Question

Find all the possible wavelengths allowed by boundary conditions for standing waves along the>x direction. Note that a standing wave is the sum of one wave travelling to the left plus an identical wave traveling to the right. At the origin there is an antinode (maximum vibration) and at x=L there is an antinode (maximum vibration)

Answer 1

Let x ( t ) propogating

in +ve x direction

and x ( t ) is propogating

in -ve x direction

They will form standing wavw if both are Identical and are travelling in opposite direction.

take x ( t ) = A sin ( wt - kx )

and x ( t ) = A sin ( wt + kx )

A - Amplitude of wave

W - Angular frequency

K - Wave Number

Resultant wave x ( t ) + x ( t )

= A sin ( wt - kx ) + A sin ( wt + kx )

= A [ sin ( wt - kx ) + sin (wt + kx ) ]

expand using

Amplitude is given by

but k is wave number given by k =

allowed wave lengths.