Amplitude normalization factor in wfs.focused_25d function

[pizqleh]

Hello. First of all, thank you for developing this tool, and for quickly answering the last issues. If possible, I would like to ask you about the 2.5D WFS foucus source case. Apparently, the sound field generated by the driving function given by wfs.focused_25d don't not ressemble correctly the sound field of it's related point source, because it's way stronger than the focused source field. Is this something that one should expect from the nature of the mathematical model? My first guess is that this might not be the case, because even in x_ref=(0,0,0) the amplitude is way stronger. Maybe there is missing an amplitude factor in the implementation? Related to this, apparently there is a missmatch on what is expressed in your pdf docs 'Theory of Sound Field Synthesis' and what is expressed in the web docs 'https://sfs-python.readthedocs.io/en/0.5.0/sfs.fd.wfs.html' (the equations for the 2.5D WFS focus source case have different amplitude factors).

import numpy as np
import matplotlib.pyplot as plt
import sfs

frequency = 343
radius = 2.5
start_x, end_x = 2 * radius * np.asarray([-1, 1])
start_y, end_y = start_x, end_x
spacing = 0.01
grid = sfs.util.xyz_grid([start_x, end_x], [start_y, end_y], 0, spacing=spacing)


### POINT SOURCE.
# Sfs way to generate point source.
position = np.asarray([0, 1/3 * radius, 0])
normalization = 4 * np.pi * np.linalg.norm(position)
point_source = sfs.fd.source.point(2 * np.pi * frequency, position, grid) * normalization
sfs.plot2d.amplitude(point_source, grid)
plt.show()


## WFS
n_loudspeakers = 100
array = sfs.array.circular(N=n_loudspeakers, R=radius)
omega = 2 * np.pi * frequency
x_ref=[0, 0, 0]
d_wfs, s_wfs, ss_wfs = sfs.fd.wfs.focused_25d(omega, array.x, array.n, position, -position / np.linalg.norm(position), xref=x_ref)
u_wfs = sfs.fd.synthesize(d_wfs, s_wfs, array, ss_wfs, grid=grid) * normalization
sfs.plot2d.amplitude(u_wfs, grid)
sfs.plot2d.loudspeakers(array.x, array.n, s_wfs * array.a, size=0.15)
plt.show()

# Plot difference.
sfs.plot2d.amplitude(u_wfs - point_source, grid)
sfs.plot2d.loudspeakers(array.x, array.n, s_wfs * array.a, size=0.15)
plt.show()

Note that I have applied the same normalization factor in the point source and WFS waves, for the sake of plots clarity. The synthetized sound field for the point source, WFS, and the difference are:

Thank you,

Pedro Izquierdo L.