IGNOU MTE 8 2018 - A tightly stretched string with fixed end points x = 0 and x = l is in

Question

A tightly stretched string with fixed end points x = 0 and x = l is initially in a position
given by y = y₀sin³ $(\frac{\pi x}{l})$ . It is released from rest from the initial position. Find the displacement y(x,t).

13 Feb 2018

Answer :

Word Count : 333

The problem describes a wave on a string with fixed ends at $x = 0$ and $x = l$. The initial position of the string is given as: \[ y(x, 0) = y_0 \sin^3\left(\frac{\pi x}{l}\right) \] The string is released from rest, which implies the initial velocity is zero, i.e., \[ \frac{\partial y(x,t)}{\partial t} \bigg|_{t=0} = 0 \] The solution for the displacement $y(x,t)$ of a vibrating string is generally found using the method of separation of variables, which leads to a solution of the form: \[ y(x,t) = \sum_{n=1}^{\infty} A_n \sin\left(\frac{n \pi x}{l}\right) \cos\left(\omega_n t + \phi_n\right) \] where: - $A_n$ are the coefficients determined by the initial condition. - \(\omega_n =

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