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Animations of a Pulsating Sphere

import sfs
import numpy as np
import matplotlib.pyplot as plt
from IPython.display import HTML

/home/docs/checkouts/readthedocs.org/user_builds/sfs-python/conda/latest/lib/python3.9/site-packages/traitlets/traitlets.py:3030: FutureWarning: --rc={'figure.dpi': 96} for dict-traits is deprecated in traitlets 5.0. You can pass --rc <key=value> ... multiple times to add items to a dict.
  warn(

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

• sfs.fd.source.pulsating_sphere() and

• sfs.fd.source.pulsating_sphere_velocity()

while the last one can be obtained by using

• sfs.fd.displacement()

which converts the particle velocity into displacement.

A couple of additional functions are implemented in

• animations_pulsating_sphere.py

in order to help creating animating pictures, which is fun!

import animations_pulsating_sphere as animation

# 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

Particle Displacement

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())

Once Loop Reflect

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.

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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  if align is 'vertical':
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  if align is 'vertical':
<ipython-input-1-17990c1f260e>:18: SyntaxWarning: "is" with a literal. Did you mean "=="?
  elif align is 'horizontal':

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())

/home/docs/checkouts/readthedocs.org/user_builds/sfs-python/checkouts/latest/doc/examples/animations_pulsating_sphere.py:18: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray.
  scat = sfs.plot2d.particles(grid + displacement, **kwargs)
/home/docs/checkouts/readthedocs.org/user_builds/sfs-python/checkouts/latest/doc/examples/animations_pulsating_sphere.py:21: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray.
  position = (grid + displacement * phasor**i).apply(np.real)
/home/docs/checkouts/readthedocs.org/user_builds/sfs-python/checkouts/latest/doc/examples/animations_pulsating_sphere.py:21: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray.
  position = (grid + displacement * phasor**i).apply(np.real)

Once Loop Reflect

Another one using a random grid.

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())

/home/docs/checkouts/readthedocs.org/user_builds/sfs-python/checkouts/latest/doc/examples/animations_pulsating_sphere.py:18: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray.
  scat = sfs.plot2d.particles(grid + displacement, **kwargs)
/home/docs/checkouts/readthedocs.org/user_builds/sfs-python/checkouts/latest/doc/examples/animations_pulsating_sphere.py:21: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray.
  position = (grid + displacement * phasor**i).apply(np.real)

Once Loop Reflect

Each grid has its strengths and weaknesses. Please refer to the on-line discussion.

Particle Velocity

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())

Once Loop Reflect

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.

Sound Pressure

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())

Once Loop Reflect

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.

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())

Once Loop Reflect

Let’s zoom in closer to the boundary of the sphere.

L = 10 * amplitude
xmin_zoom, xmax_zoom = radius - L, radius + L
ymin_zoom, ymax_zoom = -L, L

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())

Once Loop Reflect

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.