Diffraction of Oblique Water Waves by Small Uneven Channel-bed in a Two-layer Fluid

Department of Mathematics, Veer Surendra Sai University of Technology, Burla 768018, Odisha, India

Diffraction of Oblique Water Waves by Small Uneven Channel-bed in a Two-layer Fluid

Smrutiranjan Mohapatra＊

Department of Mathematics, Veer Surendra Sai University of Technology, Burla 768018, Odisha, India

Obliquely incident water wave scattering by an uneven channel-bed in the form of a small bottom undulation in a two-layer fluid is investigated within the frame work of three-dimensional linear water wave theory. The upper fluid is assumed to be bounded above by a rigid lid, while the lower one is bounded below by a bottom surface having a small deformation and the channel is unbounded in the horizontal directions. Assuming irrotational motion, perturbation technique is employed to calculate the first-order corrections to the velocity potentials in the two fluids by using Fourier transform approximately, and also to calculate the reflection and transmission coefficients in terms of integrals involving the shape function representing the bottom deformation. Consideration of a patch of sinusoidal ripples shows that the reflection coefficient is an oscillatory function of the ratio of twice the component of the wave number alongx-axis and the ripple wave number. When this ratio approaches one, the theory predicts a resonant interaction between the bed and interface, and the reflection coefficient becomes a multiple of the number of ripples. High reflection of incident wave energy occurs if this number is large.

oblique waves; two-layer fluid; bottom undulation; linear water wave theory; reflection coefficient; transmission coefficient; perturbation technique; Fourier transform

1 Introduction

In two superposed inviscid fluids, separated by a common interface with the total fluid region bounded above and below by rigid horizontal walls, only one wave mode can exist and for a given frequency, time-harmonic gravity waves can propagate in either direction at the interface. A train of progressive interface waves traveling over the bottom surface of a channel, without any obstacle, experiences no reflection when the channel is of uniform finite depth. If the bed of the channel has a deformation, the wave train is partially reflected by it, and is partially transmitted over it. However, there exists a class of natural physical form of bottom standing obstacles such as sand ripples. These ripples can be assumed to be small in somesense, for which some sort of perturbation technique can be employed for obtaining the first order corrections to the reflection and transmission coefficients. Chamberlain and Porter (2005) examined the scattering of waves in a two-layer fluid of varying mean depth in a three-dimensional context by using linear theory. A variational technique was used to construct a particular type of approximation which had the effect of removing the vertical coordinate and reducing the problem to two coupled partial differential equations in two independent variables. The behavior of water waves over periodic beds was considered by Porter and Porter (2003) in a two-dimensional context using linear water wave theory. They developed a transfer matrix method incorporating evanescent modes for the scattering problem, which reduced the computation to that required for a single period, without compromising full linear theory. Later on, Porter and Porter (2004) investigated the three-dimensional wave scattering by an ice sheet of varying thickness floating on sea water which had undulating bed topography. They obtained a simplified form of the problem by deriving a variational principle equivalent to the governing equations of linear theory and invoking the mild-slope approximation in respect of the ice thickness and water depth variations. Bhatta and Debnath (2006) analyzed a transient two-layer fluid flow over a viscoelastic ocean bed by using Laplace transform and Fourier transform. Maity and Mandal (2006) employed Green′s function technique to study the reflection of oblique surface waves over small deformations in a two-layer fluid which had a free surface. Barthe’lemyet al. (2000) investigated the scattering of a surface long wave by a step bottom in a two-layer fluid theoretically by the use of Wentzel Kramers Brillouin Jeffreys (WKBJ) theory through the nonlinear analysis. A series of quantitative laboratory studies was carried out by Sveenet al. (2002) to determine the spatial and temporal development of the velocity, vorticity, and density fields associated with the flow of an internal solitary wave of depression over a bottom ridge in a stably-stratified, two-layer system.

