Scattering of surface waves by submerged
,cylinders
- F. G. LEPPINGTON
DepartMentof Mathematics, Imperial College, London SW7, UK
and P. F. SIEW
Department of Mathematics, University of Western Australia; Perth, WA, Australia
A train of surface waves is normally incident upon a cylinder that is totally submerged in a body of deep water. Details are given for the cases of circular or elliptic cross sections with
estimates for the transmission and reflection constants when the cylinder is many wavelengths below the surface. Corresponding results are suggested for arbitrary smooth cylinders.
INTRODUCTION
A submerged cylinder is held fixed and parallel to the free surface of a body of deep water and is irradiated by a train
of sinusoidal waves whose direction is normal to the cylinder. In this two-dimensional problem, let the smooth curve S(x, y) denote the cross-section of the body and choose Cartesian coordinates (:c, y) with origin at some convenient point inside S, y pointing vertically upwards with the free surface t y=h (Fig. 1).
It is assumed that the fluid is inviscid and incom-pressible and that the motion is irrotational. For small time-periodic disturbances the velocity potential RIcD(x, y)exp ( icor)) is harmonic, has zero normal derivative on S and satisfies the linearized free surface condition:
(1)7= 0 at y=h
where the subscript denotes partial differentiation and K =co2/g is the wave number for travelling waves. The incident wave has potential:
q,=g exp /Kx+ Ky icor]
= R exp iKz /cot] (1)
which represents a wave travelling in the negative x-direction. Henceforth the time factor exp ( will be suppressed. The condition of outgoing waves at infinity requires (1) to have the form:
0cp1i-Rexp(iK.+Ky) as x--.00
andID Tex p ( iKx Ky) as x oo
where T and R are unknown (complex) constants to be found.
It is convenient to work with the scattered potential
cp(x. y). defined
by:-1)141- I 187:80;030129.0-952.00 CN1L
.van
DwaliminummisamaMinvg
R.
The specifications for cp are therefore
It is required to estimate tp, and particularly the trans-mission and reflection constants T and R, when the cylinder is many wavelengths below the surface.
It is a simple matter to relate Tand R to the values of the potential co on S. An application of Green's formula to the functions co(x, y) and 91(x, y) in the fluid region leads to the identity: R.=ie1-2-P (cp+9,)eZds -
.19e-dS
cn (8) inc. trans refl. TECHNISCHE UNIVERSITET Laboratorium voorScheepshydromechanica
Archief Mekelweg 2, 2628 CD Delft Tel.: 015 - 786873 - Fax: 015 - 781838Applicd Ocean Res-earch, 1980, l'ul. 2, No. 3 129
V2cp =0 y<h, outside S (3)
Kcp coy= 0 y= h (4)
(aien)(9 + 9,) = 0 on S where n is the outward normal from S, and
cp--Rexp(iKx+ Ky) as x+ cc
(5)
(6)
cp(T 1) exp (iKx+Ky) as x--*
co (7)Scattering of surface troces_by_subpierged cylinders: F. G. Leppingtthi and P. F. Siew
since Evidently pt. corresponds to the potential generated
by
a surface pressure distribution, proportional to P(x), in the presence of a fixed cylinder. The significantproperty of
j
eq) -, , thp,
vi s = coi(z)-.--,-dz =0
i..n
P(x) is that it is exponentially small compared with the
i
di. incident potential 9i(x,h), when K(h2-a2)/h is large.
Thus
S s '
-we anticipate that 90 will be a good approximation to 9 when K(h-a)> I: that is, when the highest pointof the cylinder is many wavelengths below the surface.
On substituting vo into formula (9), and noting that90
=pi -1 on S. we obtain the corresponding approxi-mation T -T0, where
To -1 =2ie--2KhR ecpder ds
A similar calculation with 9(x, y) and the complex
conjugate Cpi(x,..y).exp( + iKx + Ky) shows that: .T- 1 = ie -2!"1
i.
tCpieen + 9 eCpi/enids =le-2.Kh { a 2 +(CP ds en e0; enDetails are given for the circular cylinder (radius a) in the next section, where an asymptotic estimate is pre-sented for T when:
K(h -a)2/h >1 (10)
The constraint (10) necessarily implies that Kh is large, though Ka may be large or small. The estimate obtained for Tconfirms and extends previous results in three special cases: (i) Ka 1, (ii) a) (K and a fixed), (iii) Ka(C 1, Kh 1. It is well known" that R 0 for the circular cylinder, and the present asymptotic estimate also gives zero value to R.
