On the derivation of boundary integral equations for scattering
by an infinite one-dimensional rough surface
J. A. DeSanto and P. A. Martina)
Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden,
Colorado 80401-1887
~Received 24 June 1996; revised 10 December 1996; accepted 29 January 1997!
A crucial ingredient in the formulation of boundary-value problems for acoustic scattering of
time-harmonic waves is the radiation condition. This is well understood when the scatterer is a
bounded obstacle. For plane-wave scattering by an infinite, rough, impenetrable surface S, the
physics of the problem suggests that all scattered waves must travel away from ~or along! the
surface. This condition is used, together with Green’s theorem and the free-space Green’s function,
to derive boundary integral equations over S. This requires careful consideration of certain integrals
over a large semicircle of radius r; it is known that these integrals vanish as r→` if the scattered
field satisfies the Sommerfeld radiation condition, but that is not the case here—reflected plane
waves must be present. The integral equations obtained are Helmholtz integral equations; they must
be modified for grazing incident waves. As such integral equations are often claimed to be exact,
and are often used to generate benchmark numerical solutions, it seems worthwhile to establish their
validity or otherwise. © 1997 Acoustical Society of America. @S0001-4966~97!01406-9#
PACS numbers: 43.20.Fn, 43.30.Hw @ANN#
INTRODUCTION
Consider the scattering of a plane time-harmonic acoustic wave by a bounded obstacle. To fix ideas here, we consider a two-dimensional obstacle, with a smooth, sound-hard
surface S. Mathematically, this is an exterior Neu mann
problem for the Helmholtz equation. In order to have a wellposed problem, we impose the Sommerfeld radiation condition,
Ar
S
D
]u
2iku →0
]r
~1!
as r→`,
uniformly in all directions. Here, u is the scattered field, r is
a plane polar coordinate, k is the wave number, and we have
assumed a time dependence of e 2i v t . Physically, the radiation condition ensures that the scattered waves propagate
outwards, away from the obstacle. Mathematically, the radiation condition also yields uniqueness and existence for the
boundary-value problem.
A familiar method for solving the above problem is to
derive a boundary integral equation for the boundary values
of u on S. In the derivation, Green’s theorem is applied to
u and a fundamental solution G, in the region bounded internally by S and externally by C r , a large circle of radius
r. It turns out that the radiation condition implies that the
integral
I ~ u;C r ! [
ES
Cr
u
D
]G
]u
2G
ds→0
]r
]r
as r→`,
~2!
and so only boundary integrals over S remain. Thus, the
radiation condition is a crucial ingredient for two results: a
well-posed boundary-value problem; and the vanishing of a
a!
Permanent address: Department of Mathematics, University of Manchester, Manchester M13 9PL, United Kingdom.
67
J. Acoust. Soc. Am. 102 (1), July 1997
standard integral over a large circle. See Colton and Kress
~1983! for more information.
For a sound-hard surface S, the procedure described
above leads to the following boundary integral equation:
u~ p !2
E
u~ q !
S
]G
~ p,q ! ds q 5
]nq
E ]]
S
u inc
G ~ p,q ! ds q ,
nq
~3!
pPS,
here, u inc is the given incident wave. One can also derive an
equation for the boundary values of the total field
u tot5u inc1u; this boundary integral equation is
u tot~ p ! 2
E
S
u tot~ q !
]G
~ p,q ! ds q 52u inc~ p ! ,
]nq
pPS.
~4!
We shall refer to ~3! and ~4! as standard Helmholtz integral
equations. Similar equations can be derived for sound-soft
surfaces ~exterior Dirichlet problem, u tot50 on S).
Now, it is known that the waves scattered by a bounded
two-dimensional obstacle have the form
u ~ r, u ! 5
e ikr
Ar
f ~ u ! 1O ~ r 23/2!
~5!
as r→`,
where (r, u ) are plane polar coordinates, and f is called the
far-field pattern. Thus, apart from being outgoing (e ikr ), the
waves decay with distance from the obstacle (1/Ar). Indeed,
the radiation condition implies that u5O(r 21/2) as r→`.
As G also satisfies the radiation condition, we find that the
integrand in ~2! is
u
S
D S
D
]G
]u
2ikG 2G
2iku 5o ~ r 21 !
]r
]r
0001-4966/97/102(1)/67/11/$10.00
as r→`,
© 1997 Acoustical Society of America
67
whence I(u;C )→0 as r→`, where C is any piece of C r .
This shows that the result ~2! is due, essentially, to the decay
of u ~and G), not to any cancellation effects induced by the
integration.
The description given above changes completely when
the obstacle is unbounded. For example, suppose that S is an
infinite, flat plane. Then, an incident plane wave will be scattered ~reflected! specularly as a single propagating plane
wave. More generally, suppose that S is an infinite corrugated surface; then, an incident plane wave will be scattered
into a spectrum of plane waves. The specification of a ‘‘radiation condition’’ for such problems continues to attract attention ~see, for example, Ramm, 1986!; clearly, the Sommerfeld radiation condition is not appropriate, as it is not
satisfied by a propagating plane wave. Nevertheless, it is
customary to proceed, assuming that the scattered field can
be represented in terms of plane waves, at least at some
distance from S. Typically, this requires the discarding of an
integral such as ~2!, but with the large circle C r replaced by
a large semicircle H r . Can this step be justified? This paper
began as an attempt to do this.
Another possible approach is to assume that u can be
written as a surface distribution of sources or dipoles; see
Sec. VI E. One might also invoke the limiting absorption
principle, in which the wave number k is replaced by
k1i«, where « is small and positive; the corresponding u is
required to decay as r→`. However, this is delicate ~compared to scattering by a bounded obstacle! as the limits
«→0 and z→` for exp $iz(k1i«)% do not commute.
The motivation behind the present work is the pervasive
view that solving a boundary integral equation gives a rigorous, exact way ~apart from numerical errors! of solving problems involving the scattering of plane waves by infinite
rough surfaces. Indeed, one can find many books and papers
setting out this view. ~References to the literature will be
given later.! However, very little attention has been given to
the derivation of the boundary integral equations themselves,
most writers being content to start by writing down a standard Helmholtz integral equation, ~3! or ~4!. We will show
that ~3! is valid for plane-wave scattering by an infinite, onedimensional, rough surface. We will also show that ~4! is
valid, except for grazing incident waves ~in which case the
right-hand side should be replaced by u inc).
