MSL shoreline. For this study, the numerical model offshore boundary 

 extended to about the 20-meter (65-foot) depth contour (MSL), about 

 15 kilometers (9.4 miles) offshore. 



A second modification to the original program was the addition of a 

 subroutine to account for energy losses due to friction. The wave 

 height, H, at any point along the wave ray can be represented by 



H = H -K r -K s -K f (7) 



where H Q is the deepwater wave height, K r is the refraction 

 coefficient, K g is the shoaling coefficient, and Kf is the friction 

 coefficient . 



Dobson's (1967) original model calculated both the refraction and 

 shoaling coefficients. The additional subroutine calculates the fric- 

 tion coefficient by integrating an expression developed by Skovgaard, 

 Jonsson, and Bertelson (1975) along the wave ray from deep water to the 

 point of interest (optionally the point of wave breaking). The integra- 

 tion is carried out using a trapezoidal integration scheme. The local 

 bottom friction factor is calculated from the local wave conditions by a 

 numerical algorithm developed by Fritsch, Shafer, and Crowley (1973). 

 The expression for the wave friction coefficient, as given by Skovgaard, 

 Jonsson, and Bertelsen, further requires a value for the equivalent 

 (Nikuradse) bottom roughness. A field observation on a sandy coast by 

 Iwagaki and Kakinuma (1963) found that the bottom roughness ranged from 

 1 to 2 centimeters. For this study, the value of equivalent bottom 

 roughness was determined from the calibration of offshore SSMO wave 

 height (wave energy) data which had been routed inshore to wave height 

 (wave energy) data measured at Johnnie Mercer's Pier gage. Although 

 some uncertainty exists with the SSMO data, as noted in Section 2(a), it 

 was used here in a simple test to determine whether or not the 

 literature values for bottom roughness were applicable on this part of 

 the coast. A value of 1.5 centimeters gave the best results for the 

 comparison of computed and measured wave energy at the beach, and this 

 value falls within Iwagaki and Kakinuma's range of values. 



The effect of including bottom friction in the wave refraction model 

 is a reduction in the wave height and, therefore, wave energy as the 

 wave ray progresses into shallow water. It has no effect, within the 

 limits of the linear theory used by Dobson (1967), on the direction of 

 wave propagation; however, reduction of the wave height does affect 

 breaking conditions, as a wave with a reduced height can propagate 

 closer to shore before breaking. For waves in shallow water, solitary 

 wave theory defines the breaking condition 



J = 0.78 (8) 



a 



where H is the local wave height, and d is the local water depth. 



The third modification to Dobson's model was a routine to stop 

 integration of the wave ray when the ratio of wave height to local water 



69 