The third section deals with the case when S is an ellipse of arbitrary orientation. An estimate is given for Twhich again agrees with previous results in the appropriate limits. An equivalent result is conjectured for R. As this coefficient is found to be negligible, compared with Tand with the order of magnitude of our approximation for 9, its validity is not rigorously established, but it does nevertheless agree with previous results in two diverse limits.
Results for arbitrary smooth cylinders are suggested in the final section.
SCATTERING BY A CIRCULAR CYLINDER The scattered potential 9(x, y) is specified by equations (3)-(7), where the boundary S is the circle x2 +y2 =a2.
When K co, it is argued here that a good approxi-mation for cp is obtained by simply ignoring the free surface condition (4) and writing down the harmonic function 90 that vanishes at infinity and satisfies the boundary condition (5). By inspection,
9- 90= exp { -iKa2/(x-iy)} -1
(11)which is not an exact solution since it fails to satisfy the free surface condition (4). Defining a harmonic correction
potential 9,=9 -90, its surface condition is
itipcley=P(x)
on y=h
(12) where (9) P(x)= _(.l[exp -iKa21(x
-1]
K ex .-130 Applied Ocean Research. 1980. li)/. 2. No. 3 (13)
_2ie-2KhR exn (iKa 2
Z
11(z)cli
=4TaKae-2Kh (Ka)2m+
o )711.0ri
---4rtiKae-2'1,(2Ka) (14)
by a straightforward residue calculation, where II denotes a modified Besse] function. The corresponding formula (8) gives Ro =0, which is consistent with the known result R -a 0. To justify the approximation T T0 it is now necessary t. for the errors arising from the.
estimate (11). Second approximation
The correction potential 9,=9-90 is harmonic and is subject to the surface condition (12), with
ecpc/er = 0 . on S (15)
and a radiation condition at infinity. Write
(Pc=97 (16)
where 91 satisfies the surface condition (12) but ignores the boundary condition (15), which has to be accounted for by 0. An exact expression for 91 follows from the known Green's function,
G(x- x1; .1', y1)=(27r) log R(x1, 371) 1Cos Kt(x- xl)eK"Y 2h)
I'
cos 2h- y;) 7rt-1
clt (17) =Go +G, say (18)where R2(x1, -x1)2 -1-(y -y1)2, and the integration path r runs from 0 to co, indented below the singularity at t = 1. The function G is the potential due to a line source at (x1, y1) with no obstacles present. In terms of G, q1 can be written as: cp,(x,..v,)=K .(19) with Pt.' v= h. Tc in (19). ( P thearfrini
t+
Ka r - It is wor value (7 plausiblc travellin 1 pressure from thy: from the the posit bution fi 91 in agree; deforma rate wh The ck by direc When 1, integrat: (if Ka is that the of order result ti with Ka (Ka)4/(1 compar uniform when th the bolith P(x) given by equation (13). In the integral (19), G G, since the logarithmic term Go vanishes on the p/ane = Ii. To evaluate p1 use the form:
dt
exp[iKt(xx1)];
t
(19). On interchanging the order of integration and erforming the x-integral by residue calculus, it is found hat
(PA 1, 1'0
_
'a
ft
+
t-"I,(2Kat1/2)exp (itKx +rKy1.-2rKh)dt
t&ay.. yadrs rotoKopso.
van).-(21)
t is worth noting that this gives a check on our predicted
alue (T0) for the transmission constant T For it
is)lausible to expect, in the high frequency limit, that the ravelling wave terms of 9, will arise primarily from the )ressure distribution .P(x), with negligible contribution rom the (fixed) sub-rerged cylinder many wavelengths -om the surface. When aD, deform the path r onto he positive imaginal), axis, picking up a residue contri-,ution from t=1, whence,
cp1--47riKa exp iKx1+Ky,-2Kh)1,(2Ka) n agreement with the previous result (14). Similarly, the
eformation onto the negative imaginary axis, approp-iate when + co, produces no wave term.