The paper is organized as follows. Section I is devoted
to formulating the problem, with some discussion on radiation conditions and some background on angular-spectrum
representations and integral representations ~using G). Estimation of integrals over the semicircle H r is carried out in
Secs. II, III, and IV. Thus, the method of stationary phase
and an expansion method are used in Secs. II and III, respectively, but only for a single plane wave. Results for
I(u;H r ) are obtained in Sec. IV, and are then used in Sec. V
to derive various boundary integral equations of Helmholtz
type. Extensive discussion of the results is given in Sec. VI.
For example, it is shown that the standard Helmholtz integral
equations are valid for a finite patch of roughness and for
finite incident beams.
68
J. Acoust. Soc. Am., Vol. 102, No. 1, July 1997
I. FORMULATION
Consider the scattering of a plane wave by an infinite
rough surface, S. In this paper, we assume that the surface is
one-dimensional, so that it can be described by
z5s ~ x ! ,
2`,x,`
with 2h,s(x)<0 and some constant h>0. The acoustic
medium occupies z.s and, for definiteness, we assume that
S is a smooth, sound-hard surface. Thus we can write the
total field as
u tot5u inc1u,
where u is the scattered field and
u inc~ r, u ! 5e 2ikr cos~ u 1 u i ! ,
u u i u < 21 p ,
~6!
is the incident plane wave; u i is the angle of incidence ~it is
the angle between the direction of propagation and the negative z axis!, and (r, u ) are plane polar coordinates:
x5r sin u and z5r cos u. All the fields u tot , u inc , and u
satisfy the Helmholtz equation
~ ¹ 2 1k 2 ! u50,
~7!
for z.s. The boundary condition is
] u tot
50
]n
~8!
on S,
where ] / ] n denotes normal differentiation out of the acoustic medium.
A. Reflection by a flat surface
If S is flat (s50), the problem is trivial. Nevertheless,
this problem can still teach us something. It is well known
that the scattered field is
u ~ r, u ! 5e ikr cos~ u 2 u i !
for u u i u , 21 p .
~9!
When u u i u 5 21p ~‘‘grazing incidence’’!, we have u[0: The
incident wave satisfies the boundary condition on S.
Thus, for u u i u , 21p ,
u tot52 e ikx sin u i cos~ kz cos u i !
and
2 e ikx sin u i cos~ kzcos u i ! 1A 1 e ikx 1A 2 e 2ikx
both ‘‘solve’’ the problem, where A 1 and A 2 are arbitrary
constants. Of course, we disallow this second solution, unless A 1 5A 2 50: but why? The answer is because of the
radiation condition ~which we have yet to specify!. Physically, we want to exclude all ‘‘incoming’’ plane waves, apart
from the incident wave. We will be more precise in Sec. I B.
B. Angular-spectrum representations
For any S, the scattered field above the corrugations,
z.0, may be written using an angular-spectrum representation,
J. A. DeSanto and P. A. Martin: Rough surface integral equations
68
u ~ x,z ! 5
5
E
E
`
F ~ m ! e ik ~ m x1mz !
2`
p /2
dm
m~ m !
~10!
A ~ a ! e ikrcos~ u 2 a ! d a
2 p /2
~11!
1 evanescent terms.
Here, F( m ) is the spectral amplitude, A( a )5F(sin a! and
m~ m !5
H
A12 m 2 ,
i Am 2 21,
u m u ,1,
1
H6
r 5 $ ~ r, u ! :0<6 u < 2 p %
u m u .1.
The integrals are superpositions of plane waves; they are
propagating, homogeneous plane waves when u m u ,1 and
they are evanescent, inhomogeneous plane waves when
u m u .1 In ~11!, we see the propagating plane waves explicitly: They propagate at an angle a to the positive z axis, with
an ~unknown! complex amplitude, A( a ); the ‘‘evanescent
terms’’ decay exponentially with z. For more information on
angular-spectrum representations, see Clemmow ~1966! and
DeSanto and Martin ~1996!.
In general, the spectral amplitude must be considered as
a generalized function, and not as a continuous or analytic
function. This simple observation is motivated by known results for particular surfaces. Thus, for a flat surface we have
u u i u , 21 p ,
F ~ m ! 5 d ~ m 2sin u i ! cos u i ,
where d is the Dirac delta function, whereas for a periodic
surface F is a discrete sum of delta functions. So, we split
the scattered field into three parts as
~12!
u5u pr1u ev1u con ,
where
(
n51
being quarter-circles, we require that in the region x>0,
z.0, we use 0< a n < 21p , so that all plane waves propagate
out through H 1
r . Similarly, in the region x<0, z.0, we use
2 21p < a n <0, so that all plane waves propagate out through
H2
r . This form of the radiation condition will be used to
derive boundary integral equations.
C. Boundary integral equations
One way to determine the scattered field is to derive a
boundary integral equation over the rough surface S. The
appropriate fundamental solution is
G ~ P,Q ! 5G ~ y,x! 5 2 21 iH ~01 ! ~ k u x2yu ! ,
where x and y are the position vectors of Q and P, respecis a Hankel
tively, with respect to the origin O, and H (1)
n
function. Apply Green’s theorem to u and G in the region
D r with boundary ] D r 5H r øS r øT r , where H r is a large
semicircle of radius r and center O,
S r 5 $ ~ x,z ! :z5s ~ x ! ,2r,x,r %
A n v~ r, u ; a n ! ,
M
u ev~ r, u ! 5
(
B m w ~ r, u ; m m ! ,
E
C ~ m ! e ikr ~ m sin u 1m cos u !
m51
u con~ r, u ! 5
`
2`
v~ r, u ; a ! 5e ikr cos~ u 2 a ! ,
~13!
dm
,
m~ m !
with u a u < 21 p ,
~14!
~15!
2u ~ P ! 5
Am 2 21
The first term in ~12! is a sum of propagating plane waves;
the coefficients A n and the angles a n are unknown in general. The second term in ~12! is a sum of evanescent waves;
the coefficients B m and m m are unknown in general. The
third term in ~12! is a continuous spectrum of plane waves;
the unknown function C is continuous. Properties and consequences of the general representation ~12! were investigated by DeSanto and Martin ~1996!.
