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"# Author: Toma Chaumont Behrs\n",
"\n",
"from IPython.display import HTML, display\n",
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"import matplotlib.widgets as mpw\n",
"import matplotlib.animation as mpan\n",
"from ipywidgets import widgets as ipw\n",
"from mpl_toolkits.mplot3d import Axes3D"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"'''Functions to be used in plots/animations'''\n",
"\n",
"def psi(x,t,a,v):\n",
" '''Bell-Shaped Disturbance: to be used in elastic string derivation and interference demos'''\n",
" '''x & t input by plotting function; a & v provided by user via widget'''\n",
" return np.exp(-a*(x - v*t)**2)\n",
" \n",
"\n",
"def spw(x,t,A1,A2,k1,k2,w1,w2):\n",
" '''Superposed Waves: to be used for standing waves and beat patterns demos'''\n",
" '''x & t input by plotting function; As, ks & ws provided by user via widget'''\n",
" return A1*np.sin(k1*x - w1*t) + A2*np.sin(k2*x + w2*t)\n",
" \n",
" \n",
"def polar(x,tht,k,w,phi,a,b):\n",
" '''Polarized wavefunction: to be used in polarization section'''\n",
" '''x, t & theta input by plotting function; k, w, phi, a & b provided by user via widget'''\n",
" Av = [a*np.cos(tht), b*np.sin(tht), 0] # x, y, or z propagation?\n",
" return Av*np.sin(k*x - w*tht + phi)\n",
" "
]
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"[<matplotlib.lines.Line2D at 0x11503b990>]"
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"[<matplotlib.lines.Line2D at 0x7f0715f6a518>]"
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"<Figure size 432x288 with 1 Axes>"
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"source": [
"# Test plot ; not part of final code.\n",
"\n",
"fig = plt.figure()\n",
"ax = plt.axes(xlim=(0,3), ylim=(-2,2))\n",
"x=np.linspace(0,3,300)\n",
"t=1\n",
"a=5\n",
"v=1\n",
"y=psi((3-x),t,a,v)+psi((x),t,a,v)\n",
"ax.plot(x,y)\n",
"\n",
"# Works!"
]
},
{
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"[<matplotlib.lines.Line2D at 0x112bbbfd0>]"
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"<matplotlib.animation.FuncAnimation at 0x7f0715c0f400>"
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"source": [
"# Test plot ; not part of final code.\n",
"\n",
"fig = plt.figure()\n",
"ax = plt.axes(xlim=(0,3), ylim=(-2,2))\n",
"x=np.linspace(0,3,300)\n",
"t=0\n",
"A1=1\n",
"A2=1\n",
"k1=5\n",
"k2=6\n",
"w1=.2\n",
"w2=.3\n",
"y=spw(x,t,A1,A2,k1,k2,w1,w2)\n",
"ax.plot(x,y)\n",
"\n",
"# Works!"
]
},
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"source": [
"# Test plot ; not part of final code.\n",
"\n",
"fig = plt.figure()\n",
"ax = plt.axes(projection='3d') # xlim=(0,3), ylim=(-2,2), zlim=(-1,1), \n",
"\n",
"x=np.linspace(0,3,300)\n",
"tht=0\n",
"k=5\n",
"w=.2\n",
"phi=.3\n",
"a=1\n",
"b=1\n",
"#y=polar(x,tht,k,w,phi,a,b)\n",
"#ax.plot(x,y)\n"
]
},
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"source": [
"# Test animation ; not part of final code.\n",
"\n",
"fig = plt.figure()\n",
"ax = plt.axes(xlim=(0, 10), ylim=(-2, 2))\n",
"line, = ax.plot([], [], lw=2)\n",
"\n",
"# Prep animation\n",
"def init():\n",
" line.set_data([], [])\n",
" return line,\n",
"\n",
"def animate(i):\n",
" x = np.linspace(0, 10, 1000)\n",
" y = np.cos(i * 0.02 * np.pi) * np.sin(x - i * 0.02 * np.pi)\n",
" line.set_data(x, y)\n",
" return line,\n",
"\n",
"anim = mpan.FuncAnimation(fig, animate, init_func=init,\n",
" frames=100, interval=30)\n",
"\n",
"plt.show()\n",
"\n",
