In the Smith and Young 2001 paper on the Barotropic tide they have the governing equations...
\begin{array}{rcl} u_t-f_0v+p_x & = & 0 \\v_t+f_0u+p_y & = & 0 \\p_z &= &b \\b_t+wN^2 & = & 0 \\u_x+v_y+w_z & =& 0 \end{array}
with BCs \begin{equation} w(x,y,0,t)=0 ; w(x,y,-H,t)=U \cdot \nabla h \end{equation}
From here they "project onto vertical modes associated with the stratification N(z)". and get the functions defined by the eigenproblem
\begin{equation} \frac{d^2a_n}{dz^2}+\frac{N^2}{\lambda_n^2 f_0^2}a_n = 0, a_n(0)=a_n(-H)=0 \end{equation}
Could someone explain to me where this has come from?