The contribution of, to the integral (9) for Tis found y direct substitution to be
T1=27ZiK2a2e-2Kt+h I
f
2 1 I'1 I,(-Kat .2)e- 3N'"dt
Vhen a)2/h is large, the significant points of the Itearation path are at t =0, the stationary point r=(12/112 Lf Ka is large) and the singular point t= 1: it can be shown hat the contribution from the latter is negligible. If Ka is ,f order Kh, the stationary' point dominates-and leads to a esult for T, of order (Ka2/17) ex p 2Kh+2Ka2/11);
-ith Ka fixed, the leading term from the origin is of order
Ka)4/(Kh)2exp(-2Kh). In either case 7-, is negligible
ompared with formula (14) for T0l. To obtain a
niformly valid estimate for interniediate values of Ka, the stationary point may be cloSe to the origin, use he bound I ,(x)< Atx exp (x). .-I= constant. to get
IT,1<.-12(Ka)` exp(--21<h) t exp(4Kat1 2_
:
0
. lkinfilin/fiche van.Scattering of surface wores by submerged cylinders:. F. G. Leppington and P. F. Siew
=,4, exp( 21:1074 s exp (47s12 2s)ds
To verify that 1 T,1- (To 1), given by equation (14), use the inequality /,(x)> /1,x(1 + x2)- exp (x) to get:
T,/(T0 1)1<A5(Ka/K11)2(1+73)e2( I ± K2a2)e-21a
(22) with 7= Ka/WM". For large Ka, in the sense that Ka (K11)1'2, i.e. I, the exponent 272 2Ka is equal to:
{K(ha)2/h}1/2 <-27
From equation (.10) and the ratio (22) is exponentially small. For moderate Ka <(Kh)112, i.e. 7 <1, (1
+ 7"3)exp (272) is bounded., so that:
1T,/(T0-1)1<A6(Ka/Kh)2 < A6(K11)-1 < 1
Thus T,<T0 1
provided thatK(ha)2/h>1, as
asserted.
Similar estimates can be obtained from the integral (21) for 9, and 179,, when (x1, yl) is on S. It is found that:
14911<A7v2(1+ v)exp v2 (23)
aid
laVpii< A 8(Ka) 11,4(1 + v3) exp v2 (24)
where
V2 =(Ka)2IK(2h- y) <(Ka)2/Kh = 72 when (x1, y,) is on S.
Error potential
The scattered potential has been partitioned as: 49=4"o +91+0
with co,) and 9, given explicitly by formulae (11) and (20).
The error potential tp is harmonic, subject to the free surface condition (4) and radiation condition (6, 7), with
etwer= E9,/Er on S
To justify the approximation (14) for T we need a bound for cli on the surface, with K(ha)2/11 large, and this can be obtained from an integral equation analysis, due essen-tially to Davis3, as follows.
Green's formula is applied, in the fluid region, to (// and the Green's function:
Mx. v
x,. y0=
y, yi)-1-GtV: +y.
-(25)
Applied Ocean Research, 1980.. Vol.. 2, No. 3 131
_pro- I
G,(x x1; h. y,)=
I
e:-Kah-yo{exp[iKox _xl)]+ (20)where 7 = Ka/(Kli)". When 7 is small the latter integralis 0(1) while, for large;', the substitution s 472.x shows the integral to be of order 73 exp (272). Thus:
Scattering of surface waves by submerged cylinders: F. G. Leppington and P. F. Siew where G=G,,+G, is given by equations (17, 18) and
go= Go(x - x1: y, y1)-4G0(x: y. 0) It is readily found that:
10(x1, y1)=
-
i
g(ecp,ler)ds- (egler)ds, (x1, y1) on S(26)
To form a more useful integral equation, witha small kernel, we follow Daviss by defining the harmonic
function H(x, y) in the fluid region, chosenso that H 0 on the plane-surface y=h and with of I /Er = egoler on S,
27401110r).(6100{- y)-(x,, 2h- y1)1+
logl(x, y)-(0, h)1} (27)
since, by construction, the contributions cancel from the source singularities at (x1, y1) and (0, 0). This cancellation
is the reason for including the
term -IG(x; y, 0) in
equation (25) and leads to a small value of expression(27),
of order 1/11, when ha. The function H can be de-termined by conformal transformation (see Davissfor a similar calculation) but crude boundsare sufficient for the present purpose. When lila is large, the surface at y=li has negligible influence and H is of order alh; the dominant length scale near the cylinder is its radius a, so that the first and second derivatives of H are of respective orders 1//i and llah. When hla is close to unity, on the other hand, the
circle S and plane y=h appear almost parallel to an observer at a point (x, y) near the top of S where the dominant length scale is h-a. In thiscase H is of order unity, with first and second derivatives of order 1/(h -a) and 11(h-a)2. All these results can be summarized by the bounds:
IHI<.Aalh, .101 lexil< A/(h-a), 1521-11ex03:eil<Ahl(h-a)2a
(28)
where A is a constant and -xi= x or y. Now define:
W=H +(i/K)el!ley (29)
which satisfies the free surface condition (4) and depends
on K only through the explicit factor (11K) in
itsdefinition.
