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Question

A string of length L fixed at both ends vibrates in its fundamental mode at a frequency v and a maximum apmplitude A. (a) Find the wavelength and the wave number k. (b) Take the origin at one end of the string and the X-axis along the string. Take the Y-axis along the direction of the displacement. Take t = 0 at the instant when the middle point of the string passes through its mean position and is going towards the positive y-direction. Write the equation describing the standing wave.

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Solution

Fundamental frequency,

v=12LTm

On Tm = velocity of wave

(a) Wavelength, λ=velocityfrequency

=Tm12LTm=2L

and wave number,

K=2πλ=2πL=πL

(b) Therefore, equation of the stationary wave is

y=A cos(2πxλ)sin(2πVt2L)

=A cos(2πx2L)sin(2πVt2L)

=A cos(πxL)sin(πvt)v

=V2L [because v =(V2L)]


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