"# Doesn't work ;_;"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Useful resources\n",
"\n",
"https://brilliant.org/wiki/wave-equation/\n",
"\n",
"https://en.wikipedia.org/wiki/Wave_equation#Derivation_of_the_wave_equation\n",
"\n",
"Desmos avi for python : https://www.desmos.com/api/v1.3/docs/index.html\n",
"\n",
"https://www.acs.psu.edu/drussell/Demos/superposition/superposition.html\n",
"\n",
"https://www.geogebra.org/m/dJrTcxYd"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# The Wave Equation\n",
"\n",
"What *is* a wave?\n",
"\n",
"## Definition of a wave\n",
"\n",
"A wave is a periodic disturbance of a field or medium such that its components oscillate repeatedly about a stable equilibrium. A wave transfers energy in the forms of motion, sound, heat, and even light! \n",
"\n",
"\n",
"### Examples: \n",
"\n",
"Wave | Field/Medium | Oscillating Attribute \n",
"--- | --- | --- \n",
"Water wave | Water | Height\n",
"Sound wave | Air | Density\n",
"Stadium wave | People | Hand Height\n",
"Light | Electromagnetic Field | Field Amplitude \n",
"\n",
"\n",
"## Important properties of a wave : \n",
"\n",
"### Frequency\n",
"\n",
"### Amplitude\n",
"\n",
"### Wavelength\n",
"\n",
"\n",
"## Derivation of the Wave Equation: \n",
"\n",
"There are several ways to derive the wave equation:\n",
"\n",
"- Using an elastic string \n",
"- From Maxwell's equations\n",
"- Starting with the intuition of $ x \\pm vt$ and then taking two partial derivatives. (See Optics by Hecht, Ch.2)\n",
"\n",
"Some are more straightforward than others, so let's start with the simplest case and build from there.\n",
"\n",
"\n",
"### Elastic String (1-Dimensional) : \n",
"\n",
"The disturbance could be any function of $x$, for example: \n",
"\n",
" - Gaussian : $ \\psi (x) = e ^{-a x^2} $\n",
" - Lorentzian : $ \\psi (x) = \\frac{1}{1+x^2}$ \n",
"\n",
"Let us think of such a disturbance moving in the positive $x$ direction with a constant speed $v$. Since the disturbance is moving, it must be described by a function of position as well as time $\\psi(x,t)$. The shape of the disturbance at a particular time, say $t=0$, can be found by plugging the value of time as a particular constant (for example $0$). In this case, $\\psi(x,t)|_{t=0} = \\psi(x,0)$ represents the **profile** of the wave at that time. In more intuitive terms, this is analogous to taking a photograph of a pulse as it travels by. We limit ourselves to a wave that does not change it's shape as it travels through space, i.e., in time $t$, it travels a distance $vt$ but remains unchanged in all other aspects. \n",
"\n",
"Let us move to a new coordinate system ($S'$) that travels in the same direction as the wave with the same velocity, and shares the origin with the frame $S$ at time $t=0$. In this frame, the wave appears to be stationary and is no longer a function of time. Here, the coordinate is $x'$ rather than $x$. It follows that: \n",
"\n",
"$$ x' = x - vt$$\n",
"\n",
"And thus, by substituting we get, \n",
"\n",
"$$ \\psi(x,t) = f(x-vt) $$\n",
"\n",
"This represents the most general form of a 1D **wavefunction**. To recap: \n",
"\n",
"- Choose a profile by choosing a function $f(x)$\n",
"- Substitute $x-vt$ instead of $x$ in $f(x)$.\n",
"\n",
"This gives a wave traveling in the positive $x$-direction with a speed $v$.\n",
"\n",
"For example : \n",
"$f(x) = e ^{-a x^2}$ gives a bell shaped profile. \n",
"$\\psi (x,t) = e ^{-a (x-vt)^2}$ gives a bell-shaped disturbance travelling in the $+x$ direction with velocity $v$."
