Interaction of waves with two-dimensional obstacles: a relation between the radiation and scattering problems

cfl 11!F

[To be published In J.

Fluid Mech.]

Revised March 1975

Interaction of waves with two-dimensional obstacles: a relation between the radiation and scattering problems

by

J. N. Newman

Department of Ocean Engineering Massachusetts nstitùte öf Technology

Cambridge, Mass. 02139

Lab.

y.

Technische Hogeschool.

Deift

A relation is derived connecting the ref lexion and transmission coef-f icients coef-for scattering ocoef-f water waves by a coef-fixed body with the coef-far-coef-field radiated waves due to forced motions of the same body. Two alternative derivations are given, including a simple argument based on the analys-is of an appropriate linear superposition of the two problems, and a more formal application of Green's theorem to the two potentials. _{Por bodies} with horizontal symmetry, the transmission and reflexion coefficients _are related to the phase angles of the far-field radiated waves, associated with symmetric and anti-symmetric forced motions of the body. Some general con-clusions f oflOw for arbitrary symmetrical bodies, and these _{are verified in} specIfic cases by cOmparison with existing solutions. _{The applicability of} these relations to other types of wave problems is noted.

2

1. Introduction

It is customary to denote the interaction of incident plane _waves with a fixed body as a 'scattering' problem, and the generation of _waves by forced oscillatory mot-ion of the body in _{an otherwise undisturbed field} as a 'radiation' problem. _{From the mathematical viewpoint these two problems} differ in terms of the. boundary condition _{on the body, and the far-field} conditions appropriate in each case. _{From the physical viewpoint the} scattering and radiation problems appear to be unrelated, except, in long-wavelength approximations where the local influence of the body on the surrounding fluid is the same for both problems.

In spite of the apparent differences between the scattering _{and radiation} problems, there exist certain relationships between them which are conse-quences of reciprocity principles and Green's theorem. _{A particularly useful} example, known as 'Haskind' s relations' in ship hydrodynamics, _{provides a} linear relation between the exciting forces exerted by _{incident waves} on a _{fixed body and the amplitude of the far-field radiated waves generated} by forced motions of the body in otherwise calm _{water. From considerations} of the energy flux at infinity, one then can obtain a relation between the damping force in calm water and. the exciting force in _{incident waves, as shown} by Newman (1962). _{A general review of this subj ect has been given by Ogilvie}

(1973).

In this paper attention is focused on the ref1eion and transmission coefficients of the scattering problem, for two-dimensional motion involving the interaction of an incident plane. progressive _{wave system with a long} cylindrical body. The body generators are horizontal, and parallel (or pos-sibly oblique) to the wave crests. _{In this situation a portion of the incident}

wave energy is reflected by the body as an 'upstream' propagating wave, and the remainder is transmitted past the body as a 'downstream' propagating wave, the reflexion and transmission coefficients being defined as the ratios of the amplitudes of these two wave systems .to-the.-ampiitude-of-. the incident wave. In order to relate these ratios to properties of the radiation problem, we shall first construct in §3 a suitable linear combination of the scattering and radiation solut:ions, such that there is no net wave system downstream of the body, linearization being assumed throughout so that super-positiön of solutions is valid. The resulting 'composite' solution satisfies a boundary-value problem with the same boundary condition on the body as for the radiation problem (since the scattering solution involves a homogeneous boundary condition on the body), zero wave, propagation at the downstream infinity (by definition of the composite solution), and a prescribed combination of incident and reflected waves at the upstream infinity.

By restricting the normal velocity of the body to be of f ied phase, it follows that the part of the composite solution with conjugate phase satisfies a boundary-value problem which is homogeneous, except for the presence of

upstream standing waves. It is then argued heruistically that, ùnless the transmission coefficient of the body is zero, the upstream waves of this

otherwise homogeneous problem must vanish. This provides a simple relationship between the far-field characteristics of the radiation and scattering problems. Thesame relation is confirmed in §4 with amore formal derivation based on Green's theorem.

We then seek to determine the ref lexion and transmission coefficients from the solution of the radiation problem. In fact, the radiation problem is non-unique without specifying the particular forced motion of the body, and it is necessary to consider two indèpendent

4

radiátion prpblems. Attention is restricted in §5 to bodies which are sym-'metrical about the. vertical y-axis, and the.. appropriate independent radiation

problems are respectively synunetrical and anti-symmetrical in x , e.g., forced vertical and horizontal motions -of thebody...*he refle'Xion and transmission coefficients can be expressed then in terms of the phase angles

('5s'5a) of the syetric and anti-syetric radiated waves.

