526 



in error unless both b and d are constant. 



The problem also is attacked by Shuto (1974) , 

 who allows for the variation of both b and d but 

 arrives at a result (which he integrates numerically) 

 that appears to be inconsistent with conservation 

 of energy. However, his result is consistent with 

 (28) in the limit U + and with (29) in the limit 

 U + °> or, more precisely, with the result obtained 

 by neglecting only terms of exponentially small 

 order in (27) , 



F - (%J)^^^{1 - 2{ki) 's} (U + ■») , 



(30) 



which is in error by less than 1% for U > 70. It 

 therefore appears that Shuto 's numerical results 

 are not significantly in error (on the scale of his 

 plots) over the entire range of (J. 



Experiment 



Shuto (1974) compares his results with his own 

 experimental observations and with those of Iwagagi 

 and Sakai (1969) for shoaling waves periods from 

 1.2 to 6 seconds on uniform slopes of 1/20 and 1/70. 

 He concludes that linear surface-wave theory (which 

 presumably accounts exactly for dispersion) is 

 superior to his cnoidal-wave results for (J < 30 and 

 conversely for (J > 30 and that the latter are good 

 for a/d as large as 0.8. 



5. SLOWLY VARYING SOLITARY WAVE 



Theory 



however, he does not obtain an explicit description 

 of the oscillatory tail, nor does he allow for the 

 possibility of expanding the slowly varying phase 

 X(x) as well as N(6,x) [see (23)]. 



Ko and Kuehl (1978) have criticized Johnson for 

 this latter omission and develop a joint expansion 

 of (the equivalents of) N and x- They conclude 

 that the solitary wave ("soliton") experiences an 

 irreversible energy loss in the sense that it does 

 not re-establish itself if the channel gradually 

 reverts to its initial, uniform breadth and depth. 

 This may be, but the proper form of the inner 

 expansion is to some extent a matter of expediency, 

 and the ultimate validity of any particular expansion 

 can be established (albeit heuristically) only 

 through matching to a proper outer expansion. Ko 

 and Kuehl appear to overlook the crucial role of 

 matching, and, at least in this important respect, 

 their results must be regarded as incomplete. 



Johnson's results are readily generalized to 

 allow for the variation of both b and d and reveal 

 that 



5 = 2(3a/d)~^/2dA(bd5/2) = (3a) "3/2 (2dAb + 9d") 



(32) 

 is an appropriate measure of the slow variation of 

 the channel (this same measure also is appropriate 

 for a cnoidal wave for U > 100) . The Boussinesq 

 equations (13) and KdV equation (16) are based on 

 the restriction & = 0(1) as a -I- [cf. (2)], whereas 

 (26) and (31) are based on the stronger assumption 

 |6| << 1 [cf. (3)]. Moreover, a consideration of 

 the special case of linearly increasing breadth and 

 constant depth [Miles (1978a)] suggests that the 

 wave ultimately ceases to be solitary and evolves 



The slowly varying solitary wave 



= asech- 



(3ga) 

 2d 



I^ 



3 V3 -2/3 -1 

 — J b d 

 4 



[g(d+a)] % 



(31a, b) 



(31c) 



is obtained by letting U + » with KB = 0(1) in (26) 

 and (27).* There is, however, a new difficulty: 

 none of the integrals I, M, and M [see {20a) and 

 (21a, b) ] , which now are proportional to b ' d ' , 

 b^' ^d, and b^' ^d^'^, respectively, is conserved ex- 

 cept for special variations of b and d. [The failure 

 of the condition <N> = in the limit U + «> is a con- 

 sequence of the loss of the displacement -a<cn > ~ 

 a/K, which cancels the mean of acn (2K9) when inte- 

 grated over -K < 2x9 < K. ] It follows that, except 

 in the special case bd ' = constant for which (31) 

 is an exact solution of (16) and M and M vary like 



and d' 



■3/2 



, respectively, (31) cannot be a uni- 



formly valid approximation to the solution of the 

 KdV channel equation (16) ; instead, it is the first 

 term in an inner expansion, which must be matched 

 to an appropriate outer expansion. 



Johnson (1973) obtains the next term in an inner 

 expansion for b = constant and finds that it can 

 be matched to an appropriate outer expansion if d 

 is increasing in the direction of propagation (the 

 solitary wave may undergo fission if d is decreasing) ; 



*The prediction that a « b~^/^d ^ appears to be due 

 originally to Saeki, Takagi, and Ozaki (1971); see 

 also Shuto (1973, 1974) and Miles (1977a). 



0.4 



0.2 



0.1 



% 



0.05 



0.02 



0.01 



10 



20 40 



Vd 



80 100 



FIGURE 1. Decay of a solitary wave in a linearly ex- 

 panding channel. The wave is propagating in the posi- 

 tive-x direction, where x is measured from the virtual 

 origin at which b = 0, and enters the diverging chan- 

 nel (from an entry section of uniform width) at 

 x/d ^ 10. The amplitudes at the transition station are 

 a/d = 0.05(x), 0.1(+), 0.2(0), and 0.4('). The dashed 

 lines have slopes of -2/3. 



