Abstract
A direct function theoretic method is employed to solve certain weakly singular integral equations arising in the study of scattering of surface water waves by vertical barriers with gaps. Such integral equations possess logarithmically singular kernel, and a direct function theoretic method is shown to produce their solutions involving singular integrals of similar types instead of the stronger Cauchy-type singular integrals used by previous workers. Two specific ranges of integration are examined in detail, which involve the following: Case(i) two disjoint finite intervals and () and Case(ii) a finite union of disjoint intervals. The connection of such integral equations for Case(i), with a particular water wave scattering problem, is explained clearly, and the important quantities of practical interest (the reflection and transmission coefficients) are determined numerically by using the solution of the associated weakly singular integral equation.
1. Introduction
The singular integral equation where is a single or a finite union of finite intervals is encountered in many branches of mathematical physics (cf. Lewin [1], Ursell [2], and Mandal and Kundu [3]).This type of integral equation is usually solved in the literature by converting the weak singularity (logarithmic type) in the kernel to a Cauchy-type singularity which is strong in nature. For an integral equation with weakly singular kernel, the associated integral is defined in the standard Riemann sense, while for an integral equation with strongly singular kernel, the singular integral has to be defined appropriately so as to give a consistent mathematical sense. Consequently, the integrals with weak singularities are more amenable to the numerical techniques as compared to the integrals with strong singularity which requires artificial methods for its numerical evaluation. Noting this difficulty with the integrals with strong singularities, Chakrabarti and Manam [4], Manam [5], Banerjea and Dutta [6], and Banerjea and Sarkar [7] used function theoretic method to solve (1.1) for various without converting it to Cauchy-type integral equation. However, the solution function in [4–7] involves Cauchy-type singularity.
Recently, Chakrabarti [8] has developed a direct function theoretic method for (1.1) with to obtain its solution. The novelty of the method is that it avoids the occurrence of strong singularity such as Cauchy-type singularity in the solution function of the integral equation.
The present paper is a generalization of the method in [8] with a little modification. We have applied the method of [8] to the integral equation (1.1), where consists of (i) two disjoint finite intervals and (ii) a finite union of disjoint intervals . As an application, we have applied the solution of (1.1) where consists of two disjoint finite intervals to the problem of water wave scattering by a partially immersed vertical barrier with a gap present in deep water. The important quantities of physical interest, that is, the reflection and transmission coefficients, are evaluated. The numerical values of the reflection coefficient are presented graphically for various values of the wave number and various lengths of the barrier. It is observed that for certain small range of small values of wave number, there occurs almost total reflection. This phenomenon was also observed by Banerjea [9]. It is also observed that this phenomenon of almost total reflection of waves decreases when the length of lower part of the barrier increases and also when the length of the gap increases. We observe that by letting , the known results available in the literature are recovered. It is emphasized here that for numerical evaluation of the reflection and transmission coefficients, computation of singular integrals with strong singularities is avoided, and this is the main advantage in the present method over other methods existing in the literature.
In Section 2, we have presented the detailed analysis to determine the solution of the integral equation (1.1) in cases (i) and (ii) mentioned in the abstract. Then, in Section 3, we have shown the connection of (1.1) with a water wave scattering problem.
2. Solution of the Integral Equation (1.1)
2.1. Case(i)
Consider that
In this section, we present a method of solution of
with and being , , or .
We introduce a sectionally analytic function in the complex plane, cut along the segment on the real axis as given by If we use the following results, valid for the limiting values of : we find the following limiting values of : Equations (2.6) give rise to the following Plemelj-type alternative formulae: where where .
By using the relation (2.9), the singular integral equation (2.2) can be expressed in the form of a Riemann-Hilbert problem as given by where Again noting the relation (2.8), we can write Hence,
Using the relation (2.13) in (2.4), we get the solution of special Riemann-Hilbert problem (2.7) as
Utilizing the above idea, we can solve the Riemann-Hilbert problem (2.10) in the following manner.
Let where is the solution of the corresponding homogeneous problem (2.10), that is, Thus, (2.10) becomes This is a Riemann-Hilbert problem which is similar to (2.7). Noting (2.7), (2.12), and (2.14), the solution of this Riemann-Hilbert problem is given by where is an entire function.
We can choose with where
Knowing from (2.18), (2.19), and (2.20), we can find the limiting values of using the formula (2.5). Substituting in (2.7) and noting the relation (2.8), the solution of the integral equation (2.2) can be obtained as where and , , , , and are arbitrary constants.
It is observed here that the original integral equation (2.2) can be solved by the aid of formula (2.22) if and only if the following consistency conditions are satisfied: We can determine the arbitrary constants , , , , and by using the relation (2.24). Thus, can be determined completely by utilizing the relation (2.22). It is observed that the solution as obtained by the relations (2.22) and (2.23) of (2.2) neither involves any strong singularity nor any arbitrary constant.
2.2. Case(ii)
Consider that In this section, we proceed to solve the integral equation where consists of finite disjoint intervals and as , being , , .
In this case, we introduce the following sectionally analytic function cut along .
