B FD equal to direction at the peak of the frequency-direction spectrum 

 S(f ,5) , and 5p IDS equal to direction at the peak of the integrated direc- 

 tion spectrum S(^) . 



62. Note that the letter "S" has been used to represent spectral 

 density in general, and the arguments inside the parentheses dictate which 

 spectrum is being referenced. That is, S(f,5) denotes the frequency- 

 direction (FD) spectrum, S(f) is the (conventional) integrated frequency 

 spectrum (IFS) , and S(^) is the integrated direction spectrum (IDS). 



Some Illustrated Examples 



63. To form an image of these spectra. Figure 2 shows four examples of 

 wave energy distribution. Three are contrived for illustration purposes, and 

 one is from a real wave field for comparison. 



64. Figure 2a illustrates the energy of a single wave train, i.e., a 

 monochromatic, unidirectional sea as might be represented by Equation 1. The 

 picture has three parts, a base, and what appear to be two walls rising 

 perpendicular to the base at its back edges. The base contains a grid which 

 represents the direction (labeled on the front right edge of the base) and 

 frequency (front left edge of the base) coordinates. The grid spacing is d6 

 = 2 deg along the direction axis and df = 0.00974 Hz along the frequency 

 axis. (These numbers were not chosen to confuse the reader but are the 

 resolution direction and frequency increments used in the analysis described 

 later in this report.) 



65. Elevations above this base grid form a three-dimensional picture of 

 S(f,5) . For a single wave train, this appears as a spike at the frequency 

 and direction (roughly, f = 0.17 Hz and ^ = 45 deg) of the wave train. 

 (Graphics packages notwithstanding, it should formally appear as a rectangular 

 solid with vertical walls and a top and base both of dimensions df x d^ .) 

 Values of zero exist at all other grid locations. The volume under the spike, 

 S(0.17 Hz, 45 deg)-df-d5 , is half the squared amplitude of the wave train, 

 following Equation 7, which is proportional to the wave energy, following 

 Equation 3. 



66. If the frequency- direction spectriom of Figure 2a is summed over all 

 directions for each frequency, the frequency spectrum is obtained, as in 

 Equation 8. This is shown as the vertical appearing panel at the right rear 



26 



