Question

A string of length L = 1 is held fixed at both ends. The string is initially deformed into a shape given by u(x, t = 0) = sin^2(πx) and released. Assume a value of c2 = 1. Find the solution u(x, t) for the vibration of the string by separation of variables.

Answer 1

Vibrating Strings with fixed ends:

Given:   

Initial Conditions:   

                                   

Boundary Conditions:    

Now, Solution u(x, t) for the Vibration of String:

Where,

           and

Here,         and                   ( for all 0 < x < ℓ )

Now,

We have,

AND,      

     

                            

                

Now,

Hence,

Since,

And,

            Ans.