Abstract
A linearized theory is developed for the derivation of an asymptotic solution of the initial value problem of axisymmetric surface waves in an infinitely deep fluid produced by an arbitrary oscillating pressure distribution. An asymptotic treatment of the problem is presented in detail to obtain the solution for the surface elevation for sufficiently large time. Finally, the main prediction of this analysis for some particular pressure distributions of physical interest is exhibited.
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Abbreviations
- R, θ, Y :
-
cylindrical polar coordinates
- ω :
-
frequency
- g :
-
acceleration due to gravity
- ρ :
-
density of fluid
- T :
-
time
- Ω(R, Y; T):
-
velocity potential
- E(R, T):
-
vertical surface elevation
- P(R, T):
-
applied surface pressure
- r, y :
-
nondimensional cylindrical polar coordinates, \((r,y) = \frac{{\omega ^2 }}{g}(R,Y)\)
- p(r, t):
-
nondimensional surface pressure
- t :
-
nondimensional time, ωT
- ϕ(r, y; t):
-
nondimensional velocity potential, \(\frac{{P\omega ^5 }}{{\rho g^4 }}\Phi\)
- η(r, t):
-
nondimensional vertical surface elevation, \(\frac{{P\omega ^4 }}{{\rho g^3 }}{\rm E}\)
- \(\bar p\)(k):
-
Hankel transform of a function p(r) with respect to r
- I 1 :
-
transient wave integral
- I 2 :
-
steady state wave integral
References
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Debnath, L., On transient wave motion in fluids, Ph. D. thesis, University of London, London 1967.
Wehausen, J. V. and E. V. Laitone, p. 446, Handbuch der Physik Vol. IX, Springer Verlag, Berlin 1960.
Miles, J. W., J. Fluid Mech. 13 (1962) 145.
Stoker, J. J., Water Waves, Interscience, New York 1957.
Sen, A. R., Proc. Nat. Inst. Sci. India 28(4–6 (1962) 612.
Sneddon, I. N., Fourier Transforms, McGraw-Hill, New York 1951.
Copson, E. T., Asymptotic Expansions, Cambridge Un. Press, Cambridge 1965.
Erdélyi, A. et al, Tables of Integral Transforms, Vol. 2, McGraw-Hill, New York 1954.
Whittaker, E. T. and G. N. Watson, A Course of Modern Analysis, Cambridge Un. Press, Cambridge 1965.
Lamb, H., Proc. Lond. Math. Soc. 2 (1905) 371.
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Debnath, L. An asymptotic treatment of the transient development of axisymmetric surface waves. Appl. sci. Res. 21, 24–36 (1969). https://doi.org/10.1007/BF00411595
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DOI: https://doi.org/10.1007/BF00411595