Application of Green's formula to 0 and g,
and subtraction from equation (26) leads to the modified integral equation:(x1, y1)= f(f.I
- g7)(e(p der*
,
1,111 - K-1 eHlayladO
aK-1(eler)(eH/ey) is of order hK' 411- n1-2.Hence the kernel is -small when K(h-a)21h.-..z. It follows that:
y1)-= I +o(max I), (x1, y,) on S where
I= -
f(g-g)(etpiler)ds
=1(x1, y1)-1-cp, (0, O) +J[
(epileocpiegler]d.s
using Green's formula on 9, and g inside S =0(hlh-a)max19,1+ 0(a maxlVcp, I)
i.e.
j/l<A(17/h -aXv2+1,3)exp(v2)+A(Ka) i(1 + v3)v4exp(v2)
from equations (23) and (24) where v2 =K2a21K(2h-y1). Finally the contribution of the error potential 0 to the integral (9) for T has the bound:
17e.1 < Ka exp (Ka -2Kh)111
which is found to be negligible compared with expression (14) for (T0 -1), using the method followingequation(22).
Thus we have the result that:
T-1 -47tiKa exp (-2Kh)11(2Ka)
as K(h - a)21h oo (31)
If Ka is large, the known asymptotics of the Bessel function I, show that
T- 1 --2i(trKa)112 exp ( -2Kh+2Ka)(1-3/(16Ka)+...) as Ka-co, wherein the first term agrees with the result due to Davis': evidently the full expression (31) gives .a sharper
bound as it provides all the terms of order (Ka)-N 2
exp ( -2Kh+2Ka) for all N. When Ka is small, on the other hand, the approximation I ,(2Ka)-,- Ka showsthat:
(T- l)-47ri(Ka)2exp( - 2Kh), Ka <41, Kh>1
which is the result established by Davis and Leppington4. Finally, it is noted that the estimate (31) has been derived by Corners in the limit li-oo with K anda fixed.
The present analysis shows the result to be valid in the much less restricted condition of large K(h-a)21h. SCATTERING BY AN ELLIPTIC CYLINDER
Here the cross-section S of the scattering cylinder is an
(30)
inclined ellipse (Fig. 2), whose equation is:
-The point of Davis analysis is that the kernel-a(?/(71)-111, K is small. For (and hence.i'll,/irris
seen from its definition (17) to be of order K- '(IF-a)-2 when K(11-a)--4, and the bounds (28) .s-hoW- that
132 Applic(1 Ocean Research. 1980. I,b/. 2. No. .3
(x702 +0,7m2 where _ ±iy)=e--"1:.. trans.-Figure 2. Proceo. function t The co. with maps the outside of Use is no incident r c2( with and where Jn p. 14). Th alternati while CI. where C
trans.
ure 2.
4
inc.Proceeding as before, c00 is defined as the harmonic ctibn thai vanishes at infinity, with
Nolen= (elen)exp ( iKz)
on ..S (33)The conformal transformation:
z = g((,)=eiNaC + flip (34)
2=-1-(a+b),
,3=1(ab)
(35)aps the region of the .7-plane outsidethe ellipse onto the tside of the unit circle C(ICI =1) in the complex C-plane,
+hi. In terms of q), pc, satisfies
Nolen= (eian)exp { iKe'NaC
+NO} (36)se is now made of the Laurent series expansion for the cident potential cp; =SE), namely
SIM= exp { iKe'°.(a( +NC)} =ao(K)+
cal Ea,,(K)C"+ Ebn(K)." (37) 1 1
=a0+
K)+11_(C.; K) (38) ith a(K)=(i)n(413)"12J(2Ke'eax1/2131/2) (39) b(K)=(i)U(13/cc)fli2.1(2Ke.°021/2p.1/2) (40)where denotes the tessel function of order n (watson6,
). 14). The function n+ is analytic for ICI < cc and has the Llternative form
ao + +()= (27a)- dc.
