The LambertW function is defined to be the multivalued inverse of the functionw →we

 w
 . It has many applications in pure and applied mathematics, some of which are briefly described here. We present a new discussion of the complex branches ofW, an asymptotic expansion valid for all branches, an efficient numerical procedure for evaluating the function to arbitrary precision, and a method for the symbolic integration of expressions containingW.

On the Lambert W function

A third-generation numerical wave model to compute random, short-crested waves in coastal regions with shallow water and ambient currents (Simulating Waves Nearshore (SWAN)) has been developed, implemented, and validated. The model is based on a Eulerian formulation of the discrete spectral balance of action density that accounts for refractive propagation over arbitrary bathymetry and current fields. It is driven by boundary conditions and local winds. As in other third-generation wave models, the processes of wind generation, whitecapping, quadruplet wave-wave interactions, and bottom dissipation are represented explicitly. In SWAN, triad wave-wave interactions and depth-induced wave breaking are added. In contrast to other third-generation wave models, the numerical propagation scheme is implicit, which implies that the computations are more economic in shallow water. The model results agree well with analytical solutions, laboratory observations, and (generalized) field observations.

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A third-generation wave model for coastal regions: 1. Model description and validation

Morphodynamic variability of surf zones and beaches: A synthesis

An analytical theory is presented to describe the combined motion of waves and currents in the vicinity of a rough bottom and the associated boundary shear stress. Characteristic shear velocities are defined for the respective wave and current boundary layer regions by using a combined wave-current friction factor, and turbulent closure is accomplished by employing a time invariant turbulent eddy viscosity model which increases linearly with height above the seabed. The resulting linearized governing equations are solved for the wave and current kinematics both inside and outside the wave boundary layer region. For the current velocity profile above the wave boundary layer, the concept of an apparent bottom roughness is introduced, which depends on the physical bottom roughness as well as the wave characteristics. The net result is that the current above the wave boundary layer feels a larger resistance due to the presence of the wave. The wave-current friction factor and the apparent roughness are found as a function of the velocity of the current relative to the wave orbital velocity, the relative bottom roughness, and the angle between the currents and the waves. In the limiting case of a pure wave motion the predictions of the velocity profile and wave friction factor from the theory have been shown to give good agreement with experimental results. The reasonable nature of the concept of the apparent bottom roughness is demonstrated by comparison with field observations of very large bottom roughnesses by previous investigators. The implications of the behavior predicted by the model on sediment transport and shelf circulation models are discussed.

Combined wave and current interaction with a rough bottom

An extension of sinc interpolation on $\mathbb{R}$ to the class of
algebraically decaying functions is developed in the paper. Similarly to the
classical sinc interpolation we establish two types of error estimates. First
covers a wider class of functions with the algebraic order of decay on
$\mathbb{R}$. The second type of error estimates governs the case when the
order of function's decay can be estimated everywhere in the horizontal strip
of complex plane around $\mathbb{R}$. The numerical examples are provided.

Sinc approximation of algebraically decaying functions

In the last two decades the problem of wave height damping due to bottom friction has received increasing attention among near-shore oceanographers. This fact is reflected in the wealth of papers on the subject; the list of references given herein presents a minor selection only. This paper is an attempt to re-evaluate and systematize the many observations and the rather few detailed measurements of the phenomenon. In nature the wave boundary layer will always be rough turbulent. This is not necessarily the case in a hydraulic model. The aim is therefore to make it possible to determine the proper flow regime for a pure short-period wave motion over a given bed. Values for the wave friction factor and the wave boundary layer thickness are also proposed. The main results of the study are presented in three diagrams giving flow regimes, friction factors and boundary layer thicknesses. Flow parameters are a-jjj/k and RE = U-|j_ ai~/v, a-]m and U-|m being maximum bottom amplitude and velocity according to first order potential wave theory, k is the Nikuradse roughness parameter.

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Wave boundary layers amd friction factors

The oscillatory flow near the sea bed under a wave motion is always rough turbulent in a coastal zone. This type of an oscillatory boundary layer (or “wave boundary layer”) was therefore chosen as a subject for detailed velocity measurements, from which characteristics such as shear stresses, eddy viscosities, energy loss, and boundary layer thickness were determined.

Experimental and theoretical investigations in an oscillatory turbulent boundary layer

Bed friction and dissipation in a combined current and wave motion

A new approach to oscillatory rough turbulent boundary layers

The distribution of the bottom shear stresses in an open channel can be found by various approximate methods. For a shallow channel, a fair approximation is obtained if the shear is assumed to correspond to the vertical depth. If the incline of the bottom is appreciable, however, it is better to calculate the shear by measuring the depth at right angles to the bottom. A more consistent method is to let the shear correspond to the area between two bottom normals. However, these methods do not consider the transfer of shear (in the direction of flow) through the "depth lines." The proposed new method overcomes this difficulty by assuming that there is no transfer of shear through the orthogonals to the lines of equal mean velocity. The other basic assumption is that the velocity distribution is logarithmic along the bottom normals, corresponding to the local friction velocity. For a plane bottom with a transverse slope of 30, the shear stresses will be 40% higher than found by conventional methods. The method cannot be used on polygonal cross sections.

Ivar G. Jonsson

Papers

Wave boundary layers amd friction factors

Experimental and theoretical investigations in an oscillatory turbulent boundary layer

Bed friction and dissipation in a combined current and wave motion

A new approach to oscillatory rough turbulent boundary layers

Shear and Velocity Distribution in Shallow Channels