Scattering of surface waves by rectangular obstacles in water of finite depth

Th obs use and doc oth 1. ¶ mal tial regt (Kr cas veri (De exa a s for Ii for I i 96L p. 4 way the usef F way effe rect clos for t junc (Tak 32-2 Entroduction

'he scattering of water waves l)y large obstacles has been well known for the ;hematical difficulties encountered within the framework of linearized poten -theory. While some general features (such as symmetry relations, bounds)

Lrding the reflexion or transmission properties have been studied previously eisel 1949), explicit calculations have been successful only in a few limiting

s. Most of the known exact solutions are in waters of infinite depth for

ical thin barriers(l)eari 1945; Ursell 1947), semi-immersed circular cylinders

an & Ursehl 1959), or a Step shelf (Newman i965a). Furthermore, the only et solution for a continuously varying depth is that of Roseau (1952) for

ecial bottom profile. Barakat (I 968) has recently given numerical solutions

symmetrical cylinders with rounded corners fixed in the free surface.

i the case of finite water (lepill some approximate solutions have been found ong waves (Kajiura I 961; Ogilvie 1960), for long bottom obstacles (Newman sb), and for low bottom ol)StaelCS (Kreisel 1949; Mci 1967). Stoker (1957, 34) dealt with the problem of a surface dock with zero submergence in long

es. While these studies point out many interesting qualitative features of wave scattering, their approximations also severely limit the quantitative

'ulness of the results.

or engineering PIIFi)OSCS it is of interest to obtain results valid for the whole

elengtli spectrum, finite water depth, and finite obstacle dimensions. rThe

t of various geometrical proportloils can best be understood by investigating

angular obstacles in a channel of finite depth. This class of problems has

e analogues in electromagnetic wave guides with a thick inductive window, sliieh a variational method of solution is most effective.t Miles (1967) recently Theoretical attempts by sttaightforward1- joining eigenfunetion expansions at ion surfaces have been unsuccessful. hi-cause of poor convergence in numerical womk ano 1960). I

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199

ecrniscne nogescnoc

Scattering of surface waves by rectangu?ar

obstacles in waters of finite depth

By CHIA NG C. ME! AND JARED L. BLACK

Hydrodynamics Laoratorv, Department of Civil Engineering, Massacthusetts i ristitti e of riechf101ogy

(Received 11 October 1968 and in revise(1 form 4 March 1969)

scattering of infinitesimal surface waves normally incident on a rectangular tacle in a channel of finite depth is considered. A variational formulation is rl as the basis of numerical computations. Scattering propert.ies for bottom

L surface obstacles of various proportions, including thin barriers and surface

ks, are presented. Comparison with experimental and theoretical results by investigators is also made.

RCHEF

(1969), vol. 38, per(3, pp. 499-511

r

.500 (J. G. Mci and J. L. Black

applied the variational method to a stej shelf and achieved remarkably good

results, as compared with the numerical solution by Newman (1 965a), for the far

field with only a cru(Ie approximation on the near field. In this paper the same formulation is adopted and numerical results for both surface (semi-immersed) and bottom obstacles of various projoitions (figures I(a), (b)) are presented.

Only the normal incidence of simple harmonic waves on two-dimensional

obstacles is treated here. Available experimental, as well as theoretical, data aré collected for comparison.

2. General formulation

In this paper explicit calculations will be done for an obstacle fixed either on the bottoni (case I) or in the free surface (case II), see figure 1, although the com-bined problem of a thick barrier having a submerged gap can be similarly treated. \Vitli the usual assumptions of a perfect fluid and small amplitudes the velocity potential,

(l)(x, y, t) = y(x, y) &", (2.1)

In addition we impose the radiation condition that there are both left- and

right-going waves atX

-

but only right-going waves at x

As is usual in problems of wave-guide discontinuities it is convenient to split the

i)oterltial into a symmetric and an antisymmetric part

=

with ç(x,y) = ç.(x,y),

¿,.,(x,y) = ç1(x,y)