6
c.
1: inside C.,. (41)
while n_, anaiytic for KI >0; has the representation:
SI_(;)= (2ni)-1
4 dC'. outside C_ (42)c.
'vhcre "C: are circles centred at =
Scattering of surface waves by subtizekged cylinders: F. G. Leppington and P. F. Siew The solution for (P0 can now be expressed as:
(p0=S-2..(1/Z; K) (43)
and in particular. on the boundary circle (C: =1): = 11.= n+(i.:: K)n-(c: K) (44)
To calculate the contribution of coo to the integral (8) for R we have: R0 e -2Khi4(d49ddz)dz -7.-e-2"
1{0.p -n_col{o, +n-}dc
47rie-2KhE na(K)b(K) (45) i.e.20
R0 =47rie.K2'/31/2e-2/ J (1 Kenc 112/31/2)J (2KeA21/2/3112)
after using the values (39, 40) for and and the sum:
Ecind (0.1(t)
E(-1rinv_
= iffo(t).! t(t)
If we write C=(La2 b2)1/2 as the semi-focal distance for the ellipse, then c =221/ 2pn1/2, hence
R0'=27riKce0e-2 h.10(Kce1e0)..11(Kce0) (46)
which obviously tends to zero (as it must) in the
limit c,0
for the circle.Similarly, the contribution of coo to the integral (9) for the transmission coefficient is To, given by:
TO ie-2Kh'2RL Cpi
ds+
_
(po (pi)ds]
enThe rust of these integrals is:
Keri a 13/C2) exp {iKe-,.(g + x/1,) i KeiNar, + 13/C)Idi, c
=2rtK(x2 /32)(22±$2-2713cos 200)-I/2 x
f2K(a2+ j32 cos 200)"1
-from the residue at the essential singularity C .In terms
of the variables a. h. we have (al = /12)= a and
(47)
.-Ipplled Ocean Research. 1980. Vol. 2. No: 3 133
Scattering of sitiface waves by submerged cylinders: F. G. Leppinyton and P. F. Slew
(22 +10 22/3 cos 200)' 2
=(a2 sin2 0 + b2 cos2 00)' 2 =d
where d is the vertical distance of the highest point of the ellipse above the x-axis (see Fig. 2). The second integral of equation (47) can be written as:
+211-WO} fil.(1,")+K)-(0;c1C. hence where (
P(x)=(l
±-)f/J;)-(1+-
1.---.)nx/a
_ _ K K To 1 = 4rtie-2Khl(Kabld)1,.(2Kd)+134 Applied Ocean Research, 1980, 1bl. 2, No. 3
where the harmonic potential cp, satisfies the surface condition:
K-! cp,Ii.:y= P(x) at y=1, i.e.
evaluated at 4-(z) = ;(x +111). The solution for (p, is given by
formula (19): thus
(PI(Y1. y,)=(K/ ..1 .7r)1r +1eKfiy,-Illf
t
r
e'`,[L,(Kr)+L3(Kt)]lidt
(1) (2) (3)
=-91 +Pi +(Pi
say, where and 1 CI.1,1:(x+ilz) - . -iKtxax. iL,+ L3 = -S2,.(1/C)eiKrxdx 7. 7 ,=
j
(Q a0 S-2_)eiKL`dxafter uSing the representation (20) for G. Each of the integrals L, and L, + L3 can now be evaluated bycontour integration in the c-plane. Thus
L (Kr)=Y2_ (C) exp (
iKt(zih))dz
evaluated over the free surface z=x < x <
= i(Kt)- Kth
_ K) c7:(C; Kt)cic
evaluated round the unit circle C,
=27r(Kt)- le-KM 2.7na(Kt)b(K) 1)- lceillne-Kth fl,(Kcte'yo)J,(Kcei°9+
J'o(Kcren)J,(Kce'0)1,- (54)
using formulae (37)(40) and evaluating thesum of Bessel functions. Similarly. dealing with the complex conjugate of (L, + L3) for convenience, we find for real t:
(L3 4- 42)= i(Kt)ie --K"J n+(lIt...:K)7f27.(c: Kt)cli,
ac
Kill
T.3+ L2=
;f1(i,c: K)
, dI Kr)dc+'
En()0/2)1J(Kce'0)12) (49)
which will be claimed to be the asymptotic value for (T-1) when K(hd)21/2-0x. When b = a (/3 = 0), then d =a and we recover the previous result (14). An alternative form for To, obtained by expressing the first integral of equation (47) in terms of the Laurent series expansion (38) for(pi, is
To _1 = glue -2Kh
F.
