Theoretical calculation of the maximum force from an ocean wave hitting
a vertical wall breakwater.

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.

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.

**Example**

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.

Calculation from
1971 thesis as PDF