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arXiv · 2509.07446

Free oscillations of a standing surface wave and its mechanical analogue

Nikhil Yewale·Sakir Amiroudine·Ratul Dasgupta
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Abstract

We present an analogy between natural oscillations of the standing wave type on a pool of liquid with an interface and a mechanical oscillator model. It is shown that the equations of motion governing both systems have qualitatively similar solutions - trivial as well as time-periodic with finite amplitude. The time-periodic solutions can be linearly unstable in both cases depending on the oscillation amplitude, thereby leading to interesting dynamics. Linear stability results of both systems are discussed in detail; a novel Mathieu-like equation is derived for the stability of the standing wave to a super-harmonic perturbation. This is obtained through a much simpler approach that yields linear stability results while also reinforcing the analogy. Analytical predictions are compared against numerical solutions to the full nonlinear governing equations for both systems. A good match is obtained in most cases with theory; mismatches are further analysed and the limitations of this analogy are also pointed out.

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Nikhil Yewale, Sakir Amiroudine, Ratul Dasgupta. 2025-09-09. Free oscillations of a standing surface wave and its mechanical analogue. https://arxiv.org/abs/2509.07446

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