Various conclusions follow,, both for general and specific body geometries. Since the reflex ion and transmission coefficients are unique properties of the body geometry and wavelength, the same must be true of the phase angles

d5' atid _a without, specification of the forced motions of the body. Thus every syetrical (anti-yetrical) forced. motion of the body will generate radiated waves with the same phase angle s6a' subject to a possible shift

f _{±180 degrees} . Since, from the Haskind relations, the corresponding

forces or moments are directly proportional to the complex amplitudes of the radiated waves, similar conclusions apply to the phase angles of the exciting :forces and moment. In particular, the horizontal exciting force is in phase

with the exciting moment about the roll axis. For specific body shapes it is possible to verify thes results by comparison with existing calculations of the reflex ion and transmission coefficients, the exciting forces and moment,'and the radiated wave characteristics. Such comparisons are

made in' §6 for a súbmerged circular cylinder, and in §7 for floating or submer&ed vertical barriers.

Our derivation is based on the consideration of two-dimensional surface waves, but since the analysis depends only on a simple superposition of the far-field waves, and is in4ependent of the governing partial differential equation, similar conclusions hold for other types of linear

a wave guide below the cut-off frequency, and two-dimensional internal waves

u it t,

at frequencies below the Brunt-Vaisala frequency. In the latter context it may be noted that our analysis is similar in certain respects to that, of

Drazin and Moore (1967), who utilize an analogous superposition of two wave systems in order to satisfy the radiation conditions of steady flow past an obstacle, and the same scheme has been exploited in a numerical solutIon of the steady-state ship-wave problem by Mei and Chen (1974).

-6

2. Thè radiation and scattering problems

Assume that a long cylindrical body, with horizontal generators, is situated on or beneath the free surface in a fluid of constant or infinite depth. For the sake of definiteness we assume that the fluid motion is

two dimensional, and confined to planes normal to the cylinder generators, but the results -hich follow are identical in the more general instance where the motion is sinusoidal in the direction parallel to the cylinder,

as in the case of a cylinder with oblique incident waves. The problem is ¿ssumed to be linearized, with oscillatory time dependence at a frequency

w/2iT Thus the velocity potential can be written in the form

iù)t

(x,y,t) = Re[(x,y)e _i (1)

with a similar representation of the free-surface elevation. Here (x,y) are Cartesian coordinates, taken in the usual sense with x horizontal and positive to the right. The potential 4(x,y) = X(x,y) + VP(x,y) is

complex-valued to represent. ,th.L magnitude and. phaa&. of. the oscillatory motion.

The velocity potential is governed by Laplace's equation throughout the interior of the fluid domain, with appropriate boundary conditions

on the free surface and lower boundary of the fluid. However, these aspects of the boundary-value problem do not concern us explicitly here, and we need consider only the remaining boundary conditions on the body and at

infinity. .

In the

radiation

problem a normal velocity Re(f eimt) is prescribed on the body surface, with f a given function of position on the body. Denoting

the radiation potent:ial by

4r' the appropriate boundary condition on the bo4y is

Suitable radiation conditions must be imposed _{at large distances from the} body, namely that the waves are outgoing or that on the free surface

-7-such that .there are no waves at x = . This potential satisfies

(3a)

the same boundary-value problem as stàted in §2 above, except that on the body, from (2) and (4),

f ₍₇₎

whereas at infinity on the free surface, from (3) and, (5),

"*

x+_.

(3b)

Here &f are complex coefficients, representing the wave amplitude and phase at infinity.

In the scattering problem, with potential , the body is stationary

and' waves are incident from infinity. _{Hence the boundary conditions are} that, on the body,

= o , ₍₄₎

atid at infinity on the free surface,

sr'J eiKx

+Re

-iK

,

x-'-.

' (5b)

iKx

Here e denotes the incident wave, assumed to be of unit amplitude and propagating from x = -foe _{, whereas R and T are the (complex) refle>ion}

and transmission coefficients as defined in §1.

