List of Physics Wave Formulas, Equations Latex Code

rockingdingo 2023-03-26 #physics #wave

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In this blog, we will introduce most popuplar formulas in Wave, Physics. We will also provide latex code of the equations. Topics include Wave Equation, Plane Waves, Spherical Waves, Cylindrical Waves, The general solution of Wave Equation, The Stationary Phase Method, Green functions for the initial-value problem, Waveguides and resonating cavities, Non-linear wave equations, etc.

1. Waves

Wave Equation

Plane Waves

Spherical Waves

Cylindrical Waves

The general solution of Wave Equation

The Stationary Phase Method

Green functions for the initial-value problem

Waveguides and resonating cavities

Non-linear wave equations

1. Waves

Wave Equation

Equation

$\Psi(t)=\hat{\Psi}(t)e^{i(\omega t \pm \phi)}=\hat{\Psi}(t)\cos (\omega t \pm \phi) \\ \sum_{i} \hat{\Psi_{i}}\cos(\alpha_{i} \pm \omega t) =\hat{\Phi}\cos (\beta \pm \omega t) \\ \tan (\beta)=\frac{\sum_{i} \hat{\Psi_{i}} \sin (\alpha_{i})}{\sum_{i} \hat{\Psi_{i}} \cos (\alpha_{i})} \\ \hat{\Phi}^{2} = \sum_{i} \hat{\Psi_{i}^{2}} + 2 \sum_{j > i} \sum_{i} \hat{\Psi_{i}} \hat{\Psi_{j}} \cos (\alpha_{i} - \alpha_{j}) \\ \int x(t) dt=\frac{x(t)}{i \omega} \\ \frac{d^{n}(x(t))}{d t^{n}}=(i \omega)^{n} x(t)$

Latex Code

\nabla^2u-\frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}=\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+\frac{\partial^2 u}{\partial z^2}-\frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}=0 \\ v=f\lambda \\ k\lambda=2\pi \\ \omega=2\pi f \\ v_{\rm g}=\frac{d\omega}{dk}=v_{\rm ph}+k\frac{dv_{\rm ph}}{dk}= v_{\rm ph}\left(1-\frac{k}{n}\frac{dn}{dk}\right) \\ v=\sqrt{\kappa/\varrho}

Explanation

Latex code for the harmonic oscillations. I will briefly introduce the notations in this formulation.

$u$ : disturbance

$v$ : propagation velocity

$v_{ph}$ : phase velocity

$v_{g}$ : group velocity

$n$ : refractive index of the medium

$v=\sqrt{\kappa/\varrho}$ : Pressure waves in a liquid or gas, $\kappa$ is the modulus of compression.

$v=\sqrt{\gamma p/\varrho}=\sqrt{\gamma RT/M}$ : Further for pressure waves in a gas

$v=\sqrt{E/\varrho}$ : Pressure waves in a thin solid bar with diameter

$v=\sqrt{F_{\rm span}l/m}$ : Waves in a string

$\displaystyle v=\sqrt{\left(\frac{g\lambda}{2\pi}+ \frac{2\pi\gamma}{\varrho\lambda}\right)\tanh\left(\frac{2\pi h}{\lambda}\right)}$ : Surface waves on a liquid, h is the depth of the liquid and \gamma the surface tension.

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Plane Waves

Equation

$u(\vec{x},t)=2^n\hat{u}\cos(\omega t)\sum_{i=1}^n\sin(k_ix_i) \\ u(\vec{x},t)=\hat{u}\cos(\vec{k}\cdot\vec{x}\pm\omega t+\varphi) \\ \frac{f}{f_0}=\frac{v_{\rm f}-v_{\rm obs}}{v_{\rm f}}$

Latex Code

            u(\vec{x},t)=2^n\hat{u}\cos(\omega t)\sum_{i=1}^n\sin(k_ix_i) \\
            u(\vec{x},t)=\hat{u}\cos(\vec{k}\cdot\vec{x}\pm\omega t+\varphi) \\
            \frac{f}{f_0}=\frac{v_{\rm f}-v_{\rm obs}}{v_{\rm f}}

Explanation

Latex code for the harmonic oscillations. I will briefly introduce the notations in this formulation.

