By Sunil Bhardwaj

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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”.

$$ { \Psi } = 2A\sin { \left( \frac { 2\pi x }{ \lambda } \right) } \cos { \left( 2\pi \nu t \right) } $$

Consider a string clamped at two points. It will be stationary wave vibrating between these two points.

1. Second harmonic: In this case, \(L = \frac { 2\lambda }{ 2 }\)

2. Third harmonic: In this case, \(L = \frac { 3\lambda }{ 2 }\)

3. When \(x = 0\) $${ \Psi } = 2A\sin { \left( \frac { 2\pi \times 0 }{ \lambda } \right) } \cos { \left( 2\pi \nu t \right) } $$ $$ { \Psi } = 2A\sin { \left( 0 \right) } \cos { \left( 2\pi \nu t \right) } $$ $$ { \Psi } = 0 $$ When \(x = a\) and \(a = \frac { \lambda }{ 2 } \) $$ { \Psi } = 2A\sin { \left( \frac { 2\pi \times \frac { \lambda }{ 2 } }{ \lambda } \right) } \cos { \left( 2\pi \nu t \right) } $$ $$ { \Psi } = 2A\sin { \left( \pi \right) } \cos { \left( 2\pi \nu t \right) } $$ $$ { \Psi } = 0 \times \cos { \left( 2\pi \nu t \right) } $$ $$ { \Psi } = 0 $$