Smrutiranjan Mohapatra

Scattering of Surface Waves by the Edge of a Small Undulation on a Porous Bed in an Ocean with Ice-cover

Smrutiranjan Mohapatra＊

Department of Mathematics, Institute of Chemical Technology, Mumbai 400019, India

Scattering of surface waves by the edge of a small undulation on a porous bed in an ocean of finite depth, where the free surface has an ice-cover being modelled as an elastic plate of very small thickness, is investigated within the framework of linearized water wave theory. The effect of surface tension at the surface below the ice-cover is neglected. There exists only one wave number propagating at just below the ice-cover. A perturbation analysis is employed to solve the boundary value problem governed by Laplace′s equation by a method based on Green′s integral theorem with the introduction of appropriate Green′s function and thereby evaluating the reflection and transmission coefficients approximately up to first order. A patch of sinusoidal ripples is considered as an example and the related coefficients are determined.

porous bed; ice-cover; surface waves; bottom undulation; Green′s function; perturbation technique; reflection and transmission coefficients

1 Introduction1

The problems of scattering of surface water waves by an obstacle or a geometrical disturbance at the bottom of an ocean, whereas on the upper surface of the ocean is bounded above by a thin uniform ice-cover modelled as a thin elastic plate, are important for their possible applications in the area of coastal and marine engineering, and as such these are being studied for a long time. In polar region, porosity of the bed becomes an extremely important aspect to be handled for marine researchers. Scattering of surface water waves by a small bottom undulation in an ocean with free surface create interesting mathematical problems drawing attention of various types for obtaining their useful solutions (Miles (1981), Davies (1982), Mei (1985), Porter and Porter (2003), Martha et al. (2007)). Moreover, various approaches were developed by many researchers to deal with the interaction of water waves with floating ice plates only or the surface wave interaction by patches of small bottom undulations in an ocean with ice-cover (Fox and Squire (1994), Chakrabarti (2000), Linton and Chung (2003), Mandal and Basu (2004)).

The above works are focused only on the wave motion of the fluid region, where the effect of porosity of the ocean-bed was not taken into account. Martha et al. (2007), Zhu (2001) and Silva et al. (2002) considered water-wave reflection and/or transmission problems where a porous medium was assumed to lie on a sea-bed of varying quiescent depth. In the present work, we consider a fluid flow in an ocean where the upper surface is bounded above by a thin uniform ice-cover modelled as a thin elastic plate, which replaces the free surface and the bottom is bounded by a porous surface which has a small undulations. The motion of the fluid below the porous ocean bed is not analyzed here and it is assumed that the fluid motions are such that the resulting boundary condition on the porous ocean bed as considered here holds good and depends on a known parameter P, called porosity parameter, in this analysis. In this case, time-harmonic waves of a particular frequency can propagate with one wave number at just below the ice-cover. By employing perturbation analysis, the original problem is reduced to a simpler boundary value problem for the first order correction of the potential. The solution of this problem is then obtained by the use of Green′s integral theorem to the potential function describing the boundary value problem. The reflection and transmission coefficients are then evaluated approximately up to the first order of δ in terms of integrals involving the shape function when a train of progressive waves propagating from negative infinity is normally incident on the porous bed having a small undulation in an ocean with ice-cover. We present a special form of bottom undulation, that is, a patch of sinusoidal ripples and the first-order reflection coefficient is depicted graphically for various values of the different parameters.

2 Mathematical formulation of the problem

We consider the irrotational motion of an inviscid incompressible fluid of relatively small amplitude under the action of gravity, neglecting any effect due to surface tension at the free surface of the fluid and is covered by a thin uniform ice sheet modelled as a thin elastic plate. The fluid is of infinite horizontal extent in x-direction while the depth is along y-direction which is considered vertically downwards with y = 0 as the mean position of the thinice-cover and y = h as the bottom surface. We further assume that the motion is time harmonic with angular frequency ω. Here, the bed has a porous type surface with a small undulation which is described by y=h+δc(x), where c(x) is a bounded and continuous function with c(x)→0 as |x|→∞and the non-dimensional number δ(＜＜1) a measure of smallness of the undulation. Under these assumptions, the velocity potential in the fluid of density ρ can be written as Re[φ(x,y)e?iωt] where Re stands for the real part, and the potential φ(x,y) satisfies

The linearized boundary conditions at the ice-cover and at the bottom surface are:

where K=ω2g,gthe acceleration due to gravity; D=L(ρg), L is the flexural rigidity of the elastic ice-cover; ε=(ρ0ρ)h0, ρ0is the density of the ice, h0is the very small thickness of the ice-cover, ??n the derivative normal to the bottom at a point (x,y) and P is the porous effect parameter on the ocean bed. The time dependence of e?iωthas been suppressed.

