spharpy.spatial#

Wave field representations in the spatial domain.

Functions:

greens_function_plane_wave(source_points, ...)

Green's function for plane waves in free space in matrix form.

greens_function_point_source(sources, ...[, ...])

Green's function for point sources in free space in matrix form.
spharpy.spatial.greens_function_plane_wave(source_points, receiver_points, wave_number, gradient=False)[source]#

Green’s function for plane waves in free space in matrix form. The matrix describing the propagation of a plane wave from a direction of arrival defined by the azimuth and elevation angles of the source points to the receiver points. The phase sign convention reflects a direction of arrival from the source position.

\[G(k) = e^{i \mathbf{k}^\mathrm{T} \mathbf{r}}\]
Parameters:

  • source_points (pyfar.Coordinates) – The source points defining the direction of incidence for the plane wave. Note that the radius on which the source is positioned has no relevance.

  • receiver_points (pyfar.Coordinates) – The receiver points.

  • wave_number (float, complex) – The wave number. A complex wave number can be used for evanescent waves.

  • gradient (bool) – If True, the gradient will be returned as well. Default is False.

Returns:

M – The plane wave propagation matrix with shape [n_receiver, n_sources].

Return type:

ndarray, complex

Examples

Plot Green’s function in the x-y plane for a plane wave with a direction of incidence defined by the vector \([x, y, z] = [2, 1, 0]\).

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from pyfar import Coordinates
>>> import spharpy
...
>>> spat_res = 30
>>> x_min, x_max = -10, 10
>>> xx, yy = np.meshgrid(
>>>     np.linspace(x_min, x_max, spat_res),
>>> np.linspace(x_min, x_max, spat_res))
>>> receivers = Coordinates(
...     xx.flatten(), yy.flatten(), np.zeros(spat_res**2))
>>> doa = Coordinates(2, 1, 0)
...
>>> k = 1
>>> plane_wave_matrix = spharpy.spatial.greens_function_plane_wave(
...     doa, receivers, k)
>>> plt.figure()
>>> plt.contourf(
...     xx, yy, np.real(plane_wave_matrix.reshape(spat_res, spat_res)),
...     cmap='RdBu_r', levels=100)
>>> plt.colorbar()
>>> ax = plt.gca()
>>> ax.set_aspect('equal')

(Source code, png, hires.png, pdf)

../_images/spharpy-spatial-1.png

For evanescent waves, a complex wave number can be used

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from pyfar import Coordinates
>>> import spharpy
...
>>> spat_res = 30
>>> x_min, x_max = -10, 10
>>> xx, yy = np.meshgrid(
>>>     np.linspace(x_min, x_max, spat_res),
>>> np.linspace(x_min, x_max, spat_res))
>>> receivers = Coordinates(
...     xx.flatten(), yy.flatten(), np.zeros(spat_res**2))
>>> doa = Coordinates(2, 1, 0)
...
>>> k = 1-.1j
>>> plane_wave_matrix = spharpy.spatial.greens_function_plane_wave(
...     doa, receivers, k, gradient=False)
>>> plt.contourf(
...     xx, yy, np.real(plane_wave_matrix.reshape(spat_res, spat_res)),
...     cmap='RdBu_r', levels=100)
>>> plt.colorbar()
>>> ax = plt.gca()
>>> ax.set_aspect('equal')

(Source code, png, hires.png, pdf)

../_images/spharpy-spatial-2.png
spharpy.spatial.greens_function_point_source(sources, receivers, k, gradient=False)[source]#

Green’s function for point sources in free space in matrix form. The phase sign convention corresponds to a direction of propagation away from the source at position \(r_s\).

\[G(k) = \frac{e^{-i k\|\mathbf{r_s} - \mathbf{r_r}\|}} {4 \pi \|\mathbf{r_s} - \mathbf{r_r}\|}\]
Parameters:

  • sources (pyfar.Coordinates) – source points as Coordinates object

  • receivers (pyfar.Coordinates) – receiver points as Coordinates object

  • k (ndarray, float) – The wave number

  • gradient (bool, optional) – Flag to indicate if the gradient should be computed and returned. The default is False.

Returns:

G – Green’s function

Return type:

ndarray, double

Examples

Plot Green’s function in the x-y plane for a point source at \([x, y, z] = [10, 15, 0]\).

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from pyfar import Coordinates
>>> import spharpy
...
>>> spat_res = 30
>>> x_min, x_max = -10, 10
>>> xx, yy = np.meshgrid(
>>>     np.linspace(x_min, x_max, spat_res),
>>> np.linspace(x_min, x_max, spat_res))
>>> receivers = Coordinates(
...     xx.flatten(), yy.flatten(), np.zeros(spat_res**2))
>>> doa = Coordinates(10, 15, 0)
>>> plane_wave_matrix = spharpy.spatial.greens_function_point_source(
...     doa, receivers, 1, gradient=False)
>>> plt.contourf(
...     xx, yy, np.real(plane_wave_matrix.reshape(spat_res, spat_res)),
...     cmap='RdBu_r', levels=100)
>>> plt.colorbar()
>>> ax = plt.gca()
>>> ax.set_aspect('equal')

(Source code, png, hires.png, pdf)

../_images/spharpy-spatial-3.png