Panda, Department of Mechanical Engineering, IIT Guwahati for his help in some of the computer works during my research work. I am indebted to the former Heads of the Department of Mathematics, IIT Guwahati, Dr.
Preamble
In the second category, it is assumed that the wavelength is much smaller than the depth of the water, so that the effect of the disturbance gradually decreases as one moves away from the free surface. Water waves (the terms surface waves and gravity waves are also used) are usually generated by the force of gravity in the presence of a free surface along which the pressure is constant.
Brief history and motivation
Evans [24] constructed a new source potential in the linearized theory of water waves and used this source potential to reduce the problem of the reflection and transmission waves through a rack of arbitrary profile to an integral equation. They applied an extended gentle slope equation for surface waves (Porter and Staziker [78]) to the inclined region in front of the porous absorber.
Basic equations in linearised theory of water waves
In the next section, a brief description of the basic equations of linearized theory of water waves for the case of uniform finite depth is presented in consultation with the classic treatise of Lamb [44], Stoker [88], Lighthill [45], Crapper [ 8] and Dean and Dalrymple [21]. Equations and (1.22) are also the basic equations of the linearized theory of water waves for time-harmonic irrotational motion in the liquid.
Outline of the thesis
The solution of the boundary value problem for the first-order correction is achieved by using the same three techniques as used in Chapter 3. The original problem is then reduced to a simpler boundary value problem for the first-order correction of the velocity potential.
Normal incidence
In this chapter, the statement and formulation for the problem of surface wave propagation with small final ripple in a laterally unbounded sea is presented, using linear water wave theory, for both normal and oblique incidence. While setting up the boundary value problem here, some of the assumptions and boundary conditions are based on idealized conditions.
Oblique incidence
The boundary condition (2.7) and the fact that a wave train propagating in an ocean of uniform finite depth experiences no reflection together suggest that φ, R and T introduced above can be expressed in terms of the small parameter ε as . Assuming very small and neglect O(ε2) terms, the boundary condition ∂ψ/∂n= 0 on y=h+εc(x) can be expressed in an appropriate form as. 2.19) Now, in view of the geometry of the problem, ie. due to the uniformity in the z−direction, ψ(x, y, z) can be written as. The form of the approximate boundary condition (2.23) and the fact that a wave train propagating in an ocean of uniform finite depth experiences no reflection together suggest that φ, Rand T introduced above can be expressed in terms of the small parameter ε as.
Solution by Green’s function technique
Solution procedure
Reflection and transmission coefficients
Equations (3.12) and (3.15) are equivalent to the equations of Miles [68] for the case of normal incidence and of Mandal and Basu [53] for the case of normal incidence without surface tension. The reflection and transmission coefficients can be evaluated from equations (3.12) and (3.15) when the shape function c(x) is known.
Solution by Fourier transform technique
Solution procedure
To solve the boundary value problem (equations we now assume that φ1 is such that the Fourier transform of φ1 with respect to tox, denoted by φ1, exists and is given by. We observe that such a Fourier transform exists if we make an artificial assumption that K possesses a small imaginary part, as given by iµ0σ/g, where µ0 > 0 is too small to be taken as zero (in the eliminating sense) at the end of the analysis. We note that equation (3.26) also has the property certain (lying on the ξ axis) except ξ= 0.
Results
The first term on the right-hand side of each of equations (3.34) and (3.35) represents the non-propagating modes which decay rapidly away from the waveform, and the second term represents a propagating mode from the region of the bearing disturbance. Comparing equations (3.34) and (3.35) with equation (2.12), the reflection and transmission coefficients can respectively be written as
Solution by finite cosine transform technique
Solution procedure
Now the reduced boundary value problem described by equations can be solved by introducing finite cosine transform with respect to toys, which is more appropriate than Fourier transform which involves contour integration and related results with respect to the path of integration which is cumbersome, as described in Davies and Heathershaw [19] and in the previous section.