Let us now return to the radiation condition. Having
chosen an origin O, arbitrarily, we consider a large semicircle H r , with radius r and center at O. We then require
J. Acoust. Soc. Am., Vol. 102, No. 1, July 1997
u~ q !
]Dr
J
]G
]u
G ~ P,q ! ds q ,
~ P,q ! 2
]nq
]nq
2u ~ P ! 5
EH
u~ q !
Sr
J
]G
] u inc
G ~ P,q ! ds q
~ P,q ! 1
]nq
]nq
~18!
1I ~ u;H r ! 1I ~ u;T r ! ,
with u m u .1.
~16!
69
E H
where PPD r , qP] D r and ] / ] n q denotes normal differentiation at q. Use of the boundary condition ~8! yields
and
w ~ r, u ; m ! 5e ikr m sin u e 2kr cos u
~17!
is a truncated rough surface, and T r consists of two line
segments joining the ends of H r and S r . Then, as both u and
G satisfy the Helmholtz equation ~7! in D r @apart from the
singularity in G( P,Q) at P5Q], we obtain
N
u pr~ r, u ! 5
that all propagating plane-wave components v (r, u ; a n ) in
u propagate outwards through H r , away from O. This is
almost built into the decomposition ~12!: we have to be careful with grazing waves ( u a n u 5 21p ). For example, if a n
5 21p , v 5e ikx ; this wave leaves the semicircle at u 5 21p but
enters at u 52 21p . A simple way to impose our radiation
condition without excluding grazing waves is to split the
half-space z.0 and the semicircle H r into two parts. Thus
with
where
I ~ u;S ! 5
EH
S
u~ q !
J
]G
]u
G ~ P,q ! ds q
~ P,q ! 2
]nq
]nq
and normal differentiation is taken in a direction away from
the origin @so that ] / ] n5 ] / ] r on H r , consistent with ~2!#.
The scattered field u and its derivative ] u/ ] x are
bounded in the neighborhood of S. This assumption together
with simple bounds and the large-argument asymptotic behavior of Hankel functions show that I(u;T r )5O(r 21/2) as
r→`, whence
I ~ u;T r ! →0
as r→`.
J. A. DeSanto and P. A. Martin: Rough surface integral equations
~19!
69
Before estimating I(u;H r ) for large r, using ~12!, we consider a single propagating plane-wave component in ~12!.
Thus, we shall evaluate I( v ;H r ) as r→`, where v is defined
by ~15!. Indeed, we shall evaluate the limit using two different methods: the method of stationary phase ~Sec. II! and an
expansion method ~Sec. III!. We shall discuss the evaluation
of I(u;H r ) itself for large r in Sec. IV. Boundary integral
equations will then be derived from ~18! in Sec. V.
We use the method of stationary phase to estimate
I( v ;H r ). There are two cases, depending on the value of
a , which can be smoothed together using a uniform approximation.
A. The method of stationary phase: z a z < 21p
We are interested in large values of r5 u xu for fixed
values of y and k. We have
5
exp$ ik ~ r2y•x̂! %
Akr
Akr
e
e
1O ~ l 21 !
as l→`, where F5l sin a2kr sin w. Hence, from ~21!,
we obtain
as kr→`, for u a u , 21 p .
~23!
B. The method of stationary phase: z a z 5 21p
Suppose that a 5 21p . In this case, F( u )5sin u is stationary at u 56 21p , which are end points of the range of
integration. We have
g ~ 21 p ! 50 and g ~ 2 21 p ! 52 e ik r sin w .
I ~ v ;H r ! 5e ik r sin w 1O„~ kr ! 21/2…
~20!
Hence, for large r,
B
]G
]v
2G .ik
@ 12 cos~ u 2 a !#
]r
]r
Akr
3e
as l→`,
~24!
as kr→`, for a 5 21 p .
~25!
When a 52 21p , we obtain the same result except that w is
replaced by 2 w . In this case, the relevant stationary-phase
point is at u 5 21p .
When r 50, we can give an independent check of the
result ~24!. In this case, we have
L ~ l ! 5 Al
ikr ~ 11 cos~ u 2 a !! 2ik r cos~ u 2 w !
E
p /2
~ 12sinu ! e il sin u d u
2 p /2
e
5 p Al $ J 0 ~ l ! 2iJ 1 ~ l ! % ,
and then
I ~ v ;H r ! .iBe ikr L ~ kr ! ,
~21!
where
L ~ l ! 5 Al
E
p /2
g~ u !e
ilF ~ u !
du,
A
2 p /2
F ~ u ! 5 cos~ u 2 a ! .
For large l[kr, the dominant contribution to L(l)
comes from those points c in the range of integration at
which the phase F is stationary: F 8 (c)50. As
F 8 ( u )5 sin(a2u) and u a u , 21p , the only stationary-phase
point is at u 5 a . Then ~Bleistein and Handelsman, 1986, p.
220!
L ~ l ! ;Bg ~ a ! e ilF ~ a !
as l→`,
~22!
where
H
S
2
1
1
cos l2 m p 2 p
pl
2
4
D
as l→`. ~26!
C. Uniform asymptotics
and
A
where J m is a Bessel function. The result follows from the
well-known asymptotic approximation,
J m~ l ! ;
g ~ u ! 5 @ 12 cos~ u 2 a !# e 2ik r cos~ u 2 w !
70
@~ 12sin a ! e iF 1 ~ 11sin a ! e 2iF #
whence
ikr 2ik r cos~ u 2 w !
B52 12 i A~ 2/p ! e 2i p /4.
B5
Alcos a
L ~ l ! ; A2 p e i p /4e 2il e ik r sin w
as r→`, where x̂5x/r, y5( r sin w,r cos w) and
v
i
It follows that
B
B
L~ l !5
I ~ v ;H r ! 5O„~ kr ! 21/2…
II. THE METHOD OF STATIONARY PHASE
G ~ P,q ! .
