[Paper Review] First- and second-order wave generation theory

N. Karjanto|arXiv (Cornell University)|Apr 27, 2017

Coastal and Marine Dynamics1 references3 citations

TL;DR

This paper develops first- and second-order wave generation theories for flap-type wavemakers in hydrodynamic tanks using a series expansion method to solve the fully nonlinear water wave equations. It derives analytical solutions for surface wave elevation, showing that first-order waves are linear superpositions of monochromatic components, while second-order waves include bound waves at sum and difference frequencies; second-order steering of the wavemaker motion suppresses unwanted free wave components, ensuring only desired bound waves are generated.

ABSTRACT

The first-order and the second-order wave generation theory is studied in this paper. The theory is based on the fully nonlinear water wave equations. The nonlinear boundary value problem (BVP) is solved using a series expansion method. Using this method, the problem becomes a set of linear, signalling problems according to the expansion order. The first-order theory leads to a homogeneous BVP. It is a system with the first-order steering of the wavemaker motion as input and the surface wave field with propagating and evanescent modes as output. The second-order theory leads to a nonhomogeneous BVP. It is a system where the second-order steering of the wavemaker motion is prescribed in such a way that the second-order part of the surface elevation far from the wavemaker contains only the bound wave component and the free wave component vanishes. The second-order surface wave elevation consists of a superposition of bichromatic frequencies.

Motivation & Objective

• To develop a theoretical framework for wave generation in wave tanks using flap-type wavemakers.
• To address the challenge of nonlinear wave interactions that generate undesired second-order free wave components.
• To design a second-order steering mechanism for the wavemaker motion that eliminates free wave components while preserving desired bound wave components.
• To extend first-order wave generation theory to include second-order effects through a systematic perturbation approach.
• To provide a transfer function for practical application in laboratory wave tank experiments.

Proposed method

• Solving the fully nonlinear water wave boundary value problem (BVP) via a series expansion in powers of a small parameter ε.
• Reducing the nonlinear BVP to a sequence of linear BVPs at each order: homogeneous for first-order, nonhomogeneous for second-order.
• Using the linear dispersion relation (LDR) to relate wavenumbers and frequencies for propagating and evanescent modes.
• Expressing first-order surface wave elevation as a superposition of monochromatic waves with amplitudes determined by the wavemaker motion.
• Deriving second-order wave elevation as a combination of bound waves at frequencies ωm±ωn, with components from interactions of first-order modes.
• Implementing second-order steering by adjusting the wavemaker motion to cancel the free wave component in the second-order solution.

Experimental results

Research questions

• RQ1 How can first-order wave generation be modeled using a nonlinear BVP and series expansion for a flap-type wavemaker?
• RQ2 What are the mathematical conditions under which second-order bound waves are generated, and how do they interact with free wave components?
• RQ3 Why is the presence of second-order free wave components undesirable in wave tank experiments, and how can they be suppressed?
• RQ4 How can the wavemaker motion be modified at second order to ensure only bound wave components are generated?
• RQ5 What is the analytical form of the second-order transfer function relating wavemaker motion to surface wave elevation?

Key findings

• The first-order surface wave elevation is a linear superposition of monochromatic waves, each corresponding to a harmonic component of the wavemaker motion.
• The second-order surface wave elevation consists of bound wave components at frequencies ωm+ωn and |ωm−ωn|, arising from nonlinear interactions of first-order modes.
• The second-order solution contains a nonhomogeneous BVP due to quadratic nonlinearities, with source terms derived from first-order solutions.
• By applying second-order steering, the free wave component in the second-order surface elevation is eliminated, leaving only bound wave components.
• The second-order transfer function can be derived explicitly, enabling precise control of wave generation in laboratory settings.
• The method successfully decouples bound wave generation from spurious free wave generation, improving wave field homogeneity in numerical and experimental wave tank setups.