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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 amplitude 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 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=12LTmTm=V×2L [Tm=velocityofwave×wavelength ]

(a) Wavelength, λ= velocityfrequency= V2LV=2L

and wave number =K=2π/λ=2π/2L=π/l

(b) Therefore, equation of the stationary wave is
y=Acos(2πxλ)sin(2πVtL)

=Acos(2πx2L)sin(2πVt2L)

=Acos(πxL)sin(πvt) [because v=(V/2l)]

1540815_1108398_ans_e9dc36d45b73426d86946626f80232f4.png

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