function [a,b] = semiinfinite_dock(alpha,h,theta,N)
% [a,b] = semiinfinite_dock(alpha,h,theta,N)
% 
% this program solves the semiinfinite dock problem by an
% eigenfunction matching. 
% alpha is the radian frequency squared
% h is the water depth, theta is the angle of incidence 
% and N+1 is the number of eigenfunctions in each region
%
% a is the vector of coefficients for the eigenfunctions
% cos(k_n(z+h))/cos(k_n h) where k_n are the roots of the dispersion
% equation
%
% b is the vector of coefficients for the eigenfunctions
% cos(kappa_n(z+h)) where kappa_n = n*pi/h
%
% the boundary value problem is described in 
% http://www.wikiwaves.org/index.php/Eigenfunction_Matching_for_a_Semi-Infinite_Dock

% first we calculate the roots of the dispersion equation for the water
k = dispersion_free_surface(alpha,N,h); 

% we need to determine the wavenumber in the y direction
k_y = sin(theta)*k(1);

% we calculate the roots of the dispersion equation for the dock covered
% region
kappa = [0:N]*pi/h;

% we now find the modified coefficients
k_hat = [sqrt(k.^2 + k_y^2)];
kappa_hat = sqrt(kappa.^2 + k_y^2);


% we calculate the matrix of inner products. 
A = zeros(N+1,N+1);
B = zeros(N+1,N+1);

for j = 1:N+1;
    A(j,j) = innerproduct_h_to_0(k(j),k(j),h)/cos(k(j)*h)^2;
	for l = 1:N+1; 
		B(j,l) = innerproduct_h_to_0(k(j),kappa(l),h)/cos(k(j)*h);
	end
end

% We now assemble the equations to solve

M = zeros(size(B));

for j = 1:N+1;
	for l = 1:N+1; 
		M(j,l) = (k_hat(j) + kappa_hat(l))*B(j,l);
	end
end

RHS = zeros(N+1,1);
RHS(1) = A(1,1);

b = M\(2*k_hat(1)*RHS);
a = A\(B*b - RHS);


%--------------------------------------------------------------------------

% a set of commands used to test this program.  We plot the potential and
% its derivative on each side and show that they match.  You can also check
% that the program is correct by making sure abs(a(1)) = 1 (from
% conservation of energy).

% uncomment these lines to run the test (except this one)
% z = -h:h/200:0;
% 
% phi_minus = cos(k(1)*(z+h))/cos(k(1)*h);
% phi_plus = zeros(size(phi_minus));
% % phi_minus = phi_plus;
% 
% for count = 1:N+1
%     phi_minus = phi_minus + a(count)*cos(k(count)*(z+h))/cos(k(count)*h);
%     phi_plus = phi_plus + b(count)*cos(kappa(count)*(z+h));
% end
% 
% plot(z,abs(phi_minus),z,abs(phi_plus))
% 
% pause
% 
% phi_minus_n = -k_hat(1)*cos(k(1)*(z+h))/cos(k(1)*h);
% phi_plus_n = zeros(size(phi_minus));
% % phi_minus = phi_plus;
% 
% for count = 1:N+1
%     phi_minus_n = phi_minus_n + a(count)*k_hat(count)*cos(k(count)*(z+h))/cos(k(count)*h);
%     phi_plus_n = phi_plus_n - b(count)*kappa_hat(count)*cos(kappa(count)*(z+h));
% end
% 
% plot(z,abs(phi_minus_n),z,abs(phi_plus_n))

%--------------------------------------------------------------------------

% this is a program to calculate the inner product of the eigenfunctions
% with each other. 

function out = innerproduct_h_to_0(k,kappa,h)
% program to calculate 
% \int_{-h}^{0}\cos \kappa\left( z+h\right) 
% \cos k\left(z+h\right) dz

% first we test for the case when \kappa and k are equal (or nearly equal)
if abs(k-kappa) > 1e-6
    out = 1/2 * (sin(- kappa*h + k*h)) / (k - kappa) ...
        + 1/2 * (sin(kappa*h + k*h)) / (kappa + k);
else
    if k == 0
        out = h;
    else
        out = (1/2)*((cos(k*h)*sin(k*h) + k*h)/k);
    end
end