Reflection and transmission coefficients
Conclusion
In Chapter 5, we will consider various examples of the shape function to evaluate these coefficients numerically. Also, the use of this technique to obtain the first-order potential and thus the reflection and transmission coefficients greatly reduces the workload. By these methods, the solution of the first-order correction to the potential involved in the reduced boundary value problem is determined, from which the quantities of physical interest, namely reflection and transmission coefficients, are evaluated up to first order of the small parameter ε in terms of integrals involving the shape function c(x).
Solution by Green’s function technique
Solution procedure
Evaluating the integral equation (4.7) over the contour following the same procedure as described in section 3.2, the results of the integral equation (4.7) will give rise to the determination of φ1 as:.
Reflection and transmission coefficients
Solution by Fourier transform technique
Solution procedure
4.23) By taking inverse Fourier transform, the solution for the velocity potential can be written in the form 4.24) We now obtain the final result of (4.24) by contour integration using the residue theorem as done before. Here we also see that the equation (4.24) has certain singularities (which lie on the ξ-axis) other than ξb= 0. Substituting Kb (as defined in Section 3.3) into equation (4.24), the singularities of ( 4.24) is displaced from the ξ-axis to the upper and lower half planes.
Results
Solution by finite cosine transform technique
Solution procedure
Reflection and transmission coefficients
Conclusion
In Chapter 5 we will consider various examples of the shape function to determine these coefficients numerically. In the last two chapters we investigated the problem of water wave propagation by small waves on an otherwise flat seabed and obtained the analytical expressions for the reflection and transmission coefficients in the form of integrals involving the shape function c(x). In this chapter we look at various examples of the shape function to evaluate these coefficients.
Examples
- Example 1
- Example 2
- Example 3
- Example 4
- Example 5
- Example 6
Although the theory breaks down where the solution is singular (l = 2k0), a large amount of reflection of the incident wave energy by the bedforms is predicted in the vicinity of this singularity. Although the theory breaks down where the solution is singular (l= 2µ), a large amount of reflection of the incident wave energy by the bedforms is predicted in the vicinity of this singularity. Although the theory breaks down where the solution is singular (l1 = 2µ and l2 = 2µ), a large amount of reflection of the incident wave energy is provided by the bedforms in the vicinity of these singularities.
Conclusion
The value of the parameter is chosen such that all conditions of the boundary value problem up to and including the first order are fully satisfied. A patch of sinusoidal undulations is considered as a sample of the bed surface to evaluate the first-order reflection and transmission coefficients. The computational results of the first order reflection coefficient are presented graphically and compared with the results existing in the literature.
Case-I: Normal incidence
- Statement and formulation
- Solution procedure
- Solution φ(x, y) in expansion form
- Comparison of expansion form of solution with the previous results
- A special bed surface
In the solution method, they applied Fourier transform to the governing equation and the boundary conditions. The problem of Davies and Heathershaw [19] is generalized to the problem of scattering of surface water waves by small undulations of the seabed, which is solved in section 3.3. However, the first order reflection coefficient R1 and the transmission coefficient T1 were explicitly obtained by Davies and Heathershaw [19] (see also Mandal and Basu [53] for the case without surface tension and θ = 0).
Case-II: Oblique incidence
- Formulation of the problem
- Solution procedure
- Solution φ(x, y) in expansion form
- A special bed surface
The problem of water wave scattering from a sinusoidal variable topography on the seabed, considered by Davies and Heathershaw [19], is generalized to the case of oblique incidence in Section 4.3 in which the solution for φ1(x, y) is given from equations (4.33) and (4.34). As in the case of normal incidence, at the critical condition l= 2µ, equations (6.74) and (6.75) reveal that there is a Bragg resonance between the surface waves and the bedforms and thus,. For such sinusoidal waves on the seabed, the first-order reflection coefficient R1 and transmission coefficient T1 were obtained by Mandal and Basu [53] (without surface tension).
Numerical results
From the graphs it is clear that for θ = π4, |R1| vanishes independently of the shape function as observed by the results of Mandal and Basu [53]. The above four figures show that the method used here produces correct numerical estimates for the reflection coefficient. For other forms of the shape function, this method is expected to give correct results without difficulty.