Mohapatra and Bora (2009, 2010) considered the scattering of normal internal wave propagation over a small deformation on the bottom of a channel for a two-layer fluid. A perturbation technique was employed to reduce the original boundary value problem to a coupled boundaryvalue problem up to the first order and the velocity potential, reflection coefficient and transmission coefficient up to the first order were obtained by using Green′s function technique and Fourier transform. Recently, Saha and Bora (2013) investigated the existence of trapped modes supported by a submerged horizontal circular cylinder in a two-layer fluid of finite depth bounded above by a rigid lid and below by an impermeable horizontal bottom by employing multipole expansion method.

In the present work, we solve the previous work of Mohapatra and Bora (2012) of oblique water wave scattering by a small bottom undulation in a two-layer fluid flowing through a channel, by using Fourier transform technique. We consider two laterally unbounded superposed fluids with the fluid domain in the form of a long cylinder extending in the lateral direction, in which upper fluid is bounded above by a rigid lid and the lower one bounded by a bottom surface which has small deformation. The free surface,i.e., the surface above the upper layer, has been replaced and approximated by a rigid lid rendering the flow to a channel flow. In such a situation, time-harmonic propagating waves can exist at only one wave number for any given frequency. Applying perturbation analysis involving a small parameter 1,δ? which measures the smallness of the undulation, we reduce the original problem to a simpler boundary value problem (BVP) for the first order correction of the potentials. The solution of this BVP is then obtained by applying Fourier transform technique, thereby obtaining the velocity potentials. The reflection and transmission coefficients are evaluated approximately up to the first order ofδin terms of integrals involving the shape function when a train of progressive waves propagating from negative infinity is obliquely incident on the channel-bed having small deformations. We present two different special forms of bottom deformation, namely, an exponentially damped deformation and a patch of sinusoidal ripples having the same wave number.

2 Mathematical formulation of the problem

We consider the irrotational motion of a two-layer inviscid incompressible fluid flow under a rigid infinite lid through a channel which is bounded by a bottom surface with small deformation. A right-handed Cartesian coordinate system is used in whichxz-plane coincides with the undisturbed surface between the two fluids. They-axis points vertically downwards withy=0 as the interface andy=-has the position of the rigid lid. Here, the bottom of the lower layer with small deformation is described byy=H+δc(x), wherec(x) is a bounded and continuous function describing the undulation of the channel-bed,Hthe uniform finite depth of the lower layer fluid far to either side of the deformation of the bottom so thatc(x) → 0asand the non-dimensional numbera measure of smallness of the deformation (Fig. 1). The rigid infinite lid aty=-hcan be considered to be an approximation to the free surface. Under the assumptions of linear water wave theory, the velocity potentials in the lower fluid of densityρ1and in the upper fluid of densityρ2＜ρ1can, respectively, be written for oblique waves asandwhere Re stands for the real part,υthe component of the incident field wave number alongz-axis andωthe angular frequency of the incoming waves.

The governing equation for the boundary value problems involving these potentialsφandψis the modified Helmholtz equation:

in the respective regions occupied by the fluids, whereis the two-dimensional Laplacian operator. The linearized boundary conditions at the bottom of the channel, on the interface and at the lid are

withρdenotes the ratio of the densities of the two fluids（＜ 1）,K=,gthe acceleration due to gravity and ? ?nthe derivative normal to the bottom at a point（x,y）. The time dependence of e-iωthas been suppressed. The boundary conditions (4) and (5) represent the continuity of the normal velocity and the pressure, respectively, at the interface.

Fig. 1 Domain definition sketch

Within this framework in a two-layer fluid, a train of progressive interface-waves, which takes the form (up to an arbitrary multiplicative constant)φ（x,y）eiυzandψ（x,y）eiυzin the lower and upper layers fluids respectively, where

are obliquely incident upon the bottom deformation from negative infinity. Here sinkυθ= , withθas the angle of oblique incidence of progressive interface waves andksatisfying the dispersion relation Δ（k）=0 , where

In the above equation, there is a positive real rootm, which indicates the propagating modes of the fluid at the interface and a countable infinity of purely imaginary rootsikn,n= 1,2,..., that relate to a set of evanescent modes, whereknare real and positive satisfying

