A long-wave multiple scattering theory
C M Linton, Dept. of Mathematical Sciences, Loughborough University, Leicestershire, LE11 3TU, UK.
E-mail: c.m.linton@lboro.ac.uk
P A Martin, Mathematical and Computer Sciences, Colorado School of Mines, Golden, CO 80401-1887,
USA. E-mail: pamartin@mines.edu
SUMMARY
known. For submerged structures we do not include
a source (i.e. Dj = 0).
One of the most successful techniques for handling
The field incident on the n-th scatterer is
multiple scattering is the so-called T-matrix approach
X
in which the full linear solution can be computed once
un (r) = uinc (r) +
Dj G0 (r − rj ; zj )
it has been determined how each individual element
j6=n
of an array scatters an arbitrary incident field. In the
+ dj · G1 (r − rj ; zj ) . (2)
context of water waves this theory was originally formulated in [1] for constant finite depth, and extended Now, let us characterise the scattering properties of
to the deep water case in [2].
the scatterers by writing
Here we propose a very simple approach to mulT
Dn = Bn un (rn ) and dTn = Cn [vn (rn )] , (3)
tiple scattering, designed to provide approximations
valid when the wavelength is large compared to the where
size of the individual scatterers. The idea is an exvn = K −1 ∇un .
(4)
tension of a very old method used in acoustics due to
The quantity Cn is a matrix. Thus Dn is proporFoldy [3] in which the scatterers are assumed to betional to the value of the exciting field at rn and dn
have like point sources in the long-wave limit. This is
is related to the gradient of the exciting field at rn .
in fact rigorously true for scatterers on whose boundIf we substitute from (2) into (3) we get
ary the velocity potential vanishes, but it is not aph
X
propriate for rigid scatterers. For our problem we asDn = Bn uinc (rn ) +
Dj G0 (rnj ; zj )
sume that each scatterer can be modelled as a combij6=n
nation of a source and a dipole in long waves and then
i
proceed to handle the multiple scattering in much the
+ dj · G1 (rnj ; zj ) , (5)
same way as in the T-matrix approach. The scattering characteristics of each individual scatterer can be where rnj = rn − rj , and
determined by appealing to specific long-wave asymph
1 X
totics, or numerically if necessary. A time depen∇ Dj G0 (r − rj ; zj )
dTn = Cn vinc (rn ) +
K
dence of exp(−iωt) is assumed throughout and we
j6=n
iT
will use K = ω 2 /g.
, (6)
+
d
·
G
(r
−
r
;
z
)
j
1
j j r=rn
The method is applied to the scattering of a plane
wave by a group of horizontal, submerged, circular
where vinc (r) = K −1 ∇uinc . Equations (5) and (6)
cylinders and by an infinite periodic row of identical
give a system of linear algebraic equations for Dn
vertical cylinders of constant cross-section.
and the components of dn . For N scatterers in three
dimensions, there are 4N equations for the 4N scalar
unknowns; in two dimensions, there are 3N equations
FORMULATION
in 3N unknowns, though for each scatterer that is
We take the x, y-plane to be the undisturbed free sur- submerged the size of the system reduces by one.
face with z pointing vertically upwards and represent
the total field by the harmonic velocity potential
Choice of Bj and Cj
X
In order to use the method described above, we have
Dj G0 (r − rj ; zj )
u(r) = uinc (r) +
to specify the coefficient Bj and the matrix Cj for
j
each scatterer. Consider the j-th scatterer and as+ dj · G1 (r − rj ; zj ) , (1)
sume, without loss of generality, that it is located at
r = (0, 0, ζ). For any incident field uinc (r), we have
where the sum is over all scatterers, the j-th scatterer j
assumed that the total field near the scatterer is given
being centred at rj = (xj , yj , zj ), and r = (x, y, z).
by
The first term inside the summation is a source at rj ;
the strength of the source (given by Dj ) is unknown.
u(r) ≃ uinc (r) + Bj uinc (0, 0, ζ) G0 (r; zj )
The second term is a dipole at rj , the direction and
+ vinc (0, 0, ζ) CTj · G1 (r; zj ). (7)
strength of which (given by dj = (dxj , dyj , dzj )) are un-
If the incident field is a plane wave travelling in the and if we define
q
x-direction, then we have
2
xnj = xn − xj , znj = zn − zj , rnj = x2nj + znj
uinc = eiKx eKz ,
(8)
and
s Z ∞
and
µ(µ + K) µ(zn +zj ) cos µxnj
Inj
=⌣
dµ
e
a
2
Inj
sin µxnj
0 K (µ − K)
u(r) ≃ eiKx eKz + Bj eKζ G0 (r; zj )
(14)
T
Kζ
then
(12)
becomes
(i, 0, 1)Cj · G1 (r; zj ). (9)
+e
" x
This can be compared with specific long-wave caldn
i
2
= δn (Kan )
eiKxn eKzn
culations such as those derived in [4] using matched
dzn
−1
asymptotic expansions.
