Flexural- And Capillary-gravity Waves Due To Fundamental Singularities In An Inviscid Fluid Of Finite Depth

International Journal of Engineering Science 46 (2008) 1183–1193

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Flexural- and capillary-gravity waves due to fundamental singularities in an inviscid fluid of finite depth D.Q. Lu *, S.Q. Dai Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Yanchang Road, Shanghai 200072, China

a r t i c l e

i n f o

Article history: Received 28 March 2008 Accepted 3 June 2008 Available online 14 July 2008

Keywords: Flexural Capillary Fundamental singularities Asymptotic representation

a b s t r a c t Wave motion due to line, point and ring sources submerged in an inviscid fluid are analytically investigated. The initially quiescent fluid of finite depth, covered by a thin elastic plate or by an inertial surface with the capillary effect, is assumed to be incompressible and homogenous. The strengths of the sources are time-dependent. The linearized initial-boundary-value problem is formulated within the framework of potential flow. The perturbed flow is decomposed into the regular and the singular components. An image system is introduced for the singular part to meet the boundary condition at the flat bottom. The solutions in integral form for the velocity potentials and the surface deflexions due to various singularities are obtained by means of a joint Laplace–Fourier transform. To analyze the dynamic characteristics of the flexural- and capillary-gravity waves due to unsteady disturbances, the asymptotic representations of the wave motion are explicitly derived for large time with a fixed distance-to-time ratio by virtue of the Stokes and Scorer methods of stationary phase. It is found that the generated waves consist of three wave systems, namely the steady-state gravity waves, the transient gravity waves and the transient flexural/capillary waves. The transient wave system observed depends on the moving speed of the observer in relation to the minimal and maximal group velocities. There exists a minimal depth of fluid for the possibility of the propagation of capillary-gravity waves on an inertial surface. Furthermore, the results for the pure gravity and capillary-gravity waves in a clean surface can also be recovered as the flexural and inertial parameters tend to zero. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction The ice-cover in the polar region [1,2] and the very large floating structures [3] in the offshore region are usually idealized as thin elastic plates floating on an inviscid incompressible fluid in the theoretical investigations. A simple linear model for the thin plate includes the effects of flexural rigidity and vertical inertia [1], Eq. (3.18). Flexural-gravity waves can be observed due to the elasticity effect of the plate. For small-amplitude waves, the kinematic and dynamic boundary conditions can be linearized, by completely or partially neglecting the nonlinear convective term in the Bernoulli equations, at the undisturbed plate-water interface. Under the assumptions of inviscidy, irrotationality, incompressibility and uniformity of density, the continuity equation for the fluid motion reduces to the Laplace equation, being taken as the governing equation for the problem considered. Such a modelling leads to a linear problem. The limits of this classical model were discussed by Pa˘ra˘u and Dias [4]. In particular, as the flexural rigidity of the plate tends to zero, the plate-covered surface reduces to the inertial surface [5] which represents the effect of a thin uniform distribution of non-interacting floating matter, for example,

* Corresponding author. Tel.: +86 21 5633 8372; fax: +86 21 3603 3287. E-mail addresses: [email protected], [email protected] (D.Q. Lu), [email protected] (S.Q. Dai). 0020-7225/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2008.06.004

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broken ice. Once the inertial parameter is given, progressive waves with a sufficiently large frequency will not be able to appear since ‘‘the inertial surface is too heavy” [6]. However, the propagation of progressive waves with any frequency is possible if the surface tension [6] or the flexural rigidity [7,8] is included in the mathematical formulation for the waves on the inertial surface. Of the few analytical approaches available for solving linear problems, one is the singularity method, of which the basic idea is to express a general solution as the convolution product of the distribution function and the fundamental solutions. Therefore, it is essential to seek the corresponding fundamental solution as the first step to solve the whole problem. In the spatial domain, the fundamental solution is due to an inhomogenous term associated with the Dirac delta function representing a concentrated disturbance. In the temporal domain, the solutions due to an instantaneous and oscillatory disturbance are of preliminary interest since they can be used to construct a general solution for a body with a prescribed unsteady motion. The present work aims to study the fundamental solution problem for the flexural- and capillary-gravity waves in an inviscid fluid of finite depth, taking the inertial effect on the surface into consideration. Although the fundamental solution problems for the fluid with an ice cover [7,8] or with an inertial surface [9–11] have been considered in the literatures, there are four remarkable differences between the previous and present studies. Firstly, non-zero initial values for the velocity potential and the plate deflexion are considered here while only zero initial values were considered previously. Secondly, Chowdhury and Mandal [7,8] introduced an image point with respect to the mean plate-water interface. Thus, three undetermined functions and three equations are directly involved to calculate the velocity potential and the surface elevation [7,8]. To make a simpler calculation, we introduce an image point with respect to the level bottom and only two undetermined functions and two equations are directly employed. Thirdly, the models for flexural- and capillary-gravity waves are considered in parallel owing to their similarity in analytic form. Finally, the solutions obtained previously [7–11] were expressed in terms of the unevaluated Fourier integrals only, from which the corresponding physical interpretation was not provided explicitly. Both exact solutions in integral form and asymptotic solutions in algebraic form are derived here to reveal some unreported features of the wave motion. This paper is organized as follows: In Section 2, the general mathematical formulation is given. Four types of concentrated disturbance are included, namely a mass source immersed in the fluid, a dynamic load on the surface, an initial impulse and an initial displacement on the surface of the fluid. In Section 3, the fundamental solutions in integral form for the velocity potentials and the wave profiles due to line, point and ring sources with a time-dependent strength are exactly derived for the flexural- and capillary-gravity wave problems. Two kinds of unsteadiness are considered, namely instantaneous or timeharmonic singularities. All the solutions can be expressed as a single integral with different integrands. In Section 4, the asymptotic representations of the wave motion are explicitly obtained for large time with a fixed distance-to-time ratio R=t by making use of the Stokes and Scorer methods of stationary phase. The ratio can be regarded as the moving speed of an observer. As R=t approaches two critical values, namely the minimal and maximal group velocities of the wave, the Stokes method predicts an infinite wave amplitude and thus the Scorer method should be employed. A combination of the Stokes and Scorer methods gives uniformly valid results over the panoramic region. In Section 5, conclusions are reached. 2. General mathematical formulation Without loss of generality, a Cartesian coordinate system is used in which the z axis points vertically upward while z ¼ 0 represents the mean ice-water interface. Therefore, the governing equation is