But g( a )50, and so L(l)5o(1) as l→`. In fact, an integration by parts gives
J
2p
1
exp i p sgn F 9 ~ a ! 5 A2 p e 2i p /4.
u F 9~ a !u
4
J. Acoust. Soc. Am., Vol. 102, No. 1, July 1997
We have seen that the results for u a u , 21p and u a u 5 21p
are different, that is, the asymptotic estimate of I( v ;H r ) is
not uniform in the parameter a . However, we can obtain a
uniform approximation ~see Appendix A!; for example, if
a is near 21p , we find that
I ~ v ;H r ! . cos ~ 21 d ! e ik r sin w erfc~ m ! ,
~27!
where
erfc~ m ! 5
E
Ap
2
`
2
e 2x dx
~28!
m
is the complementary error function,
m 5 A2le 2i p /4 sin 21 d and d 5 21 p 2 a .
J. A. DeSanto and P. A. Martin: Rough surface integral equations
70
Note that if a 5 21p ( d 50), we recover ~25!. On the other
hand, if a , 21p ( d .0), we recover ~23!, since
erfc( m ); p 21/2m 21 exp(2m2) as m →`.
22i
G ~ u !5
p
(j
e i ~ 2 j11 ! u 4
5
2 j11
p
`
(
j50
sin~ 2 j11 ! u
,
2 j11
~33!
this is a familiar Fourier series:
III. AN EXPANSION METHOD
G ~ u !5
The integral I( v ;H r ) is over a semicircle of radius r,
between u 52 21p and u 51 21p . We can evaluate this integral explicitly, using appropriate expansions of v and G.
Thus
`
v~ r, u ; a ! 5e ikr cos~ u 2 a ! 5
(
m52`
i m J m ~ kr ! e 2im ~ u 2 a ! ,
~29!
henceforth, we suppress the limits when the summation is
over all integers. Similarly,
1
G ~ P,q ! 5 2 i
2
(n H n~ kr ! J n~ k r ! e
in ~ u 2 w !
~30!
for r. r , where H n [H (1)
n . Hence
I ~ v ;H r ! 5
H
1,
0,u,p,
21, 2p,u,0,
0,
and is defined by periodicity for other values of u . Hence,
for large kr, F n ;1 for 0< u a u , 21p but F n 5o(1) for
u a u 5 21p . Thus we obtain the same ~nonuniform! results as
derived in Secs. II A and II B. The drawbacks with this
method are that it does not yield results that are uniform in
a for a near 6 21p , and it is very complicated to use for
three-dimensional problems.
IV. ASYMPTOTIC BEHAVIOR OF I ( u ; H r )
When a plane wave is reflected by a rough surface S, we
can use the angular-spectrum representation ~12! for the reflected field above the corrugations. Thus we have
I ~ u;H r ! 5I ~ u pr ;H r ! 1I ~ u ev ;H r ! 1I ~ u con ;H r ! .
(m (n i m J n~ k r ! W mn~ kr ! A mn e i~ m a 2n w !,
For I(u ev ;H r ), with u ev defined by ~13!, we have
where
I ~ w;H r ! .iB Akre ikr
1
2
8 ~ w ! H n~ w !%
W mn ~ w ! 5 2 i p w $ J m ~ w ! H n8 ~ w ! 2J m
E
p /2
g ~ u ! e ikrF ~ u ! d u ,
2 p /2
where w is defined by ~16!,
and
1
A mn 5
p
5
E
H
p /2
e
i ~ n2m ! u
F ~ u ! 5 m sin u 1i Am 2 21cos u
du
and
2 p /2
1
g ~ u ! 5 @ 12F ~ u !# e 2ik r cos~ u 2 w ! ,
if m5n,
2 ~ 21 ! j
~ 2 j11 ! p
0
as u m u .1, integration by parts shows that
if m5n12 j11,
I ~ w;H r ! 5O„~ kr ! 21/2…
otherwise,
I ~ v ;H r ! 5
I ~ u ev ;H r ! →0
(n i n J n~ k r ! $ W nn~ kr ! 2F n~ kr, a ! % e in~ a 2 w !,
22i
p
1
(j 2 j11 W n12 j11,n e i~ 2 j11 !a .
~31!
We want to estimate I( v ;H r ) for large r. We can evaluate
the first term in the braces exactly: W nn is essentially a
Wronskian, given by W nn 51. For F n , we have
W mn ~ w ! ;exp$ i ~ m2n ! p /2%
as w→`, for fixed m and n.
~32!
We proceed formally, and use this approximation in ~31!.
~This procedure can be justified; see Appendix B.! The result
is
F n ~ kr, a ! ;G ~ a 1 21 p !
as kr→`,
independently of n, where
J. Acoust. Soc. Am., Vol. 102, No. 1, July 1997
as r→`.
For u con , we have
u con~ r, u ! ;
where
F n ~ kr, a ! 5
as kr→`.
Hence, from ~13!,
here, j is an arbitrary integer. It follows that
71
~34!
u50,6 p ,
A
2 p i ~ kr2 p /4!
e
C ~ sin u !
kr
as kr→`.
This result makes essential use of the continuity of C( m )
~see Clemmow, 1966, Sec. 3.2!. Thus, u con satisfies the Sommerfeld radiation condition ~1!, whence
I ~ u con ;H r ! →0
as r→`.
@A direct derivation of this result, based on ~14! and the
method of stationary phase, is given in Appendix C.#
Finally, consider I(u pr ;H r ). If u a n u , 21p , the results of
the previous sections are immediately applicable, and show
that I(u pr ;H r )→0 as r→`. Next, consider grazing waves,
u a n u 5 21p , and write
v 6 5 v~ r, u ;6 21 p ! 5e 6ikx .
We have
2
I ~ u pr ;H r ! 5I ~ u pr ;H 1
r ! 1I ~ u pr ;H r ! ,
J. A. DeSanto and P. A. Martin: Rough surface integral equations
71
where H 6
r are quarter-circles defined by ~17!; the radiation
condition ~all plane waves must propagate outwards through
H r , away from O) implies that we limit our attention to
7
I( v 6 ;H 6
r ), as v 6 propagates inwards through H r . So, from
Sec. II A, we have
il
I ~ v 1 ;H 1
r ! .iBe Al
E
p /2
~ 12sin u !
0
3e 2ik r cos~ u 2 w ! e il sin u d u
for large l[kr. There is one point of stationary phase ~cf.
