tk  is complex and  
 = - Tk  cos( + i  t), where - < t < . 
 



IIUM Engineering Journal, Vol. 2, No. 2, 2001  M Azram 

 31

 

Fig. 2: Change of contour in the -plane. 

Consequently, the solution was  
(1)
0 ( )2 T

i
H k r  if Tk  has small positive imaginary part 

and  
(2)
0 ( )2 T

i
H k r  if Tk has small negative imaginary part.  

Both Achenbach[1] as well as Noble [22] searched the 
solution(s) by simplifying the inversion integral.  
Solution was very much dependent on the choice of 
branch cuts, branches, and their behaviors along the 
appropriate selected contour.  Different modes of 
solution obtained for the same problem required further 
investigations of the inversion integral and the 
corresponding solution. 

3. INVESTIGATION AND DISCUSSION 

Consider the multiple valued function 
2 2( ) Tf kx x= -  with two of its branches 1( )f x and 

2 ( )f x  defined by 

1 2)(
2

1 1 2( )
i

f r r e
q q

x
+

=  (10) 

Where  
1 2

1 1

2 2

0, 0,

arg( ); 0 2 ,

arg( ); .

T T

T

T

r k r k

k

k

x x

q x q p

q x p q p

= - > = + >

= - < <

= + - < <

 

1 2 )(
2

2 1 2( )
i

f r r e
q q

x
+

=  (11) 

Where  
1 2

1 1

2 2

0, 0,

arg( ); 2 4 ,

arg( ); .

T T

T

T

r k r k

k

k

x x

q x p q p

q x p q p

= - > = + >

= - < <

= + - < <

 

Now, let us investigate the mapping properties of f1 (): 
If  lies in the first quadrant then; 1 2max( )q q p+ =  

and 1 2min( ) 0q q+ = , which yields that 
( )1 20 2 2q q p< + < , concluding that 1

st quadrant 

maps into 1 st quadrant.  If  lies in the second quarter 
then; 1 2max( ) 2q q p+ = and 1 2min( )q q p+ =  

which yields ( )1 22 2p q q p< + < , concluding that 
2nd quadrant maps into 2nd quadrant.  If  lies in the 
third quarter then; 1 2max( )q q p+ =  and 

1 2min ( ) 0q q+ = , which yields that 

( )1 20 2 2pq q< + < , concluding that 3
rd quadrant  

maps into 1st quadrant.  Similarly; If   lies in the fourth 
quarter then; 1 2max( ) 2q q p+ = and 

1 2min( )q q p+ = , which yields 
( )1 2 22p q q p< + < , concluding that 4

th quadrant 

maps into 2nd quadrant.  

Line segment -k < Re()  0 and 0  Re () < k maps 
into 0 < Im()   k (twice) i.e., -k < Re() < k maps into 
0 < Im()  k. The positive and negative imaginary axes 
map into the ray Im() > k.  

It is interesting to observe that 1 2( ) ( )f fx x= - ; hence, 
we may also conclude that 2( )f x  maps; 1

st  quadrant 
into 3rd quadrant, 2nd quadrant into 4th  quadrant, 3rd  
quadrant  into 3rd quadrant, and  4th  quadrant  into 4th  
quadrant.  
Line segment -k < Re()  0 and 0  Re () < k maps 
into -k  Im() <  0 (twice) i.e., -k < Re() < k maps 
into -k  Im() <  0  The positive and negative 
imaginary axes map into the ray Im() > -k.  
To evaluate 

2 2

2 2

1 1
( , )

2
Ti x k y

c T

w x y d
k

e x x x
p x

- - -= -
-ò

,  (12) 

we need a branch of multiple-valued function 
2 2( ) Tf kx x= - , which has positive real part along 

the contour of integration.  The only possible contour 
for the first branch 1( )f x  with the condition that 

( )2 2Re 0Tkx - > is the contour as shown in figure 3.  

 

Fig. 3: Contour of 1( )f x in the  -plane. 

The only contour for the second branch 2 ( )f x  with the 

condition that ( )2 2Re 0Tkx - >   is the contour as 
shown in Fig. 4. 



IIUM Engineering Journal, Vol. 2, No. 2, 2001  M Azram 

 32

 

 

Fig. 4: Contour of 2 ( )f x in the x -plane. 