Within this framework in the fluid, a train of two-dimensional normal incident progressive waves takes the form (up to an arbitrary multiplicative constant):

with coshuh?(Pu)sinhuh ≠0 and u satisfies the dispersion relation Δ(u)=0,where

In the above dispersion equation, there is a positive real root k that indicates the propagating modes; a complex conjugate pair of roots corresponding to the damped propagating modes; and a countable infinity of purely imaginary roots iun(n=1,2,…) that relate to a set of evanescent modes, where un′s are real and positive. The negatives of all of these are also roots, being wave numbers of the waves travelling in the opposite direction. Since dispersion equation has exactly one nonzero positive real root k, so only one nonzero wave number k can exist and the wave can propagate at just below the ice-cover along the positive x-direction.

Assuming, for small bottom undulation, δ to be very small and neglecting the second order terms, the boundary condition (3) on the bottom surface y=h+δc(x) can be expressed in an appropriate form as

on yh=.

Since the wave train is partially reflected and partially transmitted over the bottom undulation, the far-field behavior of φ is given by

The unknown coefficients R and T are the reflection and transmission coefficients, respectively, and are to be determined. Here the perturbation method can be employed to obtain these coefficients up to first order. By using the perturbation technique, the entire fluid region 0＜y＜h+δc(x),?∞＜x ＜∞, is reduced to the uniform finite strip 0＜y＜h,?∞＜x＜∞, in the following mathematical analysis.

3 Method of solution

3.1 Perturbation technique

Let us first consider a train of progressive waves to be normally incident on the bottom undulation. If there is no bottom undulation, then the normally incident wave train will propagate without any hindrance and there will be only transmission. This, along with the appropriate form of the boundary condition (6), suggest that ,φR and T which were introduced in the last section, can be expressed in terms of the small parameter δ as:

where0φ is given by Eq. (4). It must be noted that such a perturbation expansion ceases to be valid at Bragg resonance when the reflection coefficient becomes much larger than the undulation parameter δ, as pointed out by Mei (1985). Also this theory is valid only for infinitesimal reflection and away from resonance. For large reflection, the perturbation series, as defined in Ep. (8), needs to be refined so that it can deal with the resonant case, which is reported in Mei (1985).

Using Eq. (8) in Eq. (1) and boundary conditions (2), (6), (7) and then comparing the first order terms of δ on both sides of the equations, we find a boundary value problem for the first order potential1(,)xyφ which satisfies the Eqs. (1) and (2) together with the following other conditions:where s(u)=coshuh?(Pu)sinh uh. To solve the above boundary value problem for φ1, we need two-dimensional source potential (in terms of Green′s function) for Laplace′s equation due to a source submerged in the fluid. Then Green′s integral theorem will be employed and the first-order coefficients R1and T1will be obtained in terms of integrals involving the shape function c(x).

3.2 Introduction of Green's functions

Suppose the source is submerged in the fluid. Then, for 0＜η＜h, the source potential in terms of Green′s function G(x,y;ξ,η) satisfies the following boundary value problem:

and G represents outgoing waves as |x?ξ|→∞. Now we try to solve the boundary value problem defined by Eqs. (11)－(14) in the form G(x,y;ξ,η), where

and the integral representation of ln()rr′ is given in Rhodes-Robinson (1970) andWith the help of the boundary conditions at the ice-cover and at the bottom surface, we find

where Δ(u) is same as in Eq. (5). It may be noted that Δ(u) has one simple non-zero root at u=k. Since u=0 and coshuh?(Pu)sinhuh =0 will indicate that there is no wave in the region, hence the terms u2and coshuh?(Pu)sinhuh can never be zero. So the integrand in Eq. (15) has one simple pole at u=k which will be from Δ(u) only. Since the source potential G behaves like outgoing waves as |x?ξ|→∞, so the path of integration is indented to pass beneath the simple pole at u=k. Solving (15) by using (16) and (17), we obtain the solution G(x,y;ξ,η) as |x?ξ|→∞, is given by

where Δ′ denotes the derivative of Δ with respect to u. To calculate the value of φ1(ξ,η), we apply the Green′s integral theorem to φ1(x,y) and G(x,y;ξ,η) in the form

where C is a closed contour in the xy-plane consisting of the lines y=0,?X≤x≤X; y=h,?X≤x≤X; 0≤y≤h,x=±X and a small circle of radius γ with center at (ξ,η) and ultimately let X→∞, γ→0. Finally the resultant integral equation (19) will give the determination of the solution φ1of the boundary value problem as given by

where Gy=?G?y. The first order reflection and transmission coefficients R1and T1, respectively, are now obtained by letting ξ→?∞, in Eq. (20) and comparing with Eq. (10) by replacing (x,y) with (ξ,η). Thus we obtain the values of R1and T1as

Therefore, the first order reflection and transmission coefficients can be evaluated from Eqs. (21) and (22), once the shape function ()cx is known. In the following section we proceed to examine the effects of reflection and transmission for a special sinusoidal form of the shape function ()cx.

