The Wave Equation is a [[Differential Equations|Differential Equation]] that pertains to [[Spatial Wave|Spatial Waves]] over [[Time Domain|Time]]. Some [[Function]] $D(x,t)$ of space $x$ and time $t$, must satisfy the following differential equation to be a wave (with $c$ being the [[Wave Speed]] of the wave): $\huge \pderiv{^{2}D}{x^{2}} = \frac{1}{c^{2}} \pderiv{^{2}D}{t^{2}} $ An important proprty of the wave equation is that it is a [[Linear Transformation|Linear]] equation, meaning that [[Wave Interference|interferance]] between multiple waves can be expressed as the sum of their wave equations (important for [[Fourier Series]]). >[!example] Consider a pulse with speed $c$. >$\huge \begin{align} >D(x,t) &= f(x-ct)\\ > >\partial ^{2}_{x}{D} &= \partial_{x}^{2}\set{f} = f'' \cdot ( 1)^{2} \\ > >\partial^{2}_{t}D &= f''(-c)^{2} \\ >D(x,t) &= f(x+ct) >\end{align}$ >[!example] >$ \huge \begin{align} >D(x,t) &= A\sin(kx \pm \omega t) \\ >\partial^{2}_{t} D &= -A \sin(kx \pm \omega t) ( \pm \omega)^{2} \\ >\partial^{2}_{x} D&= -A\sin (kx \pm \omega t)k^{2} \\ >\frac{(\pm\omega)^{2}}{c^{2}} &= k^{2 }\\ >c^{2} &= \frac{\omega^{2}}{k^{2}} \\ >c &= \frac{\omega}{k} >\end{align} $ $\huge \begin{align} \end{align}$