waves, whose amplitude is not small compared with the mean depth, have 

 an inherent tendency to propagate "by developing harmonics, even in a 

 channel of uniform depth. Airy first drew attention to this (cf. Lamh , 

 1932, § 188), and the phenomenon is a feature of the propagation of tides 

 in narrow channels. The reason for the suppression of even harmonics, 

 in situations of sudden change, is not known at this time. Recently, 

 Groves and Harvey (1967) have made enquiry into nonlinear effects of 

 transformation. 



7. The Relationship of Runup to Coastal Resonance 



Even a perfectly straight coastline with a uniformly sloping 

 (inclined plane) shelf can provide a resonating platform for normally in- 

 cident wave trains of the right period. In this case the shelf responds 

 as a broad canal similar to Port Alberni Inlet or the Inlet of Port 

 Lyttelton. The problem has been treated by Lamb (1932) (cf . Wilson, I966) 

 with the same result, for wave amplification, already stated in Equation 

 (11). 



An important factor in this form of resonance is the value of the 

 term KL of which K is a wave number expressed by Equation (9) and L is 

 the length of the shelf from its edge to the coastline. It is readily 

 shown that 



KL - (^^2 /g) (di/T2) (l/s2) (33) 



where d-|_ is the depth at the shelf edge, T the period of the incident 

 wave train and s = (d-]_/L) the shelf slope. Consequently the critical 

 parameter governing amplification is the value of the quantity (d^^/gT^s^) 

 or simply (d-i/T )/s and the amplification a at the shore may be written 

 as 



a = H^/H = f [ (d^/T2)/s2] (3U) 



where H is the wave height at the coastal boundary, H the wave height 

 (assumed sinusoidal) at the shelf edge and f symbolizes a function. The 

 latter involves a zero-order Bessel function of the variable (2K ^ L ^ ), 

 which in turn, through Equation (33), Involves the variable (d]_/T )/s . 



Although Equation (3^) is based strictly on linear, long-wave theory 

 (cf. Lamb, 1932, §185), we may stretch a point and derive the amplification 

 for a range of wave situations in which it is known that Hj^/H^ > 0.5 where 

 rip is the crest height of the waves above still water at the shore. For 

 example, it is known that for a solitary wave rip/Hp ~ 1.0 and that for an 

 Airy, small-amplitude, sinusoidal wave n^/H^ =0.5. A whole range of 

 intermediate values is thus possible according to the prevailing relative 

 depth (di/T2) and wave steepness (H/T ) of the waves at the shelf edge. 

 If then we allow for this contingency by writing 



n^/Hp = Y (35) 



we have from Equations (3*+) and (35) 



114 