(2.6)

stich that the analysis can be restricted to x < O only. Physically s()

corre-sponds to the scattering of two waves incident from x

-

and x towards

a symmetrical obstacle, with equal amplitude and equal (opposite) phase. The symmetrical case has also been studied by Miles (1967). Since the energy flux across the plane X= O is

E() =

_J

p]

dy

J{bi( )b()]

dy, l)() = or and since (0, y, 1) Dl4 (0, y, t) = 0, it follows that ß,. = E1 = O at x = 0.

Thus the reflexion coeflicient is of unit magnitude in either case;

lI =

lIAI = 1. (2.7)

satisfies tlìe following conditions:

V2ç = 0, x,y in fluid; (2.2)

j__O9=O

U=U)/(J 1H < : (1) kIxI

>1: (II)'

(2.3)

y=h,

(

>L(Ç]};

_y=Ii,

_{xi <1: (i)and(I1);}

_(2.4)

= °

Dx ' 1x1 =i,

h <y < li: (l))

H<y<0:

(11) (2.5)

caiierinq of urfae waves 501

owever, it is the phase difTerenees between the complex reflexion coefficients

J and RA that contri hufe to (lie non-trivial character of the resultant scattering.

Ñ)ecifically, the total reflexion and transmission coefficients are I?

'I

e°u = ,(J?+R4),

(2.8) Rellected T _TI = (a) H I ncdcnt Trjnsinittcd

-Ii (I,)

J"IcuuE 1. DefluitI ion ski ¡1 i. (a) Siulanerged o1)st.aclo. (b) Surface øbtaeIe.

When the complex coefficients R and R4 are represented by unit vectors in

a phasor diagram, one can deduce the following known results (Kreisel 1949):

IRI+V11I2= i,

R.T = 0.

The phase angles 0/? and 0, djffer by i; however, their respective values depend

on the quadrants in which the vectors R, and R4 lie.

3. Case I. Bottom obstacle

(a) The anhiRyn?me/r,c pail and the variational form ulalion

Although the variational formulation to be employed is a standard 1iiocehiie in

wave-guide problems (Collin 1960) and has also been demonstrated by Miles

(1967, 1968) for byclrodynamical problems, we shall outline it bere for the. sake

of convenience.

502 G. C. Mci (111(1 .1. L. lilac/C

The potent ial for dif1rent regions will be expressed by appropriate

eigen-f utictioli CXJ ansioiis; eigen-for cxi licitness we separate the propagating mode eigen-from the evanescent modes

'i) = b1[ + r4 eiIc +t)]f + b,, 1q,,(XI)f

(x< I, h<y<0) (3.la)

=JJ1l'siiiA..v+'b',,l',sin1iQ,,x

(j2j<l, ll<y<0);

(3.lb) wiicre J1() = .,I2(h + r-' sinhl2kh)H cosh kU+h);

J,(y) = .j2(h

sin2q,k) cosq,,(y+h) (n = 2. 3, ...);

and L. q aie tue real positive roots of the following equations:

k tanli Ich = o-, q,, tan qh = - o- (n = 2, 3,.. .). (3.3)

It eau be readily shown that

_{fJ

form an orthonorinal set over the interval - h < y < 0). For the shallow region the quantities 1', K, and Qn are defined sinmilamly by replaeingf1, k, and q,, above vit1i k changed to II. We note that the reflexion coefficient is given by

11., = r (3.4)

(ontii,uitv of ç., at x = - i reqUires that

i +

r4)f +

b,,f,, = - li 1F1 sin Ki B,, J', sinh Q,l. (3.5)

Contiiiuitv of the horizontal velocity

at z = 1 requires that

U,1(y) - ilcb1(1 - r4)j1 + co 2

lJ<y<0);

= B1 KF1 eos Kl + B,, Q,,J, cosh Ql,

=0 (h<y<--II).

