Cauchy Poisson Problem for Water with a Porous Bottom

Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 834)

Abstract

This paper is concerned with generation of surface waves in an ocean with porous bottom due to initial disturbances at free surface. Assuming linear theory the problem is formulated as an initial value problem for the velocity potential describing the motion in the fluid. Laplace transform in time and Fourier transform in space have been utilized in the mathematical analysis to obtain the form of the free surface in terms of an integral. This integral is then evaluated asymptotically for large time and distance by the method of stationary phase for prescribed initial disturbance at the free surface in the form of depression of the free surface or an impulse at the free surface concentrated at the origin. The form of the free surface is depicted graphically for these two types of initial conditions in a number of figures to demonstrate the effect of the porosity at the bottom.

Keywords

Cauchy Poisson problem Porous bottom Laplace and Fourier transform Method of stationary phase Free surface depression 

Notes

Acknowledgments

The authors thank the reviewers for their comments to modify the paper in the present form. This work is carried out under CSIR research project No. 25(0253)/16/EMR-II.

References

  1. Baek, H.M., Kim, Y.J., Lee, I.J., Kwon, S.H.: Revisit of Cauchy Poisson problem in unsteady water wave problem. In: 32 IWWWFB, Dalian, China, pp. 13–16 (2017)Google Scholar
  2. Chaudhuri, K.: Waves in shallow water due to arbitrary surface disturbances. App. Sci. Res. 19, 274–284 (1968)CrossRefGoogle Scholar
  3. Gangopadhyay, S., Basu, U.: Water wave generation due to initial disturbance at the free surface in an ocean with porous bed. Int. J. Sci. Eng. Res. 4(2), 1–4 (2013)Google Scholar
  4. Kranzer, H.C., Keller, J.B.: Water waves produced by explosions. J. Appl. Phys. 30, 398–407 (1959)MathSciNetCrossRefGoogle Scholar
  5. Lamb, H.: Hydrodynamics. Dover, New York (1945)MATHGoogle Scholar
  6. Maiti, P., Mandal, B.N.: Water wave scattering by an elastic plate floating in an ocean with a porous bed. Appl. Ocean Res. 47, 73–84 (2014)CrossRefGoogle Scholar
  7. Martha, S.C., Bora, S.N., Chakraborti, A.: Oblique water wave scattering by small undulation on a porous sea bed. Appl. Ocean Res. 29, 86–90 (2007)CrossRefGoogle Scholar
  8. Stoker, J.J.: Water Waves. Interscience Publishers, New York (1957)MATHGoogle Scholar
  9. Wen, S.L.: Int. J. Math. Educ. Sci. Technol. 13, 55–58 (1982)CrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • Piyali Kundu
    • 1
  • Sudeshna Banerjea
    • 1
  • B. N. Mandal
    • 2
  1. 1.Jadavpur UniversityKolkataIndia
  2. 2.Physics and Applied Mathematics UnitIndian Statistical InstituteKolkataIndia

Personalised recommendations