Wave pressure on a vertical face breakwater.

For protection against the fierce waves from the ocean we build breakwaters around our harbors.

It can be a vertical face breakwater of big prefabricated concrete caissons, and to design the breakwater and its foundation we need to know the design wave pressure. We shall here consider both the biggest shock force of very short duration and also the more long term maximum pressure from regular waves (duration: a couple of seconds).

Wave pressure magnitude on the upper part of a breakwater, from the top of the wave is important:

Vertical acceleration reduces wave pressure, and the thin air slit reduces the maximum shock force.

Incoming waves can be regular or irregular of different shapes. We shall here consider 3 different waves hitting a vertical wall:

Shock force from wave with air compression: When a wave from the ocean hits a vertical wall there can be a high impressive splash of water. This can be caused by a breaking wave, enclosing by the wall a pocket of air, being compressed by a lot of wave energy, energy that then sends the overlying water upwards. See photo in my leaflet http://mejlhede.dk/waves/waveintr.pdf .

Shock force from air-ventilated wave front: A bigger shock pressure on the wall but of shorter duration will occur when a vertical wave waterfront hits the wall. This may seem like a so called hammer shock, an enormous force decided by the sound velocity of water, but it is not a hammer shock, the shock that a solid hammer will give. Because just before the fluent “water hammer” hits the wall a thin slit of air has to be squeezed out, and this big air pressure propagates into the fluent water and makes the nearest water deflect upwards, by which the water pressure on the wall is reduced to about 1/10 – 1/5 of the possible hammer shock (according to my theory, and according to tests in USA). Most often by such a plane vertical wave front hitting a vertical plane wall the shock pressure will be even substantial less, because the fast windflow in the air slit will cause a random wave uneven water front (as also seen by the USA tests).

Pressure from regular standing waves: A more moderate pressure of longer duration will occur when, without a shock force, the incoming wave energy over a few seconds is transformed to a regular standing wave, with water moving up and down by the wall with a wave period of about 10 seconds, which e.g. for 2-3 seconds gives wave pressure from a 5 meter high wave crest part on a breakwater on 10 meter water depth. Such a wave crest has big downwards vertical surface acceleration, up to 10 m/sec² (= acceleration of gravity). So the water pressure in this wave top is much less than hydrostatic pressure which would give a pressure of 5 m water pressure at the mean water level, which is the pressure the classical potential theory gives, a theory based on small wave heights, but used practically also for normal waves.

Breakwater design: When designing the stability of a vertical face breakwater, its overturning, then the wave pressure on the upper part of the breakwater is of most importance. If it is the short duration shock force or the longer duration standing wave pressure that is most important depends on the type of breakwater, its elasticity etc. and how critical minor deformations are considered. For design of the breakwater wall the maximum shock force may be the most important.

Maximum shock force: air ventilated shock, from a vertical wave front.

Hammer shock: a big steel hammer or a steel bar or rod hits a hard wall: the steel front will be elastically compressed and the compression propagates along the bar with the speed of sound. Scientific literature proposed the maximum possible wave shock force by considering in the same way a hammer of water: with the sound velocity and compression of water giving the hammer shock: pressure = water velocity multiplied by water sound velocity and unit mass.

But ocean water is different from solid steel so there will not be any hammer shock from a big wave top water hammer. The waterfront close to the wall will escape upwards and thereby reduce the elastic pressure, a reduction that starts just before the water reaches the wall because of the growing pressure of squeezing the air out of the thin air slit between the wall and the water, an air pressure that will reduce the oncoming water speed. (If some of the air penetrates into the water this air will “soften” the shock pressure). It is not the whole volume of the wave top that suddenly gives the shock force. In my calculations I only considered the practical so called much less hydrodynamic mass. (The considerate momentum energy of the behind coming wave water will then just cause a general rise of water somewhat like for a standing wave).

I have in 1969 thus proposed a theory with very approximate calculations of the maximum shock force: http://mejlhede.dk/waves/shocke.htm , which for an oncoming water velocity of 5 m/sec gives a shock pressure of 70 m water pressure = 0.7 MPa, a big pressure. But USA Waterways Experiment Station used the formula for max (hammer) shock force (Kamel 1968) using the elasticity and sound velocity of water which gives a shock pressure 10 times as big: 7 MPa. To investigate the formula they made experiments with hitting a horizontal water surface with a horizontal steel plate. The experiments gave in a very few cases a maximum shock pressure equivalent to my theory, but mostly much less pressure. Less pressure because the plane water front most often started to make “ripples” because of the effect of the blow from the out pressing air.

Wave pressure from regular standing waves, using reduction from vertical acceleration

I have described a hydrodynamic wave theory of 1’ order and of higher order, a theory I find more simple and more direct than the potential wave theory, and which includes the effect of Newton’s 2 law (momentum) e.g. in the wave top all the way up to the surface. The formulas can be seen on page 3 in my pdf leaflet: Wave pressure in a standing wave; see also page 8 for pressure according to the potential theory of 1’ and 2’ order.

The classical potential Airy wave theory will here give a wave pressure at mean water level equal to hydrostatic pressure, so no reducing effect of the vertical acceleration is included in this 1’ order theory for waves of small height. Is the acceleration effect then included in a 2’ order theory for not so small waves? It is seen that here hydrostatic pressure is also used.

A 1’ order wave theory considers all 2’ and higher order terms as negligible small. So these terms can be dismissed. Or they can be included where wanted. So where the formulas get more convincing for practical use necessary higher order terms should be included, e.g. for the vertical velocity and acceleration and pressure of the surface.

The complete wave theory can be seen on: http://mejlhede.dk/waves/wavese.htm