(3.6)

When the orthonormnal conditions are used the coefficients b, B can be expressed as integrals of U,, and f, F from (3.6); substitution into (3.5) yields the following

ititegral equation for L(y):

(0 (.0 S4f1 I

Ufdy'

= I UG4(y/y')dy', (3.7)

J-Il

J-Il

where S'4 (I + r.,) (ik)-' (1 - r1)', (3.8) U4(y'), etc. (3.9) 11(1 G4(y/y') =

jf/q,,+

-1J'F+

(3.10) 2 2

Multiplying both sides of (3.7) by U4 and integrating vitli respect to y froni

II to 0, we obtain finally

O

¡II

2

= U1 (x'4(ij/y') U' dy d!/' U., f dy (3.11)

-Il -II

caI/ïq of snrface waves ₅₀₃ t is easy to show that, in _{he form (3.1 1), S, is stationary with respect lo}

(lepefldent variations of L, _{1',. By virtue of (3.8), ?S is oi,viously related to}

he reflexion characteristics. \Ve _{shalt lust assume a fònn fòr (L, with many} parameters {ur}; t1e 'l)cst _{va hies fr 'r} will then be chosen by invoking the}

fact that 8 is stationary.

(h) CaT,i1atíon of Ñ

We follow the steps explained in Cofl iii (I 900, p. 352) and expand U4 in terms of eigenfunctions .i.(y),

= U;.J.(?J), _(3.I

with tIie coefficients u,.yet unknown. Substituting (3.12) into (3.11) and invoking

the conditions

= O (r =

a set of homogeneous equations for Vr results. For non-trivial solutions the

vanish-ing of the coefficient determinant (J! x JI) finally yields an expression for

=

/

.qj I Uil. (/ i / A A ¡ UL (3.13) ro

where 1

fFd/

_{(/Isin2q/)_(íJ_o._1sin2QrI)_I}

J - Ji

x f(q1, + Qm) Fsm (q h. +Q11, JI) - sin q(h - il)]

+(q11Q,5)[sin(q5h--Q,,,1J)_sinq5,(h_I1)]}, (3.14) arid co

tinJ'7

co tul1hQ i =_{'n2 "} 1r1s

₂

₁₎ 8nr'n.s (n, 1,2, 3 ,...,JI). (3.15)

In order not to repeat (3.14) for

_{Pi,,, and 1, we mention that q}

should he replaced by ik for n _{i, and Q,,, by iK for m = 1.}

Thus the problem reduces to the numerical evaluation of the determinant.s where each term is ColllpOse(1 of inliiìite series. Computational aspects are

discussed in § 6.

(c) The s,ìiì metric part

Excpt for those indicated hielow, most of the relations in § 3(a) and 3(h) remain

applicable with tile mere ellange of subscripts _{from ( ), to ( ). The potential}

for the shallow region should be changed _to

co

= B1F1eosKx+BFcoshQx (ki <i, II<y<O).

(3.16)

(pj)

_{\ li, Is} J)

(J. C. Mei and J. L. Black

Formulas replacing (3.10) an(1 (3.15) are

G5.(y/y') cot Kl

J'F'

coth Q

¡n Qn

i

'

'l"fl

eotkl8

,

cotiiQl8

. r- jr 1,

,ì

nr ne

2 n 2

4. Case II. Surface obstacles

The tillience from the ease ut botte in obstacles arises solely from the different

iindarv conditions in the region xf < i. Without introducing new iìotatioris,

shall only identify those formulas in § 3 that iìeed liIo(hihcations and givethe

Irespon(hn( replacements:

= B1Fx + lJI" sinh QnX, _{Ix < i,}

k.

(4.1) for (3.Ib)

=

B,JcoshQnx,

h<y< Iij'

(4.2) for (:1.16)

where

= D1, F =

(2/D)i cus Q,(1+h), (4.3)

Q,, = (n I)n/D, D

hII.