Using the formula (2.5), we get following limiting values of : This gives the following Plemelj-type alternative formulae for the integral equation (2.26): where Hence, from the relation (2.30), we get Thus, noting the relation (2.32), the equation (2.27) becomes Utilizing the above idea, we can solve the Riemann-Hilbert problem (2.31) in the following form: where is an entire function, and we can choose , the solution of the homogeneous equation of (2.31) as with where
Using (2.35) in (2.34), is known. Noting the relation (2.5), one can find the limiting values of . Using the limiting values , we obtain, from (2.29) and (2.30), the solution of the integral equation (2.26) as where We observe that the original integral equation (2.26) can be solved by using the above relations if and only if the following consistency conditions are satisfied: From these conditions, we find the arbitrary constant , and from the order condition, the arbitrary constant can be determined, and finally the solution of the singular integral equation (2.26) can be obtained from (2.38).
In the next section, we give the genesis of the integral equation (2.2) with and use its solution to the problem of scattering of water waves by a partially immersed vertical barrier with gap.
3. Genesis of the Integral Equation (2.2) Connection with a Water Wave Problem
3.1. The Detailed Analysis and the Analytical Solution
In this section, we show the connection of integral equation (2.2) with the problem of scattering of surface water waves by a thin partially immersed vertical barrier with a gap in deep water occupying the position , , and , -axis being directed downwards and -axis along the mean free surface. We assume the motion to be two dimensional, irrotational, and time harmonic. The velocity potential is represented by , being circular frequency. Here, satisfies the following boundary value problem: where , being the acceleration of gravity, where and are reflection and transmission coefficients. A suitable representation of satisfying the equations (3.1), (3.4), and (3.5) is given by where, By Havelock's expansion theorem, Then, using the condition (3.2) along with the condition of continuity of across the gap , along , we arrive at the following multiple integral equations: The Relation (3.9) can be recast in the form where and are two arbitrary constants. Then, applying to the equations (3.10) and (3.11), we obtain where It is observed from the relation (3.13) that we must choose for consistency. Setting where is an unknown function and using sine inversion formula, to the relations (3.12) and (3.15), we obtain from (3.13) with The solution of the integral equation (3.16) is given by the equations (2.22), (2.23), and (2.24) with given by relation (3.17).
Thus, knowing , we can obtain by using Fourier Sine inversion formulae to the relations (3.12) and (3.15). Hence, can be obtained from equation (3.14). Now, contains unknown constants , , and . To obtain the unknowns , , , and , we substitute in (3.10) and (3.11) to get the following four equations: where and , , , , and are arbitrary constants and Thus, the relations (3.18) can be solved to determine four unknowns , , , and . This completes the description of the analytical solution of the water wave problem.
3.2. Numerical Results
The reflection coefficient has been numerically evaluated for various values of wave number for four cases, namely, , ; , ; , ; , .
Numerical computation of involves the solution of four equations (3.18). It may be noted from the relation (3.19) that for the solution of (3.18), one does not need to compute any integral with strong singularity. The integrals occurring in (3.18) are defined in Riemann sense. This is the main advantage in the present method as compared to the other methods based on strongly singular integral equation formulation available in the literature. Since there is no integral with strong singularity, so for numerical computation, we have used Mathematica 5.2 to evaluate various integrals in (3.18) and (3.19) and to solve the system of (3.18).
In Table 1, we have presented the values of reflection coefficient computed from our result when and and , the reflection coefficients corresponding to the problem of scattering by a submerged vertical barrier , as given in the literature (cf. Banerjea and Dutta [6]) for various values of wave number . It is observed that the values reflection coefficient coincides with upto two or three places of decimal. This shows that in limiting case, the reflection coefficient computed from the present paper agrees with the exact value of as given in [6].
In Figures 1, 2(a), 2(b), and 3 (f), is depicted against for various values of and . In each case, we have verified that for all values of .
(a)
(b)
In Figure 1, we have plotted against various values of when and . For , it is observed that rises steeply, becomes almost a unity, and then falls rapidly forming a sharp peak. This indicates that for this range of values of , there occurs almost total reflection of wave energy. Also for , there occurs a small oscillation in . In Figures 2(a) and 2(b), is depicted against for and . In Figure 2(a), it is observed that for very small values of , that is, for , increases at first and then decreases almost to zero indicating that most of the wave energy is transmitted. If the values of are further increased, that is, , it is observed from Figure 2(b) that increases. For , sharply increases almost to unity and then decreases sharply. The peak value in this case is lower than the peak value when as observed in Figure 1. Thus, when the length of the lower part of the barrier increases, the peak value in decreases. In Figure 3, is plotted against for and . In this figure, it is observed that when the length of the gap is increased keeping the length of the lower part of the barrier the same, the sharp peak in the values of occurs for . In this case the peak value is much lower than the peak value observed in Figures 1 and 2(b). Thus, an increase in the length of the gap decreases the peak value in . Thus, it is observed from Figures 1, 2, and 3 that for certain range of values of , there occurs a sharp peak in showing that reaches a maximum value. In these three cases, the maximum value is almost unity, that is, the incident wave experiences almost total reflection for certain range of small values of . This type of behaviour of is also observed by Banerjea in [9].
4. Conclusion
A direct function theoretic method is applied here to solve the integral equations (2.2) and (2.26) without converting them to the integral equation with strong singularity. It may be noted here that the solution functions given in (2.22) and (2.38) do not involve any strong singularity. Also using the presently detailed solution of (2.2), the problem of scattering of water waves by a partially immersed barrier with a gap is examined fully.
Acknowledgments
This work is also partially supported by DST through PURSE scheme (no. SR/S9/Z-23/2008/5) at Jadavpur University. A. Chakrabarti is thankful to NASI, Allahabad for offering Senior Scientist Platinum Jubilee Fellowship which helped him in pursuing this research.