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. jensen@dadlnet.dk, www.mejlhede.dk

Introduction to wave pressure and shock force on a vertical wall, as PDF.

Thesis from 1977, 12 chapters as 12 PDF files: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

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.

0 p 0-18,
319-323 Contents, abstracts, introduction, references

1 ch.I p 19-30 Practical considerations on regular waves

2 ch.II p 31-49 Progressive first order deep water wave

3 ch.III p 50-61 Historical background

4 ch.IV p 62-80 Progressive
first order wave on arbitrary depth

5 ch.V p 81-99 Standing first order wave and wave pressure

6 ch.VI p 100-142 Progressive cnoidal deep water wave

7 ch.VII p 143-161 Progressive and standing second order sinusoidal waves

8 ch.VIII p 162-215 Progressive
cnoidal shallow water waves

9 ch.IX p 216-247 Progressive cnoidal wave on arbitrary depth

10 ch.X p 248-288 Formulas and tables for the progressive
cnoidal wave on arbitrary depth

11 ch.XI p 289-307 Progressive third order sinusoidal wave

12 ch.XII p 308-318 Standing cnoidal deep water wave