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**Water wave propagation over uneven bottoms. Part 1: Linear wave propagation. Part 2: Non-linear wave propagation. (2 vol.).**
*(English)*
Zbl 0908.76002

Advanced Series on Ocean Engineering. 13. Singapore: World Scientific. xxv, xvii, 967 p. (1997).

The contents of this splendid book are as follows: Part 1 – Linear wave propagation (Ch. 1. Basic equations; Ch. 2. Wave propagation formulation; Ch. 3. The mild-slope equation; Ch. 4. Practical aspects of linear wave propagation models); Part 2 – Nonlinear wave propagation (Ch. 5. Boussinesq-type models for uneven bottoms; Ch. 6. KdV-type models; Ch. 7. Harmonic generation, and Ch. 8. Nonlinear wave propagation of Stokes’ waves over uneven bottoms). Each chapter is concluded with notes containing additional material related to that chapter. Both volumes end with a large list of references (372 entries in the first volume and 279 entries in the second one, some of the titles are common). There exist also an author index and a subject index.

The equations for incompressible (water) wave propagation cannot be solved in a closed form, the principal difficulty being the fact that the domain in which equations have to be solved is a part of the solution, i.e., the free surfaces itself is one of the unknowns. To overcome this difficulty, the author analyses gradually some approximate models which can be solved analytically and numerically. The way in which such models are considered depends upon the spatial and temporal scales involved, the amount of variation in water depth within a wave length, the presence of structures etc. All in all, the main aim of these two volumes is to provide a large review of techniques available to solve the above-mentioned models of wave propagation in regions with uneven beds, as they are encountered in coastal areas. Moreover, the author makes important efforts to furnish these techniques with mathematical foundations, and sometimes even with mathematical details. In this connection we remark that the mathematics is kept at an elementary level. The author uses only usual differential and integral calculus, the perturbation theory and elementary numerical analysis and partial differential equations. Thus, when he considers the existence of the variational principles for Navier-Stokes equations, or tries to solve the Helmholtz equation, he does not use the very productive concept of weak solution or weak formulation. The same situation happens when he deals with some conservation laws.

Beyond these remarks, the volumes are well produced and well written with clear and precise statements. The author is an expert in the field, and his exposition has elegance and rigour. The work is richly illustrated with numerical examples which are carried out by solving boundary value problems or by integrating dynamical systems. There are also some open and unsolved problems which could be taken into account for future developments.

The equations for incompressible (water) wave propagation cannot be solved in a closed form, the principal difficulty being the fact that the domain in which equations have to be solved is a part of the solution, i.e., the free surfaces itself is one of the unknowns. To overcome this difficulty, the author analyses gradually some approximate models which can be solved analytically and numerically. The way in which such models are considered depends upon the spatial and temporal scales involved, the amount of variation in water depth within a wave length, the presence of structures etc. All in all, the main aim of these two volumes is to provide a large review of techniques available to solve the above-mentioned models of wave propagation in regions with uneven beds, as they are encountered in coastal areas. Moreover, the author makes important efforts to furnish these techniques with mathematical foundations, and sometimes even with mathematical details. In this connection we remark that the mathematics is kept at an elementary level. The author uses only usual differential and integral calculus, the perturbation theory and elementary numerical analysis and partial differential equations. Thus, when he considers the existence of the variational principles for Navier-Stokes equations, or tries to solve the Helmholtz equation, he does not use the very productive concept of weak solution or weak formulation. The same situation happens when he deals with some conservation laws.

Beyond these remarks, the volumes are well produced and well written with clear and precise statements. The author is an expert in the field, and his exposition has elegance and rigour. The work is richly illustrated with numerical examples which are carried out by solving boundary value problems or by integrating dynamical systems. There are also some open and unsolved problems which could be taken into account for future developments.

Reviewer: C.I.Gheorghiu (Cluj-Napoca)

### MSC:

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |

35Q53 | KdV equations (Korteweg-de Vries equations) |

86A05 | Hydrology, hydrography, oceanography |