Sec. II B! at u 5 21p , but the integrand vanishes there whence
I( v 1 ;H 1
r )→0 as r→`. A similar argument succeeds for
I( v 2 ;H 2
r ).
In summary, we find that our radiation condition ensures
that
I ~ u;H r ! →0
~35!
as r→`.
This is ~formally! the standard Helmholtz integral equation
for the boundary values of u on S.
It is common to not work with ~37! but with an integral
equation for the total field, u tot . To obtain such an equation,
we start by defining a region D̃ r with boundary ] D̃ r . Given
r, let H̃ r denote the semicircle in z<0, with radius r and
center O; then, D̃ r is the bounded region in z,s enclosed by
H̃ r and the rough surface. The boundary ] D̃ r consists of a
piece of S r , namely S t with t(r),r, and a piece of H̃ r ,
namely H̃ r \ T̃ r where T̃ r consists of two circular arcs joining
the ends of H̃ r and the ends of S t . Now, apply Green’s
theorem to u inc and G in D̃ r . As both fields satisfy ~7! in
D̃ r , the result is
05
EH
St
u inc~ q !
J
]G
] u inc
G ~ P,q ! ds q
~ P,q ! 2
]nq
]nq
~38!
2I ~ u inc ;H̃ r ! 1I ~ u inc ; T̃ r !
V. BOUNDARY INTEGRAL EQUATIONS
In Sec. I, we used Green’s theorem to obtain the integral
representation
2u ~ P ! 5
EH
u~ q !
Sr
J
]G
] u inc
G ~ P,q ! ds q
~ P,q ! 1
]nq
]nq
I ~ u inc ;H̃ r ! →0
when PPD r , the region bounded by the semicircle H r , the
truncated rough surface S r , and the two line segments T r .
Note that the left-hand side of this equation does not depend
on r, so that the right-hand side of the equation must have a
limit as r→`. Taking this limit, using ~19! and ~35!, we
obtain
EH
u~ q !
S
J
]G
] u inc
G ~ P,q ! ds q ,
~ P,q ! 1
]nq
]nq
5 lim
r→`
E
as u x u →`,
where U(x) is only required to be bounded.
We remark that Beckmann and Spizzichino ~1963, p.
180! and Ogilvy ~1991, p. 75! discard I(u;H r ) by assuming
erroneously that u50 on H r .
Letting P→ pPS in ~36! gives
S
u~ q !
]G
~ p,q ! ds q 5
]nq
~39!
Letting r→` in ~38!, and adding the result to ~36!, we obtain
E
S
u tot~ q !
]G
~ P,q ! ds q 2U i ~ P ! ,
]nq
PPD ` .
E
J. Acoust. Soc. Am., Vol. 102, No. 1, July 1997
~40!
Then, letting P→p P S gives
E
S
u tot~ q !
]G
~ p,q ! ds q 52u inc~ p ! 2U i ~ p ! ,
]nq
~41!
Now, the standard Helmholtz integral equation for the total
field is
w~ p !2
E
w~ q !
S
]G
~ p,q ! ds q 52u inc~ p ! ,
]nq
pPS. ~42!
Thus, for nongrazing incident waves (U i 50), we see that
u tot does satisfy the standard Helmholtz integral equation.
However, for grazing incident waves, u tot does not satisfy the
Helmholtz integral equation ~42!, but 2u tot does.
VI. DISCUSSION
A. Previous work: Helmholtz integral equations
] u inc
G ~ p,q ! ds q ,
S ]nq
pPS.
72
as r→` ~ u u i u < 21 p ! .
pPS.
U ~ x ! e ik u x u u x u 21/2
E
I ~ u inc ;H̃ r ! →Ui ~ P !
,
Sr
which is the standard definition of a principal-value integral
at infinity. In fact, the integral over S exists as an ordinary
improper integral; to see this, we note that the integrand
behaves like
u~ p !2
We combine these formulas and write
u tot~ p ! 2
where D ` is the unbounded region z.s and
S
I ~ u inc ;H̃ r ! 5u inc~ P !~ u u i u 5 21 p ! .
2u ~ P ! 5
~36!
PPD ` ,
E
as r→` ~ u u i u , 21 p ! .
For grazing incidence, we have
1I ~ u;H r ! 1I ~ u;T r !
2u ~ P ! 5
when PPD r , taking into account the direction of the normal
vector on S. As before, simple bounds show that
I(u inc ; T̃ r )→0 as r→`. If u u i u , 21p , u inc @given by ~6!# is a
plane wave propagating outwards through H̃ r , whence
~37!
The idea that a boundary integral equation may be used
to solve the problem of plane-wave scattering by an infinite
rough surface is familiar. It is discussed in books on such
J. A. DeSanto and P. A. Martin: Rough surface integral equations
72
problems; see for example Maystre and Dainty ~1991!,
Ogilvy ~1991, Secs. 4.1.1 and 6.3! and Voronovich ~1994,
Sec. 3.1!. In particular, the standard Helmholtz integral equation ~4! is equation ~6.53! in Ogilvy’s book and equation
~3.1.37! in Voronovich’s book.
Many recent authors refer to the paper by Holford
~1981! on the scattering of a plane wave by a periodic soundsoft surface. He obtains the integral representation @his equation ~20!#
u tot~ P ! 5u inc~ P ! 1
1
2
E
S
u tot~ q !
]G
~ P,q ! ds q ,
]nq
~43!
PPD ` ,
as in Sec. I C, by applying Green’s theorem to u tot and G in
the semicircular region D r @using our notation and the
boundary condition ~8!#. He claims that ‘‘the term u inc( P) is
the contribution from the large semicircle’’ H r as r→`. He
does not prove this statement and, moreover, it is not true for
grazing incident waves. To see this, we note that applying
Green’s theorem to u tot and G in D r gives
2u tot~ P ! 5
E
Sr
u tot~ q !
]G
~ P,q ! ds q 1I ~ u tot ;H r !
]nq
that we can use the large-argument approximation for
H (1)
1 (kR). Thus, assuming for simplicity that U5U i 50,
~40! gives
2u ~ x,z ! 5
I ~ u tot ;H r ! 5I ~ u;H r ! 1I ~ u inc ;H r !
