4.0 Longitudinal stationary wave in an organ pipe

  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 same frequency. Standing wave equation can be written as SHM equation, $$\Delta P = P(x)\cos \omega t$$ where, $P(x) = 2{\left( {\Delta P} \right)_{\max }}\sin kx$ and $\omega $ is the frequency of oscillation.
• The amplitude of oscillation varies with the location $x$ of the particle. $$P(x) = 2{\left( {\Delta P} \right)_{\max }}\sin kx$$
• In longitudinal wave, the points having maximum amplitude are known as antinodes or pressure antinodes. $$\begin{equation} \begin{aligned} P(x) = 2{\left( {\Delta P} \right)_{\max }}\sin kx \\ P{(x)_{\max }} = 2{\left( {\Delta P} \right)_{\max }} \\\end{aligned} \end{equation} $$ 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 longitudinal wave, the points having zero amplitude are known as nodes or pressure nodes. $$\begin{equation} \begin{aligned} P(x) = 2{\left( {\Delta P} \right)_{\max }}\sin kx \\ P{(x)_{\min }} = 0 \\\end{aligned} \end{equation} $$ 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.

• As we know pressure wave is $90^\circ $ out of phase with the displacement wave. So,
• the displacement node behaves as pressure antinode.
• and the displacement antinode behaves as a pressure node.

Types of organ pipe

• Open organ pipe
• Close organ pipe