]
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"source": [
"%matplotlib inline\n",
"'''Bell-shaped disturbance animation'''\n",
"\n",
"# Create plot\n",
"fig = plt.figure()\n",
"ax = plt.axes(xlim=(0,3), ylim=(-2,2))\n",
"\n",
"line, = ax.plot([],[],lw=2)\n",
"\n",
"def a_val(a_slide): # user-defined amplitude\n",
" a = a_slide\n",
"\n",
"def v_val(v_slide): # user-defined velocity\n",
" v = v_slide\n",
" \n",
"# Display sliders\n",
"ipw.interact(a_val, a_slide=ipw.FloatSlider(min=0, max=10, step=0.1, value=1.0, description = '$a$'))\n",
"ipw.interact(v_val, v_slide=ipw.FloatSlider(min=0, max=3, step=0.1, value=1.0, description = '$v$'))\n",
"\n",
"# Prep animation\n",
"def init():\n",
" line.set_data([],[])\n",
" return line,\n",
"\n",
"def animate(t):\n",
" '''To be called by FuncAnimation at each time increment'''\n",
" x = np.linspace(0,3,300)\n",
" a = a_slide\n",
" v = v_slide\n",
" y = psi(x,t,a,v)\n",
" line.set_data(x,y)\n",
" return line,\n",
"\n",
"# anim\n",
"mpan.FuncAnimation(fig, animate, init_func=init, frames=500, interval=50, blit=True)\n",
"\n",
"# plt.show()\n",
"\n",
"# HTML(anim.to_html5_video())\n",
"# widget and anim separate but linked; slider passes value to anim function. (Press play again but nbd)\n",
"# ipw.interact"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Wave equation\n",
"\n",
"We will now derive the wave equation using the information that the most basic form of waves traveling at a fixed velocity requires two constants to specify it (amplitude and frequency/wavelength), which suggests second derivatives. To relate the space and time dynamics of $\\psi(x,t)$, we take the partial derivative of $\\psi(x,t) = f(x')$ with respect to $x$. Using $x' = x \\pm vt$: \n",
"\n",
"$$ \\frac{\\partial \\psi}{\\partial x} = \\frac{\\partial f}{\\partial x} = \\frac{\\partial f}{\\partial x'} \\frac{\\partial x'}{\\partial x} = \\frac{\\partial f}{\\partial x'}$$ \n",
"\n",
"The partial derivative with respect to time is: \n",
"\n",
"$$ \\frac{\\partial \\psi}{\\partial t} = \\frac{\\partial f}{\\partial x'} \\frac{\\partial x'}{\\partial t} = \\frac{\\partial f}{\\partial x'} ( \\pm v) = \\pm v \\frac{\\partial f}{\\partial x'}$$ \n",
"\n",
"Taking a second derivative with respect to $x$, we get: \n",
"\n",
"$$ \\frac{\\partial^2 \\psi }{\\partial x^2} = \\frac{\\partial^2 f}{\\partial x'^2}$$ \n",
"\n",
"Similarly, taking a second derivative with respect to time: \n",
"\n",
"$$ \\frac{\\partial^{2} \\psi}{\\partial t^{2}}=\\frac{\\partial}{\\partial t}\\left(\\mp v \\frac{\\partial f}{\\partial x^{\\prime}}\\right)=\\mp v \\frac{\\partial}{\\partial x^{\\prime}}\\left(\\frac{\\partial f}{\\partial t}\\right) $$ \n",
"\n",
"$$ \\frac{\\partial^{2} \\psi}{\\partial t^{2}}=\\mp v \\frac{\\partial}{\\partial x^{\\prime}}\\left(\\frac{\\partial \\psi}{\\partial t}\\right) =v^{2} \\frac{\\partial^{2} f}{\\partial x^{\\prime 2}}$$ \n",
"\n",
"Finally, we combine the two equations for second order derivatives to obtain: \n",
"\n",
"$$ \\frac{\\partial^{2} \\psi}{\\partial x^{2}}=\\frac{1}{v^{2}} \\frac{\\partial^{2} \\psi}{\\partial t^{2}} $$ \n",
"\n",
"which is the 1D wave equation.\n",
"\n",
"\n",
"### Showing the Wave Equation for $\\vec{\\mathbf{E}}$ and $\\vec{\\mathbf{B}}$ from Maxwell's Equations\n",
"\n",
"Maxwell's equations in free space in the absence of charges and currents are given by: \n",
"\n",
"$$\\begin{align}\n",
"\\nabla \\cdot \\vec{\\mathbf{E}} &= 0 \\\\\n",
"\\nabla \\times \\vec{\\mathbf{E}} &= - \\frac{\\partial \\vec{\\mathbf{B}}}{\\partial t} \\\\\n",
"\\nabla \\cdot \\vec{\\mathbf{B}} &= 0 \\\\\n",
