(7) Directional spectra.

(a) The wave spectra described so far have been one-dimensional frequency spectra. Wave direction

does not appear in these representations, and thus variation of wave energy with wave direction was not

considered. However, the sea surface is often composed of many waves coming from different directions.

In addition to wave frequency, the mathematical form of the sea state spectrum corresponding to this situation

should therefore include the wave direction *θ*. Each wave frequency may then consist of waves from different

directions *θ*. The wave spectra so obtained are *two-dimensional*, and are denoted by *E(f,θ)*. Figures II-1-33

and II-1-34 display directional spectra.

(b) Measurement of a directional spectrum typically involves measurement of either the same

hydrodynamic parameter (such as surface elevation or pressure) at a series of nearby locations (within one

to tens of meters) or different parameters (such as pressure and two components of horizontal velocity) at the

same point. These records are then cross-correlated through a cross-spectral analysis and a directional

spectrum is estimated. In general, the more parameters or more locations involved, the higher the quality of

the directional spectrum obtained. The procedures for converting measurements into estimates of the

directional spectrum are outside the scope of this chapter. Part VII-3 of the CEM and Dean and Dalrymple

(1991) provide some additional details on this subject.

(c) The major systems routinely employed at the present time for measuring directional spectra include

directional buoys, arrays of pressure or velocity gauges, and the p-U-V technique. With directional buoys,

pitch-roll-and-heave or heave-and-tilt methods are used. Most directional buoys are emplaced in deeper

water. Arrays of pressure gauges or velocity gauges arranged in a variety of shapes (linear, cross, star,

pentagon, triangle, rectangle, etc.) are also used, but these are usually restricted to shallower water. The p-U-

V technique uses a pressure gauge and a horizontal component current meter almost co-located to measure

the wave field. This can be used in shallow or in deeper water if there is something to attach it to near the

surface. Other techniques include arrays of surface-piercing wires, triaxial current meters, acoustic doppler

current meters, and radars.

(d) A mathematical description of the directional sea `state is feasible by assuming that the sea state can

be considered as a superposition of a large number of regular sinusoidal wave components with different

frequencies and directions. With this assumption, the representation of a spectrum in frequency and direction

becomes a direct extension of the frequency spectrum alone, allowing the use of *FFT *method. It is often

convenient to express the wave spectrum *E(F,θ) *describing the angular distribution of wave energy at

respective frequencies by

(II-1-162)

where the function *G(f,θ) *is a dimensionless quantity, and is known as the *directional spreading function*.

Other acronyms for *G(f,θ) *are the *spreading function*, *angular distribution function*, and the *directional*

(e) The one-dimensional spectra may be obtained by integrating the associated directional spectra over

π

m

(II-1-163)

&π

Water Wave Mechanics

II-1-93

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