r2 U ¼ MðtÞdðx  x0 Þ;

ð1Þ

where Uðx; t; x0 Þ is the velocity potential for the perturbed flow, MðtÞ time-dependant strength of the simple mass source, dðÞ the Dirac delta function, and x an observation point, x0 the source point, t the time. For two-dimensional cases, x ¼ ðx; zÞ and x0 ¼ ðx0 ; z0 Þ with z0 6 0 while for three-dimensional cases, x ¼ ðx; y; zÞ and x0 ¼ ðx0 ; y0 ; z0 Þ. The linearized kinematical and dynamical ice-cover conditions at z ¼ 0 are given by

of oU  ¼ 0; ot oz oU o2 f q þ qgf þ Dr4 f þ qe h 2 ¼ PðtÞdðz  z0 Þ; ot ot

ð2Þ ð3Þ

where f is the vertical deflexion of the ice-water interface; q and qe are the uniform densities of the fluid and the plate, 3 respectively; g is the acceleration of gravity; D ¼ Eh =½12ð1  m2 Þ is the flexural rigidity of the plate; E; h and m are Young’s modulus, the thickness and Poisson’s ratio of the plate, respectively. PðtÞ is the time-dependant strength of the applied load. z and z0 are the field and source points at z ¼ 0, respectively. z ¼ ðx; 0Þ and z0 ¼ ðx0 ; 0Þ for two-dimensional cases while z ¼ ðx; y; 0Þ and z0 ¼ ðx0 ; y0 ; 0Þ for three-dimensional cases. The bottom condition at z ¼ H is given by

oU ¼ 0; j oz z¼H

ð4Þ

D.Q. Lu, S.Q. Dai / International Journal of Engineering Science 46 (2008) 1183–1193

1185

where H is a positive constant. It is assumed that the entire fluid is at rest for t < 0. Therefore, the initial conditions at z ¼ 0 are

Ujt¼0 ¼ 

I0

q

dðz  z0 Þ;

ð5Þ

fjt¼0 ¼ E0 dðz  z0 Þ; of j ¼ 0: ot t¼0

ð6Þ ð7Þ

Eq. (7) represents the initial vertical velocity of the inertial surface is zero, which is consistent with the assumption that the entire fluid is at rest for t < 0. Moreover, since the finite disturbance must die out at infinity, it is required that rU ! 0 as z ! 1, which imposes the uniqueness on the problem considered. It should be noted that the problems considered by Chowdhury and Mandal [7,8] are special cases in the present formulation with P ¼ I0 ¼ E0 ¼ 0. In this case, elimination of f between Eqs. (2) and (3) yields one single boundary condition at z ¼ 0 for U only and Eqs. (5)–(7) reduce to two combined initial conditions, as given by Chowdhury and Mandal [7], Eqs. (2.4) and (2.5). The elimination of f is correct if and only if the pressure at the instant t ¼ 0 is equal to zero, as stated by Miles [12]. Now let the entire solution be written as

U ¼ US ðx; t; x0 ; t0 Þ þ UI ðx; t; x1 ; t0 Þ þ UR ðx; tÞ; S

I

ð8Þ S

where U and U are, respectively, the potentials due to the simple source at x0 and at x1 . U is the known fundamental solution for Eq. (1) in an unbounded domain. In order to satisfy the requirement of Eq. (4), UI is introduced in such a way that

o ðUS þ UI Þjz¼H ¼ 0: oz

ð9Þ

Thus, x1 should be the image point of x0 with respect to the flat bottom (z ¼ HÞ, and x1 ¼ ð0; z1 Þ for two-dimensional case while x1 ¼ ð0; 0; z1 Þ for three-dimensional one, where z1 ¼ 2H  z0 . UR in Eq. (8), an undetermined continuous function everywhere in the corresponding domain, represents the effect of surface boundary. For the singular components it holds that

fr2 US ; r2 UI g ¼ MðtÞfdðx  x0 Þ; dðx  x1 Þg:

ð10Þ

For the regular component, we have

r2 UR ¼ 0:

ð11Þ

From Eqs. (4), (8) and (9) it follows that

oUR ¼ 0: j oz z¼H

ð12Þ

Therefore, the boundary conditions at z ¼ 0 can be re-written as

of oUR o  ¼ ðUS þ UI Þ; ot oz oz oUR o2 f o þ gf þ cg r4 f þ r 2 ¼  ðUS þ UI Þ  PðtÞdðz  z0 Þ; ot ot ot

ð13Þ ð14Þ

where c ¼ D=qg and r ¼ qe h=q. Thus, Eqs. (11)–(14) along with the initial conditions (5)–(7), hereinafter referred to as Model (I), constitute a well-posed problem for UR and f, mathematically as well as physically. A special case of the present formulation with r–0 and D ¼ 0 corresponds to the gravity waves on an inertial surface. When the surface tension is further taken into consideration, Eqs. (3) and (14) are, respectively, replaced by

oU o2 f þ qgf  T r2 f þ qe h 2 ¼ PðtÞdðz  z0 Þ; ot ot oUR o2 f o 2 þ gf  sg r f þ r 2 ¼  ðUS þ UI Þ  PðtÞdðz  z0 Þ; ot ot ot

q

ð15Þ ð16Þ

where T is the surface tension and s ¼ T=qg. Eqs. (11)–(13), Eq. (16) and the initial conditions (5)–(7) are hereinafter referred to as Model (II). A special case of Eq. (16) with r ¼ 0 and s–0 corresponds to the capillary-gravity waves in a clean free surface, which was considered by Chen and Duan [13]. It is evident that Eqs. (14) and (16) have the similar form. Therefore, the solutions for Models (I) and (II) will be derived in parallel. Model (I) is concerned with the flexural-gravity waves in an inviscid fluid of finite depth with an elastic plate cover. Model (II) is concerned with the capillary-gravity waves in an inviscid fluid of finite depth with an inertial surface. The discrepancy between Eqs. (14) and (16) yields two different dispersion relations for the two models.

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3. Fundamental solutions in integral form 3.1. Instantaneous singularities First we consider MðtÞ ¼ M0 dðt 0 Þ and PðtÞ ¼ P0 dðt 0 Þ with t 0 ¼ t  t0 , where t0 the instant at which the disturbances are applied, M0 and P0 are constant. Then US and UI in Eq. (8) for line and point singularities can be given as

  Z cþi1 Z þ1 M 0 dðt 0 Þ 1 1 1 log ; log ds da  expðf ÞfS0 ; S1 g; ¼ 2p r0 r1 4p2 i ci1 1   Z cþi1 Z þ1 Z þ1 M 0 dðt 0 Þ 1 1 1 ; ds dadb  expðf ÞfS0 ; S1 g; fUS ; UI g ¼  ¼ 3 4p r0 r1 8p i ci1 1 1

fUS ; UI g ¼ 

ð17Þ ð18Þ

respectively, where

fS0 ; S1 g ¼ 

M0 fexpðkjz  z0 jÞ; expðkjz  z1 jÞg; 2k

ð19Þ

c the Laplace convergence abscissa, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifx0 ; y0 g ¼ fx  x0 ; y  y0 g, and fr 0 ; r 1 g ¼ fjjx  x0 jj; jjx  x1 jjg. k ¼ jaj and f ¼ iax0 þ st 0 for line singularities while k ¼ a2 þ b2 and f ¼ iax0 þ iby0 þ st 0 for point singularities. With a change of variables

fx; yg ¼ Rfcos h; sin hg; fx0 ; y0 g ¼ R0 fcos h0 ; sin h0 g; fa; bg ¼ kfcos u; sin ug;

ð20Þ

Eq. (18) can be represented as

fUS ; UI g ¼

Z

1 4p 2 i

cþi1

ds

Z

ci1

þ1

dk  exp½ikR0 cosðh0  uÞ þ st0 kJ 0 ðkRÞfS0 ; S1 g;

ð21Þ

0

where J 0 ðkRÞ is the zeroth-order Bessel function of the first kind. For a single point source we may move the origin of the coordinate system to the source point to simplify Eq. (21) by setting R0 ¼ 0. The solution for a ring source with a constant qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi radius R0 ¼ x20 þ y20 can be constructed by integrating fUS ; UI g in Eq. (21) with respect to the source point along the ring path as follows:

fUS ; UI g ¼ ¼

Z

1 4p

2i

1 2pi

cþi1

ds

ci1 cþi1

Z

ds

ci1

Z 2p

R0 dh0

þ1

dk  exp½ikR0 cosðh0  uÞ þ st 0 kJ 0 ðkRÞfS0 ; S1 g

0

0 þ1

Z

Z

dk  expðst0 ÞkR0 J 0 ðkR0 ÞJ 0 ðkRÞfS0 ; S1 g:

