Cress Help - Wind waves and swell - Basic of waves - Z10.1

Orbital movement (also in higher order) 

Short description

This routines calculates basic wave parameters with first order (linear) and second order theory.

Used equations
First order wave theory

The deep-water wavelength is calculated with:

    equation 1

The shallow water wavelength is calculated with a first approximation of VISSER [1982], followed by a five-step iteration process. The numerical accuracy of this process is much higher than the physical accuracy of the dispersion relation:

    equation 2

The Visser approximation is:

    equation 3

The iteration loop is:

    equation 4

The wave steepness in deep water is defined as:

    equation 5

As usual in wave theory the following definitions of k and are used:

    equation 6

The maximum near bed orbital velocity is in first order theory calculated with:

    equation 3

The maximum near-bed excursion is in first order theory calculated with:

    equation 8

In case the H/h > the wave should be considered as broken; in that case neither first nor second order theory are valid
Second order wave theory

    figure 1

First order linear wave theory assumes an infinitely small amplitude, which is realistic for deep water. In shallow water this is not correct. In 1847 Stokes has presented his second order theory for finite amplitude waves. He defines the water surface profile as:

    equation 9

in which:

    equation 10

The physical meaning of dh is the rise of the mean water level. The wave-height is consequently:

    equation 11

The second order wave celerity is given by:

    equation 12

The maximum velocity under the crest is:

    equation 13

Under the trough the maximum velocity is:

    equation 14

For calculation of sediment transport, stability of bed material, etc. it is very useful to know the root-mean-square average of the velocity under the crest or under the trough. This can be calculated by determining the rms-elevation above and below the water surface:

    equation 15

    equation 16

The rms-average velocities are now given by:

    equation 17

    equation 18

The duration of the crest-passage is:

    equation 19

and the length of the crest above SWL is consequently:

    equation 20

Stokes second order approach is valid for:

    equation 21

References


IPPEN, A.T., ED. [1966] Estuarine and coastline hydrodynamics (chapter 2, Finite amplitude waves, by R.G. DEAN AND P.S. EAGLESON, pp 105 – 108) McGraw-Hill, New York.
VISSER, P.J. [1984] A mathematical model of uniform longshore currents and comparison with lab data, rep. nr. 82-1, fac. Civil Eng, Delft University.