3. The composite problem

Hereafter, we shall assume that T O and define thé conrpoeite potential

= r - (A_/T)q5 , (6)

"y (A+ - A_R/T)e - (A_/T)e

X-

oe _(8a)

,

+

-The fOrced motion of the body súrfáce i arbitrary, but we shall

restrict f to be real, so that the normal velocity of the body is in phase

with ±cosWt . It then fllows that the imaginary part _tP of the composite

potential satisfies a homogeneous boundary condition on the body, as well as at infinity downstream, and the only non-homogeneity in the boundary-value problem for is that at infinity úpstream it must be equal to a standing wave given by the imaginary part of (8a). Thus the potential corresponds

to the problem where a stnding wave is present at x = -foe _,

cnd.the body is

stationary. Equivalently, is the solution of the scattering problem with zero transmission coefficient, T=O , and complete reflexion, RI=l , in

contradiction to our assumption that TO . This leads us to one of. two possible conclusions: either the transmission cOefficient is zero, or the potential

must vanish at both infinities.

The occurrence of complete reflexion and zero transmission is generally presumed to be impossible except for pathological body geometries and special wavelengths of the incident wav. The only known examples of complete reflexion

(excluding the trivial case where the body completely blocks off the flow) in-volve situations with a resonant cavity of some sort; hence complete reflexion

is known to occtir at discrete eigenfrequencies for water wave scattering by a pair of parallel obstacles where the fluid region between the two obstacles is resonant. This phenomenon is discussed by Evans and Morris (1972a), Evans

-9

In the sequel we shall ignore special cases of this nature, and assume that the body geometry and wavelength are such that complete reflexion is impossible. _{This can be reconciled with the boundary-value problem for}

iPc = Im() if and. otilyif the upstream standing wave of that problem

vanishes, and hence the imaginary part of

_(8a)

must be zero.t Thus, it follows that

Im[(A+ - A_R/T)e

- (A_/T)e] = o

(9)

or, for this relation to be independent of x ,

- A_RiT + A...*/T* = O , (10)

where an asterisk (*) denotes the complex conjugate. _{Equation (10) is the} principal result of our ànalysis, and provides a relation between the

far-f ield radiated wave amplitudes A. and the refar-fleXion and transmission coefficients (R,T)

An alternative relation to (10) can be obtained by táking the conjugate of (10), multiplying it by R , and adding this product to (10). _{It f liows}

that

+ A+* R + A_* (l_RR*)/T* = o

From conservation of energy in the scattering problem,

(12)

Using

(12) to replace the factor in parénthesis in (il) gives the desired

expression

A+ + A+* R + A_* T = O (13)

'tIn general must vanish throughout the fluid, but this additional con-clusion is not required here, and depends on a uniqueness proof which has only been established for special body geometries.

ppppp__

_{-. lo}

-4. Derivation based on Green's theorem

While the derivation in §3 is very simple, its heruistic nature is obvious and one 'nay prefer a more formal mathematical approach. For this purpose we define the operator

(

/n -

/n) d

(14)

where C is the closed contour including the free surface, body surface, fluid bottom, and two vertical closures at x

± .

By Green's theorem

I E

O for any pair of. functions

_,4

which are harmonic in the fluid region

bounded by C , ànd if these functions satisfy the appropriate boundary

conditions on the free surface and bottom there is no contribution from those portions.of C . By straightforward reduction using the radiation and

boundary conditions (25), it follows that

= O = -2IMK(l = RR* - TT*) , (15)

I(4,

) = O = 21MKA

- ICB

dP , (16)

I(c*,) = O

= 2iMK(A+* R + A_* T)

'CB

f* _dP . (17)

Here. _CB denotes the body contour, and M is a. positive real constant., defined for water waves as

. M = (sech2 Kh)

jo

cosh2 Ky dy =

2i

. (18)

Equation (15) gives the energy relation (12), expressing conservation of:energy flux in the scattering problem. Equation (16) is the general two-dimensional form of Haskind's relations and, by appropriate choice of

the function f , can be used to find the scattered wave force

=

J

p n dR, =

_{-imp J}

n (19)

B

itt terms of the radiated wave amplitude A+ . (Hère p is the fluid density, p the pressure in the scattering problem, and the linearized Bernoulli

equation has been utilized.)

Subtracting (17) from (16) gives the relation

-

= O = 21(A + A+*

+ A_* T)

f

(f - f *) d

(20)

Now if f is restricted to be real, as in §3, the. integral in (20) vanishes and (20) reduces to (13). This derivation, of (13) remains valid if T O

in contrast to the analysis of §3.