The intensity of evanescent mode of waves decays exponentially with distance from the interface at which they are formed. Due to this evanescent mode of waves appearing in the fluid region, a part of the incident interfacial wave becomes trapped and leads to a standing wave pattern over the bottom irregularities, when the incident wave is scattered by the bottom undulation. This phenomenon is calledAnderson localization. More precisely, this implies that a periodic plane wave of finite wavelength coming on to the part of the channel with a random bottom will eventually be totally reflected, i.e., the amplitude of the disturbance created by the wave will die off exponentially with distance, with a typical length which is called the localization length. But these waves do not affect the asymptotic behavior of the resultant reflected and transmitted waves. In that sense, any sort of localization is not considered while formulating and solving the present problem.

The negatives of all roots of (10) are also roots, being wave numbers of the waves traveling in the opposite direction. As equation (10) has exactly one nonzero positive simple zero atkm= , say, on the real axis ofk, so only one mode of waves can exist at the interface and the wave can propagate in either direction. Note that if 0m= , then there is no wave in the respective regions.

A train of progressive waves of modemis of the form

where we must have inmυ＜ , in order that these progressive waves exist.

An incident plane wave of modemmaking an angleθ, 0 π 2θ≤ ＜ , with the positivex-axis is of the form:

where υ=msinθ.

Since the wave train, given byφ0（x,y）eiυzandψ0（x,y）eiυz, is partially reflected by and partially transmitted over the bottom deformation, the far-field behaviors ofφandψare given by

whereRandT, respectively, represent the reflection and transmission coefficients due to an oblique incident wave, defined to be the ratio of amplitudes of the reflected and transmitted waves, respectively, to that of the incident wave and are to be determined.

Assuming, for small bottom deformation,δto be very small and neglecting the second order terms, the boundary condition 0nφ? ? = on the bottom surfacey=H+δc(x) can be converted to the following appropriate form

The lower layery=H+δc（x）,- ∞ ＜x＜ ∞ , reduces to the uniform strip 0 ＜y＜H,- ∞ ＜x＜ ∞ in the following mathematical analysis where a perturbation technique is used.

3 Method of solution

3.1 Perturbation technique

Let us consider a train of progressive interface waves to be obliquely incident at an angleθ, 0 ≤θ＜ π 2 on the bottom deformation. If there is no bottom deformation, then the incident wave train will propagate without any hindrance and there will be only transmission. This, along with the appropriate form of the boundary condition (18), suggests thatφ,ψ,RandTcan be expressed in terms of the small parameterδas

where00,φ ψare given by equations (14) and (15) respectively.

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 parameterδ. Also this theory is valid only for infinitesimal reflection and away from resonance. For large reflection, the perturbation series, as defined in (19), needs to be refined so that it can deal with the resonant case. Because of the fact that the obstacles in the form of undulations are small, our work here does not concern large reflection and hence we will not take into account resonance while deriving the results. Using equations (19) in equations (1), (2), (18), (4), (5), (6), (16), (17) and equating the first order terms ofδin both sides of the equations, we find that the first order potentials1φand1ψsatisfy a coupled boundary value problem described by

3.2 Formation of the BVPs

Now the above BVP for1φand1ψ, described by equations (20) to (27), can be decomposed into two independent BVPs for1φand1ψas follows:

BVP-I, corresponding to1φ, is

where （）xηis assumed to be known on 0y= and1φhas the asymptotic behavior

BVP-II, corresponding to1ψ, is

Note that1ψhas the asymptotic behavior

Now using equations (29) and (33), equation (24) can be written in the following form:

3.3 Fourier transform technique

To solve BVP-I and BVP-II, we now assume the potentialsφ1andψ1are such that their Fourier transforms with respect tox, denoted byand, respectively, exist and are defined by

Applying Fourier transform to equations (28), (29) and (30), we get the following boundary value problem forφ1（ξ,y）:

whereα2=ξ2+υ2andare the Fourier transform ofη（x） andp（x） respectively. The solutionof the above boundary value problem is obtained as

Similarly, applying Fourier transform to equations (32), (33) and (34), we get the following boundary value problem for

The solution（ξ,y）of the above boundary value problem is obtained as

Now substituting equations (42) and (46) in equation (47), we obtain the value of（ξ） as

where

By Eq. (10), here Δ （α） has only one non-zero positive rootmon the real axis ofα. Substituting the value of（ξ） in equations (42) and (46), and taking the inverse Fourier transform, the solution for the velocity potentialsφ1（x,y）andψ1（x,y） are, respectively, obtained as follows:

SinceΔ(α) has one non-zero zero atα=m, correspondingly atξ1=, on the positive real axis ofξso the above integral on both cases contains a pole atξ1. Therefore we make the path for each integral in both cases indented below the pole atξ1.

