The solutions to Stokes' equation that are used are the modified Hankel functions of order 

 1/3, hi(f) and h2(f). The depth function is a linear combination of these two independent 

 solutions: 



Fn/^)-An,ihl(fn) + B„,ih2(rn), (9) 



where Fj^ is the unnormalized form of U^. The coefficients a^, j and Bj^ j for mode n in layer 

 i are determined to satisfy boundary conditions, which will be listed below. Values of Aj, for 

 which the boundary conditions can be satisfied are the eigenvalues. 



The first boundary condition is the radiation condition. It is satisfied by using a nega- 

 tive sound-speed gradient in the deepest layer, which extends to infinite depth, and by letting 

 the depth function there be proportional to h2 only. That is, 



F„(z) = B„h2(rn)- (10) 



At the surface the depth function is zero: 



Fn(0) = 0, (11) 



and at layer interfaces, pU and its depth derivative are continuous: 



PjFj^j(z) = Pj+iFj^j+l(z); (12) 



dFj^j(z)/dz = dFj^j+l(z)/dz. (13) 



Here pj is the density in layer i, and the excess acoustic pressure, p, is given by 



p = pU. 



If U is assumed to be the vertical component of the velocity potential, eq (12) and (13) are 

 equivalent to requiring that the pressure and the vertical component of particle velocity be 

 continuous across the layer interface. 



Applying these boundary conditions to a sound-speed profile consisting of M layers 

 results in 2M - 1 linear equations in h| and h2. They are homogeneous in that the constant 

 is zero in each equation. There are M-1 coefficients Aj to be determined and M coefficients 

 Bj. These coefficients can therefore be determined within a constant of proportionality D, 

 provided the system of equations is linearly dependent. That is, the 2M - 1 square matrix of 

 coefficients of Aj and Bj must be of rank 2M - 2 or less. Its determinant will then be zero. 

 This is the eigenvalue condition. Values of X must be found which make the determinant 

 zero. This determinant, G, is discussed in more detail in a later section. 



Zeroes of the determinant, G, are found by using the secant method. The variable in 

 this iterative method can as well be some function of A as X itself, and we use the following 

 complex phase velocity (v): 



Xn = co/Vj^. (14) 



To find a v that is a root of G requires an initial guess, vj , where the subscript 1 refers to the 

 step in the iteration and a small increment, 5 j . Each succeeding estimate is given by the 

 relationship 



