Journal of Applied Mathematics and Physics, 2013, 1, 54-57
Published Online November 2013 (http://www.scirp.org/journal/jamp)
http://dx.doi.org/10.4236/jamp.2013.15008
Open Access JAMP
A Semi-Analytical Method for the Solution
of Helmholtz Equation
M. Tadi
Department of Mechanical Engineering, University of Colorado at Denver, Denver, USA
Received October 2, 2013; revised October 30, 2013; accepted November 5, 2013
Copyright © 2013 M. Tadi. This is an open access article distributed under the Creative Commons Attribution License, which per-
mits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
This note is concerned with a semi-analytical method for the solution of 2-D Helmholtz equation in unit square. The
method uses orthogonal functions to project the problem down to finite dimensional space. After the projection, the
problem simplifies to that of obtaining solutions for second order constant coefficient differential equations which can
be done analytically. Numerical results indicate that the method is particularly useful for very high wave numbers.
Keywords: Helmholtz Equation; Elliptic Systems; High Wavenumber
1. Introduction
In this note we consider a numerical method for the solu-
tion of 2-D Helmholtz equation. The goal is provide a
solution method that is more suitable for Helmholtz
equation at high frequencies and can be applied to the
Helmholtz equation in 3-D.
Helmholtz equation appears very naturally in the study
of wave propagation [1], after assuming a harmonic field.
It is well-known that the numerical solution of the
Helmholtz equation is a challenging problem for high
frequencies. Higher order finite difference schemes have
been developed for Helmholtz equation with homogene-
ous domain [2-4]. Additional methods such as boundary
element [5], discontinuous Galerkin method [6], multi-
level multi-integral algorithm [7], iterative methods [8],
and methods based on parallel computing [9] have also
been developed for the numerical solution of Helmholtz
equation. Recent results also include an iterative method
based on ADI [10], a finite-element based semi-analytic
method [11], and a method based on discrete singular
convolution [12].
The method presented in this note is based on or-
thogonal functions. In addition to good accuracy for high
wave numbers, it has the following specific new fea-
tures.
● The actual calculation is performed analytically in
1-D only.
● It can also be applied to 3-D domains.
Section 2 introduces the basic principles of the method.
Section 3 presents the numerical results and Section 4 is
devoted to the concluding remarks.
2. Mathematical Formulations
Consider a 2-D Helmholtz equation given by


20,,0,1 ,
xx yy
uuku xy  (1)








 
0,, 1,,
,0, ,1,
we
sn
uyfyuyfy
uxfx uxfx


where, for simplicity, the domain is assumed to be a unit
square. A set of Dirichlet boundary conditions are also
given. In the present method, the solution to the above
system is obtained as a linear combination of two similar
problems. This is only to satisfy nonzero boundary con-
ditions (Figure 1).
The two problems can now be similarly treated. Con-
sider the first problem, where the boundary conditions at
0x

and 1x

are set equal to zero. Consider an ex-
pansion of the solution in the form
 
1
,sin
i
j
uxyf yjx





 (2)
where, the zero boundary conditions are satisfied auto-
matically. A similar formulation can be used for the
Helmholtz equation in 3-D (Appendix). Multiplying both
sides of Equation (2) by , integrating both
sides fover the domain

sin jx



0:1 , and using the orthogonal-

M. TADI 55
Figure 1. The decomposition of the elliptic problem to two elliptic problems with apporipriate boundary conditions.
ity condition
 
1
0
1if
sinsind 2
0otherwise
ij
ixjx x







it is possible to obtain a relationship for the functions

j
f
y given by
 
1
0
2,sin d
j.
f
yuxyjx
x (3)
Differentiating the above equation with respect to
twice and using Equation (1) leads to
y
  
211
2
200
d2sin d2sin d
djxx .
f
yujxxkujx
y 

x
(4)
Integrating the first integral on the right-hand side by
parts twice leads to
 

2
11
20
11
2
00
d2sin cos
d
sind2sind .
jx
fujxjujx
y
ju jxxku jxx

 





 



0
After applying the boundary conditions, the above
equation simplifies to
 


22
2
2
d0.
dj
fy kjfy
y 
j
(5)
The above equation is now a simple constant coeffi-
cient second-order differential equation. The appropriate
boundary conditions are given by
 
 
1
0
1
0
02 ,0sind,
12 ,1sind.
j
j
f
uxjx x
f
uxjx x



 (6)
The second-order differential equation can be solved.
There are three separate cases.
Case 1: , in this case the solution is
given by


2
2
kj

 
 
10cos
0cossin ,
sin
jj
jj
ff
f
yf yy




 
(7)
where,

2
2.kj
 
1
Case 2:



2
2
kj

, in this case the solution is
given in terms of exponential function. One needs to
consider two separate cases.
●



22smalljk , or . In this case
the solution is given by


22
ejk 







1e01e 0
ee
eee e
jj jj
,
y
y
j
ff ff
fy


 






(8)
where,



22
jk  
.