$u(\vec{x},t)$ : harmonic plane wave is defined as u(x,t)
$\frac{f}{f_0}$ : Doppler effect

The general solution of Wave Equation in one dimension

Equation

$\frac{\partial^2 u(x,t)}{\partial t^2}=\sum_{m=0}^{N}\left(b_m\frac{\partial ^m}{\partial x^m}\right)u(x,t) \\ u(x,t)=\int\limits_{-\infty}^{\infty}\left(a(k){\rm e}^{i(kx-\omega_1(k)t)}+ b(k){\rm e}^{i(kx-\omega_2(k)t)}\right)dk \\ u(x,t)=A{\rm e}^{i(kx-\omega t)} \\ \omega_j=\omega_j(k)$

Latex Code

            \frac{\partial^2 u(x,t)}{\partial t^2}=\sum_{m=0}^{N}\left(b_m\frac{\partial ^m}{\partial x^m}\right)u(x,t) \\ 
            u(x,t)=\int\limits_{-\infty}^{\infty}\left(a(k){\rm e}^{i(kx-\omega_1(k)t)}+ b(k){\rm e}^{i(kx-\omega_2(k)t)}\right)dk \\
            u(x,t)=A{\rm e}^{i(kx-\omega t)} \\
            \omega_j=\omega_j(k)

Explanation

The general solution of is given by above.

The stationary phase method

Equation

$\int\limits_{-\infty}^\infty a(k){\rm e}^{i(kx-\omega(k)t)}dk\approx \sum_{i=1}^{N}\sqrt{\frac{2\pi}{\frac{d^2\omega(k_i)}{dk_i^2}}} \exp\left[-i\mbox{$\frac{1}{4}$}\pi+i(k_ix-\omega(k_i)t)\right]$

Latex Code

            \int\limits_{-\infty}^\infty a(k){\rm e}^{i(kx-\omega(k)t)}dk\approx \sum_{i=1}^{N}\sqrt{\frac{2\pi}{\frac{d^2\omega(k_i)}{dk_i^2}}} \exp\left[-i\mbox{$\frac{1}{4}$}\pi+i(k_ix-\omega(k_i)t)\right]

Explanation

Green functions for the initial-value problem

Equation

$u(x,t)=\int\limits_{-\infty}^\infty f(x')Q(x,x',t)dx'+ \int\limits_{-\infty}^\infty g(x')P(x,x',t)dx' \\ \begin{aligned} Q(x,x',t)&=&\frac{1}{2} [\delta(x-x'-vt)+\delta(x-x'+vt)]\\ P(x,x',t)&=& \left\{\begin{array}{ll} \displaystyle \frac{1}{2v}&~~~\mbox{if}~~|x-x'|<vt\\ 0&~~~\mbox{if}~~|x-x'|>vt \end{array}\right.\end{aligned} \\ \displaystyle Q(x,x',t)=\frac{\partial P(x,x',t)}{\partial t}$

Latex Code

            u(x,t)=\int\limits_{-\infty}^\infty f(x')Q(x,x',t)dx'+ \int\limits_{-\infty}^\infty g(x')P(x,x',t)dx' \\
            \begin{aligned} Q(x,x',t)&=&\frac{1}{2} [\delta(x-x'-vt)+\delta(x-x'+vt)]\\ P(x,x',t)&=& \left\{\begin{array}{ll} \displaystyle \frac{1}{2v}&~~~\mbox{if}~~|x-x'|vt \end{array}\right.\end{aligned} \\
            \displaystyle Q(x,x',t)=\frac{\partial P(x,x',t)}{\partial t}

