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import sfs
import numpy as np
import matplotlib.pyplot as plt
from IPython.display import HTML
In this example, the sound field of a pulsating sphere is visualized. Different acoustic variables, such as sound pressure, particle velocity, and particle displacement, are simulated. The first two quantities are computed with
while the last one can be obtained by using
which converts the particle velocity into displacement.
A couple of additional functions are implemented in
in order to help creating animating pictures, which is fun!
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import animations_pulsating_sphere as animation
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# Pulsating sphere
center = [0, 0, 0]
radius = 0.25
amplitude = 0.05
f = 1000 # frequency
omega = 2 * np.pi * f # angular frequency
# Axis limits
figsize = (6, 6)
xmin, xmax = -1, 1
ymin, ymax = -1, 1
# Animations
frames = 20 # frames per period
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grid = sfs.util.xyz_grid([xmin, xmax], [ymin, ymax], 0, spacing=0.025)
ani = animation.particle_displacement(
omega, center, radius, amplitude, grid, frames, figsize, c='Gray')
plt.close()
HTML(ani.to_jshtml())
Click the arrow button to start the animation.
to_jshtml() allows you to play with the animation,
e.g. speed up/down the animation (+/- button).
Try to reverse the playback by clicking the left arrow.
You'll see a sound sink.
You can also show the animation by using to_html5_video().
See the documentation for more detail.
Of course, different types of grid can be chosen. Below is the particle animation using the same parameters but with a hexagonal grid.
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def hex_grid(xlim, ylim, hex_edge, align='horizontal'):
if align is 'vertical':
umin, umax = ylim
vmin, vmax = xlim
else:
umin, umax = xlim
vmin, vmax = ylim
du = np.sqrt(3) * hex_edge
dv = 1.5 * hex_edge
num_u = int((umax - umin) / du)
num_v = int((vmax - vmin) / dv)
u, v = np.meshgrid(np.linspace(umin, umax, num_u),
np.linspace(vmin, vmax, num_v))
u[::2] += 0.5 * du
if align is 'vertical':
grid = v, u, 0
elif align is 'horizontal':
grid = u, v, 0
return grid
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grid = hex_grid([xmin, xmax], [ymin, ymax], 0.0125, 'vertical')
ani = animation.particle_displacement(
omega, center, radius, amplitude, grid, frames, figsize, c='Gray')
plt.close()
HTML(ani.to_jshtml())
Another one using a random grid.
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grid = [np.random.uniform(xmin, xmax, 4000),
np.random.uniform(ymin, ymax, 4000), 0]
ani = animation.particle_displacement(
omega, center, radius, amplitude, grid, frames, figsize, c='Gray')
plt.close()
HTML(ani.to_jshtml())
Each grid has its strengths and weaknesses. Please refer to the on-line discussion.
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amplitude = 1e-3
grid = sfs.util.xyz_grid([xmin, xmax], [ymin, ymax], 0, spacing=0.04)
ani = animation.particle_velocity(
omega, center, radius, amplitude, grid, frames, figsize)
plt.close()
HTML(ani.to_jshtml())
Please notice that the amplitude of the pulsating motion is adjusted so that the arrows are neither too short nor too long. This kind of compromise is inevitable since
$$ \text{(particle velocity)} = \text{i} \omega \times (\text{amplitude}), $$thus the absolute value of particle velocity is usually much larger than that of amplitude. It should be also kept in mind that the hole in the middle does not visualizes the exact motion of the pulsating sphere. According to the above equation, the actual amplitude should be much smaller than the arrow lengths. The changing rate of its size is also two times higher than the original frequency.
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amplitude = 0.05
impedance_pw = sfs.default.rho0 * sfs.default.c
max_pressure = omega * impedance_pw * amplitude
grid = sfs.util.xyz_grid([xmin, xmax], [ymin, ymax], 0, spacing=0.005)
ani = animation.sound_pressure(
omega, center, radius, amplitude, grid, frames, pulsate=True,
figsize=figsize, vmin=-max_pressure, vmax=max_pressure)
plt.close()
HTML(ani.to_jshtml())
Notice that the sound pressure exceeds the atmospheric pressure ($\approx 10^5$ Pa), which of course makes no sense. This is due to the large amplitude (50 mm) of the pulsating motion. It was chosen to better visualize the particle movements in the earlier animations.
For 1 kHz, the amplitude corresponding to a moderate sound pressure, let say 1 Pa, is in the order of micrometer. As it is very small compared to the corresponding wavelength (0.343 m), the movement of the particles and the spatial structure of the sound field cannot be observed simultaneously. Furthermore, at high frequencies, the sound pressure for a given particle displacement scales with the frequency. The smaller wavelength (higher frequency) we choose, it is more likely to end up with a prohibitively high sound pressure.
In the following examples, the amplitude is set to a realistic value 1 $\mu$m. Notice that the pulsating motion of the sphere is no more visible.
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amplitude = 1e-6
impedance_pw = sfs.default.rho0 * sfs.default.c
max_pressure = omega * impedance_pw * amplitude
grid = sfs.util.xyz_grid([xmin, xmax], [ymin, ymax], 0, spacing=0.005)
ani = animation.sound_pressure(
omega, center, radius, amplitude, grid, frames, pulsate=True,
figsize=figsize, vmin=-max_pressure, vmax=max_pressure)
plt.close()
HTML(ani.to_jshtml())
Let's zoom in closer to the boundary of the sphere.
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L = 10 * amplitude
xmin_zoom, xmax_zoom = radius - L, radius + L
ymin_zoom, ymax_zoom = -L, L
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grid = sfs.util.xyz_grid([xmin_zoom, xmax_zoom], [ymin_zoom, ymax_zoom], 0, spacing=L / 100)
ani = animation.sound_pressure(
omega, center, radius, amplitude, grid, frames, pulsate=True,
figsize=figsize, vmin=-max_pressure, vmax=max_pressure)
plt.close()
HTML(ani.to_jshtml())
This shows how the vibrating motion of the sphere (left half) changes the sound pressure of the surrounding air (right half). Notice that the sound pressure increases/decreases (more red/blue) when the surface accelerates/decelerates.