# Third-order Stokes wave

These functions return the shape of a third-order Stokes wave with the wavenumber and steepness given by the parameters ak and k_.

double wave (double x, double y)
{
double a_ = ak/k_;
double eta1 = a_*cos(k_*x);
double alpa = 1./tanh(k_*h_);
double eta2 = 1./4.*alpa*(3.*sq(alpa) - 1.)*sq(a_)*k_*cos(2.*k_*x);
double eta3 = -3./8.*(cube(alpa)*alpa -
3.*sq(alpa) + 3.)*cube(a_)*sq(k_)*cos(k_*x) +
3./64.*(8.*cube(alpa)*cube(alpa) +
(sq(alpa) - 1.)*(sq(alpa) - 1.))*cube(a_)*sq(k_)*cos(3.*k_*x);
return eta1 + ak*eta2 + sq(ak)*eta3 - y;
}

double u_x (double x, double y)
{
double alpa = 1./tanh(k_*h_);
double a_ = ak/k_;
double sgma = sqrt(g_*k_*tanh(k_*h_)*
(1. + k_*k_*a_*a_*(9./8.*(sq(alpa) - 1.)*
(sq(alpa) - 1.) + sq(alpa))));
double A_ = a_*g_/sgma;
return A_*cosh(k_*(y + h_))/cosh(k_*h_)*k_*cos(k_*x) +
ak*3.*ak*A_/(8.*alpa)*(sq(alpa) - 1.)*(sq(alpa) - 1.)*
cosh(2.0*k_*(y + h_))*2.*k_*cos(2.0*k_*x)/cosh(2.0*k_*h_) +
ak*ak*1./64.*(sq(alpa) - 1.)*(sq(alpa) + 3.)*
(9.*sq(alpa) - 13.)*
cosh(3.*k_*(y + h_))/cosh(3.*k_*h_)*a_*a_*k_*k_*A_*3.*k_*cos(3.*k_*x);
}

double u_y (double x, double y)
{
double alpa = 1./tanh(k_*h_);
double a_ = ak/k_;
double sgma = sqrt(g_*k_*tanh(k_*h_)*
(1. + k_*k_*a_*a_*(9./8.*(sq(alpa) - 1.)*
(sq(alpa) - 1.) + sq(alpa))));
double A_ = a_*g_/sgma;
return A_*k_*sinh(k_*(y + h_))/cosh(k_*h_)*sin(k_*x) +
ak*3.*ak*A_/(8.*alpa)*(sq(alpa) - 1.)*(sq(alpa) - 1.)*
2.*k_*sinh(2.0*k_*(y + h_))*sin(2.0*k_*x)/cosh(2.0*k_*h_) +
ak*ak*1./64.*(sq(alpa) - 1.)*(sq(alpa) + 3.)*
(9.*sq(alpa) - 13.)*
3.*k_*sinh(3.*k_*(y + h_))/cosh(3.*k_*h_)*a_*a_*k_*k_*A_*sin(3.*k_*x);
}