To simplify profile representation, many engineers fit a smooth curve to 

 their data. Several possible physical mechanisms that would give rise to 

 equilibrium profiles of the power-curve type have been described (e.g., Bruun, 

 1964; Dean 1977). Other forms that sometimes fit profile data (e.g., log- 

 arithmic, parabolic, etc.) are, like the power curve, everywhere concave 

 upward. There is a problem inherent in the use of such curves to represent 

 profile response to changes in water level; e.g., Bruun (1964) found that the 

 expression for offshore deposition, based on his adopted power curve, indi- 

 cated an unrealistic thickening of the deposit offshore. Figure 23 and the 

 following paragraph show why this and similar problems occur when a curve 

 which is everywhere concave upward is adjusted according to equation (1). 



Figure 23. Limitations of analytical models with profiles everywhere concave up. Tangents 

 will have only one point of intersection. On the other hand, the idea of ex- 

 posing a trailing edge implies that the offshore slope equals Z/X, i.e., the 

 extension of the offshore slope must intersect the profile above the water 

 surface. This is impossible if the profile is everywhere concave upward. 



Adjusting an equilibrium curve to higher water leaves a trailing edge off- 

 shore. By assumptions, the projection of this surface toward the shore must 

 intersect the profile again at the highest point of wave adjustment (see Fig. 

 21), but the tangent of any concave-upward profile will intersect it at only 

 one point and everywhere else will be below the curve. 



If it is assumed that the offshore slope gradually approaches Z/X near 

 the closure depth, then concave shapes only represent the inner part of the 

 active profile. Manipulating such curves to represent adjustments to higher 

 water levels inevitably leads to unrealistic consequences offshore. 



d. Inferring Angle of Profile Adjustment from Offshore Slope . As dis- 

 cussed previously, a uniformly sloped trailing edge suggests steady-state con- 

 ditions (i.e., no" significant change in wave climate, profile dimension, or 

 sediment type). In such cases, direct inference from slope to retreat 

 (X/Z ~ tan a) is risky because forces other than wave-induced currents may 

 have modified bottom slopes over the long timespan of profile recession. 

 Furthermore, where the retreat is small relative to total width of the 

 responding profile, the mean slope over this short distance is difficult to 

 measure precisely. Lastly, errors in estimating the critical depth would lead 

 to measuring the slope at the wrong place. Nevertheless, it may be useful to 

 consider the types of geometry implied by idealized profile adjustment, com- 

 pare them with actual profile shapes, examine alternate explanations for 



43 



