Wave Scattering by Small Undulation on the Porous Bottom of an Ocean in the Presence of Surface Tension

e scattering of incident surface water waves due to small bottom undulation on the porous bed of a laterally unbounded ocean in the presence of surface tension at the free surface is investigated within the framework of two-dimensional linearized water wave theory. Perturbation analysis in con�unction with the �ourier transform technique is employed to derive the �rst-order re�ection and transmission coefficients in terms of integrals involving the shape function cccccc representing the bottom undulation. One special type of bottom topography is considered as an example and the related coefficients are determined in detail. ese coefficients are presented in graphical forms.e theoretical observations are validated computationally.e results for the problem involving scattering of water waves by bottom deformations on an impermeable ocean bed are obtained as a particular case.


Introduction
e interaction problems of surface waves with a preexisting (�xed) pattern of undulation on an otherwise �at bed are important for their possible applications in the areas of coastal and marine engineering, and as such these are being studied for a long time.e problem of re�ection of surface waves by patches of bottom undulations has received an increasing amount of attention as its mechanism is important in the development of shore-parallel bars.
e problems of water wave diffraction by undulating bottom topography of a sea bed have been considered by many scientists and several papers have been published in this direction (see Davies and Heathershaw [1], Mandal and Basu [2], Martha and Bora [3], Miles [4], and many others).
All the above works focused only on the wave motion over the impermeable bottom topography of sea bed in absence of surface tension at free surface.Nowadays, due to many interesting applications in the theory of water waves scattering, many researchers have turned their attention to the problems related to porous bed rather than an impermeable one.Mase and Takeba [5], Zhu [6], and Silva et al. [7] considered waterwave re�ection and/or transmission problems where a porous medium was assumed to lie on a sea bed of varying quiescent depth.Martha et al. [8] focused on the problem involving wave scattering by small undulation on a porous bottom topography without considering the surface tension at the free surface.
Here, we investigate the scattering problem of surface water waves by small undulation on an porous ocean-bed of �nite depth in the presence of surface tension at the free surface.e motion of the �uid below the porous ocean bed is not analyzed here and it is assumed that the �uid motions are such that the resulting boundary condition on the ocean bed as considered here holds good and depends on a known parameter , called porosity parameter, in this analysis.Due to the small parameter  (≪1) being present in the representation of the small undulation of the bottom, the perturbation analysis can be employed.By this analysis, the velocity potential, the re�ection, and transmission coefficients appearing in the governing boundary value problem (BVP) can be expanded as a power series involving this parameter .Equating the coefficients of identical powers of the parameter  from both sides of all equations appearing in the coupled BVP, a number of BVPs can be obtained.However, the BVP up to �rst order is considered here for its solutions as the BVP of higher order can be solved successively in principle.e solution of the zeroth order BVP is obvious.Using Fourier transform technique, the solution involving the velocity potential for the �rst order BVP is obtained.By comparing this solution with far-�eld behaviour of the velocity potential, the �rst-order re�ection and transmission coefficients are derived in terms of integrals involving the shape function  representing the bottom undulation.ese coefficients are computed numerically for a special form of bottom topography, namely, sinusoidal bottom, and depicted graphically.e results for problem involving scattering of water waves by bottom deformations on an impermeable ocean bed are obtained as a particular case.