"
x 2
X
1
dj (znj − x2nj ) + 2dzj xnj znj
+
2
4
2dxj xnj znj + dzj (x2nj − znj
)
K 2 rnj
SUBMERGED HORIZONTAL CYLINj6=n
##
x s
DERS
a
dj Inj + dzj Inj
+
. (15)
a
s
−dxj Inj
+ dzj Inj
As an application of the theory we will consider a
two-dimensional problem with a plane wave normally
system of equaincident on an array of submerged horizontal circu- For N cylinders this is ax 2N × 2N
z
tions
for
the
unknown
d
and
d
.
An
approximate
n
n
lar cylinders in deep water. Since the cylinders are
solution
can
be
determined
if
we
assume
that both
submerged, there are no source terms. Symmetry
Ka
≪
1
and
a
/r
≪
1.
We
obtain
n
n nj
considerations show that the vertical fluid velocity
" cannot lead to a horizontal dipole, so Czx = 0. We
x
i
dn
2
also assume that Cxz = 0; though this is an approxieiKxn eKzn
≃ δn (Kan )
−1
dzn
mation which is strictly only valid when the depth of
X
submergence of each cylinder, zn , is much larger than
+
δj (Kaj )2 eiKxj eKzj
the cylinder radius, an . It follows from the asympj6=n
totic analysis in [5], that for a circular cylinder of
"
2
radius a and submergence depth f > 0,
1
i(znj − x2nj ) − 2xnj znj
2
4
)
2ixnj znj − (x2nj − znj
K 2 rnj
Cxx = δ(Ka)2 ,
Czz = −δ(Ka)2 ,
(10)
##
s
a
iInj − Inj
. (16)
+
a
s
where
−iInj
− Inj
∞
2n
X
ns
,
(11)
δ = 4 (f /a)2 − 1
The scattered field is
1 − s2n
n=1
X
p
usc =
dj · G1 (r − rn ; zn )
(17)
with s = (f /a)− (f /a)2 − 1, is a factor which tends
n
to one as a/f → 0. For deeply submerged cylinders
X
∼ 2π
e±iK(x−xn) eK(z+zn ) (±dxn − idzn ) (18)
we have δ ≈ 1 and different cross-sections can easily
n
be accommodated by including an appropriate dipole
coefficient in (10) as shown in [4].
as x → ±∞. The reflection coefficient is thus
Equation (6) is thus
X
R = 2π
(19)
eiKxn eKzn (−dxn − idzn )
"
x
n
1 0
dn
(i, 1) eiKxn eKzn
= δn (Kan )2
which becomes
0 −1
dzn
XX
#T
R = −4π
eiK(xn +xj ) eK(zn +zj )
1 X
∇ dj · G1 (r − rj ; zj ) r=rn . (12)
+
n j6=n
K
j6=n
δn δj (Kaj an )2
2
(20)
i(znj
− x2nj ) − 2xnj znj
×
4
rnj
Now
if we insert the approximations given by (16). If all
the cylinders are at the same depth zn = ζ and have
Z
∞
dxj (x − xj ) + dzj (zj − z)
(µ + K) µ(z+zj )
the same radius an = a, then
+⌣
e
XX
Kr2
0 K(µ − K)
iK(xn +xj )
R = 4πiK 2 a4 δ 2 e2Kζ
x−2
. (21)
x
z
nj e
(dj sin µ(x − xj ) − dj cos µ(x − xj )) dµ (13)
dj · G1 (r − rj ; zj ) =
n j6=n
Furthermore, if the wavelength is large compared is symmetric with respect to both the x- and y-axes
with all the other length scales in the problem we the matrix C will be diagonal and we will make this
get
assumption here.
XX
−2
2 4 2
As an example, we consider the scattering of a
R = 4πiK a δ
xnj .
(22)
plane wave
n j6=n
uinc = ei(βx+αy) ,
(26)
This is slightly different to the equivalent expression
in [6], where the factor δ 2 is replaced by δ δ̃, δ̃ being where α = k sin ψ and β = k cos ψ, by an infinite
a different factor which also tends to 1 as a/ζ → 0. periodic row of identical cylinders. The scatterers
This discrepancy, which makes little difference to the are located at r = rm for m = 0, ±1, ±2, . . ., where
numerical results, may well be due to the neglect of r = (x, y), rm = (ms, 0) and s is the spacing. We
will use polar coordinates (rm , θm ) centred at the
the Cxz terms in the matrix C.