Explanation

Waveguides and resonating cavities

Equation

$\begin{array}{ll} \vec{n}\cdot(\vec{D}_2-\vec{D}_1)=\sigma~~&~~\vec{n}\times(\vec{E}_2-\vec{E}_1)=0\\ \vec{n}\cdot(\vec{B}_2-\vec{B}_1)=0~~&~~\vec{n}\times(\vec{H}_2-\vec{H}_1)=\vec{K} \end{array} \\ \vec{E}(\vec{x},t)=\vec{\cal E}(x,y){e}^{i(kz-\omega t)} \\ \vec{B}(\vec{x},t)=\vec{\cal{B}}(x,y){e}^{i(kz-\omega t)}$

Latex Code

            \begin{array}{ll} \vec{n}\cdot(\vec{D}_2-\vec{D}_1)=\sigma~~&~~\vec{n}\times(\vec{E}_2-\vec{E}_1)=0\\ \vec{n}\cdot(\vec{B}_2-\vec{B}_1)=0~~&~~\vec{n}\times(\vec{H}_2-\vec{H}_1)=\vec{K} \end{array} \\
            \vec{E}(\vec{x},t)=\vec{\cal E}(x,y){e}^{i(kz-\omega t)} \\ 
            \vec{B}(\vec{x},t)=\vec{\cal{B}}(x,y){e}^{i(kz-\omega t)} \\
            \begin{array}{ll} \displaystyle {\cal B}_x=\frac{i}{\varepsilon\mu\omega^2-k^2}\left(k\frac{\partial {\cal B}_z}{\partial x}-\varepsilon\mu\omega\frac{\partial {\cal E}_z}{\partial y}\right)~~&~~ \displaystyle {\cal B}_y=\frac{i}{\varepsilon\mu\omega^2-k^2}\left(k\frac{\partial {\cal B}_z}{\partial y}+\varepsilon\mu\omega\frac{\partial {\cal E}_z}{\partial x}\right)\\ \displaystyle {\cal E}_x=\frac{i}{\varepsilon\mu\omega^2-k^2}\left(k\frac{\partial {\cal E}_z}{\partial x}+\varepsilon\mu\omega\frac{\partial {\cal B}_z}{\partial y}\right)~~&~~ \displaystyle {\cal E}_y=\frac{i}{\varepsilon\mu\omega^2-k^2}\left(k\frac{\partial {\cal E}_z}{\partial y}-\varepsilon\mu\omega\frac{\partial {\cal B}_z}{\partial x}\right) \end{array}

Explanation

Non-linear wave equations

Equation

$\frac{d^2x}{dt^2}-\varepsilon\omega_0(1-\beta x^2)\frac{dx}{dt}+\omega_0^2x=0 \\ \frac{d}{dt}\left\{\frac{1}{2}\left(\frac{dx}{dt}\right)^2+\frac{1}{2} \omega_0^2x^2\right\}= \varepsilon\omega_0(1-\beta x^2)\left(\frac{dx}{dt}\right)^2 \\ \frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}-\underbrace{au\frac{\partial u}{\partial x}}_{\rm non-lin}+ \underbrace{b^2\frac{\partial ^3u}{\partial x^3}}_{\rm dispersive}=0 \\ u(x-ct)=\frac{-d}{\cosh^2(e(x-ct))}$

Latex Code

            \frac{d^2x}{dt^2}-\varepsilon\omega_0(1-\beta x^2)\frac{dx}{dt}+\omega_0^2x=0 \\
            \frac{d}{dt}\left\{\frac{1}{2}\left(\frac{dx}{dt}\right)^2+\frac{1}{2} \omega_0^2x^2\right\}= \varepsilon\omega_0(1-\beta x^2)\left(\frac{dx}{dt}\right)^2 \\
            \frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}-\underbrace{au\frac{\partial u}{\partial x}}_{\rm non-lin}+ \underbrace{b^2\frac{\partial ^3u}{\partial x^3}}_{\rm dispersive}=0 \\
            u(x-ct)=\frac{-d}{\cosh^2(e(x-ct))}

Navigation

1. Waves

Wave Equation

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Plane Waves

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Spherical Waves

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Cylindrical Waves

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The general solution of Wave Equation in one dimension

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The stationary phase method

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Green functions for the initial-value problem

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Waveguides and resonating cavities

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Non-linear wave equations

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