"\\nabla \\times \\vec{\\mathbf{B}} &= \\mu_0 \\epsilon_0 \\frac{\\partial \\vec{\\mathbf{E}}}{\\partial t}\n",
"\\end{align} $$\n",
"\n",
"Taking the curl of the second equation and using a well-known vector identity, we get: \n",
"\n",
"$$ \\nabla \\times (\\nabla \\times \\vec{\\mathbf{E}}) = - \\nabla \\times (\\frac{\\partial \\vec{\\mathbf{B}}}{\\partial t}) = \\nabla (\\nabla \\cdot \\vec{\\mathbf{E}} ) - \\nabla^2 \\vec{\\mathbf{E}} = - \\frac{ \\partial (\\nabla \\times \\vec{\\mathbf{B}})}{\\partial t}$$ \n",
"\n",
"Plugging in the 1st and 4th equations yields: \n",
"\n",
"$$ \\nabla^2 \\vec{\\mathbf{E}} = \\mu_0 \\epsilon_0 \\frac{\\partial^2 \\vec{\\mathbf{E}}}{\\partial t^2} $$ \n",
"\n",
"which is the 3-dimensional wave equation for the electric field.\n",
"\n",
"#### Exercise:\n",
"\n",
"Analogously derive the wave equation for the magnetic field.\n",
"\n",
"\n",
"### Why any function of $(x-ct)$ and $(x+ct)$ is a solution\n",
"\n",
"Staring from the 1D wave equation: \n",
"\n",
"$$ \\frac{\\partial^{2} \\psi}{\\partial x^{2}}=\\frac{1}{v^{2}} \\frac{\\partial^{2} \\psi}{\\partial t^{2}} $$ \n",
"\n",
"and making the change of variables: \n",
"\n",
"$$ u = x - vt ; v = x + vt $$ \n",
"\n",
"gives:\n",
"\n",
"$$ \\frac{\\partial^2 \\psi}{\\partial u \\partial v} = 0$$\n",
"\n",
"which leads to the general solution: \n",
"\n",
"$$ \\psi(u,v) = F(u) + G(v) $$ \n",
"\n",
"Or equivalently,\n",
"\n",
"$$ \\psi(x,t) = F(x-ct) + G(x+ct) $$\n",
"\n",
"Which shows that any general solution is a linear combination of left-moving and right-moving waves. \n",
" \n",
" ## Good demonstration of this with funny shapes (possibly from desmos) + Gaussian and other standard shapes whose functional form is known\n",
" \n",
"## Extension to 3 Dimensions - (Wave Equation in 3D)\n",
"\n",
"The wave equation in 3D is given by: \n",
"\n",
"$$ \\frac{\\partial^2 \\psi}{\\partial t^2} = v^2 \\nabla^2 \\psi $$"
]
},
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"cell_type": "markdown",
"metadata": {},
"source": [
"## 3D \"Plane Wave\" and why it is called so\n",
"\n",
"A plane wave in 3D refers to a special case of a wave whose amplitude takes a constant value over a plane. In its most general form, a plane wave can be written as: \n",
"\n",
"$$ F(\\vec{x}, t) = G(\\vec{x} . \\vec{n} , t) $$ \n",
"\n",
"Where $\\vec{n}$ is a unit vector. Thus, the value of the field is dependant only on $t$ and $\\vec{x} \\cdot \\vec{n}$ which represents the displacement in the direction of the unit vector $\\vec{n}$. For the puposes of illustration, let us consider a sinusoidal plane wave: \n",
"\n",
"$$ F(\\vec{x}, t) = sin( 2 \\pi f (\\vec{x} . \\vec{n} - ct) + \\varphi ) $$ \n",
"\n",
"--- \n",
"\n",
"(because it takes the same value in the entire plane). What may be easier mathematically is not always physical (infinitely extended wave in this case)\n",
"\n",
" # Demo of plane wave in 3D (try to implement plane_wave1 and plane_wave2 side by side for easier understanding)\n",
"\n",
"Here's a 3D plot of plane wave, showing its oscillations in both the horizontal and vertical directions:\n",
"![A Plane Wave](../static/img/plane_wave2.png)\n",
"\n",
"Now here's what it looks like propagating through space:\n",
"\n",
"![Plane Wave Propagating](../static/img/plane_wave1.gif) \n",
"\n",
" # Shift to explaining polarisation.\n",
" \n",
" # Equations and animations for circular, radial and azimuthal polarisation."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Superposition of waves\n",
"\n",
"The superposition principle for waves states that when multiple waves are travelling through a medium, the net displacement/amplitude at any point is equal to the algebraic sum of the contribution by all individual waves. \n",
