where θ is the angle between the sailing line and the direction of wave propagation (Figure II-7-40). Thus,

the *transverse *waves travel at the same speed as the vessel and, in deep water, θ has a value of 35E16' for the

(4) At increasing distances from the vessel, diffraction causes the wave crest lengths to continually

increase and the resulting wave heights to continually decrease. It can be shown (Havelock 1908) that the

wave heights at the cusp points decrease at a rate that is inversely proportional to the cube root of the distance

from the vessel's bow (or stern). *Transverse *wave heights at the sailing line decrease at a rate proportional

to the square root of the distance aft of the bow (or stern). Consequently, the *diverging *waves become more

pronounced with distance from the vessel.

(5) The above discussion applies to deep water, i.e. water depths where the particle motion in the vessel-

generated waves does not reach to the bottom. This condition holds for a Froude number less than

approximately 0.7, where the Froude number F is defined by

(II-7-22)

(6) As the Froude number increases from 0.7 to 1.0, wave motion is affected by the water depth and the

wave crest pattern changes. The cusp locus line angle increases from 19E28' to 90E at a Froude number of

one. The *diverging *wave heights increase more slowly than do the *transverse *wave heights, so the latter

become more prominent as the Froude number approaches unity. At a Froude number of one, the *transverse*

and *diverging *waves have coalesced and are oriented with their crest perpendicular to the sailing line. Most

of the wave energy is concentrated in a single large wave at the bow. Owing to propulsion limits (Schofield

1974), most self-propelled vessels can only operate at maximum Froude numbers of about 0.9. Also, as a

vessel's speed increases, if the vessel is sufficiently light (i.e. has a shallow draft), hydrodynamic lift may

cause the vessel to plane so that there is no significant increase in the height of generated waves for vessel

speeds in excess of the speed when planing commences.

(7) For harbor design purposes, one would like to know the direction, period, and height of the waves

generated by a design vessel moving at the design speed. For Froude numbers up to unity, Weggel and

Sorensen (1986) show that the direction of wave propagation θ (in degrees) is given by

θ ' 35.27 ( 1 & *e * 12 (*F*&1) )

(II-7-23)

(8) Then, from Equation II-7-21, the *diverging *wave celerity can be calculated, and the wave period can

be determined from the linear wave theory dispersion equation.

(9) Figure II-7-41 is a typical wave record produced by a moving vessel. Most field and laboratory

investigations of vessel-generated waves (Sorensen and Weggel 1984; Weggel and Sorensen 1986) report

the maximum wave height (*H*m, see Figure II-7-41) as a function of vessel speed and type, water depth, and

distance from the sailing line to where the wave measurement was made. Table II-7-5 (from Sorensen

(1973b)) tabulates selected *H*m values for a range of vessel characteristics and speeds at different distances

from the sailing line. These data indicate the range of typical wave heights that might occur for common

vessels and show that vessel speed is more important than vessel dimensions in determining the height of the

wave generated.

Harbor Hydrodynamics

II-7-55

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