Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Waves produced by a pulsating source travelling beneath a free surface

Author: H. S. Tan
Journal: Quart. Appl. Math. 15 (1957), 249-255
MSC: Primary 76.0X
DOI: https://doi.org/10.1090/qam/89643
MathSciNet review: 89643
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Abstract: The propagation of surface waves induced by a pulsating source travelling beneath a free surface is investigated by introducing Rayleigh's dissipative force. By defining a disturbance velocity potential $ R\varphi \left( {x,y} \right) exp \left( {i\omega t} \right)$, it is shown that $ \varphi \left( {x,y} \right)$ has to satisfy throughout the flow field not only the Laplace equation, but also a differential equation resulting from the analytic extension of the free surface boundary condition. The solution is obtained in the form

$\displaystyle \varphi \left( {x,y} \right) = f\left( z \right) + g\left( {\bar z} \right),\qquad z = x + iy$

, i.e., $ \varphi \left( {x,y} \right)$ is a complex harmonic function of $ x$, $ y$ but is not an analytic function of $ z$. The wave propagation is found to depend on a parameter $ \tau $, which is the ratio of the pulsation frequency $ \omega $ of the source strength to the fundamental frequency $ {\omega _0}$ of the surface wave produced by a constant strength source travelling at the same speed. The case $ \tau = 0$, corresponding to the travelling source of constant strength, does give a single undamped harmonic wave train on the downstream side, of wave length $ 2\pi {c^2}/g$, (or frequency $ {\omega _0}$), as is expected. For $ 0 < \tau < 1/4$, there are four, and for $ \tau > 1/4$, two undamped harmonic wave trains of different wave lengths on the downstream side. It is further observed that there exists a critical frequency at $ \tau = 1/4$, at which frequency resonance phenomena occur. Thus violent disturbance is predicted at $ \tau = 1/4$ by the present analysis. No disturbance is found to propagate infinitely upstream. This result evidently justifies the imposition of an asymptotic upstream condition of ``vanishing disturbance at infinity'' to replace the effect of the dissipative force in rendering the solution unique.$ ^{1}$

References [Enhancements On Off] (What's this?)

  • [1] H. S. Tan, On source and vortex of fluctuating strength travelling beneath a free surface, Quart. Appl. Math. 13, (1955) MR 0072604
  • [2] Lord Rayleigh, The form of standing waves on the surface of running water, Proc. Lond. Math. Soc. 15, 69 (1883) MR 1575241
  • [3] H. Lamb, On some cases of wave-motion on deep water, Ann. di Matematica 21, 237 (1913)
  • [4] T. H. Havelock, The forces on a circular cylinder submerged in a uniform stream, Proc. Roy. Soc. A157, 526-534 (1936)
  • [5] M. D. Haskind, The hydrodynamical theory of the oscillation of a ship in waves, Prinkladnaya Mat. Mekh. 10, 33-66 (1946) MR 0031927

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DOI: https://doi.org/10.1090/qam/89643
Article copyright: © Copyright 1957 American Mathematical Society

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