"\n",
"\n",
"## Important cases\n",
"\n",
"### Standing Waves\n",
"\n",
"These are seen when two waves having the same frequency travel in opposite directions:\n",
"\n",
"![Standing Wave](../static/img/Standing_wave_2.gif)\n",
"\n",
"And can be described mathematically by:\n",
"\n",
"$$ A(x, t)= A \\sin (\\mathrm{kx}-\\omega \\mathrm{t})+A \\sin (\\mathrm{kx}+\\omega \\mathrm{t})=2 A \\sin (\\mathrm{kx}) \\cos (\\omega \\mathrm{t}) $$\n",
"\n",
"\n",
"### Beat Patterns\n",
"\n",
"This is seen when two waves having slightly different frequencies travel in the same direction. \n",
"\n",
"<img src=\"../static/img/Beating_Frequency.png\" alt=\"Drawing\" style=\"width: 350px;\"/>\n",
"\n",
"And can be described mathematically by:\n",
"\n",
"$$ A(x, t)=A \\sin \\left(k_{1} x-\\omega_{1} t\\right)+A \\sin \\left(k_{2} x+\\omega_{2} t\\right)=2 A \\cos \\left[\\frac{\\left(k_{1}-k_{2}\\right)}{2} x-\\frac{\\left(\\omega_{1}-\\omega_{2}\\right)}{2} t\\right] \\sin \\left[\\frac{\\left(k_{1}+k_{2}\\right)}{2} x-\\frac{\\left(\\omega_{1}+\\omega_{2}\\right)}{2} t\\right] $$\n",
"\n",
"#### Try producing standing waves and beat patterns yourself:"
]
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"data": {
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\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
},
{
"data": {
"application/vnd.jupyter.widget-view+json": {
<<<<<<< HEAD
"model_id": "daddb6d8f5df4bf181e9a9503491e192",
=======
"model_id": "cd65b0eebc3d404ca3c1b278b850b35f",
>>>>>>> c5ad7f0fcb506c2d229e87443c7ab5d482a065d9
"version_major": 2,
"version_minor": 0
},
"text/plain": [
"HBox(children=(FloatSlider(value=1.0, description='$A_1$', max=2.0, min=-2.0, step=0.2), FloatSlider(value=1.0…"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"application/vnd.jupyter.widget-view+json": {
<<<<<<< HEAD
"model_id": "3b41a9b257dc49a48c8fa06d75ccb364",
=======
"model_id": "6c26ec5ed25d4add958499566371b3e3",
>>>>>>> c5ad7f0fcb506c2d229e87443c7ab5d482a065d9
"version_major": 2,
"version_minor": 0
},
"text/plain": [
"HBox(children=(FloatSlider(value=1.0, description='$A_2$', max=2.0, min=-2.0, step=0.2), FloatSlider(value=1.0…"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"application/vnd.jupyter.widget-view+json": {
<<<<<<< HEAD
"model_id": "aa6a4506b27140dfbf9f533b0c34144e",
=======
"model_id": "cba8e7855dc84b8b88ad065286d72ba6",
>>>>>>> c5ad7f0fcb506c2d229e87443c7ab5d482a065d9
"version_major": 2,
"version_minor": 0
},
"text/plain": [
"Output()"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
<<<<<<< HEAD
"<matplotlib.animation.FuncAnimation at 0x112a69450>"
=======
"<matplotlib.animation.FuncAnimation at 0x7f0715a772e8>"
>>>>>>> c5ad7f0fcb506c2d229e87443c7ab5d482a065d9
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"'''Superposed waves animation'''\n",
"\n",
"# Create plot\n",
"fig = plt.figure()\n",
"ax = plt.axes(xlim=(0,3), ylim=(-2,2))\n",
"\n",
"line, = ax.plot([],[],lw=2)\n",
" \n",
"# user-defined amplitudes, wavenumbers and frequencies of first and second waves\n",
"A1_slide=ipw.FloatSlider(min=-2, max=2, step=0.2, value=1.0, description = '$A_1$')\n",
"A2_slide=ipw.FloatSlider(min=-2, max=2, step=0.2, value=1.0, description = '$A_2$')\n",
"k1_slide=ipw.FloatSlider(min=-2, max=2, step=0.2, value=1.0, description = '$k_1$')\n",
"k2_slide=ipw.FloatSlider(min=-2, max=2, step=0.2, value=1.0, description = '$k_2$')\n",
"w1_slide=ipw.FloatSlider(min=-2, max=2, step=0.2, value=1.0, description = '$\\\\omega_1$')\n",