4 Reflection and transmission coefficients

The first-order reflection and transmission coefficientsR1andT1, respectively, due to obliquely incident wave of modem(hereυ=msinθ), are now obtained by lettingξ→in equation (50) or (51) and comparing with equation (31) or (35) . To calculate the first order reflection coefficient, we makeξ→ -∞ in either (50) or (51). Asξ→ -∞ , the behavior ofφ1（x,y） orψ1（x,y） can be obtained by rotating the path of the integral, involving the term） into a contour in the first quadrant, so that we must include the residue term at the poleξ1. The path of the integral involving the term） in equation (50) or (51) is rotated into a contour in the fourth quadrant so that the integral involving the termdoes not contribute asξ→ -∞ . Then comparing the resultant integral value with the equation (31) or (35), we obtain the value ofR1as

where Δ′ （m）cosθis the value ofat the poleξ=ξ1.

Similarly to find the first order transmission coefficient, we letξ→ ∞ in either (50) or (51). Asξ→ ∞ , the behavior ofφ1（x,y） orψ1（x,y） can be obtained by rotating the path of the integral involving the terminto a contour in the first quadrant so that we must include the residue term at the poleξ1. The path of the integral involving the term） in equation (50) or (51) is rotated into a contour in the fourth quadrant, so that the integral involving the term）, does not contribute asξ→ ∞ . Then comparing the resultant integral value with the equation (31) or (35), we obtain the value ofT1as

Therefore, the first order reflection and transmission coefficients can be evaluated from equations (52) and (53), respectively, once the shape functionc(x) is known.

Here, if we take 0θ= (i.e., the case of normal incidence), then the above results (52) and (53) coincide with the corresponding results in Mohapatra and Bora (2009, 2010).

In the following section we proceed to examine the effects of reflection and transmission for some special forms of the shape functionc(x).

5 Special forms of the bottom surfaces

Here, we consider different special forms of shape functionc(x) for the uneven bottom surface. As mentioned earlier, these functional forms of the bottom disturbance closely resemble some naturally occurring obstacles formed at the bottom due to sedimentation and ripple growth of sands.

5.1 Example-1

Consider the following shape function:

This shape functionc(x) corresponds to an exponentially damped deformation on the bottom surface. In this example, the top of the elevation lies at the point（0,a0）, and on either side it decreases exponentially. In order to calculate the reflection coefficientR1, substituting the value ofc(x) from equation (54) into equation (52), we obtain

In a similar way, we can calculate the transmission coefficient1Tby substituting the value ofc(x) from equation (54) into equation (53):

It is known 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 application point of view, we place significant importance on them and subsequently the example is considered.

5.2 Example-2

We consider a special form of the shape functionc(x) in the form of a patch of sinusoidal bottom ripples on an otherwise flat bottom:

For continuity of the bed elevation one can take

wheren1andn2are positive integers andδ′ is a constant phase angle. This patch of sinusoidal ripples on the bottom surface with amplitudeaconsists of （n1+n2）2 ripples having the same wave numberl. For this case, the reflection and transmission coefficientsR1andT1, respectively, are obtained as

For the special case when there is an integer number of ripples wavelengths in the patchL1≤x≤L2such thatn1=n2=nandδ′= 0, we find the reflection and transmission coefficients, respectively, as

whereβ= 2mcosθl. Equation (60) illustrates that for a given number ofnripples, the first order reflection coefficientR1is an oscillatory function ofβwhich is the ratio of twice the component of the interface wave number alongx-axis and the ripple wave number. Furthermore, when the bed wave number is approximately twice the component of the interface wave number alongx-axis, that is2mcosθl≈ 1, the theory predicts a resonant interaction between the bed and interface. Hence, we find from equation (60) that near resonance the limiting value of the reflection coefficient assumes the value

