"""Compute SDM driving functions.
.. include:: math-definitions.rst
.. plot::
:context: reset
import matplotlib.pyplot as plt
import numpy as np
import sfs
plt.rcParams['figure.figsize'] = 6, 6
xs = -1.5, 1.5, 0
# normal vector for plane wave:
npw = sfs.util.direction_vector(np.radians(-45))
f = 300 # Hz
omega = 2 * np.pi * f
grid = sfs.util.xyz_grid([-2, 2], [-2, 2], 0, spacing=0.02)
array = sfs.array.linear(32, 0.2, orientation=[0, -1, 0])
def plot(d, selection, secondary_source):
p = sfs.fd.synthesize(d, selection, array, secondary_source, grid=grid)
sfs.plot2d.amplitude(p, grid)
sfs.plot2d.loudspeakers(array.x, array.n, selection * array.a, size=0.15)
"""
import numpy as _np
from scipy.special import hankel2 as _hankel2
from . import secondary_source_line as _secondary_source_line
from . import secondary_source_point as _secondary_source_point
from .. import util as _util
[docs]def line_2d(omega, x0, n0, xs, *, c=None):
r"""Driving function for 2-dimensional SDM for a virtual line source.
Parameters
----------
omega : float
Angular frequency of line source.
x0 : (N, 3) array_like
Sequence of secondary source positions.
n0 : (N, 3) array_like
Sequence of normal vectors of secondary sources.
xs : (3,) array_like
Position of line source.
c : float, optional
Speed of sound.
Returns
-------
d : (N,) numpy.ndarray
Complex weights of secondary sources.
selection : (N,) numpy.ndarray
Boolean array containing ``True`` or ``False`` depending on
whether the corresponding secondary source is "active" or not.
secondary_source_function : callable
A function that can be used to create the sound field of a
single secondary source. See `sfs.fd.synthesize()`.
Notes
-----
The secondary sources have to be located on the x-axis (y0=0).
Derived from :cite:`Spors2009`, Eq.(9), Eq.(4).
Examples
--------
.. plot::
:context: close-figs
d, selection, secondary_source = sfs.fd.sdm.line_2d(
omega, array.x, array.n, xs)
plot(d, selection, secondary_source)
"""
x0 = _util.asarray_of_rows(x0)
n0 = _util.asarray_of_rows(n0)
xs = _util.asarray_1d(xs)
k = _util.wavenumber(omega, c)
ds = x0 - xs
r = _np.linalg.norm(ds, axis=1)
d = - 1j/2 * k * xs[1] / r * _hankel2(1, k * r)
selection = _util.source_selection_all(len(x0))
return d, selection, _secondary_source_line(omega, c)
[docs]def plane_2d(omega, x0, n0, n=[0, 1, 0], *, c=None):
r"""Driving function for 2-dimensional SDM for a virtual plane wave.
Parameters
----------
omega : float
Angular frequency of plane wave.
x0 : (N, 3) array_like
Sequence of secondary source positions.
n0 : (N, 3) array_like
Sequence of normal vectors of secondary sources.
n: (3,) array_like, optional
Normal vector (traveling direction) of plane wave.
c : float, optional
Speed of sound.
Returns
-------
d : (N,) numpy.ndarray
Complex weights of secondary sources.
selection : (N,) numpy.ndarray
Boolean array containing ``True`` or ``False`` depending on
whether the corresponding secondary source is "active" or not.
secondary_source_function : callable
A function that can be used to create the sound field of a
single secondary source. See `sfs.fd.synthesize()`.
Notes
-----
The secondary sources have to be located on the x-axis (y0=0).