Conclusion
In Chapters 2, 3, 4 and 6, the problem of surface wave propagation over small undulations on the seabed is investigated assuming that the bed is impermeable. In this chapter we investigate the problem of small ripple surface water wave propagation in a seabed of finite depth assuming that the seabed is composed of porous material of a specific type. Fluid movement within the porous bed is not analyzed here and it is assumed that the fluid movements are such that the resulting boundary condition in the seabed as used here holds well and depends on a known parameter G0, called porosity. parameter, in the analysis.
Case-I: Normal incidence
Formulation of the problem
In recent times, due to the many interesting applications of the theory of surface water wave scattering, many researchers have turned their attention to problems related to porous rather than impermeable beds. A Fourier transform method is used to obtain the full solution of the mixed boundary value problem under the assumption that the porous seabed undulation is small enough to apply a regular perturbation expansion in terms of a parameter of small wavy. A small undulation seabed composed of porous material of a specific type is described by ngay=h+εc(x) where c(x) is a function with compact support and describes the end undulation, denotes the uniform finite depth of the sea . away on either side of the bottom wavelet so c(x)→0 as|x| → ∞and the nondimensional number ε(¿1) a measure of the smallness of the wave.
Solution procedure
Ksinhk0h−k0coshk0heik0x, (7.15) kuk0, the wavenumber of the incident wave, is the unique positive root (for a given G0) of Eq. By replacing K by Kb = (σ2 +iµ0σ)/g in equation (7.26), the singularities are shifted from the ξ axis to the upper and lower half-planes. 7.31) Here the contour consists of the part −R at R on the real ξ axis and a semicircle centered at the origin and having a large radius R. The first term on the right-hand side of each equation represents the non-propagating modes which decay rapidly away from the wave and the second term represents a propagating mode from the region of bed disturbance.
A special bed surface
In the situation in which there is an integer number of ripple wavelengths in the patch L1 ≤x≤L2 such that m=n and δ0 = 0, we find R1 and T1, respectively, as Equation (7.42) illustrates that for a given number m of ripples, the first-order wave reflection coefficient is an oscillatory function in the quotient of twice the surface wavenumber to the ripple wavenumber. A possible consequence of this is a link between ripple growth and wave reflection, which may be important in the problems of coastal protection.
Case-II: Oblique incidence
Formulation of the problem
Solution procedure
Ksinhk0h−k0coshk0heiµx, (7.63) kuk0, the wavenumber of the incident wave, is the unique positive root of equation (7.16) and. Now, taking the Fourier transform of the governing equation (7.59) and the boundary conditions (7.60) and (7.61) with respect to the horizontal space variable x, we get 7.70) Taking the inverse Fourier transform, the solution for the velocity potential can be written in the form Comparing equations (7.80) and (7.81) with equation (7.62), the reflection and transmission coefficients can be written as, respectively.
A special bed surface
Numerical results
Conclusion
In Chapter 2, using a perturbation analysis that includes the small parametersε present in the representation of small undulations of the seabed, we pose boundary value problems that the velocity potential must satisfy for wave scattering with small undulation topography for normal and oblique incidences. The Fourier transform technique is used to obtain the complete solution of the mixed boundary value problem, from which the reflection and transmission coefficients involving the shape function c(x) are determined. These methods lead to a more computationally acceptable form of solution for the scattered field.
Scope for future work
Belibassakis, A consistent pair mode theory for the propagation of small amplitude water waves over variable bathymetry regions, J. Johnson, Propagation of axisymmetric nonlinear shallow water waves over slowly varying depth, Mathematics and Computers in Simulation no. Nachari, Note on Bragg scattering of water waves by parallel bars in the seabed, J.
Roots of k tanh kh = K when Kh = 1
The problem domain
Contour Integration
Reflection and transmission coefficients against the wave number k 0 h for θ =
Reflection and transmission coefficients against the wave number k 0 h for θ =
A patch of sinusoidal ripples
