HSC Science (Computer Science) 12th Board ExamMaharashtra State Board
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Explain Analytically How the Stationary Waves Are Formed - HSC Science (Computer Science) 12th Board Exam - Physics

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Question

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)

 

  Is there an error in this question or solution?

APPEARS IN

 2015-2016 (March) (with solutions)
Question 3.2 | 7.00 marks
 2016-2017 (March) (with solutions)
Question 4.2 | 7.00 marks

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Solution Explain Analytically How the Stationary Waves Are Formed Concept: Formation of Stationary Waves on String.
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