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. 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.

Answer

a) `lambda = 2L , K = pi/L`

Explanation

Fundamental frequency
`v=1/(2L)sqrt(T/m)`
on `sqrt(T/m)`= velocity of wave`
(a).` Wavelength `(lambda)`=(velocity)/(frequency)
`=sqrt((T/m))/(1/(2L)sqrt(T/m))2L`
and wave number
`=K=(2pi)/lamda=(2pi)/(2L)=pi/L`
b. Therfore, equation of the stationary wave is
`y=acos((2pi)/lamda)sin((2piVt)/(2L))`
`=Acos((2pix)/(2L))sin((2piVt)/(2L))`
`A cos((pix)/L)sin(pivt)v`
`=V/(2L)[`because `v=(V/(2L))]`