A string of length L fixed at both ends vibrates in its fundamental node at a frequency v and a maximum amplitude A. 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.
To find such an equation, let's go to the middle point of string. We know it would be in SHM. It will have amplitude of A. It is in it's mean position moving upwards, it's equation will
be y=Asin(ω t), since it is at y=0 at t=0
Also ω = 2 π f
y=Asin(2 π ft)
Different points have different amplitudes so to make a complete standing wave equation we need to take into account how the amplitude varies.
A(x) = A sin (kx)
=A sin (ωvx)
=A sin (2 π f2Lfx)[asv2L =‘f,fundamentalnode]
=A sin (π xL)
so the actual equation will be
y = A sin (π xL) sin (2π ft)