5. Symmetrical obstacles

In order to simplify the application of (13) we shâli restrict our atten-tion to bodies which are symmetrical, about the plane x0 . It follows that

two radiation problems may be considered separately in which the normal velocity distribution on the body, represented by the functIon f , is

respectively symmetrical and anti-symmetrical. _{In the symmetric case,} exemplified by vertical oscillations of the body, the radiation _potential

is an even function of x , and thus

(21)

In the anti-symmetric case, exemplified by horizontal oscillations _{of the} body, the radiation potential is an odd function of x , and thus

A+ = A_ A

₍₂₂₎

7pp]ying (13) separately in each case gives the equations

R+T_A/A*_e25

s-s

R - T = -A 'IA *

2iSa

a a

12

-Hence the ref lexion and transmission coefficients can be expressed in terms of the phase angles of the syetric and anti-syetric radiated waves,

6 = arg(A ) , in the form

s,a s,a

R =

1 (2i6a+ e25)

, (25)

T

=

4

(e2

-

e28)

₍₂₆₎

Equations (25) and (26) can be replaced by parametric relations

R = - cos

,

(27)

T=isincxe

, (28) where

a=6

-a s

=6 +6

_a s

In this form the energy relation (12) is obvious, as is the phase-angle relation arg (R) - arg

(T)I

= Tr/2 which was derived by Newman (1965,

eq. 2.20). From (29-30) it is apparent that the parameters (a,) , which

determine the magnitude and phase,, respectively, of the reflexion and trans-mission coefficients, depend uniquely on the phase angles

6S'6a

of the radiation problem, and vice versa. Thus a knowledge of

both

the synnnetric and anti-symmetric radiation phase angles is

necessary, in general, to determine R or T This limits the utility of our relations to those problems where solutions

òf both the Symmetric and aùti-symmetric radiation problems have been found,.

13

-On the other hand, two general conclusions follow which add signi-ficantly tO the value of these relations. First note that we have dealt rather abstractly with a syetr±c and anti-sy=etric radiation

potential, specifying only that the normal velocity on the body surface should be respectively even or odd in x , and with the added restriction that this normal velocity must be of constant phase proportional tö ±cost

(i.e., f must be real). There are an infinite number of modes of possible body motions in each case, and yet it is clear from (27-30) that,

since (R,T) are uniquely specified by the body shape and frequency, the same must be true of the radiation phase angles

s'6a , with a possible

ambiquityof ±1800. _{Thus it follows that the phase of the radiate4 waves}

takes the same value, 6 , for all possible symmetric modes of body motion,

and for all possible anti-symmetric modes, modulo 180°. In particular, vertical oscillations of the body and syrnmetricàl source-like dilations of the body will produce radiated waves of the same phase & ; likewise,

horizontal oscillations and rolling oscillations will produçe radiated waves of the same phase angle ±&a at x = ±o Moreover, we recall that the Haskind relations (16,19) relate the amplitude

and phase

of the exciting force or moment, exerted on the body in the scattering

problem, to the amplitude

and phase of

the radiated waves forj forced motions of the body in otherwise calm water, the mode of the forced motions corres-ponding to the appropriate component of the exciting force or moment. In fact, the complex exciting force or moment is directly proportional to the complex amplitude of the radiated wave, with a constant of proportionality that is real if the phase is related to the incident wave amplitude at the

-14-origin. Thus the phase angles

s'6a are identical to the phase angles of the exciting forces and moments. In particular, is the phase of

the vertical exciting force, and both the horizontal force and rolling moment must have the same phase angle

a

6. The submerged circular cylinder

A particular scattering pröblem of interest is the refleXion and transmission of surface waves by a submerged circular cylinder in water of

infinite depth. Dean (1948) showed that for this case the refleXion coef-ficient R=0 , the waves being totally transmitted but with a phase shift

Ursell (1950) re-examined the problem, setting it on a more rigorous basis and outlining a practical scheme for comput-ing the velocity potential. Ogilvie (1963) has presented a complete numerical solution, including the magnitudes and phases of the vertical and horizontal forces acting upon the cylinder in both the scattering and radiation problems.

Since R=0 , equations (27) through (30) imply the relations

71

a

s2

T= 2a

From (31) the phase of the anti-symmetrical radiated waves, and hence also the horizontal exciting force, will lead the symmetrical radiated waves and vertical exciting force, by one-quarter period. This is consistent with Ogilvie's (1963) equation 27. From (32) (and the Haskind relations), the phase of thetransmittéd wave is twice the phase of the horizontäl exciting force, and this is confirmed by Ogilvie's (1963) analysis (page 467), which includes a physical explanation concerning the plausibility of this result. Thus specific results of Ogilvie (1963) confirm our relations (27-30),

15

-but the completeness of his treatment of the problem precludes additional conclusions or extensions here.

