Question

A taut string of length L has its ends x=0, x=L fixed. The midpoint of the string is drawn aside a small distance h and released at time t=0. Obtain Obtain the displacement of the string at any subsequent time t>0. Leave your answer in terms of sin(nπ/2).

Answer 1

The displacement of a string at any time t>0 after being plucked at the midpoint can be modeled using a sinusoidal function. The midpoint oscillates up and down in simple harmonic motion with the displacement described by sinusoidal functions of time and position on the string.

Step-by-step explanation:

To obtain the displacement of the string at any subsequent time t > 0 after it is plucked at the midpoint and released, consider that the string is fixed at both ends (x=0 and x=L). Given that the string has been deflected by a small distance h, it forms a shape that is akin to two right-angled triangles. The sides opposite the right angle are of length h, and the displacement y(x, t) can be described using a sinusoidal function:

Y(x, t) = A cos(kx) cos(ωt),

where A is the initial amplitude (related to h), k is the wave number (k = nπ/L with n being the mode of vibration), and ω is the angular frequency. We focus on the displacement at the midpoint, hence substituting x = L/2 into the wave function. For the fundamental mode (n = 1), this yields:

Y(L/2, t) = A cos(π/2) cos(ωt) = 0, since cos(π/2) = 0.

This indicates that at the fundamental mode, there is no displacement at the midpoint since it remains at its equilibrium position (y = 0). However, considering higher harmonics (where n is an odd integer), the midpoint will oscillate up and down. Applying simple harmonic motion principles, we can find the displacement function for any n applying the boundary conditions.