function [a,b,c] = submerged_semiinfinite_dock(alpha,h,d,theta,N)
% [a,b,c] = submerged_semiinfinite_dock(alpha,h,d,theta,N)
% 
% this program solves the submerged semiinfinite dock problem by an
% eigenfunction matching. 
% alpha is the radian frequency squared
% h is the water depth, d is the depth of the submerged plate,
% 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(kh_n(z+h))/cos(kh_n h) where kh_n are the roots of the dispersion
% equation for depth h
%
% b is the vector of coefficients for the eigenfunctions
% cos(kd_n(z+h))/cos(kd_n h) where kd_n are the roots of the dispersion
% equation for depth d (b has size N_above)
%
% c is the vector of coefficients for the eigenfunctions
% cos(kappa_n(z+h) where kappa_n = n*pi/(h-d) (c has size N_below)
%
% the boundary value problem is described in 
% http://www.wikiwaves.org/index.php/Eigenfunction_Matching_for_a_Submerged_Semi-Infinite_Dock


% we need to allocate the modes on the right so that they are roughly in
% the same ratio as d and (h-d) while allowing at least one. 
% remember that there is actually N + 1 modes.

N_above = round(d/h*(N+1)) -1; % number of modes above is N_above + 1;
N_below = N-1-N_above; % numebr of modes below is N_below +1;
   if N_above <  0
       N_above = 0;
       N_below = N-1;
   elseif N_below < 0
       N_above = N-1;
       N_below = 0;
   end
       


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

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

% we calculate the roots of the dispersion equation for the dock covered
% region
kappa = [0:N_below]*pi/(h-d);

% we now find the modified coefficients
kh_hat = sqrt(kh.^2 - k_y^2);
kd_hat = sqrt(kh.^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_d(kh(j),kh(j),h,0)/cos(kh(j)*h)^2;
    % first we do the modes above
	for l = 1:N_above+1; 
		B(j,l) = innerproduct_d_to_0(kh(j),kd(l),h,d)/(cos(kh(j)*h)*cos(kd(l)*d));
    end
    % then the modes below
    for l = 1:N_below+1; 
		B(j,l+N_above+1) = innerproduct_h_to_d(kh(j),kappa(l),h,d)/cos(kh(j)*h);
	end
end

% we do not calculate in the most efficient manner but we want the code to
% be easy to follow. We now assemble the equations to solve

M = zeros(size(B));

for j = 1:N+1;
	for l = 1:N_above+1; 
		M(j,l) = (kh_hat(j) + kd_hat(l))*B(j,l);
    end
    for l = 1:N_below+1; 
		M(j,l+N_above+1) = (kh_hat(j) + kappa_hat(l))*B(j,l+N_above+1);
	end
end

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

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

c = b(N_above+2:end);b = b(1:N_above+1);
%--------------------------------------------------------------------------

% 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_full = -h:h/200:0;
% z_above = -d:h/200:0;
% z_below = -h:h/200:-d;
% 
% phi_minus = cos(kh(1)*(z_full+h))/cos(kh(1)*h);
% phi_plus_above = zeros(size(z_above));
% phi_plus_below = zeros(size(z_below));
% % phi_minus = phi_plus;
% 
% for count = 1:N+1
%     phi_minus = phi_minus + a(count)*cos(kh(count)*(z_full+h))/cos(kh(count)*h);
% end
% 
% for count = 1:N_above+1
%     phi_plus_above = phi_plus_above + b(count)*cos(kd(count)*(z_above+d))/cos(kd(count)*d);
% end
% 
% for count = 1:N_below+1
%     phi_plus_below = phi_plus_below + c(count)*cos(kappa(count)*(z_below+h));
% end
% 
% plot(z_full,abs(phi_minus),z_above,abs(phi_plus_above),z_below,abs(phi_plus_below))
% 
% pause
% 
% phi_minus_n = -kh_hat(1)*cos(kh(1)*(z_full+h))/cos(kh(1)*h);
% phi_plus_above_n = zeros(size(z_above));
% phi_plus_below_n = zeros(size(z_below));
% % phi_minus = phi_plus;
% 
% for count = 1:N+1
%     phi_minus_n = phi_minus_n + kh_hat(count)*a(count)*cos(kh(count)*(z_full+h))/cos(kh(count)*h);
% end
% 
% for count = 1:N_above+1
%     phi_plus_above_n = phi_plus_above_n - kd_hat(count)*b(count)*cos(kd(count)*(z_above+d))/cos(kd(count)*d);
% end
% 
% for count = 1:N_below+1
%     phi_plus_below_n = phi_plus_below_n - kappa_hat(count)*c(count)*cos(kappa(count)*(z_below+h));
% end
% 
% plot(z_full,abs(phi_minus_n),z_above,abs(phi_plus_above_n),z_below,abs(phi_plus_below_n))

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

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

function out = innerproduct_h_to_d(k,kappa,h,d)
% program to calculate 
% \int_{-h}^{-d}\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)/(h-d)) > 1e-6
    out = 1/2 * (sin(d*kappa - d*k - kappa*h + k*h)) / (k - kappa) ...
        + 1/2 * (sin(-d*kappa - d*k + kappa*h + k*h)) / (kappa + k);
else
    if k == 0
        out = h-d;
    else
        out = (1/2)*((cos(-k*d+k*h)*sin(-k*d+k*h)-k*d+k*h)/k);
    end
end

function out = innerproduct_d_to_0(kh,kd,h,d)
% program to calculate 
% \int_{-d}^{0}\cos kh\left( z+h\right) \cos kd\left(z+d\right) dz

% first we test for the case when kh and kd are equal (or nearly equal)
if abs((kd-kh)/(h-d)) > 1e-6
    out = (1/2)*(((sin(kh*h-d*kd))*kh+(sin(kh*h-d*kd))*kd ...
        + (sin(kh*h+d*kd))*kh -(sin(kh*h+d*kd))*kd-2*(sin(-d*kh+kh*h))*kh)/(kh^2-kd^2));
else
    if kd == 0
        out = d;
    else
        out = (1/4)*((sin(kd*h+d*kd)+2*(cos(-d*kd+kd*h))*d*kd-sin(-d*kd+kd*h))/kd);
    end
end