Due to its forward speed, the vessel experiences a modified wave spectrum, this is explained in the following sections.

When studying the behaviour of vessels in a sea way it is important to consider the frequency with which the vessel encounters waves (the encounter frequency f_e). The encounter frequency depends on the velocity of the waves, the velocity of the vessel and the heading of the waves in relation to the vessel.

The wave heading, m, is the angle between the vessel track and the wave direction. Thus following seas correspond to m=0; since both vessel and waves are travelling in the same direction. Head seas corresponds to m=180°, and beam seas m=90° (starboard) or 270° (port).

Encounter frequency

The encounter frequency may be calculated by considering the component of vessel velocity in the direction of the waves and subtracting the wave phase velocity c, this is shown visually above. The relative velocity is given by the equation below.

Hence the encounter period T_e is given by:

Recalling that: , it follows that the encounter frequency is given by:

Often the vessel is assumed to be operating in deep water. Thus using the deep water wave speed relationship, the encounter frequency simplifies to:

An important concept when calculating vessel motions is that of the encountered wave spectrum. This is a transformation of the wave spectrum which describes the waves encountered by a vessel travelling through the ocean at a certain speed. This is effectively a Doppler shift of the spectrum.

Given the relationship between the encounter frequency and the wave frequency (from above), and since the total energy in the ocean must be constant whether experienced by a stationary or moving observer, the transformation of the wave spectrum to the encountered spectrum is shown in the following equation. Examples of the same spectrum encountered at different speeds are shown in the figure below.

for the stationary observer: , whilst

for the moving observer:

Thus:  or

recall that for deep water , thus .

Hence the spectral ordinates in the encounter frequency domain are given by:

Comparison of encountered wave spectra at different speeds