Problem Formulation
A right-handed rectangular Cartesian coordinate system is considered in which -axis is the position of the undisturbed free surface of the ocean and -axis is positive vertically downward from the undisturbed free surface (see Figure 1).e ocean bed with small undulation, composed of porous material of speci�c type, is described by     , where  is a bounded and continuous function, describing the undulation of the ocean bed and    as ||  ∞ so that the ocean is of uniform �nite depth  far away from the undulation on either side; the nondimensional number  (≪1) is a small parameter giving a measure of the smallness of the undulation.It is also assumed that the �uid is incompressible and inviscid, and the motion is irrotational.e usual assumptions of linear water wave theory and the removal of the harmonic time independence exp− lead to the following governing equations for the time-independent complex-valued velocity potential function  : where    2 / and   /,  is the angular frequency of the incoming water-wave train,  is acceleration due to gravity,  is the coefficient of surface tension at the free surface of the ocean,  is the density of water, / denotes the normal derivative at a point   on the bottom, and  is the porous effect parameter corresponding to the ocean bed under consideration.
It is assumed that a train of progressive wave is incident to the bottom undulation from the direction of   −∞, then it is partially re�ected by and partially transmitted over the bottom undulation so that  has the far-�eld behaviour given by where  and  represent, respectively, the un�nown re�ection and transmission coefficients which are to be determined here.

Perturbation Technique.
Assuming  to be very small, the bottom boundary condition (3) can be expressed as If there is no undulation at the bottom, the surface wave train propagates without any hindrance and there is total transmission.In view of this along with the approximate form of relation ( 5),    can be expressed in terms of the small parameter  as follows: Substituting the expansions (6a)-(6c) into ( 1)-( 2) and ( 4)-( 5 e solution of the BVP-I corresponds to the incident wave over �nite depth  and can be written as where  0 , the wave number of the normal incident wave, is the unique positive root of the following dispersion relation: Note that,  0 is the negative root of the dispersion relation (11).

Fourier Transform
Technique.e solution of BVP-II is obtained by using Fourier transform technique.To solve this BVP, we now assume  0 to have a small positive imaginary part so that  1 decreases exponentially as ||  .is ensures the existence of Fourier transform   1 (  of  1 (  with respect to  and de�ned as follows: with the following inverse transform: Applying Fourier transform de�ned in relation (12), to (8a), (8b) and (8c) and solving them, we obtain   1 (  as given by   1   =  +  3  cosh    snh   (  + 3 +snh+ + 3 cosh where ( is the fourier transform of (. Taking the inverse fourier transform, the solution for the �rst order velocity potential  1 (  can be written as To calculate the �rst order re�ection coe�cient  1 , we let    in relation (15).As   , the behaviour of  1 (  can be obtained by rotating the path of the integral involving the term ( into a contour in the �rst quadrant so that we must include the residue term at the pole  =  0 .e path of the integral involving the term ( in the relation ( 15) is rotated into a contour in the fourth quadrant so that the integral involving the term ( does not contribute as   .en comparing the resultant integral value with (9), we obtain the value of  1 as where  =  0 +  3 0  cosh  0    snh  0   =  0 +  3 0  snh  0    cosh  0 and ( 0  is given in the relation ( 16) with  � denoting the derivative of  with respect to .
Similarly, to calculate the �rst order transmission coe�cient  1 , we take    in relation (15).As   , the behaviour of  1 (  can be obtained by rotating the path of the integral involving the term ( into a contour in the �rst quadrant so that we must include the residue term at the pole  =  0 .e path of the integral involving the term ( in (15) is rotated into a contour in the fourth quadrant so that the integral involving the term ( does not contribute as   .en comparing the resultant integral value with (9), we obtain the value of  1 as ese resulting integrals  1 and  1 can be evaluated once the bottom pro�le ( is known.In the Section 5, we consider a special form for the shape function (.
In the absence of the surface tension, that is, when   0, the above results coincide exactly with the results of Martha et al. [8].