Some preliminary calculations of exciting forces m-th scatterer and defined by x − ms = rm cos θm ,
have been performed based on solving (15) and then y = rm sin θm . In terms of (rm , θm ), we have
numerically integrating the potential (1) around each
uinc = Im eikrm cos(θm −ψ) with Im = eiβms . (27)
cylinder. These results have been compared with the
full linear solution computed using the multipole exThe representation (1) becomes
pansion method described in [7]. As expected, the
X
two approaches agree in the long wave limit.
Dj H0 (krj )
u(r) = uinc (r) +
j
VERTICAL CYLINDERS
In the case of vertical cylinders of constant cross section extending throughout the depth (h say) we can
develop a very similar theory. A depth dependence of
cosh k(z+h)/ cosh kh, where k the positive root of the
dispersion relation k tanh kh = K, can be factored
out in the usual way and then the reduced potential
(which we still call u) satisfies the Helmholtz equation
(∇2 + k 2 )u = 0. The definition of vn is changed to
k −1 ∇un . Infinite depth is treated by setting k = K
with the depth factor as exp(−Kz).
Equation (1) remains the same, but now G0 (r) =
(1)
(1)
H0 (kr) and G1 (r) = r̂H1 (kr), with r = |r| and
r̂ = r/r. Equations (5) and (6) become (dropping
the superscripts on the Hankel functions)
h
X
Dj H0 (krnj )+
Dn = Bn uinc (rn ) +
+ (dxj cos θj + dyj sin θj )H1 (krj ) . (28)
As uinc (ns, 0) = In , the scalar system (23) becomes
h
X
Dn = B In +
Dj H0 (ks|n − j|)
j6=n
i
+ dxj H1 (ks|n − j|) sgn(n − j) . (29)
Note that r̂nj = (1, 0) for n > j and r̂nj = (−1, 0) for
n < j. The vector system (24) reduces to two scalar
systems. They are
Cxx dxn = iIn cos ψ +
X
dxj H1′ (ks|n − j|)
j6=n
and
− Dj H1 (ks|n − j|) sgn(n − j)
j6=n
i
dj · r̂nj H1 (krnj )
and
(23)
dyj
H1 (ks|n − j|)
.
ks|n − j|
(31)
The periodicity of the geometry and the quasiperiodicity of the incident plane wave imply
Dj = Ij D0 ,
j6=n
oiT
− r̂nj (dj · r̂nj ) H2 (krnj ) − Dj r̂nj H1 (krnj )
.
(24)
Standard low frequency approximations show that for
circular cylinders
Cj = −2Bj I,
X
j6=n
h
X n H1 (krnj )
dj
dTn = Cn vinc (rn ) +
krnj
Bj = − 41 iπ(kaj )2 ,
Cyy dyn = iIn sin ψ +
(30)
(25)
where I is the identity matrix. Equivalent quantities
can easily be determined for cylinders of arbitrary
cross-section. For cylinders with a cross-section that
dxj = Ij dx0 ,
dyj = Ij dy0 .
(32)
When these relations are used in (29), (30)and (31)
we obtain a coupled system for D0 and dx0 , representing the component of the solution which is symmetric
with respect to the x-axis, and a separate equation
for the antisymmetric component dy0 . The solutions
are
D0 = Cxx − 12 (σ0 − σ2 ) + iσ1 cos ψ /∆,
dx0 = −σ1 + (B −1 − σ0 )i cos ψ /∆,
(33)
−1
y
1
,
d0 = i sin ψ Cyy − 2 (σ0 + σ2 )
where ∆ = (B −1 − σ0 )[Cxx − 12 (σ0 − σ2 )] + σ12 , and
σp (ψ) =
∞
X
(I−j + (−1)p Ij )Hp (kjs).
(34)
j=1
The efficient computation of these sums is non-trivial,
but integral representations for σp exist which greatly
facilitate the process. If we define
Sn± =
∞
X
I±j Hn (kjs)
(35)
assume that ks < π in which case M = {0} and
we have just one reflected and one transmitted wave,
with reflection and transmission coefficients given by
2 D0 − idx0 cos ψ + idy0 sin ψ ,
(41)
ks sin ψ
2 D0 − idx0 cos ψ − idy0 sin ψ . (42)
T =1+
ks sin ψ
R=
DISCUSSION
Calculations based on this long-wave multiple scattering theory will be presented at the workshop for
so that σn = (−1)n Sn+ + Sn− , then
the two problems considered above. Comparisons will
Z
−in arccos t
be made with the full linear solution in each case.
i
e
Sn± = −
dt,
(36) We will also discuss the three-dimensional water wave
ksγ∓iβs
π C γ(e
− 1)
problem involving an array of floating hemispheres.
where the contour C lies on the real axis but is indented above the poles for which t < 0 and below
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γ(t) =
(37)
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|t| > 1.
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