{ "cells": [ { "cell_type": "markdown", "id": "b4a3fb69", "metadata": {}, "source": [ "# Attenuation\n", "In this notebook, the different types of attenuation will be discussed. Group velocity and dispersion (the other topics of lecture 7 of Y2 waves) are discussed *here* (link?).\n", "\n", "Attenuation is the loss of intensity as waves travel through a medium." ] }, { "cell_type": "code", "execution_count": null, "id": "9e1b35c1", "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt" ] }, { "cell_type": "markdown", "id": "ac2c0293", "metadata": {}, "source": [ "## 1. Dispersion\n", "In dispersion, higher frequency waves travel faster than lower frequency waves. The energy spreads out thus leaving less intensity for the pulses.\n", "\"image\n" ] }, { "cell_type": "markdown", "id": "a5b01876", "metadata": {}, "source": [ "## 2. Geometric spreading\n", "Energy spreads out as a function of area. Thus, intensity decreases away from the source. It is important to look at the dimensions of spreading, i.e. water waves pread in 2D so the area is $2\\pi r$. Body waves spread out in 3D, so the area to consider is $2 \\pi r^2$. Geometric spreading is defined as initial intensity divided by the area as a function of distance from the source so $$\\frac{I_0}{area}$$ For example, $\\frac{I_0}{2\\pi r}$ for water waves." ] }, { "cell_type": "code", "execution_count": 41, "id": "9cb6b16a", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "x = np.arange(-0.5,0.5,0.1)\n", "y = np.arange(-0.5,0.5,0.1)\n", "X, Y = np.meshgrid(x,y)\n", "\n", "I_0 = 10\n", "r = (X**2+Y**2)**0.5 # calculate radius\n", "z = I_0/(2 * np.pi * r**2) # geometric spreading \n", "fig, ax= plt.subplots(figsize = (5,5))\n", "CS = plt.contourf(X,Y,z)\n", "plt.xlim(-0.2, 0.2)\n", "plt.ylim(-0.2, 0.2)\n", "cbar = plt.colorbar()\n", "cbar.set_label('Intensity', rotation=90)\n", "plt.title('Geometric spreading')\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "73ab532d", "metadata": {}, "source": [ "## 3. Scattering\n", "Scattering occurs when a medium's impedance has length scales that are smaller than the wavelength of the wave. These changes can also cause reflection and transmission.\n", "\"image" ] }, { "cell_type": "markdown", "id": "4f0b0bd8", "metadata": {}, "source": [ "## 4. Intrinsic attenuation\n", "Gradual loss of energy from the wave, due to materials not being perfectly elastic. This energy is converted into heat. Attenuation can cause damping of particle motion.\n", "If there is a friction that opposes the motion of oscillation, the equation of motion for the particle becomes $$m \\frac{d^2 y}{d t^2} = -sy -r \\frac{dy}{dt}$$\n", "This equation has three solutions:\n", "- Heavy damping: strongest case of attenuation, prohibiting the particle to oscillate. Solution of the form $y=Ce^{-at}$. So damping in this situation is stronger than the restoring force.\n", "- Critical damping: solution of the form $y=(A+Bt)e^{-pt}$. Less attenuation than in heavy damping case, but returns to rest position more quickly still if it has zero velocity at the start.\n", "- Light damping is of the form $y=y_0e^{-bt}cos(\\omega t + \\phi)$." ] }, { "cell_type": "code", "execution_count": 65, "id": "ba0f10d5", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "t = np.linspace(0,10,1001)\n", "a = 2\n", "w = 2\n", "damping_factor = (1+1*t)*np.exp(-t)\n", "y = damping_factor * a \n", "plt.plot(t,y, label = 'critically damped')\n", "\n", "damping_factor2 = np.exp(-0.2*x)\n", "y2 = damping_factor2 * a \n", "plt.plot(x,y2, label = 'heavy damped')\n", "\n", "damping_factor3 = np.exp(-0.2*x)\n", "y2 = damping_factor2 * a * np.cos(w * x)\n", "plt.plot(x,y2, label = 'lightly damped')\n", "# plt.legend()\n", "\n", "plt.legend()\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "2059d9ad", "metadata": {}, "source": [ "## Quality factor\n", "The quality factor is of the form $$ Q = \\frac{m \\omega_0}{r}$$ where $\\omega_0 =\\sqrt{\\frac{s}{m}}$, the natural frequency." ] }, { "cell_type": "code", "execution_count": null, "id": "e3666d4f", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "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": "3.9.12" } }, "nbformat": 4, "nbformat_minor": 5 }