nian12
= 47rie-2KhE n(2/fi)'l-/,,(Kce'°912 (50)
This has a more compact appearance than its counterpart (49) but is less convenient to estimate when Kcocc or when )6.0. It can be shown that the sum in formula (49) is of exponential order exp (Kcsin00) where here and henceforth the notation f-- 0 exp (Kp) means there exist numbers A, n such that If < AK" exp (Kp) as Kco,p
fixed. Thus the sum is negligible compared with the Bessel function when Kc and Kd are large. Thus
To 1 47ri(Kubld)exp( 2K11).1,(2Kd) (51)
as Kcox,
For small values of Kd, on the other hand (but with h large to ensure Kh).a)), the two tert-ns of equation (49) are comparable and equation (50) is slightly more convenient. Second approximation
Proceeding as in the previous section for the circle,we
introduce the improved potential:
4it (48)
(PVo+.9i
(52) which inteer, functic where The Apper. expon on S. respot conch simila contri reflect Thi limitec is of e expres ensurc has th +2K, of the allow theref Ro. the le. with I (49). Fiat reduc and which = 0, s large for-01:112111ZEZEINM1=4=0Erz5715
which can now be regarded as two separate integrals. The integral for L3 can be evaluated explicitly as a Bessel
function, while the other can be expressed as a sum to get:
L3
L, =27(22 ne-
/
I11(2Ki.)+27r(Kr)- le-Kzh2.7nbn(K)r).(Kt) (56)
where
;.2
e -2i0(21._
flei°9(fit
2e'.)
(57)The solution (53) is now complete. It is shown in the Appendix that this potential and its derivatives have the exponential bound:
jVcp11 0 exp(KE) (58)
on S. where E<d. An integral equation analysis, cor-responding to that in the second section, leads to the conclusion that the error potential IP =9 90-9, has a similar bound when K(hd)2111 3o. In particular, the
contributions of 91 and cfr to the integrals (8) and (9) for the reflection and transmission constants have the bounds:
T, and R1-0 exp {K(E+d 2h)] (59)
This is sufficient to justify the estimate (51) for Tin the limited circumstance of large Kd. For then expression (51) is of order 0 exp (2Kd 2Kh) and large compared with expression (59) (the overall requirement K(h d)z lh
ensures KEKd). On the other hand, formula (46) for Ro has the much smaller exponential behaviour
exp 2Kh
+2Kc sin Bo}, because of the high degree of cancellation of the integral. Our crude bound (59), which does not allow for the possibility of such effective cancellation, is therefore not sufficiently sharp to justify the estimate R --Ro. There is, however, some evidence that Ro is indeed the leading term, and it is conjectured here that:R R0 and To for KO d)2111>1 (60)
with R, given by equation (46) and T0 given by equation
(49).
For small ellipses (a </z, Ka <1) the results (46) and (50) reduce to:
and
Scattering of surface.waves by submerged cylinders: F. G. Leppington and P. F. Siew
R 2i exp ( 2K11) cos 2Ka.
ie-Kzh df2
Kt
K)((: Kt)d(:
dC (53)rriK2(a2_b2)e-2Kh+.1i0.
T triK2(a + b)2e2h
which agree with those due to Davis and Leppingtoe. If h =0. so that the ellipse degenerates on to a strip, and for large values of Kasin 0, we have:
R exp 2K(11J, sill 0o) 2i Ka cos 00;
_
for 0 <0, <
_
17. mormagsnwaffs=mezn=maggiga
-for 0, = 0, Ktt I. which agree with the results obtained by Tengara7. The trarislnission coefficient for thedegenerate ellipse (1)=0. hence 11=0. c =a) requires evaluation of the infinite sum (50) and this can be done explicitly when 0,
=in (the x7ertical plane barrier). Then
T, 1 = 4rrie' E n1(Ka)2 =27riKae'10(Ka)1,(Ka) and the leading term for large Ka is:
1 4exp(-2Kh+
2Ka)-which agrees with the results of Tengara. The next term does not agree, nor could it be expected to since our
results have been derived for smooth bodies, and the limit of ellipse to strip is not strictly allowable.
