Mysak 



waves" for the fi- ^'s. These waves correspond to Robinson's plane -traveling 

 shelf waves (5). 



2. The eigenfrequencies jj ^ form a doubly discrete set. In the Cartesian 

 geometry the eigenfrequencies are given by ^.^^^ = -4f^^k v.^j, where k is a N-S 

 wave number which lies in the range k^V « 1 (5). It is clear that in the case 

 of the circular geometry, the effect of the periodicity requirement is to quantize 

 the wave number m. 



3. In both geometries the waves are nondispersive and the phase velocity 

 (wave speed) of the jth mode is given by 



c. = -4f^V^j . (16) 



From Eq. (16) we note that the wave speed is proportional to the sheK width and 

 is independent of the depth and acceleration of gravity, which is quite unlike the 

 case for gravity waves and edgewaves. In the circular geometry the waves travel 

 counterclockwise in the southern hemisphere and clockwise in the northern 

 hemisphere. 



Forced Solution 



To solve the forced problem, we expand tj on the shelf in terms of the eigen- 

 f unctions given in Eq. (15) with a ^ in the exponent replaced by oj. The unknown 

 coefficients are then determined from Eq. (11). The complete solution takes the 

 form 



^(^,'A, t) = eacpg 2_^ — 



\%]m J 1 ( 2v%]m) m (7 - y^ ■ ) 



X Jo (2V7^~^) exp[i(m^-ajt)] (m M) • (17) 



Equation (17) explicitly shows that unless the forcing frequency ^ lies within a 

 "resonant" neighborhood of x ^ ^ defined by 



|m(7-7oj)l < ^^^0 IJm(l^P)/v'7V^) Ji (2vT^)l , (18) 



the sea level behavior on the shelf is very nearly barometric. However, if ^^ 

 lies within a resonant neighborhood of m. ^, then v has an amplitude of o(lOcm), 

 the leading contribution being n ^, the j,ni the term of Eq. (17). In such a case 

 the theoretical sea-level amplitude on the shelf is given by ' ; + n J with , in 

 the neighborhood specified above. However, since the eigenfrequencies corre- 

 sponding to j = 2, 3, ... in the neighborhood of ,^ or ^ have large values of m 

 (m = o(40), 0(10-^), . . .) and since J„(l<^0 rapidly decreases with increasing m, 

 the resonant neighborhood (Eq. (18)) will be largest for the eigenfrequencies 

 corresponding to j = l. Hence we conclude that an anomalous sea-level behav- 

 ior on the shelf is essentially determined by U + Nj J with within the reso- 

 nant neighborhood of ^j ^ given by Eq. (18) with j = i. The following question 



482 



