Kernel representation of long-wave dynamics on a uniform slope

Area of Science:

• Fluid dynamics
• Oceanography
• Coastal engineering

Background:

• Long-wave propagation on beaches is a complex phenomenon crucial for coastal processes.
• Existing models often struggle with accurately capturing transient dynamics and nonlinear effects.

Purpose of the Study:

• To develop a novel transient-response formulation for long-wave propagation on uniformly sloping beaches.
• To provide exact solutions for linear shallow water equations and extendable to nonlinear dynamics.
• To offer new insights into the long-studied problem of near-shore wave behavior.

Main Methods:

• Formulating long-wave propagation as a transient-response problem with initially stationary water.
• Representing water surface elevation and flow velocity as convolutions with a singular kernel function.
• Utilizing double exponential formulas for numerical implementation to handle kernel singularity.
• Extending the kernel formulation to nonlinear dynamics via the hodograph transform.

Main Results:

• The kernel function accurately accommodates dynamic processes like shoaling, reflection, and multiple reflections.
• Exact solutions of linear shallow water equations are obtained for any smooth incident wave.
• The method enables instantaneous prediction of nonlinear wave properties and wave breaking.
• The formulation provides a unified framework for understanding long-wave dynamics.

Conclusions:

• The developed transient-response formulation offers a powerful and exact method for analyzing long-wave dynamics on beaches.
• The kernel convolution approach successfully models both linear and nonlinear wave behaviors.
• This work provides significant new insights into coastal wave dynamics and prediction.