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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.07446 |
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| _version_ | 1866910164321304576 |
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| author | Yewale, Nikhil Amiroudine, Sakir Dasgupta, Ratul |
| author_facet | Yewale, Nikhil Amiroudine, Sakir Dasgupta, Ratul |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_07446 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Free oscillations of a standing surface wave and its mechanical analogue Yewale, Nikhil Amiroudine, Sakir Dasgupta, Ratul Fluid Dynamics 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. |
| title | Free oscillations of a standing surface wave and its mechanical analogue |
| topic | Fluid Dynamics |
| url | https://arxiv.org/abs/2509.07446 |