Theoretical calculation of the maximum force from an ocean wave hitting a vertical wall breakwater.
Niels Mejlhede Jensen, Bogelovsvej 4, 2830 Virum. email@example.com, www.mejlhede.dk
Introduction to wave pressure and shock force on a vertical wall, as PDF.
On www 2011 with this summary:
When waves from the sea hit a coast with a vertical wall or a vertical harbour breakwater then we occasionally can experience a fierce shock. This brief big shock force on the wall is important in designing the wall and harbour structures. In my thesis from 1971 I have given a theoretical calculation of the maximum pressure from water hitting the vertical wall.
Usually when a wave comes from the ocean e.g. with the wave height Hi = 1 meter, as a "normal" wave of the same form on the front and the rear, coming perpendicular to a vertical wall, then the wave will stop at the wall with a double wave height Hs = 2 m (approximately), and the wave will reflect and travel out into the ocean again in some wave form. If we have identical regular waves with Hi = 1 m coming perpendicular to the wall e.g. every 10 seconds, the whole area in front of the wall will get a pattern of standing waves with Hs = 2 m. The water by the wall will move up and down and the wall will get a wave pressure decided by the height of the water surface and its vertical acceleration and the acceleration distribution down by the wall. (The theory of regular waves and their pressure on the vertical wall breakwater is considered in my thesis from 1977).
If the waves arrive more irregularly, e.g. a big incoming wave arrives at the wall just after the previous wave is reflected, then the big wave may break in a manner so a vertical waterfront hits the wall. If the top of the breaking wave reaches the wall early enough to confine pockets of air between the wall and the water, then these air pockets will soften the shock force by compression of the air. This is called a compression shock.
If also the wave top stays in place so the whole waterfront is vertical as it moves towards the wall, then air can escape from the slit through the opening at the top resulting in a ventilated shock. This gives a bigger force on the wall than the compression shock, a force so big that it has been believed it is decided by the elasticity of the water and the wall structure like a hammer shock. But in the situation where a vertical front of water hits a tight vertical wall then the air in the slit in between must be pressed out of the slit, and this gives a reactive force from the air which will prolong and soften the shock to a substantial less maximum value, as can be seen from my theoretical calculations included here from my thesis of 1971.
We have a harbour breakwater with vertical sides on 10 m water depth. Waves from the ocean hit the breakwater perpendicularly and occasionally a big wave breaks right onto the breakwater. We here consider the case where an incoming wave breaks just in front of the vertical wall so that a waterfront like a vertical wall of water hits the breakwater wall and gives a shock force.
The vertical waterfront has a height of R = 1 m and a horizontal velocity of U = 3 m/sec.
From fig.3 with the graph (it is before page 1 in the thesis) we get from the top graph (water without air bubbles):
P/U2 = 3.2 that is P = 3.2 x 3x3 = 29 m of water or app. 30 m (= 30 MP/m2 = 300 kN/m2).
The shock pressure increases and decreases within app. 1/100 sec. (equation 49). When the waterfront is 10 mm close to the wall the pressure (in the air slit and on the wall) is 10 m of water = 10 MP/m2 = 100 kN/m2 = 1 atm above atmospheric pressure (11).
The calculations are based on an approximate theory:
Between the vertical wall and the vertical waterfront there is a narrow slit of air from which the air is squeezed vertically out (fig. 1). The out flowing air gives a reactive force to the air slit so the pressure of the air increases (4) and (5). In my calculations I consider the air as incompressible during the first part of the shock, until a wave pressure of 10 MP/m2 = 1 atm. Then in the second part of the shock I change to close the slit and consider the air as compressible according to the equation for isothermal compression, as a practical solution to the combined air outflow and adiabatic compression of the slit.
The shock force can be said to be created by a hydrodynamic mass Sm = 0.45R = app. 0.5 m (23) which comes at the velocity U and is slowed down by 0.65 m/sec. (25) and from then on the air in the slit starts being compressed. The increasing air pressure in the slit propagates to the water creating a vertical water velocity (26). The water squeezed upwards means that the back of the hydrodynamic mass can have a little higher horizontal velocity than the front so that the front is slowed down by 0.85 m/sec. and the back by 0.45 m/sec. (27) and (29) when reaching the pressure 10 MP/m2.
From then on the hydrodynamic mass compresses the air slit and in (30) we use that kinetic energy from the horizontal velocity equals the energy of the isothermal compression of the air slit. By this we could find how narrow the slit will become, delta-min, to find the maximum pressure pm (p-star) of the shock force from (12). But to include the effect of the generated vertical velocity in the hydrodynamic mass again, changing the slow down a little (35) and (37), we change from the equation of energy (30) to the equation of momentum (32). This gives equation (38) to determine delta-min, by which the shock force is determined by (39), and resulting in the graph on the mentioned fig. 3.
So the air is not ventilated out of the air slit without giving a reactive force that softens the wave shock force, and the water of the wave top is not to be considered like a long solid mass (bar) that hits the wall as a hammer shock determined by the elasticity of the water. It is only a relative small hydrodynamic mass that creates the very shock force. The shock force in the example here could be from a big incoming approximate solitary wave with a wave height of Hi = 6 m breaking at the wall with a horizontal water velocity of 3 - 6 m/sec. The wave top reaches about 30 m out from the wall, but it is then only 0.5 m hydrodynamic mass that creates the actual shock force. The rest of the wave becomes a kind of an irregular standing wave reaching a maximum height of maybe 13 m.
The hydrodynamic mass is determined approximately. The shock pressure slows the water down, much at the waterfront and less further out. So the horizontal acceleration decreases from the waterfront and out somehow, and we approximate the decrease to be parabolic as shown in fig. 2. The total horizontal acceleration is determined by the horizontal dynamic equation (18). In the air slit we get a pressure distribution with increasing air pressure down in the slit. This vertical pressure distribution is also found in the water by the waterfront (14). Using the equation of continuity (the conservation of mass) we then get an approximate expression to be used for the horizontal acceleration (16) and (17), giving the tangent to the graph for the horizontal acceleration. This horizontal acceleration decreasing from the waterfront and out is then totalled in a constant acceleration over a short length: the so-called hydrodynamic mass (22) and (23).
By model tests the waves and structures are scaled down to e.g. or less while the air pressure of the atmosphere is the same 1 atm, so model tests are treated separately in my theses here from 1971. Further there is a consideration of the influence of air bubbles in the water.