Note that when 2 cosmlθapproaches one andnbecomes large, the reflection coefficient becomes unbounded contrary to our assumption thatR1is 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 coefficientR1, in this case, becomes a constant multiple ofn, the number of ripples in the patch. Hence, the reflection coefficientR1increases linearly withn. Although the theory breaks down when 1β= , that is 2 cosmlθ= , 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 1β= .

As this is a non-dissipative system and since10T= andR1may be large, it is likely to witness a violation in the conservation of energy in the solution for the potentials. Actually the solution is required to satisfy a condition with respect to wave energy flux,i.e., the incident component of wave energy flux on the undulating part is to be balancedapproximately by the sum of the reflected and transmitted components. This requirement may not be fulfilled by the expressions for the reflection and transmission coefficients derived here. Here the reason for the imbalance is that the linearized analysis does not permit any attenuation of the incident interface waves as it travels over the regionL1≤x≤L2, which causes the predicted reflected wave in the perturbation solution to be overestimated and the transmitted wave to be very small or zero. In practice, if the reflection wave is non-zero, there must be a progressive attenuation of the incident interface waves over the regionL1≤x≤L2.

6 Numerical results and discussions

In this section, the numerical computation, related to the two special forms of bottom profiles mentioned in the previous section: the exponentially damped undulation and the sinusoidal ripple bed, is shown for the first order reflection and transmission coefficients.

In Figs. 2-8, the non-dimensionalized first order reflection and transmission coefficients are shown for the case of a wave train of wave number (an interfacial mode) obliquely incident at an angleθto the positivex-axis on the bottom undulation. In Figs. 2 and 3, which correspond to the exponentially damped undulation in Example-1, the numerical results for the non-dimensionalized first-order reflection coefficientand transmission coefficientdue to an obliquely incident interface wave of wave numberma0for different heights of the hump, calculated from equations (55) and (56) respectively, are plotted againstKa0. In this case we consider angle of incidenceθas π 6, depth of the lower layerHas05a, depth of the upper layerhas02aand density ratioρa(bǔ)s 0.5. For a two-layer fluid consisting of fresh water and salt water, the value ofρwould ideally be around 0.97. The same qualitative features are observed for such a density ratio, but the effects of the interface are not observed clearly. Therefore, for clear observation of the features, the density ratio of the fluids is chosen as 0.5. From the figures, it is evident that the values of reflection and transmission coefficients decrease when the height of the hump of the bottom undulation increases. This means that when an obliquely incident water wave propagates over a small bottom undulation in a two-layer fluid, a substantial amount of reflected and transmitted energy can be produced. Moreover, the increasing rate of the reflected and transmitted energy is higher for this case. Again, it is also observed that the reflection coefficientis comparatively much smaller than the transmission coefficient. It happens due to the small undulation at the bottom surface. Since the uneven bed (like a hump) decreases exponentially on either side, so both the first order reflected and transmitted waves exist, and propagate only up to some wave numbers.

Fig. 2 Reflection coefficientplotted againstKa0forθ=π/6

Fig. 3 Transmission coefficientplotted againstKa0forθ=π/6

In Example-2, a patch of sinusoidal bottom undulations on the channel bed is considered because of its considerable physical significance in the ability of an undulating bed to reflect incident wave energy which is important for a channel flow consisting of a two-layer fluid. We consider the numerical computations for the non-dimensionalized first order reflection coefficient, which is calculated from equation (60), due to an oblique incident wave at an angleθon the undulating bed of ripple wave numberlahavingnnumber of ripples in the patch. In Figs. 4-8, different curves ofare shown againstKaforH=5aandh=2a. Figs. 4 and 5, respectively, show the first order reflection coefficient due to an obliquely incident wave of wave numbermafor four different sets of angles of incidence and a particular angleθ= π 4, whilenis fixed at 3, density ratioρis at 0.5 and ripple wave numberlaat 0.52 for these figures.