ð22Þ

0

As MðtÞ ¼ 4p; US in Eq. (22) reduces to the cases considered by Hulme [14] for an inviscid fluid and Chowdhury & Mandal [7] for an ice-covered fluid. A combination of the Laplace transform with respect to t and the Fourier transform with respect to the spatial variables is introduced for fUR ; fg as

fUR ; fg ¼ fUR ; fg ¼

1 4p

2i

1 8p

3i

Z

cþi1

ci1 Z cþi1 ci1

ds ds

Z

þ1

1 Z þ1 1

~ R cosh½kðz þ HÞ; ~fg; da  expðf ÞfU Z

þ1

~ R cosh½kðz þ HÞ; ~fg; dadb  expðf ÞfU

ð23Þ ð24Þ

1

for two- and three-dimensional problems, respectively. Then Eqs. (11) and (12) are automatically satisfied. By substituting the integral representations of fUS ; UI ; UR g into the Laplace–Fourier transforms of kinematical and dynamical boundary con~ R and ~f, which can readily be solved ditions, two simultaneous algebraic equations are set up for two unknown functions, U for the two models with a line/point/ring singularity. Consequently, the formal integral expression for fUR ; fg can be written as

n o P n o I n o 0 fUR ; fg ¼ M0 URm ðx; t; x0 ; t 0 Þ; fm ðx; t; x0 ; t0 Þ þ URp ðz; t; z0 ; t0 Þ; fp ðz; t; z0 ; t 0 Þ þ 0 URi ðz; t; z0 Þ; fi ðz; t; z0 Þ q q n o R þ E0 Ue ðz; t; z0 Þ; fe ðz; t; z0 Þ :

ð25Þ

For line, point, and ring singularities, we have, respectively, 2 Z þ1 1 X 0 dk  exp½ð1Þmþ1 ikx  b L; 2p m¼1 0 Z þ1 1 L¼ dk  exp½ikR0 cosðh0  uÞkJ 0 ðkRÞ b L; 2p 0 Z þ1 dk  kR0 J 0 ðkR0 ÞJ 0 ðkRÞ b L; L¼

L¼

0

ð26Þ ð27Þ ð28Þ

D.Q. Lu, S.Q. Dai / International Journal of Engineering Science 46 (2008) 1183–1193

where

n o L ¼ URm ; URp ; URi ; URe ; fm ; fp ; fi ; fe ; n o b bR ; U b R; U b R; U b R ; bf m ; bf p ; bf i ; bf e : L¼ U m p i e

1187

ð29Þ ð30Þ

For the instantaneous sources associated with the Dirac delta function, the right-hand side of Eq. (29) can be expressed as

n

o

1 U  n o bf m ; bf p ; bf i ; bf e ¼ 1 C cosðxt0 Þ;  k sinðxt 0 Þ;  k sinðxtÞ; V cosðxtÞ ; V x x 1 G ¼ ½expðkz0 Þ þ expðkz1 Þ; 2 B ¼Gð1  rkÞ;

bR ; U b R; U b R; U b R ¼ fBdðt 0 Þ  C x sinðxt 0 Þ; k cosðxt 0 Þ; k cosðxtÞ; V x sinðxtÞg; U m p i e

ð31Þ ð32Þ ð33Þ ð34Þ

C ¼G½1 þ cothðkHÞ;

ð35Þ

U ¼k½coshðkHÞ þ rk sinhðkHÞ; V ¼ cothðkHÞ þ rk:

ð36Þ ð37Þ

The dispersion relation for Model (I) reads

xðk; r; cÞ ¼

!1=2 4 1 þ ck : gk V

ð38Þ

The dispersion relation for Model (II) reads 2

xðk; r; sÞ ¼

1 þ sk gk V

!1=2

:

ð39Þ

The fundamental solutions for the pure gravity waves on an inertial surface and on a clean surface can be obtained from Eqs. (26)–(28) by taking c ¼ 0 and c ¼ r ¼ 0, respectively. 3.2. Oscillating singularities As MðtÞ ¼ M 0 expðiltÞ and PðtÞ ¼ P 0 expðimtÞ; US and UI for line and point singularities can be obtained from Eqs. (17) and R cþi1 R cþi1 (18) by replacing dðt 0 Þ and ci1 ds with expðiltÞ and ci1 ðs  ilÞ1 ds, respectively. US and UI for a ring singularity can be R cþi1 R cþ i1 obtained from Eq. (22) by replacing ci1 ds with ci1 ðs  ilÞ1 ds. The mathematical procedure for the solutions is similar to that for an instantaneous disturbance. The corresponding solutions can n be given in the o same form as Eqs. (26)–(28). Howb R; b bR ; U b in Eq. (29) reads ever, the integrands involved are different. For oscillating disturbances U m p fm; fp

n

o

1 n S S o 1 n T To /m ; /p þ /m ; /p ; U U n o 1n o 1n o bf m ; bf p ¼ nSm ; nSp þ nTm ; nTp ; V V bR ; U bR ¼ U m p

