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Explain Analytically How the Stationary Waves Are Formed - Physics

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Explain analytically how the stationary waves are formed

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Solution

Consider two simple harmonic progressive waves of equal amplitude and frequency propagating on a long uniform string in opposite directions.

If wave of frequency ‘n’ and wavelength ‘l’ is travelling along the positive X axis, then

`y_1=Asin((2pi)/lambda)(vt-x)` ..........(1)

If wave of frequency ‘n’ and wavelength ‘l’ is travelling along the negative X-axis, then

`y_2=Asin((2pi)/lambda)(vt+x)`............ (2)

These waves interfere to produce stationary waves. The resultant displacement of stationary waves is given by the principle of superposition of waves.

y=y1+y2    .....(3)

`y=Asin((2pi)/lambda)(vt-x)+Asin((2pi)/lambda)(vt+x)`

By Using

`sinC+sinD=2sin[(C+D)/2]cos[(C-D)/2]`

 We get

`therefore y=2Asin[((2pi)/lambda)((vt-x+vt+x)/2)]cos[((2pi)/lambda)((vt-x-vt-x)/2)]`

`therefore y = 2Asin((2pivt)/lambda)cos((2pi)/lambda(-x))`

`therefore y=2Asin(2pint) cos((2pix)/lambda)`    `(because n=v/lambda) [because cos(-theta)=costheta]`

`therefore y=2Acos((2pix)/lambda)sin2pint`

Let Equetion of stationary wave 

`y=2Acos((2pix)/lambda)sin2pint`

Let `R=2Acos((2pix)/lambda)`

`therefore y=Rsin(2pint )` ......(4)

But, `omega=2pin`

`therefore y=Rsinomegat`........(5)

 

Concept: Formation of Stationary Waves on String
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