527 



0.8 



0.4 



0.2 



% 



0.1 



0.05 



0.025 



10 



20 



Vd 



40 



80 100 



FIGURE 2. Growth of a solitary wave in a linearly con- 

 tracting channel. The wave is propagating in the nega- 

 tive-x direction (right to left) , where x is measured 

 from the virtual origin at which b = 0, and enters the 

 converging channel (from an entry section of uniform 

 width) at x/d i 94. The amplitudes at the transition 

 station are a/d = 0.05(x), 0.1(+), 0.2(o), and 0.4(-); 

 the corresponding slopes of the dashed lines are 

 -0.39, -0.34, -0.41, and -0.42. 



into a dispersive wave train for which the first 

 peak closely approximates a solitary wave in shape 

 but is followed by successive peaks of only gradually 

 diminishing amplitude. There remains, however, the 

 difficulty of nonconservation of mass, and the 

 general problem of an aperiodic wave (in particular, 

 an initially solitary wave) in a gradually varying 

 channel is unresolved at this time. 



Experiment 



Shuto (1973) compares Green's law, a = d ", and the 

 present prediction a = d~l, with the experimental 

 observations of Camfield and Street (1969) and Ippen 

 and Kulin (1954) for shoaling of solitary waves on a 

 uniform slope. He concludes that the range of valid- 

 ity of the "-1 power" law decreases with increasing 

 slope and that the "h power" law holds for slopes in 

 excess of 0.045 and a/d as large as 2.0. A more 

 precise comparison can be made on the basis of (32) , 

 which reduces to 



6 = 9(3a/d)"3/2d' 



(b 



constant) 



(33) 



for a channel of constant breadth. The estimated 

 critical values of 6, such that a = d~ or d~ < pro- 

 vide better fits to the data for 6 < 6* or 6 > 6*, 

 respectively, are 6* = 0.10, 0.10, and 0.09 for 

 slopes of .01, .02, and .03, and a « d~ is typically 

 within the experimental scatter for 6 < 0.01. 



Chang and Melville (unpublished) have recently 

 measured a(x) in linearly diverging and converging 

 channels. Their results for a diverging channel 

 (Figure 1) tend to confirm the prediction a = b~^/3 

 for initial values (at the transition from a uniform 

 channel) of 0.05 < a/d < 0.2 [the corresponding 

 values of 6 = 2 (3a) ~^/2 (db'/b) are in the range 

 (0.01, 0.07], although the decay ultimately exceeds 

 this inviscid prediction — presumably in consequence 

 of viscous or other dissipation — and exceeds it 

 after only a rather brief section for an initial 



value of a/d =0.4. Their results for a converging 

 channel (Figure 2) predict a growth that is roughly 

 approximated by a " b-o.4. Dissipation in the 

 converging channel would tend to decrease the magni- 

 tude of the exponent, but why this decrease should 

 be so much larger than the corresponding increase 

 for the diverging channel is not clear at this time 

 (intuition suggests that reflection could be more 

 significant in a converging than in a diverging 

 channel, but neither analytical nor experimental 

 evidence is available to support this conjecture) . 



ACKNOWLEDGMENT 



This work was partially supported by the Physical 

 Oceanography Division, National Science Foundation, 

 NSF Grant OCE74-23791, and by the Office of Naval 

 Research under Contract N00014-76-C-0025 . Most of 

 the material in Sections 3 and 4 has been published 

 elsewhere [Miles (1978b)] in slightly different 

 form. 



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