Fig.4 Reflection coefficientplotted againstKaforla=0.52 andn=3

Fig.5 Reflection coefficientplotted againstKaforla=0.52,n=3 andθ=π/4

It may be noted from Fig. 4 that forθ= π 6 (the case of an oblique incidence), the maximum value ofis 0.067736, attained atmacosθ= 0.2536, correspondingly atKa= 0.071, that is, when the ripple wave numberlaof the bottom undulation becomes approximately twice as large as the interface wave numbermacosθ. The same can be observed when the value ofθare 0, π 10 and π 5. Another common feature in Fig. 4 is the oscillating nature of the absolute values of the first order coefficients as functions of the wave numberKa. For the value ofθ= π 4, the reflection coefficientin Fig. 5 is much less (almost negligible) compared to the other angles of oblique incidence,e.g.,θ= π 10,π 6, π 5. As the angle of incidenceθincreases, the peak value ofdecreases. For the case of normal incidence, the peak value ofis the largest.

In Fig. 6, different curves correspond to different number of ripplesn=2, 3, 4, 5 in the patch of the undulation. In this figure, for all curves, we considerθ= π 6,ρ= 0.5,la= 0.52. The curve which corresponds ton=2, the maximum value ofis 0.045816, attained atmacosθ= 0.241944 (whenKa=0.066). Similarly for the curve corresponding ton=3, the maximum value ofis 0.067736, attained atmacosθ= 0.2536, correspondingly atKa=0.071, for the curve corresponding ton=4, the maximum value ofis 0.089684 attained atmacosθ= 0.2536 correspondingly atKa=0.071, and for the curve corresponding ton=5, the maximum value ofis 0.111096, attained atmacosθ= 0.2536 (whenKa=0.071). From Fig. 6, it is clear from each curve that when the values ofnincreases, the peak value ofis attained atmacosθ= 0.2536, that is, when the ripple wave numberlaof the bottom undulation becomes approximately twice as large as the interface wave numbermacosθ. Asnincreases, the value ofmacosθconverges to a number in the neighborhood of 0.26,i.e., forla/2, whereattains its maximum, and also the peak value of non-dimensionalized reflection coefficientincreases. Its oscillatory nature againstKais more noticeable with the number of zeros ofincreased but the general feature ofremains the same.

Fig.6 Reflection coefficientplotted againstKaforθ=π/6 andla=0.52

In Fig. 7, different curves correspond to different density ratiosρ= 0.1,0.4,0.5,0.6,0.9 in a two-layer fluid flow region. In this figure, for all curves, we considern=5,θ=π/6 andla=0.52. For the curve corresponding toρ=0.1 (i.e., in the case of density of the lower fluid being much heavier than the upper fluid), the maximum value ofis 0.196721, attained atmacosθ= 0.256962, correspondingly atKa=0.206. Similarly, for the curve corresponding toρ=0.4, the maximum value ofis 0.125127, attained atmacosθ= 0.259241, correspondingly atKa=0.097. It has been cleared from these two curves that the peak values of the reflection coefficient are attained at different values ofKa(also the same observation can be made from the other curves in Fig. 7). This is due to the effect of density ratios in the dispersion relation, which gives the wave number for the fluid region. Since the wave number will be different for the change in density ratios, therefore the peak values of the reflection coefficient will be attained at different values ofmacosθ, correspondingly at different value ofKa. Here it is observed that as the density ratioρincreases, the peakvalues ofdecreases so that the first order reflection coefficient is quite sensitive to the density ratio due to an obliquely incident wave at modemapropagating at an angleθto the positivex-axis on the bottom undulation. When the density ratio approaches to one, the first order reflection coefficient becomes smaller than those for the low density ratios.