"w2_slide=ipw.FloatSlider(min=-2, max=2, step=0.2, value=1.0, description = '$\\\\omega_2$')\n",
"\n",
"# Set up slider layout\n",
"ui1 = ipw.HBox([A1_slide, k1_slide, w1_slide]) # top row\n",
"ui2 = ipw.HBox([A2_slide, k2_slide, w2_slide]) # bottom row\n",
"\n",
"def slide_vals(A1_slide,A2_slide,k1_slide,k2_slide,w1_slide,w2_slide):\n",
" A1 = A1_slide\n",
" A2 = A2_slide\n",
" k1 = k1_slide\n",
" k2 = k2_slide\n",
" w1 = w1_slide\n",
" w2 = w2_slide \n",
"\n",
"# Display sliders\n",
"out = ipw.interactive_output(slide_vals, {'A1_slide': A1_slide, 'k1_slide': k1_slide, 'w1_slide': w1_slide, \n",
" 'A2_slide': A2_slide, 'k2_slide': k2_slide, 'w2_slide': w2_slide})\n",
"display(ui1, ui2, out)\n",
"\n",
"# Prep animation\n",
"def init():\n",
" line.set_data([],[])\n",
" return line,\n",
"\n",
"def animate(t):\n",
" '''To be called by FuncAnimation at each time increment'''\n",
" x = np.linspace(0,3,300)\n",
" A1 = A1_slide\n",
" A2 = A2_slide\n",
" k1 = k1_slide\n",
" k2 = k2_slide\n",
" w1 = w1_slide\n",
" w2 = w2_slide\n",
" y = spw(x,t,A1,A2,k1,k2,w1,w2)\n",
" line.set_data(x,y)\n",
" return line,\n",
"\n",
"# Plot animation\n",
"# anim =\n",
"mpan.FuncAnimation(fig, animate, init_func=init, frames=500, interval=50, blit=True)\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Interference\n",
"\n",
"All of the above are examples wave *interference*, which can be divided into two categories:\n",
"\n",
"#### Constructive\n",
"\n",
"Two waves reinforce each other. This effect is maximized when they are perfectly in sync.\n",
"\n",
"![Constructive Interference](../static/img/Constructive_interference.svg)\n",
"\n",
"#### Destructive\n",
"\n",
"Two waves negate each other. This effect is maximized when they are exactly out of sync.\n",
"\n",
"![Destructive Interference](../static/img/Destructive_interference1.svg)\n",
"\n",
"##### Try replicating these two different kinds of interference!"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
},
{
"data": {
"application/vnd.jupyter.widget-view+json": {
<<<<<<< HEAD
"model_id": "7a73954c313745adb9a71c4e0265855b",
=======
"model_id": "67ea3be175614b62b8cd022fb30171fc",
>>>>>>> c5ad7f0fcb506c2d229e87443c7ab5d482a065d9
"version_major": 2,
"version_minor": 0
},
"text/plain": [
"HBox(children=(FloatSlider(value=1.0, description='$a_1$', max=10.0), FloatSlider(value=1.0, description='$a_2…"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"application/vnd.jupyter.widget-view+json": {
<<<<<<< HEAD
"model_id": "ab80de393afd4839acea329cee13cf55",
=======
"model_id": "bed89905b968443191c93b55c6fb1a8f",
>>>>>>> c5ad7f0fcb506c2d229e87443c7ab5d482a065d9
"version_major": 2,
"version_minor": 0
},
"text/plain": [
"HBox(children=(FloatSlider(value=1.0, description='$v_1$', max=3.0), FloatSlider(value=1.0, description='$v_2$…"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"application/vnd.jupyter.widget-view+json": {
<<<<<<< HEAD
"model_id": "4dfadbf5b7f54e42911790ab87b09775",
=======
"model_id": "df0e22e26f5a4f0abd5e9f9bbc2f1336",
>>>>>>> c5ad7f0fcb506c2d229e87443c7ab5d482a065d9
"version_major": 2,
"version_minor": 0
},
"text/plain": [
"Output()"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"'''Interference animation'''\n",
"\n",
"# Create plot\n",
"fig = plt.figure()\n",
"ax = plt.axes(xlim=(0,3), ylim=(-2,2))\n",
"\n",
"line, = ax.plot([],[],lw=2)\n",
"\n",
"# user-defined amplitudes and velocities of first and second pulses\n",
"a1_slide=ipw.FloatSlider(min=0, max=10, step=0.1, value=1.0, description = '$a_1$')\n",
"a2_slide=ipw.FloatSlider(min=0, max=10, step=0.1, value=1.0, description = '$a_2$')\n",
"v1_slide=ipw.FloatSlider(min=0, max=3, step=0.1, value=1.0, description = '$v_1$')\n",
