# Research

## Research

The main research activity of **SchäfferWaves** is to develop an efficient, deterministic model for surface wave propagation spanning the whole range from high-end, academic investigations of basic wave phenomena to practical engineering applications covering millions of computational points in horizontal space and hours of real-time modelling

## The general idea

The basic difficulty of solving water wave problems is associated with the elliptic nature of the physics: At a given instant the motion of all parts of a fluid domain depend on each other. This contradicts the practical observation that throwing a stone in the water will not immediately affect the other end of the ocean. The reason for this disparity is a combination of the presence of a free surface and a limited water depth. (Wave compressibility is another explanation, but irrelevant in the present context). These effectively introduce an exponential spatial decay of the response to a disturbance. Capturing this feature in a numerical model eliminates the need for the solution of huge systems of equations. The global mutual dependence is effectively reduced to a local one. This is the overall idea behind project.

## The specific method

The classical wave transformation problem may be formulated in terms of the dynamic and the kinematic free surface boundary conditions by which the surface elevation and surface velocity may be stepped forward in time. The key point is that this requires a closure of the system by essentially establishing a relation between the vertical and horizontal velocity. This basically involves a solution to the Laplace equation and it can be done in many different ways. Two prominent examples are Boundary Integral Equation Methods using free-space Greens functions and Boussinesq formulations of various types and orders using polynomial and Padé expansions.

The method behind the present work employs expansions from shallow water, but finally departs from Boussinesq-type theory and eliminates the depth limitation. In the final formulation the vertical particle velocity at still-water-level (SWL) is expressed explicitly as a convolution integral involving a depth-dependent kernel (impulse response function) and the horizontal particle velocity at SWL. This is exactly what is needed for time stepping the surface boundary conditions and thus the usual task of solving large systems of algebraic equations is avoided. This is for linear waves, but procedures for tackling non-linearity exist, see e.g. Schäffer (2008) under publications.

A thorough account for the progress of the convolution approach is referenced in the publications section. Overall aspects of the associated numerical model can be found under SWIFT.

The accomplishment of this work opens up a wealth of potential applications ranging from practical engineering problems to academic investigations of basic physics

- Wave-wave interactions and energy transfer in deep and shallow water
- Wave disturbance in ports and harbours
- Wave kinematics
- Interactions with and wave forces on offshore structures, ocean energy devices, wind turbine foundations etc.
- Transformation of ship waves
- Bragg reflection of various types
- Freak waves (rogue waves)
- Wave instabilities (crescent waves, Benjamin-Feir instability)
- Wave breaking and run-up
- Generation of coastal currents and infragravity waves

## Acknowledgement

Support from the Danish Research Council for Technology and Production Sciences (FTP grant No 26-04-0191) is greatly appreciated.