Sound field of a pulsating sphere

.travis.yml

@@ -9,6 +9,7 @@ addons:
   apt:
     packages:
       - pandoc
+      - ffmpeg
 install:
   - pip install .
   - pip install -r tests/requirements.txt

doc/examples.rst

@@ -15,4 +15,5 @@ Examples
 
     examples/modal-room-acoustics
     examples/mirror-image-source-model
+    examples/animations-pulsating-sphere
     example-python-scripts

doc/examples/animations-pulsating-sphere.ipynb

@@ -0,0 +1,366 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Animations of a Pulsating Sphere"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import sfs\n",
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "from IPython.display import HTML"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "In this example, the sound field of a pulsating sphere is visualized.\n",
+    "Different acoustic variables, such as sound pressure,\n",
+    "particle velocity, and particle displacement, are simulated.\n",
+    "The first two quantities are computed with\n",
+    "\n",
+    "- [sfs.mono.source.pulsating_sphere()](../sfs.mono.source.rst#sfs.mono.source.pulsating_sphere) and \n",
+    "- [sfs.mono.source.pulsating_sphere_velocity()](../sfs.mono.source.rst#sfs.mono.source.pulsating_sphere_velocity)\n",
+    "\n",
+    "while the last one can be obtained by using\n",
+    "\n",
+    "- [sfs.util.displacement()](../sfs.util.rst#sfs.util.displacement)\n",
+    "\n",
+    "which converts the particle velocity into displacement.\n",
+    "\n",
+    "A couple of additional functions are implemented in\n",
+    "\n",
+    "- [animations_pulsating_sphere.py](animations_pulsating_sphere.py)\n",
+    "\n",
+    "in order to help creating animating pictures, which is fun!"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import animations_pulsating_sphere as animation"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Pulsating sphere\n",
+    "center = [0, 0, 0]\n",
+    "radius = 0.25\n",
+    "amplitude = 0.05\n",
+    "f = 1000  # frequency\n",
+    "omega = 2 * np.pi * f  # angular frequency\n",
+    "\n",
+    "# Axis limits\n",
+    "figsize = (6, 6)\n",
+    "xmin, xmax = -1, 1\n",
+    "ymin, ymax = -1, 1\n",
+    "\n",
+    "# Animations\n",
+    "frames = 20  # frames per period"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Particle Displacement"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "grid = sfs.util.xyz_grid([xmin, xmax], [ymin, ymax], 0, spacing=0.025)\n",
+    "ani = animation.particle_displacement(\n",
+    "        omega, center, radius, amplitude, grid, frames, figsize, c='Gray')\n",
+    "plt.close()\n",
+    "HTML(ani.to_html5_video())"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "You can play with the animation more interactively by using `.to_jshtml`."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "HTML(ani.to_jshtml())"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Of course, different types of grid can be chosen.\n",
+    "Below is the particle animation using the same parameters\n",
+    "but with a [hexagonal grid](https://www.redblobgames.com/grids/hexagons/)."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def hex_grid(xlim, ylim, hex_edge, align='horizontal'):\n",
+    "    if align is 'vertical':\n",
+    "        umin, umax = ylim\n",
+    "        vmin, vmax = xlim\n",
+    "    else:\n",
+    "        umin, umax = xlim\n",
+    "        vmin, vmax = ylim\n",
+    "    du = np.sqrt(3) * hex_edge\n",
+    "    dv = 1.5 * hex_edge\n",
+    "    num_u = int((umax - umin) / du)\n",
+    "    num_v = int((vmax - vmin) / dv)\n",
+    "    u, v = np.meshgrid(np.linspace(umin, umax, num_u),\n",
+    "                       np.linspace(vmin, vmax, num_v))\n",
+    "    u[::2] += 0.5 * du\n",
+    "\n",
+    "    if align is 'vertical':\n",
+    "        grid = v, u, 0\n",
+    "    elif align is 'horizontal':\n",
+    "        grid = u, v, 0\n",
+    "    return  grid"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "grid = hex_grid([xmin, xmax], [ymin, ymax], 0.0125, 'vertical')\n",
+    "ani = animation.particle_displacement(\n",
+    "        omega, center, radius, amplitude, grid, frames, figsize, c='Gray')\n",
+    "plt.close()\n",
+    "HTML(ani.to_html5_video())"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Another one using a random grid."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "grid = [np.random.uniform(xmin, xmax, 4000),\n",
+    "        np.random.uniform(ymin, ymax, 4000), 0]\n",
+    "ani = animation.particle_displacement(\n",
+    "        omega, center, radius, amplitude, grid, frames, figsize, c='Gray')\n",
+    "plt.close()\n",
+    "HTML(ani.to_html5_video())"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Each grid has its strengths and weaknesses. Please refer to the\n",
