[Paper Review] Water Waves from General, Time-Dependent Surface Pressure Distribution in the Presence of a Shear Current

Yan Li, Simen Å. Ellingsen|arXiv (Cornell University)|Nov 9, 2015

Ocean Waves and Remote Sensing18 references3 citations

TL;DR

This paper presents an analytical solution for three-dimensional water waves generated by a time-dependent surface pressure distribution in the presence of a uniform vorticity shear current. It demonstrates that shear currents induce strong asymmetry in transient wave patterns, with upstream waves decaying significantly slower than downstream waves, and shows that transient wave resistance diminishes over time while steady ship-wave patterns persist, especially for vessels moving against the current.

ABSTRACT

We obtain a general solution for the water waves resulting from a general, time-dependent surface pressure distribution, in the presence of a shear current of uniform vorticity beneath the surface, in three dimensions. Linearized governing equations and boundary conditions including the effects of gravity, a distributed external pressure disturbance, and constant finite depth, are solved analytically, and particular attention is paid to classic initial value problems: an initial pressure impulse and a steady pressure distribution which appears suddenly. In the present paper, good agreement with previous results is demonstrated. We subsequently show both analytically and numerically how transient waves from a suddenly appearing steady pressure distribution vanis for large times, and steady ship waves remain. The transient contribution to wave resistance was derived. The results show that a shear current has significant impact on the transient wave motions, resulting in asymmetry between upstream and downstream waves. The case of the suddenly appearing steady pressure is an intermediate case between ring waves and ship waves, starting out as the former and evolving gradually into the latter. The ship's direction of motion relative to the current is found to be crucial in determining how quickly effects of transient waves die out. Transient effects take much longer to doe out for a ship going against the shear flow than for one going downstream, when the two ships have the same velocity relative to the water surface. For ship motion against the shear current, thus, wave effects of transients, e.g., due to maneuvering could accumulate and be far more significant than on a uniform current.

Motivation & Objective

• To derive a general analytical solution for water waves generated by arbitrary time-dependent surface pressure distributions in the presence of a shear current.
• To investigate the transient wave behavior arising from a suddenly applied steady pressure distribution, bridging ring wave and ship wave dynamics.
• To quantify the impact of shear current on wave resistance, particularly the transient contribution.
• To analyze the asymmetry in wave patterns between upstream and downstream propagation due to current-induced dispersion effects.
• To determine how the direction of ship motion relative to the shear current affects the decay rate of transient wave effects.

Proposed method

• Solving the linearized Euler equations with gravity, constant depth, and a distributed external pressure disturbance.
• Applying boundary conditions that include the surface pressure distribution and the presence of a uniform vorticity shear current.
• Using analytical techniques to solve initial value problems, particularly the impulse and sudden-onset pressure cases.
• Employing Fourier and Laplace transforms to handle time-dependent forcing and derive closed-form solutions.
• Conducting numerical simulations to validate analytical results and visualize wave evolution.
• Deriving expressions for wave resistance, separating transient and steady-state contributions.

Experimental results

Research questions

• RQ1 How does a shear current affect the transient wave pattern generated by a suddenly applied pressure distribution?
• RQ2 What is the decay behavior of transient waves in the presence of a shear current, and how does it differ between upstream and downstream propagation?
• RQ3 To what extent does the direction of ship motion relative to the shear current influence the persistence of transient wave effects?
• RQ4 How does the wave resistance decompose into transient and steady-state components under time-dependent forcing?
• RQ5 In what way does the system transition from ring-wave-like to ship-wave-like behavior under a suddenly applied steady pressure?

Key findings

• Transient waves from a suddenly appearing steady pressure distribution evolve from ring-wave-like patterns into ship-wave-like patterns over time.
• The transient wave contribution to wave resistance decays over time, but the decay is significantly slower when a ship moves against the shear current compared to downstream.
• Shear currents induce strong asymmetry: upstream waves persist much longer than downstream waves due to altered group velocity and dispersion relations.
• For a ship moving against the current, transient wave effects—such as those from maneuvering—can accumulate and remain significant over extended periods.
• The analytical solution shows excellent agreement with previous results in the limit of zero vorticity, validating the model.
• The steady-state wave pattern persists indefinitely, while the transient component vanishes asymptotically, confirming the existence of a permanent wave signature for steady forcing.