ARBITRARY SMOOTH CYLINDERS
The form of the solutions in the previous section suggests
an immediate generalisation for the scattering by an arbitrary, submerged, smooth cylinder of cross section S.
Suppose the conformal transformation:
z=g(C)=2(+1, +72/C+23/C2 + (61)
maps the outside of Sonto the outside of the unit circle in the (-plane, with 7 real and g(() analytic for p <ICI< ce(p <1). Then the incident potential
swo=exp iKg(()),
is also analytic for p <ICI< oo and can be expressed as a Laurent series
=a, + E cn+ Eb.c"
(67)where the first sum f2+ is analytic for ICI < 7o and the second sum Q. is analytic for ICI> p.
Proceeding as in the previous section it is now con-jectured that the reflection and transmission coefficients
are given by T.-- T0 and R Ro where
R0 =- 4taexp(-2Kh)E nab
(63)and T, has the equivalent forms:
T0 1 -=47ti exp ( 2Kh)Enkt12 (64)
= 4tri ex p ( .2Kh)Enlbj2 T,(71-71(pil2d5 (65)
These estimates are ex p-ected to be valid for K1-12/D>1
.Where His the depth of the highest point of S and D is its Maximum diameter.
Scattering of surface wares by submerged cylinders: F. G. Leppington and P. F. Siew REFERENCES
1 Dean, W. R. On the reflexion ofsurface waves by a submerged
circular cylinder. Prix. Camh. Phil. Soc. 1948, 44. 483
lirsell. F. Surface waves on deep Nvater in the presence of a submerged circular cylinder.1.Proc. Coln!). Phil. Soc. 1950. 46. 141
3 Davis.A. NL.J.Short surface waves in the presenceofa subthereed circular cylinder. SIAM J. Appl. Muth. 1974. 27. 479
4 Davis. A. M. J. and Lepriington, F. G. The scattering of elec-tromagnetic wavesbydm-War or elliptic cylinders, Proc. R.Soc.1977. A353.,55
5 Corner. I. E. Small amplitude surface waves over submerged
cylinders and channels with periodic floors. Ph.D. Thesis, Lancaster
University(1978)
6 Watson. G. N. Besse! Functions. 2nd Edn.. Cambridge University
Press. London. 1944
7 Tenggra. I. Ph.D. Thesis. Imperial College.London. (1980)
APPENDIX
The potential co = (of' + 4' is defined by formulae (53H56) terms of three functions Li(Kt), L2(Kt) and L3(Kt). An estimate for 9, and its derivatives is now derived for points (x ):1) on the ellipse S.
It is a simple matter to estimate For the known asymptotics of the Bessel functions at large argument show that the expression (54) has the form:
L (Ki)-0 exp 21(21/2/3'121sin 001(t +1) Kthl (Al) and
$ 1sin 001= clsn 001a2
2.2112
I '2i
D )2s1, 001is the vertical distance of the highest focal point above the x-axis. Since h>d.(a2 sin= 00 + b2 cos2 00)" >cisin001, it follows that 2/1y > cisin 001 for all points on the ellipse. Thus the integrand of expression (53) for q,"i" decreases exponentially as t increases, and
integrand 0 exp (Kcisin (A2)
In view of the infinite range of integration and the
singularity at t = 1, a little more is required to show that the integral 0,1) has the same bound. Write the integralas:1)- 'F(t)dt .r
F(t)
F(1d
) =f
- r+ F(1) -if
(t clt + t 0+0 (t - 1)- I nlidtwhere Tis.some large positive number. The first integral has the .bound (A2), while .the.secorid and third integrals have the respective bounds
exp 1:(211 y1+ 2Kcisin 001 and
e:s_p K71211-11+ Kcisin 0(T + I
136 Applied Ocean Research, 1980, 1i)1. 2. Nit 3
Which are negligible. Thus V," and its
derivatives ha vC the bound:
0 exp(Kcjsin 0I)
f..13
To derive a similar estimate for 0,2) itis helpful to use the representation (55) for L2:
T-2(Kt)=i(Kt)le-Kt5 fLog:
KW;
(A4;
Now from formula (42), with C_ chosento be the circular path =(/3/2)'12e°,
1 exp( iKei°02ai 213"cos 0)
12_(1/C.:
j
(fl/(fl/'2e1_2)o
(/3/2)1 2ie'd whe.-7E
The exponent has greatest real part Kelsin 001, with = 7211?fl", so that:
Al-g% K) 0 exp(Kcisin001)
for 1c1-< (2//3)1' 2 (AS)
Now deform the path C, in integral (A4),on to the circle : =66/2)"e10,
.0r, and use the bound
(A5) to get:L,(Kt)-0 exp (KcIsin 001(1 + t)Kilt) whence (p(12) has the same bound (A3)as (pi,".
