Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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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}$

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


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