Derived from :cite:`Ahrens2012`, Eq.(3.73), Eq.(C.5), Eq.(C.11):
.. math::
D(\x_0,k) = k_\text{pw,y} \e{-\i k_\text{pw,x} x}
Examples
--------
.. plot::
:context: close-figs
d, selection, secondary_source = sfs.fd.sdm.plane_2d(
omega, array.x, array.n, npw)
plot(d, selection, secondary_source)
"""
x0 = _util.asarray_of_rows(x0)
n0 = _util.asarray_of_rows(n0)
n = _util.normalize_vector(n)
k = _util.wavenumber(omega, c)
d = k * n[1] * _np.exp(-1j * k * n[0] * x0[:, 0])
selection = _util.source_selection_all(len(x0))
return d, selection, _secondary_source_line(omega, c)
[docs]def plane_25d(omega, x0, n0, n=[0, 1, 0], *, xref=[0, 0, 0], c=None):
r"""Driving function for 2.5-dimensional SDM for a virtual plane wave.
Parameters
----------
omega : float
Angular frequency of plane wave.
x0 : (N, 3) array_like
Sequence of secondary source positions.
n0 : (N, 3) array_like
Sequence of normal vectors of secondary sources.
n: (3,) array_like, optional
Normal vector (traveling direction) of plane wave.
xref : (3,) array_like, optional
Reference point for synthesized sound field.
c : float, optional
Speed of sound.
Returns
-------
d : (N,) numpy.ndarray
Complex weights of secondary sources.
selection : (N,) numpy.ndarray
Boolean array containing ``True`` or ``False`` depending on
whether the corresponding secondary source is "active" or not.
secondary_source_function : callable
A function that can be used to create the sound field of a
single secondary source. See `sfs.fd.synthesize()`.
Notes
-----
The secondary sources have to be located on the x-axis (y0=0).
Eq.(3.79) from :cite:`Ahrens2012`.
Examples
--------
.. plot::
:context: close-figs
d, selection, secondary_source = sfs.fd.sdm.plane_25d(
omega, array.x, array.n, npw, xref=[0, -1, 0])
plot(d, selection, secondary_source)
"""
x0 = _util.asarray_of_rows(x0)
n0 = _util.asarray_of_rows(n0)
n = _util.normalize_vector(n)
xref = _util.asarray_1d(xref)
k = _util.wavenumber(omega, c)
d = 4j * _np.exp(-1j*k*n[1]*xref[1]) / _hankel2(0, k*n[1]*xref[1]) * \
_np.exp(-1j*k*n[0]*x0[:, 0])
selection = _util.source_selection_all(len(x0))
return d, selection, _secondary_source_point(omega, c)
[docs]def point_25d(omega, x0, n0, xs, *, xref=[0, 0, 0], c=None):
r"""Driving function for 2.5-dimensional SDM for a virtual point source.
Parameters
----------
omega : float
Angular frequency of point source.
x0 : (N, 3) array_like
Sequence of secondary source positions.
n0 : (N, 3) array_like
Sequence of normal vectors of secondary sources.
xs: (3,) array_like
Position of virtual point source.
xref : (3,) array_like, optional
Reference point for synthesized sound field.
c : float, optional
Speed of sound.
Returns
-------
d : (N,) numpy.ndarray
Complex weights of secondary sources.
selection : (N,) numpy.ndarray
Boolean array containing ``True`` or ``False`` depending on
whether the corresponding secondary source is "active" or not.
secondary_source_function : callable
A function that can be used to create the sound field of a
single secondary source. See `sfs.fd.synthesize()`.
Notes
-----
The secondary sources have to be located on the x-axis (y0=0).
Driving function from :cite:`Spors2010`, Eq.(24).
Examples
--------
.. plot::
:context: close-figs
d, selection, secondary_source = sfs.fd.sdm.point_25d(
omega, array.x, array.n, xs, xref=[0, -1, 0])
plot(d, selection, secondary_source)
"""
x0 = _util.asarray_of_rows(x0)
n0 = _util.asarray_of_rows(n0)
xs = _util.asarray_1d(xs)
xref = _util.asarray_1d(xref)
k = _util.wavenumber(omega, c)
ds = x0 - xs
r = _np.linalg.norm(ds, axis=1)
d = 1/2 * 1j * k * _np.sqrt(xref[1] / (xref[1] - xs[1])) * \
xs[1] / r * _hankel2(1, k * r)
selection = _util.source_selection_all(len(x0))
return d, selection, _secondary_source_point(omega, c)