"v2_slide=ipw.FloatSlider(min=0, max=3, step=0.1, value=1.0, description = '$v_2$')\n",
"\n",
"# Set up slider layout\n",
"ui1 = ipw.HBox([a1_slide, a2_slide]) # top row\n",
"ui2 = ipw.HBox([v1_slide, v2_slide]) # bottom row\n",
"\n",
"def slide_vals(a1_slide, a2_slide, v1_slide, v2_slide):\n",
" a1 = a1_slide\n",
" a2 = a2_slide\n",
" v1 = v1_slide\n",
" v2 = v2_slide\n",
"\n",
"# Display sliders\n",
"out = ipw.interactive_output(slide_vals, {'a1_slide': a1_slide, 'a2_slide': a2_slide,\n",
" 'v1_slide': v1_slide, 'v2_slide': v2_slide})\n",
"display(ui1, ui2, out)\n",
"\n",
"# Prep animation\n",
"def init():\n",
" line.set_data([],[])\n",
" return line,\n",
"\n",
"def animate(t):\n",
" '''To be called by FuncAnimation at each time increment'''\n",
" x = np.linspace(0,3,300)\n",
" a1 = a1_slide\n",
" a2 = a2_slide\n",
" v1 = v1_slide\n",
" v2 = v2_slide\n",
" y = psi(x,t,a1,v2) + psi((3-x),t,a2,v2)\n",
" line.set_data(x,y)\n",
" return line,\n",
"\n",
"# Plot animation\n",
"anim = mpan.FuncAnimation(fig, animate, init_func=init, frames=500, interval=50, blit=True)\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Reflection and Transmission\n",
"\n",
" \n",
" \n",
"Check out the notebook on Fresnel coefficients for more details."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Polarisation \n",
"\n",
"Polarisation is property of transverse waves which specifies the geometric orientation of the oscillation.\n",
"\n",
"For electromagnetic radiation, this describes the time-varying direction and relative magnitude of the electric field vector; the trace and magnitude of the electric field vector are observed in the direction of light propagation. This indicates the plane of the oscillation.\n",
"\n",
"The polarisation direction of a wave is indicated by using vector notation for the amplitude of a wave. This will be clarified by the examples given below.\n",
"\n",
"\n",
"## Types of Polarisation\n",
"\n",
"There are three different types of polarisation:\n",
"\n",
"### Linear\n",
"\n",
"$$ A(x,t) = \\vec{A} sin(kx - \\omega t + \\varphi) $$ with $$ \\vec{A} = (a, 0, 0) $$\n",
"\n",
"### Circular\n",
"\n",
"$$ A(x,t) = \\vec{A} sin(kx - \\omega t + \\varphi) $$ with $$ \\vec{A} = (cos(\\theta t), sin(\\theta t), 0) $$\n",
"\n",
"### Elliptical\n",
"\n",
"$$ A(x,t) = \\vec{A} sin(kx - \\omega t + \\varphi) $$ with $$ \\vec{A} = (a cos(\\theta t), b sin(\\theta t), 0) $$\n",
"\n",
"\n",
"#### Try creating the different types of polarisation yourself!"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
},
{
"data": {
"application/vnd.jupyter.widget-view+json": {
"model_id": "332c5db89f974066bf77914c6c48b65a",
"version_major": 2,
"version_minor": 0
},
"text/plain": [
"HBox(children=(FloatSlider(value=1.0, description='$k$', max=2.0, min=-2.0), FloatSlider(value=1.0, descriptio…"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"application/vnd.jupyter.widget-view+json": {
"model_id": "248e05894e5640cfa5dc0dbd18b63273",
"version_major": 2,
"version_minor": 0
},
"text/plain": [
"HBox(children=(FloatSlider(value=1.0, description='$a$', max=3.0, min=-3.0), FloatSlider(value=1.0, descriptio…"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"application/vnd.jupyter.widget-view+json": {
"model_id": "d0cf2b9ac83749c28276f57eb990259d",
"version_major": 2,
"version_minor": 0
},
"text/plain": [
"Output()"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"<matplotlib.animation.FuncAnimation at 0x112b686d0>"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"'''Polarization animation (with projections)'''\n",
"\n",
"fig = plt.figure()\n",
