Small waves, big waves, deep water, shallow water, 1’ order, 2’ order, progressive waves and standing waves, theory of sinusoidal and of cnoidal waves on arbitrary depth.
Niels Mejlhede Jensen, Bogelovsvej 4, 2830 Virum. firstname.lastname@example.org, www.mejlhede.dk
Introduction to wave pressure and shock force on a vertical wall, as PDF.
On www 2011 with this summary:
When regular ocean waves of moderate height hit a vertical wall breakwater then the water at the wall will move up and down, will oscillate like a regular simple harmonic function. The velocity up and down can then be calculated by differentiation and the acceleration by the 2' differentiation, to give simple harmonic sine and cosine functions, as described in my thesis from 1977. This vertical acceleration of the water above is of noticeable influence on the wave pressure on the wall.
The simple wave theory of Airy here gives a little more complicated expressions for the velocity and acceleration of the surface. Airy's classical 1' order theory is good and much used and gives an approximate description of the whole wave which often is surprisingly good for even bigger waves, even though the theory assumes small waves. But it is possible to get some deviations of practical interest in numerical results when using wave theories developed after different guidelines.
These deviations in 1' order expressions from two slightly different wave theories are shown graphically with practical examples of wave pressure on the vertical wall page 89 in ch.V in my 1977 thesis, where the direct application of the vertical acceleration is seen to give a noticeable less wave pressure than the Airy theory. For progressive waves the same type of difference is seen p. 42 and 44 for wave pressure and p 45 in ch.II for horizontal velocity. Wave pressure on the vertical wall according to the 1' order theory of the thesis is shown in the example p. 57 ch.III to be in good agreement with model tests and employed theories of higher order. On p. 98 and 99 ch.V there is compared to model tests I performed in 1968 at the Technical University of Denmark. It is seen that there is a reasonable agreement for wave pressure from the wave crest, while the 1' order theory for the negative pressure from the wave trough is too big for high long waves. This is due to that high long waves (with big H/D) have a tall short crest and a long not so deep trough. A not so deep trough makes the negative pressure less. In contrast the crest is taller but with a short crest the crest gets an increased negative vertical acceleration which reduces the pressure the taller crest otherwise would give.
These waves with a tall crest and a long flat trough appears in the theory of the cnoidal waves, a classical theory for shallow water waves, and which I have expanded to a 2' order theory for waves even to infinite depth. The cnoi function is the Jacobian elliptic cosine function, cn, which is well suitable to give a variable tall short crest and a long flat trough, as it is observed by regular waves in the nature and in the laboratory.
In ch.VI the cnoidal progressive wave on infinite deep water is described with a summary on p. 138 of the most important resulting formulas. E.g. for the wave steepness of H/L = 0.14 we get a wave crest of 0.6H and a trough of 0.4H, the same as from other 2' order waves mentioned here in ch.VI, see p. 129 and 139 with wave profiles. Wave pressure below the crest and trough is shown on p. 120 and 121 ch.VI for the cnoidal wave and the 1' order wave. P. 120 can be compared to the Airy wave p. 44 ch.II. When this cnoidal wave hits a vertical wall and becomes a standing cnoidal wave the wave pressure will be more special.
The standing cnoidal regular wave on infinite deep water is described as a "cnoi product" of t and x (time and coordinate) in ch.XII, with a summary of the most important resulting formulas on p.317. (When progressive cnoidal waves on less deep water are the incoming waves to a vertical wall it will not necessarily result in such regular standing cnoidal waves). The wave steepness H/L = 0.14 will again result in a wave crest of 0.6H. The course of the wave profile during the time of a wave period is shown on p. 313 for H/L = 0.18. The wave pressure on the vertical wall is shown on p. 314 and 316 and 318 ch.XII. We see that the maximum positive pressure mostly does not appear when there is a crest by the wall, but up to 1/4 of a period before and after. This is due to the effect of the negative vertical acceleration.
On p. 58 and 61 ch.III there is shown the pressure on the vertical wall from Stokes' 2' order wave where it is proposed (The Technical University of Denmark 1973) to use hydrostatic pressure above mean water level and Stokes pressure below. This is mathematically correct in a 2' order wave theory. But neglecting the big vertical acceleration in the wave crest gives a leap of wave pressure at the mean water level of up to 100% for steep waves.
The simple 1' order regular sinusoidal wave is developed in ch.II and ch.IV for the progressive wave and in ch.V for the standing wave. On p. 49 ch.II there is shown a general outline of the procedure in finding the wave equation: the horizontal pressure gradient is decided in 2 different ways: directly from the horizontal water velocity and indirectly from the vertical water velocity. By this we get a wave equation with terms that all directly describe the physical conditions in the water and can be evaluated mathematically and numerically.
All the examples of waves in the thesis are theoretically correct within the 1' order (that includes the 2' order wave examples). E.g. p. 75 ch.IV shows the horizontal velocity u according to 3 different wave theories that all mathematically fulfil the wave theory of 1' order, i.e. all terms of 1' order are included. The cnoidal wave includes additionally all terms of 2' order, but this still makes it a wave theory also correct of 1' order. And even if it included some terms of 3' and higher order it would still be a wave theory correct of both 1' and 2' order. It is different which 2' order terms and higher order terms wave theories of 1' order include, so this gives the shown differences in numerical results. Then those higher order terms that are most important for a specific purpose should be included, e.g. the vertical acceleration at the vertical wall should fit the surface condition here. In a 2' order theory all terms of 2' order must be included, and additionally some 3' order and higher order terms can be included.
The cnoidal wave on arbitrary depth in the thesis ch.IX is a 2' order wave. For this it can occasionally be relevant to use the wave celerity obtained by the 3' order sinusoidal wave ch.XI. This is correct in a 2' order wave theory, but the wave is still only of 2' order. In the same way part of a 1' order wave can be improved e.g. using the cnoidal wave profile with the tall short crest and long less deep trough and then in an appropriate way use 1' order expressions for e.g. velocity distribution and pressure distribution. By this it is not a 2' order wave but one of the possible 1' order waves.