4 Special form of the bottom surface

We consider a special sinusoidal form of the shape function ()cx for the uneven bottom surface of the porous bed. This functional form of the bottom disturbance closely resembles some naturally occurring obstacles formed at the bottom due to sedimentation and ripple growth of sands. Davies (1982) studied sinusoidal form of undulations in an ocean bed by using Fourier transform technique and found that an undulating bed has the ability to reflect incident wave energy which has important implications in respect of coastal protection as well as possible ripple growth if the bed is erodible. Because of the importance of the bed topographies with sinusoidal ripples from the applicationpoint of view, significant emphasis is laid upon them and subsequently consideration of the following example is deemed appropriate.

Now we consider the shape function ()cx in the form:

Here, we can take1πLnl=? and2πLml=, where m and n are positive integers. This patch of sinusoidal ripples on the surface of porous bed with amplitude a consists of ()2nm+ ripples having the same wave number l. Substituting the value of ()cx from equation (23) into Eqs. (21) and (22), we obtain the reflection coefficient1R and transmission coefficient1T, respectively, as follows:

When the sinusoidal ripples wave number is approximately twice the surface wave number (i.e.,2kl≈), the theory points towards the possibility of a resonant interaction taking place between the bed and the surface waves. Hence, we find from Eq. (24) that near resonance, i.e., 2kl≈, the limiting value of the reflection coefficient assumes the value

Note that when 2k approaches l and the number of ripples in the patch of the undulation on the porous bed ()2nm+ become large, the reflection coefficient becomes unbounded contrary to our assumption that1R is a small quantity, being the first-order correction of the infinitesimal reflection. Consequently, we consider only the cases excluding these two conditions in order to avoid the contradiction arising out of resonant cases.

Thus the reflection coefficient R1, in this case, becomes a constant multiple of (n+m)2, the total number of ripples in the patch. Hence, the reflection coefficient R1increases linearly with n and m. Although the theory breaks down when l=2k, a large amount of reflection of the incident wave energy by this special form of bed surface will be generated in the neighborhood of the singularity at l=2k.

5 Numerical results

We consider the numerical computations for the non-dimensionalized first order reflection coefficient1||R due to an incident surface wave of wave number k propagating along the ice-cover and a ripple bed with wave number l having (n+m)2 number of ripple wavelengths in the patch of the bottom of the porous ocean bed, which is calculated from Eq. (24). In this case, we again consider the ratio of the amplitude of the ripples and the depth of the fluid (ah) as 0.1. In Fig. 1, the first order reflection coefficient R1is plotted against Kh for different porous effect parameters Ph of the ocean bed, while we fixed the ice parameters as Dh4=1,εh=0.01, the ripple wave number lh as 1 and the number of ripples as 5(m=2,n=3). This is most evident in the curves that the peak value of |R1| increases as the porous effect parameter increases. This shows that the first-order correction to the reflection coefficient is somewhat sensitive to the changes in the porous effect parameter of the ocean bed. Computations show that the peak values of the first-order reflection coefficient corresponding to the porous effect parameters Ph = 0, 0.01, 0.05 and 0.1 are attained at Kh = 0.498 099, 0.498 495, 0.497 771 and 0.497 922, respectively. It is clear that its peak value is attained when the ripple wave number lh of the bottom undulation becomes approximately twice as large as the surface wave number Kh. In this figure one feature that is common to all the curves is the oscillating nature of the absolute values of the first-order coefficients as a function of Kh.

Fig.1 Reflection coefficient |R1| plotted against Kh for Dh4=1,εh=0.01,m =2 and n=3

The different curves in Fig. 2 correspond to different sets of ice parameters(Dh4=0;εh=0),(Dh4=1;εh=0.01), (Dh4=1.5;εh=0.01) and (Dh4=2;εh=0.1), while Ph=0.1, lh=1 and the number of ripples as 5 (m=2,n=3) are fixed for these curves. This figure shows that the peak values of the reflection coefficient |R1| for the normally incident wave of wave number Kh, decrease as the values of the flexural rigidity of the ice-cover Dh4and εh increase. Here also it is observed that the peakvalues of1||R are attained when the wave number of the bottom undulations on the porous ocean bed becomes approximately twice the surface wave number. It may be noted that when4Dh and hε are taken to be zero (i.e. the ice-cover is absent), the solutions for problems with free surface can be obtained as particular cases.