5I ~ u;H r ! 1I ~ u inc ;C r ! 2I ~ u inc ;H̃ r ! ,
where C r 5H r øH̃ r is a large circle. But, for an incident
plane wave,
I ~ u inc ;C r ! 52u inc ,
E
S
u tot~ q !
E
;iB Ak
S
H ~11 ! ~ kR !
u tot~ q !
R
n~ q ! • ~ q2y! ds q
e ikR
n~ q ! • ~ q2y! ds q
R 3/2
~44!
as kz→`, where B is defined by ~20! and n(q) is the unit
normal vector at q pointing out of D ` .
If u tot(q) has a compact support ~so that it vanishes for
u qu .L, say!, or if S is finite ~bounded scatterer!, we can
make a second approximation:
R5 $ r 2 22y•q1q 2 % 1/25r2ŷ•q1O ~ q 2 /r !
as r→`,
~45!
where r5 u yu , ŷ5y/r, and q5 u qu . ~The notation used here
differs from that used in Secs. I C and II A.! To the same
order, we can also replace (q2y) in ~44! by (2y). Hence,
we find that u is given by ~5!, where the far-field pattern is
f ~ u !5 2
1I ~ u tot ;T r ! .
Now, I(u tot ;T r )→0 as r→` and
ik
2
iB Ak
2
E
S
u tot~ q ! n~ q ! •ŷ exp$ 2ikŷ•q% ds q .
However, one cannot justify the use of the approximation ~45! for plane-wave incidence and unbounded surfaces
~see the discussion by Ogilvy, 1991, p. 78!. This is immediately clear, because there must be a reflected plane wave,
whereas ~45! leads to a cylindrical wave. For explicit confirmation, consider the reflection of a plane wave by a flat
surface, so that u tot(q)52 exp$ikj sin ui%; the integral in ~44!
can then be estimated using the method of stationary phase
@and yields the correct u, given by ~9!#, whereas the integral
for f ( u ) diverges.
exactly, whence
I ~ u tot ;H r ! →2u inc2Ui
as r→`,
where Ui is defined by ~39!. Thus ~43! is correct whenever
Ui [0.
Holford himself refers to earlier papers by Urosovskii
~1960!, who in turn refers to Lysanov ~1956!. For more recent work, we can cite Thorsos ~1988!, Bishop and Smith
~1992! and McSharry et al. ~1995!. All these papers start
from the Helmholtz integral equation for u tot , ~4!, or the
analogous equation for a sound-soft surface. Moreover, all
but one of these papers are concerned with plane-wave incidence, the exception being the paper by Thorsos ~1988!. He
considers a ‘tapered’ plane wave; we will discuss beams of
finite extent in Sec. VI D.
B. Far-field asymptotics
Care is needed when approximating the scattered field at
large distances from an infinite surface. To illustrate this,
consider the integral representation ~40!. Let P[(x,z)
PD ` and q[( j ,s( j ))PS have position vectors y and q,
respectively. In the far field, kR@1, where R5 u y2qu , so
73
J. Acoust. Soc. Am., Vol. 102, No. 1, July 1997
C. A finite patch of roughness
Suppose that the infinite surface S is flat, apart from a
finite patch of roughness, S patch , confined to u x u ,L, say. A
plane-wave incident on such a patch will generate a specular
plane wave and a cylindrical wave. Thus, for nongrazing
incidence, the standard Helmholtz integral equation for the
total field, ~4!, is valid.
To see that the decomposition itself is valid, write
u tot5u flat1u cyl ,
where
u flat5e 2ikrcos~ u 1 u i ! 1e ikr cos~ u 2 u i ! ~ u u i u , 21 p !
is the total field for reflection by an infinite flat sound-hard
surface. Thus
] u cyl
] u flat
52
]n
]n
on S patch.
~46!
There are now three cases to consider, namely, ridges,
grooves, and a combination thereof.
J. A. DeSanto and P. A. Martin: Rough surface integral equations
73
1. Ridges
Suppose that S patch consists of a finite number of ridges
~‘‘bosses’’!, so that s(x)>0. Then, the scattering problem is
equivalent to the scattering of two plane waves, u flat , by a
finite bounded obstacle ~a ‘‘double-body’’! with boundary
S patchøS 8patch , where S 8patch is the reflection of S patch in the
line z50. Thus u cyl is a cylindrical wave, satisfying the Sommerfeld radiation condition ~1!.
The use of bosses to model rough surfaces is well
known ~Ogilvy, 1991, Sec. 6.1!. The use of images to treat
scatterers near flat impenetrable boundaries is also well
known; for a recent application, see Chao et al. ~1996!. Indeed, if we introduce the exact Green’s function for the halfplane z>0,
5 2 21 i $ H ~01 ! ~ k A~ x2 j ! 2 1 ~ z2 z ! 2 !
1H ~01 ! ~ k A~ x2 j ! 2 1 ~ z1 z ! 2 ! % ,
we find that
E H
S ridge
1
u cyl~ q !
]GE
~ P,q !
]nq
J
E
] u flat E
G ~ P,q ! ds q ,
]nq
S ridge
From the discussion above, we see that if S patch consists
of a finite number of ridges and grooves, then u cyl can be
represented using
u cyl~ P ! 5
E
S1
n ~ q ! G E ~ P,q ! ds q ,
u tot~ p ! 52u inc~ p ! 1
PPD ` ,
where S ridge is the union of all the ridge surfaces
(S patch\ S ridge is part of z50). Letting P→pPS ridge yields a
boundary integral equation for u cyl(p).
Alternatively, we can write
u cyl~ P ! 5
3. Ridges and grooves
where S 1 5S ridgeøS mouth , P[(x,z) and z.max$s(x),0% .
The determination of n on S 1 is complicated, although S 1 is
a finite surface. If we use the Helmholtz integral equation for
u tot , which we know is legitimate, we have to solve an integral equation over an infinite surface. However, this can be
reduced to an integral equation over S patch as follows. Since
] G(p,q)/ ] n q 50 when both p and q are on the flat part of
S, S flat5S \ S patch , ~4! gives
G E ~ P,Q ! [G E ~ x,z; j , z !
