One of the primary goals is to be able to compare the ability of different design alternatives to carry out the required mission. We have seen how to compute the motion energy spectrum and the variance of the spectrum. Now we will investigate how to compute the probability of exceeding the defined design criteria. This probability may then be used to guide the selection of the appropriate vessel.

By assuming that the probability density function of the motions follow that of a Rayleigh distribution, it is possible to evaluate the probability of exceeding some critical value z_crit given the variance of the motion energy spectrum, m_0z. This is given by:

The above equation assumes that the sea state and hence the variance of the motion energy spectrum is constant. These are described as short-term statistics. Over longer periods of time an alternative method is required. If the probability of encountering the seas states is known, long-term statistics may be calculated using the equation below.

where q_i is the probability of sea state i occurring, and p_i is the probability of exceeding the criterion in sea state i.

These types of statistics may be used to evaluate the down time over a year or the life cycle of the vessel.

In the previous section we have touched on the subject of joint probability; the probability of two independent events occurring is the product of the probabilities of the two individual events. Thus the joint probability of two independent events (x,y) is described by:

This method may be used to compute the probability of a slam occurring. For a slam to occur two events must occur:

§   The bow must emerge from the water; the relative motion at the bow must be greater than the draught at the bow.

§   The relative velocity of the bow with respect to the water surface, at the time of impact, must exceed a certain value. This value is referred to as the threshold velocity and is given the symbol u_*. Experiments for a wide variety of hull forms have shown that the threshold velocity is given by:

where B is the total beam of the vessel at amidships; b is the half beam at a station 0.25L from the bow, at a point 0.03B above the keel; L is the length of the vessel.

It is usual to compute the probability of a slam at station 90 to 95% of the length of the vessel.

It may be shown that these two events are statistically independent, hence the joint probability law may be applied and the probability of a slam occurring is the product of the probabilities of the two events occurring.

A summary of events relevant to seakeeping is given below: