3.1 Transverse stationary wave on a stretched string

  Superposition of Waves

• Equation of standing wave is not of the form $f(x \pm vt)\;{\text{or}}\;f(ax \pm bt)$. Therefore, it does not describe a travelling wave.
  
• In standing wave, the particle at any particular point $x$ executes simple harmonic motion and all the particles vibrate with the same frequency. Standing wave equation can be written as SHM equation, $$y = A(x)\cos \omega t$$ where, $A(x) = 2A\sin kx$ and $\omega $ is the frequency of oscillation.
• The amplitude of oscillation varies with the location $x$ of the particle. $$A(x) = 2A\sin kx$$
• In a transverse wave, the points having maximum amplitude are known as antinodes or displacement antinodes. $$\begin{equation} \begin{aligned} A(x) = 2A\sin kx \\ A{(x)_{\max }} = 2A \\\end{aligned} \end{equation} $$ So, $$\begin{equation} \begin{aligned} \sin kx = 1 \\ kx = (2n + 1)\frac{\pi }{2} \\\end{aligned} \end{equation} $$ As, $k = \frac{{2\pi }}{\lambda }$, therefore, $$\begin{equation} \begin{aligned} \left( {\frac{{2\pi }}{\lambda }} \right)x = (2n + 1)\frac{\pi }{2} \\ x = (2n + 1)\frac{\lambda }{4}\quad {\text{where,}}\;n = 0,1,2,3... \\\end{aligned} \end{equation} $$or $$x = \frac{\lambda }{4},\frac{{3\lambda }}{4},\frac{{5\lambda }}{4}...\frac{{(2n + 1)\lambda }}{4}$$
• Distance between two consecutive antinode is, $$\begin{equation} \begin{aligned} {d_{A - A}} = \frac{{[2(n + 1) + 1]\lambda }}{4} - \frac{{(2n + 1)\lambda }}{4} \\ {d_{A - A}} = \frac{\lambda }{2} \\\end{aligned} \end{equation} $$
• In transverse wave, the points having zero amplitude are known as nodes or displacement nodes. $$\begin{equation} \begin{aligned} A(x) = 2A\sin kx \\ A{(x)_{\min }} = 0 \\\end{aligned} \end{equation} $$ So, $$\begin{equation} \begin{aligned} \sin kx = 0 \\ kx = n\pi \\\end{aligned} \end{equation} $$ As, $k = \frac{{2\pi }}{\lambda }$, therefore, $$\begin{equation} \begin{aligned} \left( {\frac{{2\pi }}{\lambda }} \right)x = n\pi \\ x = \frac{{n\lambda }}{2}\quad {\text{where,}}\;n = 0,1,2,3... \\\end{aligned} \end{equation} $$ or $$x = 0,\frac{\lambda }{2},\lambda ,\frac{{3\lambda }}{2},2\lambda ,\frac{{5\lambda }}{2}......\frac{{n\lambda }}{2}$$
• Distance between two consecutive nodes is, $$\begin{equation} \begin{aligned} {d_{N - N}} = \frac{{(n + 1)\lambda }}{2} - \frac{{n\lambda }}{2} \\ {d_{N - N}} = \frac{\lambda }{2} \\\end{aligned} \end{equation} $$
• Distance between a node and the subsequent antinode is, $$\begin{equation} \begin{aligned} {d_{A - N}} = \frac{{(2n + 1)\lambda }}{4} - \frac{{n\lambda }}{2} \\ {d_{A - N}} = \frac{\lambda }{4} \\\end{aligned} \end{equation} $$
• Energy is not transported along the string because energy cannot flow past the nodal point in the string which are permanently at rest. However, energy oscillates between vibrational kinetic energy and elastic potential energy.
• Differences between traveling wave and stationary wave.

S. No.	Travelling wave	Stationary wave
1	Travelling wave moves in a medium with definite velocity	Standing waves remains stationary between two boundaries in the medium
2	All the particles of the medium oscillate with same frequency and amplitude	All the particles except nodes oscillate with same frequency but different amplitude. Amplitude is zero at nodes and maximum at antinodes
3	All the particles of the medium never pass through their mean position simultaneously. $$y = A\sin \omega t$$	All the particles of the medium pass through their mean position simultaneously twice in each time period. $$y = A(x)\cos \omega t$$
4	Phase difference between two particles can have any value 0 and $2\pi $	Phase difference between two particles can be either 0 or $\pi $
5	Energy is transported along the medium	No energy is transported along the medium