![[Waventerference.gif|invert_B]] A [[Standing Wave]] is a type of [[Wave]] that appears to be not be oscillating in the [[Spatial Domain]]. ### Derivation from the [[Superposition Principle]] of two [[Travelling Wave|travelling waves]] Suppose we have two [[Travelling Wave|Travelling waves]], $D_1, D_2$ who have the same [[Wavenumber]] and negated [[Angular Frequency|angular frequencies]]: $\huge \begin{align} D_{1}(x,t) &= A\sin(kx-\omega t) \\ D_{2}(x,t) &= A\sin(kx+\omega t) \\ \end{align}$ The [[Superposition Principle]] of the waves are: $\huge \begin{align} D_{1}+D_{2} &= A\pa{ \sin(kx-\omega t) + \sin(kx+\omega t) } \end{align} $ Using the trig identity: $\begin{align} \sin \theta_{1} + \sin \theta_{1} &= 2\sin \pa{\frac{1}{2}(\theta_{1}+\theta_{2})} \cos \pa{\frac{1}{2}(\theta_{1}-\theta_{2})} \\ D(x,t) &= 2A \sin \pa{ \frac{1}{2}(kx-\omega t + kx + \omega t) } \cos \pa{ \frac{1}{2}(kx-\omega t - kx - \omega t) } \\ &= 2A\sin(kx) \cos( -\omega t ) \end{align}$ $\huge D(x,t) = 2A\sin(kx) \cos( -\omega t ) $ Note that this is identicial to a [[Spatial Wave]] with a constant [[Phase]] that is simply being scaled in a sinusoidal pattern throughout the [[Time Domain]]. ### Standing Wave on a fixed length string Given a standing wave of a string with range $x\in [0, L]$. $\huge \begin{align} k &= \frac{h \pi}{L} \\ \lambda &= \frac{2L}{h} \\ f &= \frac{c}{\lambda} = \frac{hc}{2L} \end{align} $ Where $h\in \Z^{+}$ and represents the $h$-th [[Harmonic]]. ![[Standing Wave Harmonic.gif|invert_Sepia]] The point on the string that reaches the highest point / oscillates with the largest amplitude is called an [[Anti-Node]], while a [[02 Areas/Physics/Node|Node]] is a point on the string that could oscillate but doesnt.