wave_gradient_and_phase.md

Wave, gradient, and phase

We consider a plane gravity wave in two dimensions; $x$ is the propagation direction, and $y$ the vertical. The complex velocity potential reads

$$ \phi = e^{ikz} $$

where $z=x+iy$.

Mesh

We first build a rectangular mesh:

L_box = 6.
H_box = 1.5

mesh_points = {}
mesh_points['surface'] = [ [L_box/2,0], [-L_box/2,0] ]
mesh_points['left_wall'] = [ mesh_points['surface'][-1], [-L_box/2,-H_box] ]
mesh_points['bottom'] = [ mesh_points['left_wall'][-1], [L_box/2,-H_box] ]
mesh_points['right_wall'] = [ mesh_points['bottom'][-1], [L_box/2,0] ]

Th = pyff.TriMesh_from_boundaries( boundaries = list( mesh_points.values() ), labels = list( mesh_points.keys() ) )

Before using this mesh, we refine it:

for _ in range(3) :
    Th = pyff.adaptmesh( Th, iso = 1, hmax = 0.1 )

Here's how the mesh looks like:

Th.plot_triangles( color = 'grey', alpha = .1, lw = .5 )
Th.plot_boundaries( )
legend()

Wave field

We define the complex velocity potential, $\phi$, and plot its real part:

k = 2

z = Th.x + 1j*Th.y
phi = exp( -1j*k*z )

fig_phi = figure()
ax_phi = gca()
ax_phi.tricontourf( Th, real(phi) )

Gradients and finite-element spaces

To compute the gradient of a field, we first define the gradient matrices:

M = pyff.gradient_matrices( Th )

M is a dictionnary that contains two matrices, which are not square:

print( shape(M['grad_x']) )

(2302, 1227)

This is because the gradient of a P1 field is naturally a P0 field. We can check this:

print( len(Th.x), len(Th.triangles) )

2302 1227

We can compute the gradient $\nabla \phi$:

dx_phi = M['grad_x']@( phi )
dy_phi = M['grad_y']@( phi )

To plot this gradient, however, we need to define the space coordinates, $x$ and $y$, on the P0 space. We thus create a projector:

proj = pyff.get_projector( Th, 'P1', 'P0' )
print(shape(proj))

(2302, 1227)

Here are our new coordinates:

x_P0 = proj@Th.x
y_P0 = proj@Th.y

We can now use them to produce a quiver plot:

quiver( x_P0, y_P0, real( dx_phi ), real( dy_phi ) )

Energy flux

The energy flux associated to our wave is the imaginary part of $\phi^* , \nabla \phi$. Let's plot this:

phi_P0 = proj@phi
quiver( x_P0, y_P0, imag( phi_P0.conj()*dx_phi ), imag( phi_P0.conj()*dy_phi ) )

As expected, the wave is propagating from right to left.