+    "[on-line discussion](https://github.com/sfstoolbox/sfs-python/pull/69#issuecomment-468405536)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Particle Velocity"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "amplitude = 1e-3\n",
+    "grid = sfs.util.xyz_grid([xmin, xmax], [ymin, ymax], 0, spacing=0.04)\n",
+    "ani = animation.particle_velocity(\n",
+    "        omega, center, radius, amplitude, grid, frames, figsize)\n",
+    "plt.close()\n",
+    "HTML(ani.to_html5_video())"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Please notice that the amplitude of the pulsating motion is adjusted\n",
+    "so that the arrows are neither too short nor too long.\n",
+    "This kind of compromise is inevitable since\n",
+    "\n",
+    "$$\n",
+    "\\text{(particle velocity)} = \\text{i} \\omega \\times (\\text{amplitude}),\n",
+    "$$\n",
+    "\n",
+    "thus the absolute value of particle velocity is usually\n",
+    "much larger than that of amplitude.\n",
+    "It should be also kept in mind that the hole in the middle\n",
+    "does not visualizes the exact motion of the pulsating sphere.\n",
+    "According to the above equation, the actual amplitude should be\n",
+    "much smaller than the arrow lengths.\n",
+    "The changing rate of its size is also two times higher than the original frequency."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Sound Pressure"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "amplitude = 0.05\n",
+    "impedance_pw = sfs.default.rho0 * sfs.default.c\n",
+    "max_pressure = omega * impedance_pw * amplitude\n",
+    "\n",
+    "grid = sfs.util.xyz_grid([xmin, xmax], [ymin, ymax], 0, spacing=0.005)\n",
+    "ani = animation.sound_pressure(\n",
+    "        omega, center, radius, amplitude, grid, frames, pulsate=True,\n",
+    "        figsize=figsize, vmin=-max_pressure, vmax=max_pressure)\n",
+    "plt.close()\n",
+    "HTML(ani.to_html5_video())"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Notice that the sound pressure exceeds\n",
+    "the atmospheric pressure ($\\approx 10^5$ Pa), which of course makes no sense.\n",
+    "This is due to the large amplitude (50 mm) of the pulsating motion.\n",
+    "It was chosen to better visualize the particle movements\n",
+    "in the earlier animations.\n",
+    "\n",
+    "For 1 kHz, the amplitude corresponding to a moderate sound pressure,\n",
+    "let say 1 Pa, is in the order of micrometer.\n",
+    "As it is very small compared to the corresponding wavelength (0.343 m),\n",
+    "the movement of the particles and the spatial structure of the sound field\n",
+    "cannot be observed simultaneously.\n",
+    "Furthermore, at high frequencies, the sound pressure\n",
+    "for a given particle displacement scales with the frequency.\n",
+    "The smaller wavelength (higher frequency) we choose,\n",
+    "it is more likely to end up with a prohibitively high sound pressure.\n",
+    "\n",
+    "In the following examples, the amplitude is set to a realistic value 1 $\\mu$m.\n",
+    "Notice that the pulsating motion of the sphere is no more visible."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "amplitude = 1e-6\n",
+    "impedance_pw = sfs.default.rho0 * sfs.default.c\n",
+    "max_pressure = omega * impedance_pw * amplitude\n",
+    "\n",
+    "grid = sfs.util.xyz_grid([xmin, xmax], [ymin, ymax], 0, spacing=0.005)\n",
+    "ani = animation.sound_pressure(\n",
+    "        omega, center, radius, amplitude, grid, frames, pulsate=True,\n",
+    "        figsize=figsize, vmin=-max_pressure, vmax=max_pressure)\n",
+    "plt.close()\n",
+    "HTML(ani.to_html5_video())"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Let's zoom in closer to the boundary of the sphere."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "L = 10 * amplitude\n",
+    "xmin_zoom, xmax_zoom = radius - L, radius + L\n",
+    "ymin_zoom, ymax_zoom = -L, L"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "grid = sfs.util.xyz_grid([xmin_zoom, xmax_zoom], [ymin_zoom, ymax_zoom], 0, spacing=L / 100)\n",
+    "ani = animation.sound_pressure(\n",
+    "        omega, center, radius, amplitude, grid, frames, pulsate=True,\n",
+    "        figsize=figsize, vmin=-max_pressure, vmax=max_pressure)\n",
+    "plt.close()\n",
+    "HTML(ani.to_html5_video())"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "This shows how the vibrating motion of the sphere (left half)\n",
+    "changes the sound pressure of the surrounding air (right half).\n",
+    "Notice that the sound pressure increases/decreases (more red/blue)\n",
+    "when the surface accelerates/decelerates."
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python [default]",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.5.6"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}