ð40Þ ð41Þ

where

 S S  expðiltÞ /m ; nm ¼ 2 fVGðx2  l2 Þ  C l2 ; ilCg; x  l2 n o expðimtÞ /Sp ; nSp ¼ 2 fimk; kg; x  m2  T T C fx½x cosðxtÞ þ il sinðxtÞ; x sinðxtÞ  il cosðxtÞg; /m ; nm ¼ 2 x  l2 n o k fx sinðxtÞ  im cosðxtÞ; cosðxtÞ  imx1 sinðxtÞg: /Tp ; nTp ¼  2 x  m2

ð42Þ ð43Þ ð44Þ ð45Þ

n o S S S S It can n be seen that the o terms involving /m ; /p ; nm ; np represent the steady-state progressive waves while the terms involving /Tm ; /Tp ; nTm ; nTp the transient dispersive waves. As l ¼ m ¼ 0, Eqs. (40) and (41) reduce to the solutions due to the sources with constant strengths. 4. Asymptotic representations for the wave profiles 4.1. Transient singularities To analyze the dynamic characteristics of the flexural- and capillary-gravity waves due to unsteady disturbances, the asymptotic solutions for the wave motion shall be considered for large time with a fixed distance-to-time ratio by making

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use of the Stokes and Scorer methods of stationary phase. This kind of asymptotic analysis has been applied to the classical Cauchy–Poisson problems for transient gravity waves [15–19], transient capillary-gravity waves [13,20], and transient flexural-gravity waves [21–23] due to line and point sources. The asymptotic schemes and solutions for the transient flexuralgravity waves due to an instantaneous line disturbance in an inviscid fluid of infinite depth were studied by Lu and Dai [21] and the corresponding results are simply quoted here and the method is extended to the three-dimensional cases. For a point/ring singularity, we may replace J 0 ðkRÞ by its asymptotic formula for large kR



J 0 ðkRÞ ’

2 pkR

1=2

p

cos kR  : 4

ð46Þ

Thus, we have an approximate formulae for fm due to instantaneous singularities as follows

fm ’

2 X 2 Z þ1 1X A expðitHmn Þdk; 2 m¼1 n¼1 0

ð47Þ

where

ð1Þmþ1 w þ ð1Þnþ1 x; t  1=2  1=2 C k C k C A¼ ; ; R0 J 0 ðkR0 Þ 2pV 2pR 2pV 2pR V

Hmn ¼

ð48Þ ð49Þ

for a ling, point and ring source, respectively. w ¼ kx for a ling source while w ¼ kR  p=4 for a point and ring one. According to the stationary phase approximation [24], Section 3.4, the dominant contribution to the integral in Eq. (47) for large t stems from the stationary points of Hmn . It is easily seen that H12 and H21 have the same stationary points. The solutions for the stationary points, denoted by kj , are determined by

oHmn ¼ 0: ok

ð50Þ

For a ling singularity, R hereinafter should be replaced with jxj. A straightforward derivation for Eq. (50) from Eqs. (38) and (39) yields

# pffiffiffi " 4 4 2 5 R 1 ð1 þ 5ck Þ cothðkHÞ þ kHð1 þ ck Þcsch ðkHÞ þ 4crk g 1 pffiffiffiffiffi  pffiffiffi  ¼0 Qðk; r; cÞ ¼  C g ¼ 4 t 2 k k0 ½cothðkHÞ þ rk3=2 ð1 þ ck Þ1=2

ð51Þ

for Model (I) while

# pffiffiffi " 2 2 2 3 g 1 R 1 ð1 þ 3sk Þ cothðkHÞ þ kHð1 þ sk Þcsch ðkHÞ þ 2srk pffiffiffiffiffi  pffiffiffi  ¼0 Qðk; r; sÞ ¼  C g ¼ 2 2 t k k0 ½cothðkHÞ þ rk3=2 ð1 þ sk Þ1=2

ð52Þ

for Model (II), where C g ¼ ox=ok is the group velocity, and k0 ¼ gt 2 =4R2 . It is noted that k0 is the exact solution of Q ðk; 0; 0Þ ¼ 0 with H ¼ þ1, corresponding to the classical Cauchy–Poisson gravity waves on the clean surface of an infinitely deep fluid. Generally speaking, the explicit analytical solutions for Eqs. (51) and (52) cannot readily be given for arbitrary h; H; c; r; s, and R=t. However, once the physical parameters are given, the nature of roots with respect to k for Eqs. (51) and (52) depends on the value of R=t only. The physical parameters given by Squire et al. [1], p. 105, 3 3 E ¼ 5 GPa;m ¼ 0:3; q ¼ 1024 kg m ; qe ¼ 917 kg m and g ¼ 9:8 ms2 can be adopted for the graphical representation of Eq. (51), as shown in Fig. 1. The surface tension coefficient is taken as T ¼ 0:074 Nm1 for the graphical representation of Eq. (52), as shown in Fig. 2. It can be seen that the group velocities for Models (I) and (II) reach their respective minimum 2 at kc , namely C g min ¼ C g jk¼kc , for which x00c ¼ o2 x=ok jk¼kc ¼ 0. It is noted that

lim ox=ok ¼ k!0

pffiffiffiffiffiffi gH ¼ C g max

and

2

lim o2 x=ok ¼ 0 k!0

ð53Þ

for the two models. A close examination of C g ðk; r; sÞ of Model (II) shows that, as k tends to infinity, there also exists a limiting group velocity for r > 0 and s > 0, denoted by C g lim ðr; sÞ, which is determined by

