By Sunil Bhardwaj

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Waves are of two types:

1. Progressive Wave

2. Standing Wave

But progressive wave is also a function of time. So the equations can be …

$${ \Psi }_{ 1 } = A \sin { 2\pi \left( \frac { x }{ \lambda } + \nu t \right) } \qquad ...(1) $$ For wave travelling in opposite direction $$ { \Psi }_{ 2 } = A \sin { 2\pi \left( \frac { x }{ \lambda } - \nu t \right) } \qquad ...(2) $$ On combining equation (1) and (2) $$ { \Psi } = { \Psi }_{ 1 } + { \Psi }_{ 2 }$$ $${ \Psi } = \left[ A \sin { 2\pi \left( \frac { x }{ \lambda } + \nu t \right) } \right] + \left[ A \sin { 2\pi \left( \frac { x }{ \lambda } - \nu t \right) } \right] $$ $$ { \Psi } = A\left[ \sin { 2\pi \left( \frac { x }{ \lambda } + \nu t \right) } + \sin { 2\pi \left( \frac { x }{ \lambda } - \nu t \right) } \right] $$ $$ { \Psi } = A\left[ \sin { \left( \frac { 2\pi x }{ \lambda } + 2\pi \nu t \right) } + \sin { \left( \frac { 2\pi x }{ \lambda } - 2\pi \nu t \right) } \right] $$ As we know from the trigonometry, $$ \sin { \left( A+B \right) } + \sin { \left( A-B \right) } = 2\sin { \left( \frac { A+B }{ 2 } \right) } \cos { \left( \frac { A-B }{ 2 } \right) } $$ Lets simplyfy accordingly, $${ \Psi } = A\left[ 2\sin { \left( \frac { \left( \frac { 2\pi x }{ \lambda } +2\pi \nu t \right) +\left( \frac { 2\pi x }{ \lambda } -2\pi \nu t \right) }{ 2 } \right) } { \left( \frac { \left( \frac { 2\pi x }{ \lambda } +2\pi \nu t \right) -\left( \frac { 2\pi x }{ \lambda } -2\pi \nu t \right) }{ 2 } \right) } \right] $$ $$ { \Psi } = A\left[ 2\sin { \left( \frac { \left( \frac { 2\pi x }{ \lambda } \right) +\left( \frac { 2\pi x }{ \lambda } \right) }{ 2 } \right) } \cos { \left( \frac { \left( 2\pi \nu t \right) +\left( 2\pi \nu t \right) }{ 2 } \right) } \right] $$ $$ { \Psi } = A\left[ 2\sin { \left( \frac { \left( \frac { 4\pi x }{ \lambda } \right) }{ 2 } \right) } \cos { \left( \frac { \left( 4\pi \nu t \right) }{ 2 } \right) } \right] $$ $$ { \Psi } = A\left[ 2\sin { \left( \frac { 2\pi x }{ \lambda } \right) } \cos { \left( 2\pi \nu t \right) } \right] $$ $$ \boxed { { \Psi } = 2A\sin { \left( \frac { 2\pi x }{ \lambda } \right) } \cos { \left( 2\pi \nu t \right) } } $$ This is the final equation for the standing wave.

Hence the standing wave can be defined as: All waves whose amplitude function can be factorised into factor independent of space coordinates and a factor independent of time are called “Stationary Wave”.

$$ { \Psi } = 2A\sin { \left( \frac { 2\pi x }{ \lambda } \right) } \cos { \left( 2\pi \nu t \right) } $$ Therefore when, \( x = 0\), $$ \frac { \lambda }{ 2 } , \frac { 2\lambda }{ 2 } , ... \frac { n\lambda }{ 2 } $$ $$ \sin { \left( \frac { 2\pi x }{ \lambda } \right) } = 0 $$ $$ \therefore { \Psi } = 0 $$ These points are known as nodes (minimum amplitude).