22
jk For this case, in order to avoid un-
bounded functions, one needs to approximate e0
 

.
For this case, the solution is given by
 

 

1
1e01ee.
y
y
jj jj
fy fff
  
 (9)
Case 3:



2
2
kj

 This is the singular case. For
these cases, it is possible to treat the dif
according to the following. For , the
differential equation leads to
ferential equation


2
2
kj
 
2
2
d0.
djj
fy fy
y


 (10)
It is possible to obtain a solution in terms of the regu-
lar perturbation according to








012233
jj jjj
fyf yfyf yf y

 
(11)
The boundary conditions are imposed at the zero-th
order. For higher-order terms, zero boundary conditions
can be imposed.







0
0
01 0
jjj
fy ffyf . (12)
Additional terms can be added as needed. This com-
pletes the solution for the first problem where only two
of the boundary conditions are accounted for. The second
problem can also be treated in a similar way.
1The case where


sin 0 

is the degenerate case [13]. We are not
treating that case here.
Open Access JAMP

M. TADI 56
3. Numerical Experiments
In this section, we use a numerical example to investigate
the applicability of the proposed method. The proposed
method is particularly useful for very high values of the
wave number. For these values none of the existing nu-
merical methods can be applied within the available
computer capacity. It is possible to use a problem for
which analytical solutions exist [2]. An exact solution for
the problem is given by
  

,coscos sinuxy kxy   (13)
where,

is the angle of the incoming wave. Using the
exact solution one can provide the boundary conditions
for the numerical method. The domain is divided into
equal intervals in both
x
and . Figure 2 presents the
numerical results for . The figure shows the
error as a function of the number of orthogonal functions.
There is little dependency to the mesh size ne. The error
is the L2 norm of the difference between the numerical
result and the exact solution divide by the number of
nodes.
y
2000k
Figures 3-5 show the same error reduction for higher
wave numbers. Figure 3 presents the reduction in the
error for . The number of intervals is equal to
. For higher wave numbers, the same reduction
in error is observed. For higher values of k, the method
15000 k
1600
e
n
Figure 2. Reduction in the error as a function of the num-
ber of orthogonal functions. Other parameters are
, 2000k4
 , , and the three maximum
number of orthogonal functions are .
e800n
600,800,1600
orth
n
Figure 3. Reduction in the error as a function of the num-
ber of orthogonal functions. Other parameters are
, 15000k4

, , and the three maximum
number of orthogonal functions are
.
e1600n
3600,
orth
n4200,4800
Figure 4． Reduction in the error as a function of the
number of orthogonal functions. Other parameters are
25000k

, 4



, e1600n

, and the three maximum
number of orthogonal functions are
6400,
orth
n7200,8400,10800


Figure 5． Reduction in the error as a function of the
number of orthogonal functions. Other parameters are
100000k

, 4

  , , and the three maximum
number of orthogonal functions are
e1600n
3620022200,
orth
n30000,8400,


simply requires the inclusion of more orthogonal func-
tions in the expansion given in Equation (2). Figure 5
presents the result for and a similar reduction in
the error is obtained.
5
10k
The present method is particularly useful for the
Helmholtz equation at higher frequency for which exist-
ing methods require a large amount of memory. The ac-
tual calculation for the present method is done analyti-
cally in one-dimension.
4. Conclusion
In this note, we presented a numerical method for the
solution of 2-D Helmholtz equation. The method can be
applied to 3-D domains. It is based on orthogonal func-
tions. Apart from the projection of the problem onto the
space of orthogonal functions, the solution is obtained
analytically. Numerical results for a number of cases
with very high wave numbers were presented.
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M. TADI
Open Access JAMP
57
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Appendix
In 3-D one can assume a form given by
 
11
,,sin sin,
ij
ij
uxyzf yixjz






d
(14)
here, after using the orthogonality conditions, the func-
tions are given by
)( yfij
   
11
00
4,,sinsind
ij .
f
yuxyzixjzx
 z
(15)
Also, for the 3-D Helmholtz equation, one needs to split
the problem into three similar problems.