C g lim ðr; sÞ ¼ lim C g ðk; r; sÞ ¼ k!þ1

pffiffiffiffiffiffiffiffiffiffiffi g s=r:

ð54Þ

Therefore the outgoing capillary-gravity waves can appear if and only if C g max > C g lim , namely

H > Hmin ¼ s=r:

ð55Þ

Accordingly, there exists a minimal depth Hmin of fluid for the possibility of the propagation of capillary-gravity waves on an inertial surface. C g lim is independent of H. As H ! 1; C g max ! 1 and C g max > C g lim always holds. It can be seen from Fig. 1 that Eq. (51) has two distinct real positive roots k1 and k2 with 0 < k1 < k2 < þ1 for C g min < R=t < C g max and one real positive root k2 for R=t P C g max . It can be seen from Fig. 2 that Eq. (52) has two distinct real

D.Q. Lu, S.Q. Dai / International Journal of Engineering Science 46 (2008) 1183–1193

8

h= h= h= h=

Cg

6

1189

1.0 -4 m 1.0 -3 m 1.0 -2 m 1.0 -1 m x/t = Cg

4

2

0

0

2

4

6

8

10

k Fig. 1. Group velocity curves for flexural-gravity waves with H ¼ 5:0 m.

(i) (ii) (iii) (iv) (v) Cg

0.2 x/t = Cg

0 0

250

k

500

750

Fig. 2. Group velocity curves for capillary-gravity waves with H ¼ 1:0 m. (i) s ¼ 0 m2 ; r ¼ 103 m; (ii) s ¼ 7:374  106 m2 ; r ¼ 103 m; (iv) s ¼ 7:374  106 m2 ; r ¼ 102 m; (v) C g lim with s ¼ 7:374  106 m2 and r ¼ 102 m.

s ¼ 0 m2 ; r ¼ 102 m; (iii)

positive roots k1 and k2 for C g min < R=t < C g lim , one real positive root k1 for C g lim < R=t 6 C g max , and no real solution for R=t > C g max . Generally, k1 and k2 can be calculated numerically from Eq. (51) for flexural-gravity waves and from Eq. (52) for capillary-gravity waves. k1 is the wavenumber of gravity waves for the two models. k2 is the wavenumber of flexural waves for Model (I) and of capillary waves for Model (II). As D ¼ 0 and r–0, Eq. (51) has only one real positive root, of which the analytical form can be exactly derived for the cases with infinite depth [19]. The asymptotic and exact solutions of k1 and k2 for capillary-gravity waves with r ¼ 0 were provided by Chen and Duan [13] and Lu and Ng [20] for the case with infinite depth, respectively. As r ¼ s ¼ 0, Eq. (52) has only one real positive root, corresponding to the pure gravity waves. According to the Stokes stationary phase approximation, the expansion for the phase function near kj is taken as

Hmn ðkÞ  Hmn ðkj Þ þ

1 o2 Hmn ðkj Þ ðk  kj Þ2 ; 2 2 ok

ðj ¼ 1; 2Þ:

ð56Þ

By a straightforward application of the method of stationary phase, the asymptotic representation of Eq. (47) on kj can be given as

fmj ’ Aj

2p jx00j jt

!1=2 cos uj ;

ð57Þ

1190

D.Q. Lu, S.Q. Dai / International Journal of Engineering Science 46 (2008) 1183–1193

where

p uj ¼ wj  xj t  sgnðx00j Þ ;

ð58Þ

4

2

fAj ; xj ; x00j ; wj g ¼ fA; x; o2 x=ok ; wgjk¼kj , and sgnðxÞ ¼ 1 as x?0. It should be noted that Eq. (57) holds for x00j –0 only. When C g min < R=t < C g max for Model (I) and C g min < R=t < C g lim for Model (II), fm ¼ fm1 þ fm2 . When C g lim < R=t < C g max for Model (II), fm ¼ fm1 . As R=t decreases to C g min ; k1 and k2 move together toward kc while x00j tends to zero. Accordingly, Eq. (57) predicts an 3 3 infinitely increasing wave amplitude. It is noted that x000 c ¼ o x=ok jk¼kc –0. In this case, according to Scorer [25], the expansion for the phase function near kc is taken as

oHmn ðkc Þ 1 o3 Hmn ðkc Þ ðk  kc Þ þ ðk  kc Þ3 : 3 ok 6 ok

Hmn ðkÞ  Hmn ðkc Þ þ

ð59Þ

Thus, Eq. (47) can be approximately given as

fm ’ 2pAc AiðZ c Þ



1=3

2

jx000 c jt

cos uc ;

ð60Þ

where

Z c ¼ ðx0c t  RÞ



1=3

2

ð61Þ

;

x000c t

uc ¼ wc  xc t;