Fig.2 Reflection coefficient R1plotted against Kh forPh=0.1, m=2 and n=3

In Fig. 3, different curves correspond to different number of ripples in the patch of the undulation on the porous bed. For all curves, we consider Ph=0.1, Dh4=1,εh=0.01 and lh=1. It is clear from this figure that as (n+m)2, the number of ripples, increases the value of Kh converges to a number in the neighborhood of 0.5 (i.e., lh2) and also the peak value of the reflection coefficient |R1| increases. But when the number of ripples, becomes very large, the reflection coefficient become unbounded. That means the perturbation expansion which is discussed in section 3.1, ceases to be valid when the reflection coefficient becomes much larger than the undulation parameter, as pointed out by Mei (1985). Its oscillatory nature against Kh is more noticeable with the number of zeros of |R1| increased but the general feature of |R1| remains the same.

Fig.3 Reflection coefficient R1plotted against Kh for Ph=0.1, Dh4=1 and εh=0.01

In Fig. 4, different curves correspond to different ripple wave numbers lh = 0.8, 1.0, 1.2 and 1.4 in the patch of the undulation on the porous bed. In this figure, for all curves, we consider Ph=0.1, Dh4=1,εh=0.01,m =2 and n=3. It has been cleared from this figure that the peak values of the reflection coefficient are attained at different values of Kh. The reason is, the values of reflection coefficient |R1| (which is calculated from Eq. (24)) become maximum, only when lh≈2Kh. It is observed from this figure that as the ripple wave numbers increase the reflection coefficient |R1| becomes smaller than those for the bigger ripple wave numbers. That means when an incident waves propagates over a small bottom undulation on the porous bed in an ice-covered ocean, a substantial amount of reflected energy can be produced.

Fig.4 Reflection coefficient R1plotted against Kh for Ph=0.1,Dh4=1,εh=0.01,m =2 and n=3

6 Conclusions

A patch of sinusoidal bottom undulations on the porous surface of an ocean bed, where the upper surface is bounded above by a thin ice-cover modelled as a thin elastic plate, which replaces the free surface, is considered because of its considerable physical significance like the ability of an undulating bed to reflect incident wave energy which is important in respect of both coastal protection and possible ripple growth if the bed is erodible. In such a situation propagating waves can exist at only one wave number for any given frequency. A perturbation analysis has been deployed and thereby finding new expressions for the first order corrections to the reflection and transmission coefficients for the problem by using a method based on Green′s integral theorem with the introduction of appropriate Green′s function. For the particular example of a patch of sinusoidal ripples, first order approximations to the reflection and transmission coefficients are obtained in terms of computable integrals and the reflection coefficient depicted graphically through a number of figures. The main result that follows is that, the resonant interaction between the bed and the surface waves attains in the neighborhood of the singularity when the ripple wave numbers of the bottomundulation become twice the surface wave number. This singularity point varies with porous effect parameters of the ocean bed, the flexural rigidity of the ice-cover and the ripple wave numbers on the bottom surface. Another main advantage of this method, demonstrated through this example, is that a very few ripples may be needed to produce a substantial amount of reflected energy. Also the theory discussed in this paper is valid only for infinitesimal reflection and away from resonance. The results obtained here are expected to be qualitatively helpful in the construction of an effective reflector of the incident wave energy for protecting coastal areas from the rough sea in arctic regions.

Acknowledgement

The author wishes to thank Prof. S.N. Bora, Department of Mathematics, Indian Institute of Technology Guwahati, India for his valuable discussions and suggestions to carry out the preparation of the manuscript. The author expresses his deep gratitude to the referee for valuable comments and suggestions which enabled the author to carry out the desired revision of the manuscript.

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Author’s biography

1671-9433(2014)02-0167-06

jan Mohapatra

his PhD from the Indian Institute of Technology, Guwahati, India in 2009. He also worked as a post doctoral fellow at the Indian Institute of Science, Bangalore, India, prior to accepting his present position of Assistant Professor in the Department of Mathematics, Institute of Chemical Technology Mumbai, India. His main areas of interest are water wave scattering problems. He has 10 research publications to his credit and is involved in number of sponsored projects.

Received date: 2013-09-04.

Accepted date: 2013-10-11.

＊Corresponding author Email: sr.mohapatra@ictmumbai.edu.in

? Harbin Engineering University and Springer-Verlag Berlin Heidelberg 2014