2u cyl~ P ! 5
with grooves ~compared to ridges! has given rise to more
sophisticated methods for solving such problems, involving
more complicated integral representations; see Willers
~1987! and Asvestas and Kleinman ~1994!. Applications of
G E to rough-surface scattering were made by Berman and
Perkins ~1985! and by Shaw and Dougan ~1995!.
n 1 ~ q ! G E ~ P,q ! ds q ,
PPD ` ,
~47!
where the boundary condition ~46! implies that the source
density n 1 solves a Fredholm integral equation of the second
kind over S ridge .
E
u tot~ p ! 2
E
S patch
E
S mouth
n 2 ~ q ! G E ~ P,q ! ds q ,
P[ ~ x,z ! and z.0.
~48!
To find the source density n 2 , we can apply Green’s theorem
inside each groove to u cyl and G; we have the boundary
condition ~46! on the surface of each groove, and we have
~transmission! conditions enforcing the continuity of u cyl and
] u cyl / ] z across the mouth of each groove.
Note that we cannot use ~48! inside the grooves because
of the image singularities in G E . This extra complication
74
J. Acoust. Soc. Am., Vol. 102, No. 1, July 1997
pPS flat .
u tot~ q ! K ~ p,q ! ds q 52 f ~ p ! ,
pPS patch ,
where
K ~ p,q ! 5
]G
~ p,q ! 1
]nq
2. Grooves
u cyl~ P ! 5
]G
~ p,q ! ds q ,
]nq
This means that u tot on the ~infinite! flat part of S is known in
terms of u tot on the rough part of S. Hence, we can write ~4!
for pPS patch as
f ~ p ! 5u inc~ p ! 1
Suppose that S patch consists of a finite number of
grooves, so that s(x)<0. Then, ] u cyl / ] z is known, in principle, for all x on z50: It is zero except across the mouth of
each groove. As there is a finite number of grooves, it follows that u cyl is a cylindrical wave; it has an angularspectrum representation with a continuous spectral amplitude.
Let S mouth be the union of all the groove mouths; it is
part of z50. We can write
S patch
u tot~ q !
E
S flat
E
]G
]G
~ p,l !
~ q,l ! ds l ,
]
n
]
nq
S flat
l
u inc~ q !
]G
~ p,q ! ds q .
]nq
D. Finite beams
So far we have taken the incident field to be a plane
wave. However, for many applications, the incident field is a
finite beam. To construct such a beam, we start by considering a single line-source at Q,
u inc~ P ! 5G ~ P,Q ! .
As u inc satisfies the Sommerfeld radiation condition, an energy argument ~DeSanto and Martin, 1996! shows that the
scattered field cannot include any reflected plane waves.
Thus, the standard Helmholtz integral equations, ~3! and ~4!,
are valid; see DeSanto and Brown ~1986, Sec. 4.1!.
Next, we distribute the line-sources over a finite curve
Q ~or a finite region!, to give
u inc~ P ! 5
En
Q
inc~ Q ! G ~ P,Q ! ds Q ,
where n inc is prescribed and can be adjusted to make u inc
beamlike. ~If Q is far from S, the asymptotic approximations
J. A. DeSanto and P. A. Martin: Rough surface integral equations
74
described in Sec. VI B can be used.! It follows that the standard Helmholtz integral equations remain valid.
Another way to generate a beam is to use an angularspectrum representation,
u inc~ r, u ! 5
E
p /2
2 p /2
A inc~ a ! e ikr cos~ u 1 a ! d a ,
~49!
where A inc is a prescribed continuous function, and is typically taken as a Gaussian ~Saillard and Maystre, 1990!. The
standard Helmholtz integral equations are valid for incident
fields of this type.
Thorsos ~1988! uses a ‘‘tapered’’ plane wave. This incident field does not satisfy the Helmholtz equation, and so the
derivation of the Helmholtz integral equation for u tot fails.
@The Helmholtz integral equation for u, ~3!, is valid without
this qualification.# Nevertheless, as Thorsos points out, the
tapered plane wave is an approximation to an actual wave
field, constructed using ~49!; see Thorsos ~1988, Sec. I B!.
E. Other integral equations
An alternative way of solving scattering problems,
touched on above, is to assume that the scattered field can be
written as
u~ P !5
Eg
S
~ q ! G ~ P,q ! ds q ,
PPD,
where the source density g is unknown. For sound-hard surfaces, the boundary condition ~8! yields an integral equation
for g ,
g~ p !2
Eg
~q!
S
]G
] u inc
,
~ p,q ! ds q 5
]np
]np
pPS.
whereas the integral equation for the boundary values of the
total field is valid for nongrazing incident waves ~a simple
modification is required for grazing incident waves!. These
results underpin the use of these integral equations for numerical computations.
The standard Helmholtz integral equations are valid if
the roughness is confined to a finite portion of an otherwise
flat but infinite surface. They are also valid for incident
beams of finite width.
Similar results may be obtained for sound-soft surfaces.
Extension to electromagnetic and elastodynamic problems,
and to penetrable interfaces, should be straightforward.
Finally, extension of these ideas to two-dimensional
rough surfaces can also be made, although the analysis is
more difficult and the results are different. Some of these
aspects are currently under investigation.
ACKNOWLEDGMENTS
PAM acknowledges receipt of a Fulbright Scholarship
Grant. He also thanks the Department of Mathematical and
Computer Sciences, Colorado School of Mines, for its kind
hospitality. Both authors are grateful to A.G. Voronovich for
his perceptive remarks on an earlier version of the paper,
which encouraged us to clarify the role of grazing waves.
APPENDIX A: UNIFORM ASYMPTOTICS
We derive a uniform approximation for I( v ;H r ), using
a method discussed by Bleistein and Handelsman ~1986, Sec
9.4!. We start by focussing on the non-uniformity at a 5 21
p ; the nonuniformity at a 52 21p can be treated similarly.
Write L5L 1 1L 2 where
For sound-soft surfaces, the corresponding integral equation
is
Eg
S
~ q ! G ~ p,q ! ds q 5 2u inc~ p ! ,
pPS;
this has been solved numerically by Lentz ~1974!, Rodrı́guez et al. ~1992!, and others.
Chandler-Wilde and Ross ~1995, 1996! use a doublelayer potential for sound-soft surfaces,
u~ P !5
Eg
S
~q!