ð62Þ

and fAc ; xc ; x ¼ fA; x; ox=ok; wgjk¼kc ; AiðÞ is the Airy integral. As R=t is sufficiently close to C g max ; k1 tends to zero, we may expand the phase function with small k as follows 0 c ; wc g

"

R t

Hmn ðkÞ ¼ ð1Þmþ1 k þ ð1Þnþ1 k

# 3 oxð0Þ k o3 xð0Þ þ    : þ ok 6 ok3

ð63Þ

In this case, we have

fm ’ pA0



2

1=3

jx000 0 jt

AiðZ 0 Þ þ fTm2 ;

ð64Þ

where

Z 0 ¼ ½R  ðgHÞ1=2 t



1=3

2

x0000 t

ð65Þ

;

3

3 A0 ¼ Ajk¼0 and x000 0 ¼ o x=ok jk¼0 . For Model (I),

x0000 ¼ ðgHÞ1=2 ðH þ rH2 Þ

ð66Þ

while for Model (II),

x0000 ¼ ðgHÞ1=2 ðH2 þ 3rH  3sÞ:

ð67Þ

Fig. 3 shows the flexural- and pure gravity waves due to an instantaneous ring source predicted by above-mentioned asymptotic schemes. It can be seen that the short flexural waves ride on long gravity waves in the region of C g min < R=t < C g max . The Stokes and Scorer methods match well near the region of R=t  C g min and R=t  C g max . 4.2. Oscillating singularities The wave motion due to an oscillating line/point/ring source with a time-harmonic strength M0 expðiltÞ is considered in this section. For the wave profile fm , Eqs. (28), (40) and (41) will be employed. The steady-state and transient components of fm are denoted by fSm and fTm , respectively. Replacing J 0 ðkRÞ by its asymptotic formula for large kR, we have the approximate representations for fSm and fTm as follows

fSm ’ il expðiltÞ

2 Z X m¼1

fTm ’

0

þ1

A

x2  l2

exp½ð1Þmþ1 iwdk;

2 X 2 Z þ1 1X A ½ð1Þn ix  il expðit Hmn Þdk; 2 m¼1 n¼1 0 x2  l2

where Hmn and A are given by Eqs. (48) and (49), respectively.

ð68Þ ð69Þ

D.Q. Lu, S.Q. Dai / International Journal of Engineering Science 46 (2008) 1183–1193

1191

(i ) (ii) (iii) (iv) (v )

ζm

0. 5

0

-0 .5 0

R min

30

R

R max 60

Fig. 3. Transient flexural-gravity waves due to an instantaneous ring source of unit strength with R0 ¼ 1:0 m; z0 ¼ 1:0 m; h ¼ 0:01 m; H ¼ 10:0 m, and t ¼ 5 s. (i) Flexural-gravity waves predicted by Eq. (57) for C g min < R=t < C g max ; (ii) Flexural-gravity waves predicted by Eq. (64) for R=t  C g max and R=t > C g max ; (iii) Flexural-gravity waves predicted by Eq. (60) for R=t  C g max and R=t > C g max ; (iv) Pure gravity waves predicted by Eq. (57) for R=t < C g max ; (v) Pure gravity waves predicted by Eq. (60) for R=t  C g max and R=t > C g max .

The asymptotic representations of Eq. (69) can be derived by the aforementioned schemes and will not be repeated. It should be noted that at a certain R=t, we have xj ¼ l and thus the asymptotic representations will break down at xj ¼ l. The failure might be caused by the linear theory and the nonlinearity might be a possible explanation for this resonance. The asymptotic representation of Eq. (68) for far-field waves can be derived by means of Lighthill’s general theory for water waves, as stated by Debnath [18], Section 3.6. The dominant contribution to the integral in Eq. (68) stems from the poles of the integrand. By the Cauchy residue theorem, the approximation of Eq. (68) for the outgoing progressive waves can be given as

fSm ’ 2plAl exp½iðlt  wl Þ;

ð70Þ

where kl is the positive root of x ¼ l , and fAl ; wl g ¼ fA; wgjk¼kl . It follows from Eqs. (68) and (70) that the frequency of steady-state response to an oscillating disturbance is the same as that of the disturbance. The steady-state wave propagates outwards with a single wave number kl . Generally, kl can be obtained numerically from x2 ¼ l2 . In some particular cases with infinite depth, kl can be analytically obtained. As D ¼ 0 or s ¼ 0, it follows that [19] 2

xlim ¼ lim x ¼ k!1

2

pffiffiffiffiffiffiffiffiffi g=r:

ð71Þ

Furthermore, it holds for H ¼ 1 that [19]

kl ¼

l2 : g  rl2

ð72Þ

As r ¼ 0, Eq. (72) reduces to what Miles [12] obtained for pure gravity waves. Since kl > 0, it follows from Eq. (72) that l < xlim . Accordingly, progressive gravity waves with l P xlim are not possible on an inertial surface (r–0). As D–0 or s–0; xlim ! 1 and then kl always exists for any l > 0. As s–0 and H ¼ 1 for Model (II), x2 ¼ l2 becomes 2

g  rl2 þ g sk ¼ l2 =k:

ð73Þ

Rhodes–Robinson [6] showed the existence of solution for Eq. (73) by asserting that there always exists an intersection be2 tween the parabolic curve f1 ðkÞ ¼ g  rl2 þ g sk and the hyperbolic curve f2 ðkÞ ¼ l2 =k for s > 0. As it is, the exact solution for Eq. (73) can be given as

kl ¼

X 1=3 g  rl2  ; 3g s X 1=3

ð74Þ

where

X ¼ ½27ðg lsÞ2 þ

pffiffiffiffi Y =2; 2

3

ð75Þ 4

Y ¼ 108½gðg  rl Þs þ 729ðg lsÞ :

ð76Þ

1192

D.Q. Lu, S.Q. Dai / International Journal of Engineering Science 46 (2008) 1183–1193

A graphical representation of Eqs. (72) and (74) is given in Fig. 4. It should be noted that as

kl 6 klcrt ¼

1 ½3q1=3 þ sð4r2 þ 9sÞ þ 3q1=3 s3 ð8r2 þ 9sÞ 4r3

ð77Þ

where

q ¼ 8r4 s4 þ 36r2 s5 þ 27s6 þ 8

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r6 s8 ðr2 þ sÞ;

ð78Þ

we have Y P 0 and Eq. (74) predicts a real positive solution. For kl > klcrt ; Y < 0 and Eq. (74) predicts a complex solution with a positive real part. One may check graphically that the real part is the solution for Eq. (73). For general cases, the numerical solution of kl shows that kl decreases with increasing h and/or H for fixed l while kl increases with increasing l for fixed h and H. It should be noted that the wave amplitude predicted by Eq. (70) is directly proportional to l and R0 . The values of l and R0 should be taken under the assumption that the amplitude is small compared with the wavelength. Fig. 5 shows that the steady-state flexural-gravity waves predicted by Eq. (70). It can be seen from Fig. 5 that with decreasing kl the flexural-gravity wave amplitudes decease while the flexural rigidity D increases. This is also true for the capillary-gravity waves with increasing s.

100

(i ) (ii) (iii)

kμ

75

50

25

0

0

10

μ

20

30

Fig. 4. Wave numbers of steady-state flexural-gravity waves on a fluid of infinite depth. (i) Eq. (52) with s ¼ 7:374  106 m2 and r ¼ 0:01 m; (ii) Eq. (51) with r ¼ 0:01 m; (iii) Eq. (51) with r ¼ 0 m.

(i) (ii) (iii)

ζ Sm

0. 5

0

-0 .5

-1

25

50

75

100

R Fig. 5. Steady-state flexural–gravity waves ðfSm Þ due to an oscillating ring source of unit strength with l ¼ 2:0 s1 ; R0 ¼ 1:0 m; z0 ¼ 1:0 m, and H ¼ 10 m. (i) h ¼ 0:01 m; kl ¼ 0:421194 m1 ; (ii) h ¼ 0:1 m; kl ¼ 0:30458 m1 ; (iii) h ¼ 0:5 m; kl ¼ 0:148061 m1 .

D.Q. Lu, S.Q. Dai / International Journal of Engineering Science 46 (2008) 1183–1193

1193

5. Conclusions Within the framework of linearized theory, the flexural- and capillary-gravity waves can be investigated in parallel in view of their analytical similarity. The wave profiles are significantly affected by the presence of the flexural rigidity or the surface capillarity. As the system is subject to an instantaneous disturbance, the waves generated consist of the transient response only. As the system is subject to an oscillating disturbance, the progressive waves generated on the surface consist of the steady-state monochromatic flexural/capillary-gravity waves and the transient waves. When the time approaches infinity, the transient components tend to zero and an ultimate steady-state can be attained. There exist maximal and minimal group velocities, for which the finite depth of the fluid and the flexural/capillary effect of the surface are, respectively, responsible. The transient wave system observed depends on the relation between the moving speed ðR=tÞ of the observer and the minimal and maximal group velocities (C g min and C g max ). As C g min < R=t < C g max , the transient response consists of two components, namely the transient flexural/capillary waves and the transient gravity waves. The short transient flexural/ capillary waves ride on the long transient gravity waves. As R=t < C g min or R=t > C g max , the transient response consists of one rapidly decaying component only. As the flexural rigidity and/or surface capillarity tends to zero, the transient response consists of transient gravity waves only and the near-field waves due to a surface singularity predicted by the potential theory go to infinity, which is incompatible with the physical reality. Such an inconsistency can be avoided by the inclusion of the flexural rigidity or the surface capillarity that are responsible for the occurrence of a calm region in the near field. As for the steady–state component, the wave amplitudes decease and the wavelengths increase due to the effect of the flexural rigidity or the surface capillarity. Another remarkable feature of the capillarity is that there exists a minimal depth of fluid for the possibility of the propagation of capillary-gravity waves on an inertial surface. Acknowledgements This research was sponsored by the National Natural Science Foundation of China under Grant No. 10602032 and the Shanghai Rising-Star Program under Grant No. 07QA14022. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

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