The third integral 0,3) of formula (53) is a little more difficult. Note firstly that L3 (given by the first term. involving the Bessel function, of equation (56)) has the behaviour:
L3 ^- 0 exp (Kc1sin 001t
Kht) as t=,x,
so that the integrand of formula (53) for 91)has the form: integrand exp {Kelsin 001t Kr(2h=y,)} (A6) as t 00, and is certainly exponentially small. It remains to bound the exponential behaviour for all real positive values oft. When 00=0 this is easily done from the known asymptotics of the Besse' function /, but the algebra is cumbersome for arbitrary values of 00. The simplest
procedure is to use the integral form (55) for L3, taking C to be the circle r, =Re°, tr-40 <it. This shows that the integrand of expression (53) for teoc,3' has the exponential bound:
where
E(00)= max [ t. +H +(///R)sin
(0
(0)
xR.sin (0 +00)1 +(a/ R) sin (0+ fiR sin (0 0)]
here II =21v1 and R is a positive
!lumber at our'integrand exp(KE(00)) (A7')
The sub.) RI< thar. R2 < if disp. SO t max pros
disposal. Now the coefficient of 1 has the least value: H 1(7R 13/R)2 +41/Isin2 00; I2
so that restricting R so that this is positive gives the maximum value for E when t =0. Thus
E..max 1(2/R) sin (8 +00) fiR sin (8 00)1 ={(13R-7/R)2 + 44 sin2 0 }112 (A8)
provided
Kr <R (A9)
where
R2, R = (22) I(H2 +42ficos2 Bo)" +
(H2 sin2 00)1/21
The best bound for E is now obtained by choosing R,
subject to the constraint (A9), so as to minimize (A8). Now R1 <(1/13)1/2, while R2 may be either greater than or less than (4)12. If R,>(2/fl)1/2, choose R=(a//3)112, while if
R2 <(21P)112 the best choice for R is R2. Thus
E2.. sin2 00 if
H2 4713 sin280 > (22 /32)2/2fl
y.o.a ....,trosubasse
ani
Scattering of surface wares by ..submerged -cylinders: F. G. Leppingtonand P. F. Slew otherwise
E2--<.:(13R.,,T2/R2)2+ 42/3sin2 00
Let tl=(a2sin2 00+ b2cos2 0)l2 denote the vertical dis-placement of the highest point of the ellipse above the x-
-axis. Since H=11 y, ..>-d on the ellipse, we can write H2
(12 ± /42, say: on writing 2: and /3 in terms of a and b
(definition 35). the bound for E can be expressed as:
E2 b2 for p2 > b2(3a2 +b-2)/(a2 b2) (A10)
and
..E2
<1
2ab(a2 ± i12)1i2 ...(a2 +b2)(b2 4. p2)1.12)2-. a2 b2
d2b2
(All)
otherwise. Since this function decreases as p increases, its largest value is at p = 0, and since p> 0, we have
E<cl (Al2)
This, together with the bound (A6) and the result (A3) for
cpC,1) and cp(,21 ensures that:
(pi. and IV'tp Li exp (KE), with E <d (A13) for large values of K. A more precise bound for E is
provided by formulae (A10) and (All) In particular, when the ellipse is sufficiently deeply submerged so that p2 =H2
d2>b2(3a2 +b2)/(a2 b2), we have
91 and IV'co,I 0 exp(Kcisin 001) (A14) Li matrotilml-ilche an