Hence, we conclude that 
( )1 2

2
2 1 2( )

i

f r r e
q q

x
+

=  is an 
appropriate choice.  Considering x = r cos  , y = r sin  
with 0 <  <  , the integral (Eq. 12) reduces as  

2 2

1 1
( , )

2
i rS

c T

w x y d
k

e x
p x

-= -
-ò

 (13) 

where 2 2cos sinTS i kx q x q= - - .  
To apply Cagniard-de Hoop method, we need a contour 
c1 in -plane such that whenever  is on c1, S must be 
real.  Let us consider 0 2q p< < , 2 ( )f x  maps the 

first  quadrant into the third quadrant and if 
1 2ix x x= + ,  where 1x  > 0, 2x  > 0, then 

2 2
3 4Tk ix x x- = - -  for some real 3x > 0, 4x  > 0, 

and 

2 2cos sinTS i kx q x q= - -  (14) 

1 4 2 3( cos sin ) ( cos sin )S ix q x q x q x q= - + + ,

 (15) 

wherein the imaginary part on right hand side can not 
be equal to zero.  Hence, we deduce that no part of c1 
can be in the 1st quadrant.  Similarly, it can also be 
shown that no part of c1 can be in the 4

th quadrant.  
Hence, we conclude that if c1 exists, it exists only in the 
left-half plane.  Similarly, if  1 0Tk k> >  and 1x x=  
in the 2nd quadrant then  

2 2
1 1 1cos sinTk i k kx q q= - + - , (16) 

which can be rewritten as  

2 2 2 2
1 1 1cos sinT Tk k k ikx q q- = - -   (17) 

thus, 

2 2
1 1 1cos sinTi k kx q x q- - = - . (18) 

Hence c1 exist in the 2
nd quadrant and S is negative 

along this contour. Similarly, taking  = 2 in the 3rd 
quadrant, it can be seen that c1 exist in the 3

rd quadrant 
and S is again negative along this contour. 

Hence, we deduce that for 0
2
p

q< < , c1 is the branch 

of the hyperbola 
( ) ( )

2 2
Re Im

1
cos sinT Tk k

x x
q q

é ù é ù
ê ú ê ú- =ê ú ê úë û ë û

, which 

lies in the left-half plane as shown in figure 5. 

 

Fig. 5: Change of contour as C1 in the  -plane. 

By considering appropriate arcs with large radii, 
contour c can be deformed as c1.  As radii , the 
integration vanishes and consequently, 

11
12 2

11

1 1
( , )

2
T

i r k

k T

w x y dk
kk

ex
p x

¥ ¶
= +

¶-ò
    

                  12 12 2
12

1 1
2

Tk
i r k

T

dk
kk

ex
p x¥

¶
¶-ò

. 

Considering 2 2 2 21 1 1cos sinT Tk k k ikx q q- = - -  and 
2 2 2 2
2 1 1cos sinT Tk k k ikx q q- = - - - , the above 

integral reduces the solution as Hankel function of first 

kind (1)0( , ) ( )2 T
i

w x y H k r= . Similarly, if 
2
p

q p< <  

then c1 will be that branch of the hyperbola  
2 2

Re Im
1

cos sinT Tk k
x x

q q
é ù é ù
ê ú ê ú- =ê ú ê úë û ë û

, which lies in the right-

half plane and the end solution is Hankel function of 

first kind (1)0( , ) ( )2 T
i

w x y H k r= . 

4. COMPARISON AND CONCLUSION 

Even earlier than 1973, Achenbach had investigated 
this problem of half-space subjected to anti-plane 
surface disturbance.  He obtained a solution 

(2)
0( , ) ( )2 T

i
w x y H k r= . This solution was obtained by 

assuming the pivotal parameter 
2 2cos sinTS i kx q x q= - -  to be real and positive. 

This parameter played significant role in applying the 
Cagniard-de Hoop method.   



IIUM Engineering Journal, Vol. 2, No. 2, 2001  M Azram 

 33

In our analysis, investigations were carried out in detail 
about the choice of branch cuts, branches and their 
behavior in different quadrants of the  - plane.  
Investigations show that the contour which simplifies 
the inversion integral, may or may not exist in some 
part of the   - plane  and is very much dependent on 
the position of the point.  To apply Cagniard-de Hoop 
method, the choice for the required contour c1 to 
simplify the inversion integral depends particularly on 

.  In the half-plane, if 
2
p

q p< < , c1 must be selected 

as that branch of hyperbola which lies in left-half of  - 
plane.  While for

2
p

q p< < , the contour c1 must be 

selected as that branch of hyperbola which lies in right-
half of -plane.  Corresponding to the contour in figure 
1, the contour c1 is as shown in figure 5 rather than as 
shown in figure 2.  The parameter S is real but negative 
for all value of 0 q p< < .  The ultimate solution is 

(1)
0( , ) ( )2 T

i
w x y H k r= . 