�. ��ecial �orm o� t�e �ottom �ro�le
In 1982 Davies [9] found that an undulating bed has the ability to re�ect incident wave energy which has an important implication in the application of coastal protection and in the case of possible ripple growth if the bed is erodible.Here we consider a special form of the shape function  in the form of a patch of sinusoidal ripples as the bottom undulation.e shape function is given by where with  as the amplitude of the sinusoidal ripples,  the wave number of the sinusoidal ripples with an arbitrary phase angle , and ,  positive integers.For this case, the re�ection coefficient  1 and the transmission coefficient  1 , respectively, are obtained as In the situation in which there is an integer number of ripple wavelengths in the patch  1 ≤  ≤  2 such that    and   0, we �nd  1 and  1 , respectively, as Relation (23) illustrates that for a given number of  ripples, the �rst order re�ection coefficient  1 is an oscillatory in nature.Furthermore, when the bed wave number is approximately twice the surface wave number (i.e., when   2 0 ), Bragg-resonance described earlier by Davies [9], Mei [10], and Davies and Heathershaw [1] takes place between the bed forms and the surface waves.e relation (23) becomes of the form 00 when   2 0 (at resonance), and in such a situation, we employ L'Hôpital's rule to compute the limiting value and �nd that the re�ection coefficient is given by the following expression: us the re�ection coefficient  1 becomes a constant multiple of , the number of ripples in the patch.Hence, the re�ection coefficient  1 increases linearly with .is indicates that relatively few bottom undulation with its wave number equal to approximately twice the surface wave number may give rise to a very substantial re�ected wave.

Numerical Results and Discussion
In this section, the �rst order re�ection coefficient | 1 | as given by ( 23) is computed numerically for various values of different dimensionless parameter and they are presented graphically.
In Figure 2, the �rst order re�ection coefficient | 1 | is depicted against wave number  0 ℎ for different values of the porous effect parameter ℎ and for the dimensionless parameters ℎ  1, ℎ  0.1,   1, ℎ 2  0. From this �gure, it is clear that | 1 | is oscillatory in nature and the peak value of | 1 | is attained when the wave number of the bottom undulation ℎ becomes approximately twice the surface wave number  0 ℎ, which validate the theoretical observation.Also the peak value of the re�ection coefficient increases as the value of the porous effect parameter increases.
In Figure 3, the �rst order re�ection coefficient | 1 | is plotted against the wave number  0 ℎ for different number of ripples of the bed (  1, 2, ) and for ℎ  1, ℎ  0.1, ℎ 2  1, ℎ  0.1.From this �gure, it is concluded that when the number of ripples  of the bed increases, the peak value of | 1 | also increases, which also validate the theoretical observation of (25).It is also clear that its oscillatory nature is more pronounced with the number of zeros of | 1 | increased but the general feature of | 1 | remains the same.e effect of the surface tension on re�ection coefficient | 1 | is shown for ℎ  1, ℎ  01,   1, ℎ  0 in Figure 4 and for the same set of values with   2 in Figure 5. From these �gures, it is clear that the value of the �rst order re�ection coefficient decreases as the surface tension increases.

Conclusion
Scattering of water waves by small undulation of the porous bottom of an ocean in the presence of surface tension is investigated here.Perturbation analysis in conjunction with the Fourier transform technique is used to derive the �rst order velocity potential, the re�ection, and transmission coefficients.e main advantage of this method is that we need to solve relatively easier ordinary differential equation to �nd the Fourier transform of the velocity potential.e derived results for re�ection and transmission coefficients are exactly coincide with the known ones obtained earlier in the case when the bed has no porous effect and when the surface tension at the free surface is neglected.e results for the re�ection and transmission coefficients for sinusoidal bottom topography are determined in detail and demonstrated graphically.From the computational results, it is clear that the peak value of re�ection coefficient increases when the porosity of the uneven ocean bed increases.Another important conclusion is that the value of the re�ection coefficient is found to be decreasing with the surface tension at the free surface.e analysis of the present work which is developed mainly for two-dimensional problems is also applicable to problems for the case of oblique incidence (three dimensions), with appropriate assumptions and modi�cations and this will be the subject matter of our future work.
), we �nd, a�er equating the coefficient of   and  from both the sides, that the functions     and  1  , respectively, satisfy the following BVPs.BVP-I.e function     satis�es (9)osh    snh   1 and  1 , due to normally incident wave, are obtained by letting    in the relation (15) and comparing with(9).