(4.4)

TIme functions 1J form tiìc complete orthonormal set appropriate for the

boundary conditions (2.4) (case II). In ç we have omitted a constant terni

J3 F1 vlmieli is immaterial to the flow field. rflle limits of integration iii formulas

correspondimig to (3.7), (3. [1), and (3. 14) are now from - h to II. Furthermore

we have

G(y/y') =

(4.5) for (3.10) 2 J,, 2 = (4.6) for (3.17) 2 q ¿n and n,

J) P

n t'uihQ i

(j = . nr +i8jròj,,+ _: nrò'n., (4.7) for (3.15) 2 n 9 _¿n

n rp ¡J

cothm i i

= V

nr n.

'

r ò',, I (4.8) for (3.18) L q

where {Jn} are given by (3.2) as before. Because of the changes in {j the modified

fòrms of arc:

P,,1 = (2/D)i (lt ' sin2 qh) sin q4D q

= 2(

I ) [I)(h

' sin2 qh)]i sifl q,,D

Q;n

5. Limiting cases

.

(a) J hut barru'ì'. ,fu tie dept/i

For a barrier of zero thickness, fixed either on t he bottonì or in the free surface,

the symmetric part., , corrcspon(ls to a vertical cliff extending for the entire depth of the fluid, and the reflexion coefficient, R. must be 1. Thisresult also

(4.9 a) (4.9 b) (3.17) (3.18)

Scattering of surface waves 505 flows from the general formulas of the preceding sections. Taking the bottom bstacics, for instance, for small i, q1 behaves like l_l while all other g (r or

s 1) remain finite; thus U' and tini /?. = 1, as is evident from (3.18) and. with ( ) changed to ( ), (3.13) and (3.14). The limit of 1- O is easily taken for the antisymmetric part. 'l'ue representation for U4(y) by (3.12) is still a useful one, although {1i} is now merely one of many ortlìonormal sets that may be

a(lopted.

(b) infinite water dept/i, arbitrary thickness

As h - cc, the cigenvalue spectrum in the zone _lxi > i for bottom obstacle con-tains a discrete value (Ic = o-, propagating) and a continuons part (O < q < cc,

evanescent). The corresponding eigenfunetions are (Miles 1967)

f1C) = /(2o-) e'', (5.!)

fC) [2/n(q2 + o)] (q cos qy + o-sin (Jy),

which are orthonormal in the following sense:

Jo

o o

fdy

= I,

f

f,fdy

=

0, f

f(y,q)f(y,q')dy

= 6(qq').

(5.2)

The series involving f, in (3.10) and (3.17) must be changed to integral forni

The variatonal expressions for S or _{are then obtained by substituting (5.3)}

into (3.11).

Although the general representation of U4 or L by (3.12) can be used, in view

of Miles' (1967) success, in the ease of a semi-infinite shelf, by using only the

first term (the plane wave approximation), we record the corresponding approxi-mation below

[SA] (K2v2)2e2 1

(1 q[qsinqH+o-cosqfJj2

Lsi -

4o-3

kJo

q _{(q2+ K)2}

+

I Ltj1]

[tan

_i

(H+o-1sinh2KH)l. (5.4)

J

The integral above has been evaluated ¡n terms of tabulated functions by

Miles (1967).

For surface obstacles, as h-* cc, D =

h ii

->c.c also and the eigenvalue

spectrum Q, becomes continuous. The limiting expressions for (ì

and (

can

be similarly written; however, a convenient and suitable trial function for

U4(q) or U4(j) has not been found by the authors. The formulas will therefore

be omitted.

6. Aspects of numerical computation

For all cases of Imite water depth. the numerical work involves the summation

of an infinite series for each q.. and the evaluation of determinants. From (3.3) and (3.14) it can be seen that, for large n, q,,h

(n i)i and

_flr q1; hence

hence GAl = Osi Joc drj,., + q [ tan L - cot J I I Qnhl _(5.3) Leoth Qlj Q

(]. Alci and ,J. L. BlacL

n for given r and s, and the series for q converges reasonably fast.

