Continental Shelf Waves 



for e = -t/R « 1 and y = -il/coYi, a = f ^R^gd, B/3^, and 'd/'^^j each of order 

 unity; the dimensionless variable ^ is defined by <f = (r - ^)/l. Hence 

 7? = 0(ea(pQ/7), which implies that the shelf sea-level behavior is also essenti- 

 ally barometric unless resonance occurs. The unforced equations obtained 

 from Eqs. (10) and (11) together with the above stated boundary and continuity 

 conditions constitute an eigenvalue problem with eigenvalue y. Since y^or^, it 

 follows that when the forcing frequency is in a small neighborhood of an eigen- 

 frequency, 17 on the sheK is amplified, thereby leading to a nonbarometric be- 

 havior. For a nonresonant response however, the Australian shelf sea level 

 behavior is also barometric to within 0(10" ^). 



Unforced Solution: Continental Shelf Waves 



The appropriate homogeneous solutions to Eqs. (10) and (11) which are 

 single-valued in are given by 



v = 



E A^K^ W{U+^)] exp(im^), ^ > 1 , 



m 



(12) 



E B^Jo (2VyJ^) exp(im0), 0<^<1. 



where m= 0, ± 1, . . . , K^ is the mth order modified Bessel function of the second 

 kind, and Jg the zeroth order Bessel function of the first kind. Application of the 

 continuity conditions to Eq. (12) implies that A^ = Bq = and implies the eigenvalue 

 equation 



Jo(2v7S) [1- A+ e(l+ e) /3RK;/7mK^] - [A( 1 + e )A/7S] J^ ( 2v'>^) - 0(m + 0) , (13) 



where K^ and K^ are evaluated at ^= 1 and A = d/D. For /3RK^/K^ =0(1) and 

 e/A = 0(1), Eq. (13) implies that to be consistent with our earlier approxima- 

 tions we must take the eigenvalues to be given by the zeros of Jg . Let k^.^ 

 (j = 1, 2, . . .) denote the roots of the equation Jo(^) = 0; then the eigenfrequen- 

 cies are given by 



^j,m = -4f'fe/R^oj ("1+0) . (14) 



Finally, in view of Eq. (14), the adjusted sea- level eigenf unctions can be written 

 in the form 



ro(A), ^ > 1 , 



V-. Jr,0,t) = \ (15) 



I Jo (2x7^7^) exp[i(m0-u;3 „,t)] + 0(A), 0<^<1, 



where y^ . is defined by 2\-"y^ = ^0 j • 



The following properties of the unforced solution should be noted: 



1. Each eigenf unction f?. ^ is in the form of a circularly traveling wave 

 almost entirely confined to the shelf; hence we adopt the term "continental shelf 



481 



