What is Standing or stationary waves?

When two harmonic waves of equal frequency and amplitude travelling through a medium (say string) in opposite directions superimpose each other, we get stationary waves. Suppose the two waves are

\(\begin{array}{l}{{y}_{1}}=Asin left( kx-omega t right)\end{array} \)
and
\(\begin{array}{l}{{y}_{2}}=Asin left( kx+omega t right)\end{array} \)

By the principle of superposition their sum is

\(\begin{array}{l}y={{y}_{1}}+{{y}_{2}}\end{array} \)

or

\(\begin{array}{l}y=Aleft[ sin left( kx-omega t right)+sin left( kx+omega t right) right]\end{array} \)

By using the identity,

\(\begin{array}{l}sin A+sin B=2sin left( frac{A+B}{2} right)cos left( frac{A-B}{2} right),\end{array} \)

We obtain

\(\begin{array}{l}y=2Asin left( kx right)cos left( omega t right)\end{array} \)

Or

\(\begin{array}{l}y=2Asin left( frac{2pi x}{lambda } right)cos left( frac{2pi t}{T} right)\end{array} \)

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