I low-ever, owing to the factor (q

_{- Q)}

11 a large (although finite) 'ir is possible. As an estimate we take n and r to be large, thenq Q,.

if

u rh/lI. (6.1)

Therefòre, the series cuinot be truncated until beyond this threshold value.

It also indicates that, as h/Il increases, ixiore terms must be summed to assue accuracy. Although these qualitative observations aie itiade with respect to the bottom obstacles, tue conclusions ai-e truc foi surfice obstacles as well.

Numerical tests were made to check the convergence of the results by varying

the number of terms summed in each q,.,. and the number of ternis used to ap-pioxiinìte U(y) (equal to the order of det.erwiiìaiits). LTsing 5 termst for U(y)

and 5h/1I + 15 ternis in the series (based on (6.1)) we obtain an accuracy within one per cent if h/Il 10 for bottonì obstacles, and ifh/ll G f» surface obstacles.

rllìe slower convergence in the latter cuse may be expected because the leading ternis in the assumed expansioll for (f(y) become increasingly unsuitable for large h/Il. All computations were donc on an i BM 1130 computer.

For bottoni obstacles and infinite h/Il calculations ere donc only with a one-term (plane wave) approximation according to (5.4). Evidence of good accuracy

is provided by comparing withì Dean's (1945) exact solution for a thin barrier. Presuniably, an even better approximation for Dean's case would be

U(y) (fJ2_ì/2)

whichi would give tue scattering coefficients iii i-niis of modified Bessel and

Struve funeti(.)ns.

7. Results and discussions

(a) Bottoth obstacles (figures 2. 3, 4, 5)

As in Newman (I 965b), the prornineiìt feature is the oscillatory nature of the

reflexion cocflicicnts resulting from the interaction between the two ends of

the obstacle (figure 2). r1lme oscillation increases with the obstacle length. We remark that the scattering of a semi-infinite shelf studied by Newman (1964 a)

and Miles (1967) cannot be obtained by taking the limit l from our treatment, since, froni the standpoint of an initial-value problem, their problem corresponds

to letting I - before letting t -* co whereas our formulation assumes the inverse.

In ligure 2 only the phase shift of the transmitted wave, O, is plotted: that of the

reflected wave_°R can be obtained through the following iclation:

= O+( l)Nir,

for k1< k < kN+j, (7.1)

where k (iV = 1, 2, 3, ...) refer to the successive nodes of the reflexion coefficient

(excluding k = 0). The sudden changes in O can be explained by imagining

a phiasor diagram with the unit vectors R5 and R1 issuing from the origin.

\Vlìeiì the relative phase 1A O changes from less than mr to greater than

iii similar electromagnetic wave-guide pi-obleilLs, a two-term approximation gives an accuracy within a few per cent (Colhiii 1960, p.359).

Rl -0 ¡ 0 -4 il fi k,! SI 02 O (12 04 i

i'

&atterinq of surface waves :507

ft6 4H

Fieuiu 2. Reflexion coefficient ami I rallslnissicin phaie anglo for a submerged obstacle,

4/11=2:

_{.1/iI = 0;----_,//Jf= 2;---,l/H}

_{4; 1/11= (L} -- A mp s1op 21//I 4/If

'!

___X\ IO 1.2 14 16

'tcnmi 3. Reflexi,,, coefficient, for a slIlmerged obstacle. For _{/ II = 443. h/li = 278:}

-, theory; O, Jolas' experiment. For i/Il = 443, h/Fl

_{: - - -, theory;}

- - -, Newman's approximation.

C. Mci and .1. L. Black

or

K

1/11=25

Lii

.F'1GuIu 4. J(ellexiozi and trarLsinission cfliciexit _{fn siibixicrged obstacles. For i/H = 25,} h/li = 5: 0, Dick's experiment, , theomv. J'or 1/H = O, h/H = 5: , Dick's expon-ment, , tliory.

