Summary
An obstacle, represented by a delta function, is placed on the bed of a three dimensional stream and, as a result, a steady “V” shaped surface wave pattern develops downstream. The rate of decay of the transient motion is determined and existence and uniqueness of the steady pattern are established for all values of the stream velocity. Asymptotic expressions for the steady state wave amplitude, valid far from the obstacle, are obtained. Near the border of the wave pattern, expressions are obtained which are uniform in the polar angle.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. J. Stoker,Water Waves, Interscience, New York 1957, Ch. 7.
H. Jeffreys and B. S. Jeffreys,Methods of Mathematical Physics, Cambridge Univ. Press, 3rd Ed., 1956, pp. 508–511.
H. Lamb,Hydrodynamics, 6th ed., Cambridge Univ. Press, 1932; Dover, New York, 1945, Ch. 9.
T. H. Havelock, The propagation of groups of waves in dispersive media,Proc. Roy. Soc., Vol. A 81 (1908) 398–430.
Lord Kelvin, On the waves produced by a single impulse in water of any depth,Proc. Roy. Soc. of London, Vol. A42 (1887) 80–85.
C. Chester, B. Friedman and F. Ursell, An extension of the method of steepest descents,Proc. Camb. Phil. Soc., Vol. 54 (1957) 599–611.
F. Ursell, On Kelvin's Ship Wave Pattern,J. Fluid Mech., Vol. 8 (1960) 418–431.
N. Bleistein, Uniform asymptotic expansions of integrals with stationary point near algebraic singularity,Comm. Pure Appl. Math., Vol. 19 No. 4 (1966) 353–370.
L. Sirovich,Techniques of Asymptotic Analysis, Springer-Verlag, New York, 1971.
E. Palm, On the formation of surface waves in a fluid flowing over a corrugated bed and on the development of mountain waves,Astrophys. Norveg. 5 (1953) 61–130.
Cherkesov, The development of ship waves on fluid of finite depth,Mekh. Zhidk. i Gaza 4 (1968) 70–76.
A. I. Smorodin, On the use of an asymptotic method for analyzing waves with unsteady motion of the source,Priklad. Mat. Mekh. 29 (1965) 62–69.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Harband, J. Three dimensional flow over a submerged object. J Eng Math 10, 1–21 (1976). https://doi.org/10.1007/BF01535423
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01535423