Explain Analytically How the Stationary Waves Are Formed - Physics

Explain analytically how the stationary waves are formed

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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