7. The vertical flat plate

A number of ànalytic solutions have been carried Out for scattering and radiation from a vertical fiat plate, which is either intersecting the free surface, or completely submerged beneath the free surface. In this case the reflexion coefficient is zero, offering a more complete opportunity to test our relationships.

For this problem the vertical force is zero (In the linear theory), and there is no wave radiation associated with vertical oscillatIons. Nevertheless, the symmetric phase angle can be Inferred, by recalling that any

syii-metric mode' of forced motion will suffice to determine . A non-trivial

- s

mode is that of symmetric expansion of the vertical plate, and In the simplest example (corresponding to a delta function mode of expansion) we may replace the plate by a-point source of oscillating strength. From the known expres-.sións for the oscillating source beneath a free surface (cf. Wehausen and

Laitone, l96O éq. 13.31 or 13.34 for infinite or finite depth, respectively), it can be verified that the phase angle = rr/2 . In fact, the Same result

follows from a more elementary calculation based upon the plane wave potential, for if f(y) Is a specified horizontal velocity on the vertical ¿xis, it

follows from our argument regarding the uniqueness of phase angles that is independent of the choice of f(y) so long as this function is real. (This fact is confirmed by the 'wavemaker theory' of Hävelock (1929).) The simplestchoice for f(y) is the horizontal velocity of a plane wave. The appropriate outward-radiating wave for positive x is proportional to and, after diffèrentiation with respect to i , we conclude that the phase

-

16

-angle relating the radiated wave to 'the normal velocity on thê flat plate is 71

With ir/2 , it follows immediately frOm (23) that

'R+Tl

. ' (31)

This property is well known for the surface-piercing vertical plate (cf. Wehausen and Laitone, 1960, page 532) and can be confirmed for the submerged case from Evans' (1970) equations 38-39. Our analysis shows that

(31) is also valid for 'the case of finite depth, and for the scattering

T

problem of oblique wäves studied by Evans and Norris (1972b). The phase angle of the horizOntal exciting force and rolling moment,

a = + 71/2

álso can be compared in the above problems, and our relations are consistent with Evans' (1970) equations 73-74 for the submerged case, and with cotres-ponding results Obtained by Kotik (1963) for the surface-piercing flat plate. 'In the light of our conclusions regarding the uniqueness of

a it is more readily understood why the forced rolling and horizontal oscillation problems for vertical obstacles have the same phase angle and why, for

example, there exists for such bodies a vertically displaced point of rotation such that rolling oscillations about this point do.not radiate waves at

infinity. In façt, we observe that these properties apply to more general bodies and in finite depth 'as well.

17

-8. Summary and conclusions

An expression has been derived relating the reflexion and transmission coefficients of the scattering problem to the phases of symmetrical and anti-symmetrical radiated waves. in conjunction with the Haskind relations it is possb1e also to relate the ref lexion and transmission coefficients to the exciting forces acting on the body in the same scattering problem. Alternatively, both the ref lexion and transmission coefficients, and the exciting forces, can be derived from the properties of the radiation problem, making it unnecessary to solve the scattering problem for these coefficients and forces. The results are limited to two-dimensional linear wave

propa-gation, but aside from this restriction they are quite general and can be applied to other problems such as the scattering of water waves with oblique incidence, where the. governing equation is a modified two-dimensional, wave equation, and to acoustic or internal waves in the regimes where only one radiating wave component is present.

Using these relations' one can verify or extend the calculations

of several specific water-wave prOblems. In §6-7 this was done for a submerged circular cylinder, and a vertical flat plate. Other body Shapes have been tráated numerically, and it 'is possible to apply our results to these too, but the list of references is already lengthy, and the general ideas which

follow from our relations should, be sufficiently clear from the examples already chosen.

The author gratefully acknowledges stimulating discussions with Professors C. C. Nei and F. Ursell, and with several members of staff at the Dept of Applied Mathematics, Ulilversity of Bristol. Financial support was provided by the National Science Foundation (Grant GK-43886X) and the

Office of Naval Research (Contract N00014-67-A-1204-0023),

19

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