deformation of long waves over a gentle slope using the nonlinear long 

 wave theory and power series expansions. They found the shoaling 

 coefficient to be a function of beach slope when S > .01. The steeper 

 the slope the smaller the shoaling coefficient, a fact which can be 

 attributed to partial reflection. In fact, due to friction effect, the 

 ratio H/Hq for a given value of d/LQ decreases instead of increases 

 (Sawaragi, Iwata, and Masayashi, 1976) . 



The first reliable experiments were conducted by Brink-Kjaer and 

 Jonsson (1973) and Flick (1978) . Figures 7 and 8 show results for different 

 values of Hq/L^. Flick separates the first, second, and third harmonics 

 from his wave data and is subsequently able to give a reliable ex- 

 perimental shoaling coefficient. Flick compares his results with 

 Le Mehaute and Webb (1964) (third-order Stokesian) and also with the 

 cnoidal solution of Svendsen and Brink-Kjaer (1972) in shallow water 

 (see Fig. 6) . 



The shoaling coefficient of a hyperbolic wave is also fairly well 

 verified by Iwagaki (1968) who gives results very close to the two 

 mentioned above. 



Svendsen and Hansen (1977) compared the shoaling of cnoidal wave 

 with a set of careful experiments and claimed that other experimenters 

 (Wiegel, 1950, Iversen, 1951; Eagleson, 1956) carried out their ex- 

 periments on too steep a slope for the shoaling theory to be valid. 

 Furthermore, they calculate the damping due to viscous friction, 

 obviously important on a gentle slope. Svendsen and Hansen concluded 

 that if the wave height at depth d/LQ = 0.10 is matched between cnoidal 

 and linear, rather than the energy flux, the cnoidal theory predicts 

 the shoaling quite well, even close to breaking with small deepwater 

 wave steepness Hq/Lq < 3 to 4 percent but not beyond. Consistently, 

 with all theories, the wave just before breaking suddenly peaks up very 

 rapidly (Le Mehaute, 1971). In this range of values, all shoaling 

 theories (third Stokes, cnoidal and hyperbolic) tend to slightly under- 

 estimate the value of the shoaling coefficient. Subsequently, the cal- 

 culated breaking wave height tends to be underestimated. The linear 

 wave theory underestimates the breaking wave height most significantly, 

 sometimes by a factor of almost 2 (Fig. 9). 



It is pertinent to remember that (a) the shoaling coefficient 

 given by the linear theory is valid beyond the limit generally con- 

 sidered as valid for a linear theory, and (b) the shoaling coefficient 

 given by third-order Stokesian wave is fairly well verified ex- 

 perimentally and actually very close to the value given for the cnoidal 

 wave, even though, as in the case of the linear wave, free-surface 

 profile, pressure, velocity, and acceleration could be significantly 

 different . 



In general, the linear theory can be applied throughout from deep 

 water to shallow water and then the linear breaking wave height is multi- 

 plied by a coefficient function of the beach slope (Koh and Le Mehaute, 

 1966) . After the wave height, H, is determined as a function of the 

 deepwater wave height, H^, and wave period, T (or deepwater wavelength 

 Lq) , all other shallow-water characteristics (free-surface profile. 