In this paper, we have discussed an immensely 
important problem from applications prospective but 
unique in nature because it has an exact solution.  Our 
investigations reflect that a good understanding of 
mathematical formulation and a considerable 
knowledge in applied mathematics are the essential 
ingredients to deal and understand the challenging 
phenomena of wave propagation. 

REFERENCES 

[1] J. D. Achenbach, “Wave Propagation in Elastic Solids”, 
North-Holland. 1984. 

[2] A. B. H. Love, “The Mathematical Theory of Elasticity”, 
N.Y, Dover Publications, Inc, 1944. 

[3] S. P. Timoshenko and J. N. Goodier, “Theory of 
Elasticity”, McGraw-Hill Co., 1987. 

[4] K. B. Bullen, “An Introduction to the Theory of 
Seismology”,  Cambridge, University Press, 1963. 

[5] W. M. Ewing, W. S. Jardetzky, and F. Press, “Elastic 
Waves in Layered Media”, NY, McGraw-Hill Co., 1957. 

[6] L. Cagniard, “Reflexion et Re fraction des ondes 
Seismimique Progressives”,  Gauthiers-Villars, 1939. 

[7] A. Schoch, “Schallreflexion, Schallbrechung und 
Schallbeugung”, Ergebnisse der Exakten 
Naturwissenschaften, Vol. 23, Springer-Verlag, 1950. 

[8] D. Gross, and Ch. Zhang, “On Wave Propagation in 
Elastic Solids with Cracks (Advance in Fracture 
Mechanics Series)”, WIT press, 1997. 

[9] J. Miklowitz, “Elastic Waves propagation, Applied 
Mechanics Surveys”, Spartan Books, 1966 

[10] B. A. Auld, “Acoustic Fields and Waves in Solids”, Vol. 
I, Krieger, 1990. 

[11] J. Billingham and A. C. King, “Wave Motion”, 
Cambridge University Press,  2001. 

[12] A. Bedford and D. S. Drumheller, “Introduction to 
Elastic Wave Propagation”, John Wiley and Sons, 1994. 

[13] H. Kolsky, “Stress Waves in Solids”, NY, Dover 
Publications,Inc., 1963. 

[14] L. Rayleigh, “The Theory of Sound”, Vol.(I & II), N.Y, 
Dover Publications, Inc., 1945. 

[15] L.M. Brekhovskikh, “Waves in Layer Media”, NY, 
Academic Press, 1960. 

[16] R. B. Lindsay, “Mechanical Radiation”, NY. McGraw-
Hill Co., 1960  

[17] P.M. Morse and K.U. Ingard, “Theoretical Acoustics”, 
NY, McGraw-Hill Co, 1968. 

[18] I. Tolstoy and C. S. Clay,  “Ocean Acoustics”, NY, 
McGraw-Hill Co., 1966. 

[19] J. L. Davis, “Mathematics of Wave Propagation”, 
Princeton University Press, 2000. 

[20] F. Moser, C. Valle, L. J. Jacobs, and J. Qu,  “Application 
of Finite Element Methods to Study Transient Wave 
Propagation in Elastic Wave Guides”, “Review of 
Progress in Quantitative NDE, Vol.16 A, 1998. 

[21] F. Moser, L. J. Jacobs, and J. Qu, “Application of Finite 
Element Methods to Study Wave Propagation in Wave 
Guides”, “NDT & E Int., Vol. 32, 1999. 

[22] B. Noble, “Wiener-Hopf Technique”, Chelsea Pub.Co., 
1988. 

[23] A. T. de Hoop, “A Modification of Cagniard’s method 
for Solving Seismic Pulse Problems”, Applied Science 
Research, B 8, 349-356, 1960. 

BIOGRAPHY 

Dr. Mohammad Azram received his Bachelor degree 
in Mathematics from University of Peshawar, Pakistan 
in 1974. He obtained his Master degree in Mathematics 
with a position from University of Peshawar, Pakistan 
in 1976. He obtained his second Master degree in 
Mathematics from University of Idaho, USA in 1985 
and then graduated with a Ph.D. from the same 
University in 1989. He started his career as a lecturer, 
Department of Mathematics, University of Peshawar, 
Pakistan in January 1977. He is now a professor in the 
Department of Mathematics, University of Peshawar, 
Pakistan. He has served the University of Peshawar in 
the capacity of Chairman, member Academic Council, 
member Board of Studies, member Affiliation 
Committee, member Peshawar University Teacher's 
Association, and Director of Postgraduate Programmes, 
Department of Mathematics, University of Peshawar, 
Pakistan. 
Currently, he is a professor at Faculty of Engineering, 
IIUM, Malaysia. In addition, he is also Deputy Dean 
(Academic Affairs), Centre for Postgraduate Studies, 
IIUM, Malaysia.