"ax = plt.axes(projection='3d') # xlim=(0,3), ylim=(-2,2), zlim=(-1,1), \n",
"line, = ax.plot([], [], lw=2)\n",
"\n",
"# user-defined amplitudes and velocities of first and second pulses\n",
"k_slide=ipw.FloatSlider(min=-2, max=2, step=0.1, value=1.0, description = '$k$')\n",
"w_slide=ipw.FloatSlider(min=-2, max=2, step=0.1, value=1.0, description = '$\\\\omega$')\n",
"phi_slide=ipw.FloatSlider(min=-np.pi, max=np.pi, step=0.1, value=0, description = '$\\\\phi$')\n",
"a_slide=ipw.FloatSlider(min=-3, max=3, step=0.1, value=1.0, description = '$a$')\n",
"b_slide=ipw.FloatSlider(min=-3, max=3, step=0.1, value=1.0, description = '$b$')\n",
"\n",
"# Set up slider layout\n",
"ui1 = ipw.HBox([k_slide, w_slide, phi_slide]) # top row\n",
"ui2 = ipw.HBox([a_slide, b_slide]) # bottom row\n",
"\n",
"def slide_vals(k_slide, w_slide, phi_slide, a_slide, b_slide):\n",
" k = k_slide\n",
" w = w_slide\n",
" phi = phi_slide\n",
" a = a_slide\n",
" b = b_slide\n",
"\n",
"# Display sliders\n",
"out = ipw.interactive_output(slide_vals, {'k_slide': k_slide, 'w_slide': w_slide, 'phi_slide': phi_slide,\n",
" 'a_slide': a_slide, 'b_slide': b_slide})\n",
"display(ui1, ui2, out)\n",
"\n",
"\n",
"def init():\n",
" line.set_data([], [])\n",
" return line,\n",
"\n",
"def animate(tht):\n",
" '''To be called by FuncAnimation at each time increment'''\n",
" x = np.linspace(0,3,300)\n",
" k = k_slide\n",
" w = w_slide\n",
" phi = phi_slide\n",
" a = a_slide\n",
" b = b_slide\n",
" y = polar(x,tht,k,w,phi,a,b)\n",
" line.set_data(x,y)\n",
" return line,\n",
"\n",
"\n",
"# Plot animation\n",
"mpan.FuncAnimation(fig, animate, init_func=init, frames=500, interval=50, blit=True)\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Special Topics\n",
"\n",
"\n",
"## Total Destructive Interference\n",
"\n",
"Earlier we discussed how superposing two waves of equal and opposite amplitude results in a moment when they cancel each other out (there is no wave; see the image above). Makes sense, but what about energy conservation?! \n",
"\n",
"\n",
"## Sound Waves and Heat\n",
"\n",
"So far we've mostly talked about light waves and string waves, but there are two other interesting kinds of waves worth mentioning:\n",
"\n",
"### Sound waves\n",
"\n",
"\n",
"\n",
"### Heat\n",
"\n",
"\n",
"\n",
"## Cherenkov Radiation\n",
"\n",
"If you've ever seen images of the inside of nuclear reactors, you've probably noticed a telltale blue-ish glow:\n",
"\n",
"<table><tr><td><img src='../static/img/ReedResearchReactor.jpg'></td><td><img src='../static/img/Advanced_Test_Reactor.jpg'></td></tr></table>\n",
"\n",
"Earlier we briefly discussed how light travels at speeds slower than $c$ when propagating through media less permeable than vaccuum (such as the water in the above image).\n",
"So what do you think happens when a highly-energetic particle (such as one emanating from a nuclear reactor) enters a medium travelling *faster* than the local speed of light?\n",
"\n",
"Well, just like a plane breaking the sound barrier creates shockwaves known as a sonic boom, a particle breaking **local** \"light-barrier\" creates *shockwaves of light*!\n",
"\n",
"![CherenkovGIF](../static/img/Cherenkov_radiation-animation.gif)\n",
"\n",
"The resulting eerie blue glow is known as *Cherenkov radiation* (after Soviet physicist Pavel A. Cherenkov, who discovered it in 1934).\n"
]
},
{
"cell_type": "raw",
"metadata": {},
"source": [
"Reminders:\n",
"Cite ATR image, or remove and shrink RRR image via \n",
"<img src=\"../static/img/ReedResearchReactor.jpg\" alt=\"Drawing\" style=\"width: 200px;\"/>"
]
}
],
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