Fig.7 Reflection coefficientR1 plotted againstKaforn=5 andla=0.52

Fig.8 Reflection coefficient plotted againstKaforρ=0.5,n=5 andθ=π/6

In Fig. 8, different curves correspond to different ripple wave numbersla=0.52, 0.7, 0.8 and 1, in the patch of the undulation. In this figure, for all curves, we considerρ=0.5,n=5 andθ=π/6. Here also, it is clear from this figure that the peak values of the reflection coefficient are attained at different values ofKa. The reason is that the values of the reflection coefficient(which is calculated from equation (60)) become maximum only whenla≈2macosθ(i.e.,β≈ 1). It is observed from this figure that as the ripple wave numbers increase, the reflection coefficientbecomes smaller than those for the larger ripple wave numbers. This means that when an obliquely incident water waves propagates over a small bottom undulation in a two-layer fluid, a substantial amount of reflected energy can be produced.

It is observed from Figs. 4-8 that in each figure and each curve, there will a peak value of reflection coefficient which either increases or decreases with the change of different physical quantity (such as density ratios, angle of incidence waves, number of ripples, and ripple wave numbers). This mainly happens due to the presence of term (1-β2) in the denominator of equation(60). When the value ofβis approximately to one （2mcosθl≈ 1）, that is, when the bed wave number is approximately twice the component of the interface wave number alongx-axis, 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 atβ=1. The entire theory will break down whenβ=1. In such a situation, the reflection coefficient becomes unbounded and it predicts a resonant interaction between the bed and the interface.

7 Conclusions

In the present paper, we revisited the work of Mohapatra and Bora (2012) of oblique water wave scattering by a small bottom undulation in a two-layer fluid flowing through a channel and solve it by using Fourier transform technique. The work described in this paper is the classical problem of oblique water wave scattering by a small bottom undulation, where the upper layer is of finite height and is bounded above by a rigid horizontal lid, which replaces the free surface. In such a situation propagating waves can exist at only one wave number for any given frequency. By developing a suitable perturbation technique, the problem reduces up to first order to a coupled boundary value problem. The determination of the first order potentials and hence of the reflection and transmission coefficients becomes easier when Fourier transform is employed. The converted BVPs are adequately solved with the help of Fourier transform and its inverse. First order approximations to the reflection and transmission coefficients are obtained in terms of computable integrals. Two special examples of bottom deformation are considered to validate the results. The main result that follows is that the resonant interaction between the bed and the interface attains in the neighborhood of the singularity when the ripple wave numbers of the bottom undulation become twice the interface wave number. This point of singularity varies with density ratios of a two-layer fluid and the ripple wave numbers on the bottom surface. Moreover, when an obliquely incident water wave propagates over a small bottom undulation in a two-layer fluid flowing through a channel, the values of the reflection coefficients become higher so that the amplitude of the generated wave increases. Another main advantage of this method, demonstrated through the example of a patch of sinusoidal ripples, is that a very few ripples may be needed to produce a substantial amount of reflected energy. Computations show that when the angle of incidence becomes smaller, the reflected and transmitted waves almost coincide with the corresponding waves for the case of normal incidence (Mohapatra and Bora(2009), (2010)). Also it is observed that for small angles of incidence, the reflected energy is more as compared to other angles of incidence up to π/4. The solution developed here is expected to be helpful in considering two-layer fluid problems in a channel with uneven bottom surface.

Acknowledgement

The author thanks Professor Swaroop Nandan Bora, Indian Institute of Technology Guwahati, India for his valuable suggestions and comments during the preparation of the manuscript.

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Author biography

Smrutiranjan Mohapatrareceived his Ph.D. from Indian Institute of Technology Guwahati, Guwahati, India in 2009. He also worked as a post doctoral fellow at Indian Institute of Science, Bangalore, India prior to joining his present position of Lecturer in the Department of Mathematics, Veer Surendra Sai University of Technology, Burla, India. His main areas of interest are water wave scattering and two-layer fluid. He has about 12 research publications to his credit and is involved in a number of sponsored projects.

1671-9433(2014)03-0255-10

Received date: 2014-01-29.

Accepted date: 2014-04-22.

Foundation item: This work is partially supported by a research grant from Department of Science and Technology (DST), India (No. SB/FTP/MS-003 /2013).

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

? Harbin Engineering University and Springer-Verlag Berlin Heidelberg 2014