1/11=0 1/11=0

OS 14

ici!

FIGJJRE 5. _{Reflexion coefficient and transmission pliaso anglo fòr a sulimergcxl thin plato.}

-, Dean's exact solution for h - co; , Ogilvie's long wave approximation. ((.4

i

i

&atiering of surface waves 509

i(Rs+RA) changes its sense by ir and hence °fl°T varies from

- ir to

ir in principal value. Since /?j = O

at O. O

ir these discontinuous changes

occur at the nodes.

Comparison with Jobs' (1960) _{experiment and Newman's (1965b)}

corre-sponding approximate theory is made in ligure 3. Newman's calculation was (lone Ofl the, basis of h - while in Ihe exl)erinlents h is finite. Hence two curves

j, T O I'O 06 / / 1111= I 04 1l1= / o

/

/

/

/7/1= O

/

/

kil

FIGURE 6. Reflexion coefficient arid reflexion phase angle for a surface ohtac1c:

theory for h/il = 2, 1/H = 0, 1, 3, a; - - -, Ur.sell's exact solution for h/H

-1/H = 0; ----, theory for h/H = O, i/H = 1. o, Kincaid's (19GO) _{experiment for}

h/H = 617, i/H =

are calculated from the present theory. For h - _{our values are based on the} plane wave approximation arid the agreement with Newman is good for long and very short wavelengths, but rather crude for intermediate values. Tt should

be remarked that, to use Newman's approach for finite h, the explicit knowledge

of the semi-infinite shelf with finite h is needed, which is not yet available, see Mites (1967).

Recent experiments by Dick (196S) are plotted against our theory in figure 4.

For the obstacle with finite length (i/Il = 25) higher harmonics are recorded

on the transmission side and the experimental values of TI are not easy to define;

direct comparison with the present theory is therefore not made here.

For submerged plate barriers (figure 5) it may be noted that the plane-wave

o

/

/

510 C. C. Mci and J. L. Black

approximation. (5.4), compares quite favourably with Dean's exact solution.

Ogilvie's (1960) long wave approximation is also included and the agreement with

the j)reseut theory is excellent.

O.:

,1/

---z z

''-z

/

/

//

/2

/

,, /

I I 04 06 05 10 Id

-

--

----Fiuuiu 7. Reflexion coefficient and reflexion phase angle for a finito clock: h/I =

1; - - -, h/I

= 3; - - - - h/I = 5; , Stoker's shallow water result.

(b) Sw jtce obstacles (figures 6 and 7)

The effect of varying obstacle length is shown in figure 6 while the case of finite dock of zero draught is shown in figure 7. We find it convenient to plot the modified

phase shift O O - 2/cl instead of 0; physically tito former amounts to

measur-ing the phase angles with respect to the incommeasur-ing edge x

= -

i. In all cases,

=O,+ iii. For a finite dock in a vater of infinite depth Ilolford (1964) lias

obtained analytically O = +O(1/Icl),which scents to diflèr from the standing-wave solution (hence O = 0) of Friedrichs & Lewv (1949, equations (39), (40),

. 146) for a semi-infinite dock. Our calculated results indicate the tendency of

-

O for large /1. Stoker's (1957. p 434) formula based on linear long wave

approximation is also plotted in figure 7. Fori/li = i it can be seen that Stoker's

result is surprisingly good for practically all values of frl.

In conclusion, the variational approach greatly facilitates the calculation of

scattering properties of rectangular obstacles, of which only limiting cases have

bcoiì treated heretofore.

This research lias been carried out with the sponsorship of the Office of Naval

Research, U.S. Navy, under Contract no. Nonr-1841 (59). The authors are

indebted to a refèree whose comments led to some revisiong of case II.

18

caIierinq of surface waves 511

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