]G1
~ P,q ! ds q ,
]nq
VII. CONCLUSIONS
We have seen that the use of standard Helmholtz integral equations for the scattering of a plane wave by an infinite, sound-hard, one-dimensional, rough surface is justified
in most circumstances. In particular, the equation for the
boundary values of the scattered field is always valid,
J. Acoust. Soc. Am., Vol. 102, No. 1, July 1997
E
0
g ~ u ! e ilF ~ u ! d u
2 p /2
and L 2 5L2L 1 . We have L 2 5o(1) as l→`, uniformly in
a , for a bounded away from 2 21p . For L 1 there is a point of
stationary phase at u 5 a 2 p ~outside the range of integration! which approaches the end point at u 52 21p as a
→ 12p . Let us make a preliminary change of variables, mapping the end point to the origin: Put u 5x2 21p and a
5 21p 2 d giving
PPD,
where G 1 satisfies an impedance condition on a line
z52h 0 (h 0 .h).
All of the formulations mentioned in this section are
indirect, in that they assume that u( P) can be represented in
a specified form. Thus the radiation condition is implicit in
the representation.
75
L 1 ~ l ! 5 Al
L 1 ~ l ! 5 Al
E
p /2
h ~ x ! e il f ~ x; d ! dx
0
with
h ~ x ! 5 @ 11cos~ x1 d !# e ik r sin ~ w 2x !
and
f ~ x; d ! 5 2cos~ x1 d ! .
Thus L 1 has a stationary-phase point at x52 d , which approaches the end-point x50 as d →0. The prototype special
function with this property is the complementary error function, defined by ~28!. In order to relate this function to L 1 ,
we change the integration variable from x to t, using
f ~ x; d ! 2 f ~ 0; d ! 5 21 t 2 1 g t,
J. A. DeSanto and P. A. Martin: Rough surface integral equations
75
requiring that x52 d is mapped to t52 g gives g 52
3 sin 21d . Hence,
L 1 ~ l ! 5 Ale il f ~ 0; d !
E
T
h̃ ~ t ! e il ~ 1/2 t
21gt !
dt,
0
where
h̃ ~ t ! 5h ~ x ~ t !!
APPENDIX C: ASYMPTOTIC BEHAVIOR OF
I ( U con ; H r )
dx
t1 g
dx
and
5
.
dt
dt f 8 ~ x; d !
For the dominant contribution, we can set the upper limit
T5` and replace h̃ (t) by h̃ (0); this gives
L 1 ~ l ! . Ale il f ~ 0; d !
gh~ 0 !
f 8 ~ 0; d !
E
`
e il @~ 1/2 ! t
21gt
#
dt.
p i p /4 2 1/2il g 2
e e
erfc~ m !
2l
with m 5 g
Disregarding the evanescent terms in ~14!, we can write
u con as the integral in ~11!, whence
I ~ u con ;H r ! 5
D~ l !5
A
5
L 1 ~ l ! . A2 p cos 21 d e i p /4e 2il e ik r sin w erfc~ m !
In order to estimate F n (kr, a ) for large kr, we substituted the asymptotic approximation ~32! into ~31!. However,
this requires some justification, as ~32! presupposes that m
and n are fixed. @A hint that nonuniform behavior might be
expected comes from ~29!: The left-hand side is a plane
wave whereas every term on the right-hand side is
O((kr) 21/2) as kr→`.# We start by writing
F n ~ w, a ! 5w $ H 8n ~ w ! F n ~ w, a !
2H n ~ w !~ ] / ] w ! F n ~ w, a ! % ,
where
~B1!
Next, we use a standard integral representation for Bessel
functions,
J m~ w ! 5
1
2p
E
p
e i ~ m u 2w sin u ! d u ,
2p
whence
F n ~ w, a ! 5
i
4
E
p
2p
G ~ u 1 a ! e i ~ n u 2w sin u ! d u
~B2!
where G ( u ) is defined by ~33!. Finally, we use the method
of stationary phase to estimate ~B2! for large w; there are
stationary-phase points at u 56 21p , whence ~22! gives
F n ~ w, a ! 5 A~ 21 p /w ! sin ~ w2 21 n p 2 41 p ! 1O ~ w 21 !
as w→`, for 0< u a u , 21p , whereas
F n ~ w,6 21 p ! 5O ~ w 21 !
76
as w→`.
J. Acoust. Soc. Am., Vol. 102, No. 1, July 1997
A ~ a ! I ~ v ;H r ! d a .iBe il AlD ~ l !
E
p /2
A~ a !
2 p /2
E
p /2
2 p /2
@ 12cos~ u 2 a !#
E
p /2
A~ a !
2 p /2
E
p /22 a
~ 12cos c !
2 p /22 a
As A is a continuous function of a , we can change the order
of integration to give
APPENDIX B: ASYMPTOTIC BEHAVIOR OF F n
1
2 p /2
3e 2ik r cos~ c 1 a 2 w ! e il cos c d c d a .
whence the final result ~27! follows.
(j 2 j11 J n12 j11~ w ! e i~ 2 j11 !a .
p /2
3e 2ik r cos~ u 2 w ! e il cos~ u 2 a ! d u d a
1 2i p /4
le
.
2
Substituting back, we obtain
F n ~ w, a ! 5
E
as l[kr→`, where we have used ~21!, and
0
Standard manipulations show that the integral can be expressed as
A
These results are exactly the same as we would have obtained if we had replaced the Bessel function in ~B1! by the
leading term in its large-argument asymptotic expansion,
namely ~26!.
D~ l !5
E
p
~ 12cos c ! Q ~ c ! e il cos c d c ,
~C1!
0
where
Q~ c !5
E
p /22 c
2 p /2
$ A ~ a ! e 2ik r cos~ c 1 a 2 w !
1A ~ 2 a ! e 2ik r cos~ c 1 a 1 w ! % d a .
Now we can estimate D(l) for large l using the
method of stationary phase. The stationary-phase points are
c 50 and c 5 p . At c 50, 12cos c50, whereas Q( p )50.
Hence, the integrand in ~C1! vanishes at both stationaryphase points, whence D(l)5O(l 21 ) as l→`. Thus, we
deduce that I(u con ;H r )→0 as r→`.
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