Fourier transform method is used to solve the problem of water wave scattering by a small undulation on an otherwise flat bottom of a porous sea-bed for normal and oblique incidence.
The presence of the singularities lead us to use the residue theorem to evaluate the integral appearing in the first-order correction of the potential. After deriving the velocity potential, the reflection and transmission coefficients up to the first order are obtained. Application of these results for a sinusoidal bottom undulations yields results which coincide exactly with the results for the same obtained earlier in this thesis when the bed has no porous effect and for normal incidence. From the computational results it is observed that the reflection coefficient increases with increasing porous effect.
0 0.5 0.7854 1 1.5 0
0.02 0.04 0.06 0.08 0.1 0.12 0.14
Angle of incidence
|R1|
G’h=0 G’h=0.05 G’h=0.1
Figure 7.4: Reflection coefficient against the angle of incidence θ for Kh = 0.1;a/h = 0.1;lh= 1;m= 1.
0 0.5 0.7854 1 1.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Angle of incidence
|R1|
G’h=0 G’h=1
Figure 7.5: Reflection coefficient against the angle of incidence θ for Kh = 0.1;a/h = 0.1;lh= 1;m= 1.
0 0.5 0.7854 1 1.5 0
0.05 0.1 0.15 0.2 0.25
Angle of incidence
|R1|
G’h=0 G’h=0.05 G’h=0.1
Figure 7.6: Reflection coefficient against the angle of incidence θ for Kh = 0.1;a/h = 0.1;lh= 1;m= 2.
Summary and Further Work
This chapter is devoted to a brief summary of the results highlighting the contributions made by this thesis and techniques used in deriving these. It also provides information for the scope of possible extensions and future investigations.
8.1 Summary of results
In this thesis the scattering of a train of small amplitude harmonic surface water waves by small undulation using linear water wave theory has been investigated.
In Chapter 2, applying perturbation analysis, which involves a small parameterεpresent in the representation of the small undulation of the sea-bed, we set up the boundary value problems to be satisfied by the velocity potential for the scattering of waves by small undu- lating topography for normal and oblique incidences.
Chapters 3 is concerned with the solution of the velocity potential for the boundary value problem established in Chapter 2 for normal incidence. The solution is obtained by three different techniques, namely, the Green’s integral theorem, Fourier transform technique and finite cosine transform. Using this solution the reflection and transmission coefficients are found which involve the shape functionc(x) and the results may be interpreted as the results obtained by Miles [68] for normal incidence and Mandal and Basu [53] for normal incidence in absence of surface tension. The Fourier transform technique employed to solve the problem has a more general approach than that employed by Davies and Heathershaw [19] to the problem of scattering of water waves by sinusoidal undulations on an otherwise flat bed.
The boundary value problem for oblique incidence in Chapter 2 is also solved by the above three techniques in Chapter 4. The solution for the velocity potential is obtained from which the reflection and transmission coefficients are evaluated which involve the shape
function c(x). These results may be interpreted as the results obtained by Miles [68] and Mandal and Basu [53] in absence of surface tension. Here by putting θ = 0 the results for normal incidence in Chapter 3 can be obtained.
To evaluate these coefficients numerically, different shape functions are considered in Chapter 5 and the results for both normal and oblique incidence are presented graphically.
Among those cases the particular case of sinusoidal ripples on the sea-bed is of considerable significance due to the ability of an undulating bed to reflect incident wave energy which is important in respect of both coastal protection, and of possible ripple growth if the bed is erodable. For this particular case we observe that a large amount of reflection of the incident wave energy is produced for Bragg resonance. This result may be useful in the construction of an effective reflector of the incident wave energy for protecting coastal areas from the rough sea in the arctic regions. The same conclusion can be observed even if all the ripples in the patch do not have the same wave number.
In Chapter 6, the same physical problem is solved for both normal and oblique incidence by a direct method, quite different from the other three methods described in Chapters 3 and 4. This method is based on an eigenfunction expansion that includes both decaying and progressive wave mode terms. The analytical results for the reflection and transmission coefficients obtained by this method are different from the results obtained in Chapters 3 and 4. This is due to the solution approach of this direct method containing an appropriate set of orthogonal eigenfunctions which depends upon a single parameter. However, for a patch of sinusoidal undulations on the bottom these results are computed and compared with the results obtained by other methods. Excellent agreement is observed between the numerical estimates obtained by the present method and those by the known method.
Chapter 7 is concerned with the investigation of the problem of scattering of surface water waves by small undulation on a sea-bed of finite depth by assuming the sea-bed to be composed of porous material of specific type. The boundary condition on the porous sea- bed is derived by taking the porosity effect into the account. Fourier transform technique is employed to obtain the complete solution of the mixed boundary value problem from which the reflection and transmission coefficients are determined which involve the shape function c(x). It is observed that with zero porosity, the results for these coefficients might be interpreted as the results obtained in Chapters 3 and 4 for normal and oblique incidence respectively. These results are applied to the case of a patch of sinusoidal undulations on the bed to evaluate the corresponding coefficients numerically and then the results are presented graphically.
It is observed that the methods presented in the thesis in obtaining the first order poten-
tial, and hence the reflection and transmission coefficients, reduce the workload to a large extent. These methods lead to a computationally more tractable form of the solution for the scattered field.