## List of Physics Wave Formulas, Equations Latex Code

rockingdingo 2023-03-26 #physics #wave 0 0

List of Physics Wave Formulas, Equations Latex Code

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

##### 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.

• : disturbance
• : propagation velocity
• : phase velocity
• : group velocity
• : refractive index of the medium
• : Pressure waves in a liquid or gas, is the modulus of compression.
• : Further for pressure waves in a gas
• : Pressure waves in a thin solid bar with diameter
• : Waves in a string
• : Surface waves on a liquid, h is the depth of the liquid and \gamma the surface tension.

• #### Plane Waves

##### 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.

• : harmonic plane wave is defined as u(x,t)
• : Doppler effect

• #### Spherical Waves

##### Latex Code
            \frac{1}{v^2}\frac{\partial^2 (ru)}{\partial t^2}-\frac{\partial^2 (ru)}{\partial r^2}=0 \\
u(r,t)=C_1\frac{f(r-vt)}{r}+C_2\frac{g(r+vt)}{r}

##### Explanation

When the wave is spherically symmetric, the homogeneous wave equation is given by:Latex code for the Spherical Waves. I will briefly introduce the notations in this formulation.

• #### Cylindrical Waves

##### Latex Code
            \frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}-\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial u}{\partial r}\right)=0 \\
u(r,t)=\frac{\hat{u}}{\sqrt{r}}\cos(k(r\pm vt))

##### Explanation

When the wave is cylindrical symmetry, the homogeneous wave equation becomes as above. This is a Bessel equation, with solutions that can be written as Hankel functions. For sufficient large values of r.

• #### The general solution of Wave Equation in one dimension

##### 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

##### 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]


• #### Green functions for the initial-value problem

##### 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}


• #### Waveguides and resonating cavities

##### 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}


• #### Non-linear wave equations

##### 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))}