of Figure 2a. Since there is energy at only one frequency in the frequency- 

 direction spectrum, there is energy at only one frequency in the frequency 

 spectrum and the spectrum appears as a spike. (Again, it should be a rec- 

 tangle of width df .) All direction information is lost, but its cumulative 

 contribution has been accounted for. In this case, the area under the 

 frequency spectrum, S(0.17 Hz)-df, is half the squared amplitude of the wave 

 train. 



67. The direction spectrum for this case is shown in the vertical 

 appearing panel at the left rear of Figure 2a. It is found by summing the 

 frequency-direction spectrum over all frequencies for each direction, as in 

 Equation 9. As with the frequency spectrum, there is energy from only one 

 direction, so the direction spectrum is also a spike (i.e., rectangle of width 

 dd ). The area under this curve, S(45 deg)-d^ , is again half the squared 

 amplitude of the wave train. 



68. The unidirectional, monochromatic case is the simplest example of 

 the different spectral types. A more complicated sea can be constructed or 

 represented by simply adding more spikes (rectangular solids) with various 

 amplitudes and at various frequencies and directions to the frequency- 

 direction spectrum. 



69. Figure 2b illustrates a case where energy is distributed in 

 frequency but not in direction. This unidirectional, irregular sea is 

 represented by a series of spikes along a line of constant direction (here, 

 again, about 45 deg) . The volume under a spike, proportional to the wave 

 energy at the frequency and direction where the spike is located in the grid, 

 can be considered as half the amplitude squared of a wave train having that 

 frequency and that direction. The energy distribution shown in Figure 2b is a 

 JONSWAP (frequency) spectrum as described by Hasselmann et al . (1973) which 

 has been assigned here to a particular direction. When summed over all 

 directions, the result is the frequency spectrum shown in the right rear 

 vertical panel. Since the directional distribution has a width of only one 

 incremental arc, the frequency spectrum is simply a shadow of the frequency- 

 direction distribution. Note that this is not the case in general; the 

 frequency spectrum is the integral with respect to direction of the frequency- 

 direction spectrum. 



70. The direction spectrum of Figure 2b, shown in the left rear 

 vertical panel, is simply a spike because all the energy is concentrated in 



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