Vj+i=Vj + 5j, 



where 



5j = -(vj-Vj_i)Gj/(Gj-Gj_i). 

 The details of this iterative process are given in a later section. 



(15) 



When an eigenvalue Vj^ is found, the coefficients are then evaluated. One coefficient 

 can be given an arbitrary value, so A] is set to Pih2[f l(0)] . From eq (1 1), Bj is then 

 -Plhl [f l(0)] . Pairs of equations (eg (12) and (13)) for each successive interface can then 

 be used to evaluate the next Aj and Bj as discussed later. 



Finally the normahzing factor, Dj^, for mode n is obtained by the relationship 



oo 



Dn= / pF^Cndz. (16) 







This equation follows from the orthogonality of the depth functions. It is not the pressure, 

 however, which is proportional to pU, but p/-U that is orthogonal (ref 7). Therefore, Dj^ 

 must be determined such that the integral of pU^ is 1. 



From Stokes' equation (eq (6)) and eq (7-9), the integral of F'- takes the form 



^i+1 



^i+1 

 j F2(f)dz 



rj(z)F^(r)/ai + F'^(r)/a;^ 



(17) 



Therefore 



Dn = -PjW2/aj +2^ Pi[?i(Zi+i)/ai -pjri+i(Zi+i)/(aj+iPj+i)] F^^(Zj+i) 

 i=l [ 



+ w^^ - ^i+ 1 / V 1 Vp^+ 1 ) 



(18) 



where eq (12) and (13) have been used to combine terms at each interface. The derivative of 

 F takes the form 



Fl(Zi+l) = ai JAih'i [fi(Zi^i)] + Bih'2[?i(Zi+i)] 



(19) 



The Wronskian, W, is an imaginary constant (see eq (85)) and is the contribution of eq (17) at 

 the surface: 



W = -1.457495441041. 



7. Some Effects of Velocity Structure on Low-Frequency Propagation in Shallow Water, by AO Williams; 

 J Acoust Soc Am, vol 32, p 363-365, March 1960. 



