2.6.1. Vibrations of a String (General Case) (**)

The movement of a string with length L that is free to vibrate can be determined from the Fourier analysis seen in this chapter. It can also be determined directly using the following method we owe to d'Alembert (1747, [ESC 01]). We already know that this movement can be expressed as

1. Show that the condition u(0, t) = 0 for any t implies that g( y) = - f(- y) and therefore

   
   

2. Show that the condition u( L, t) = 0 for any t implies that f is 2 L-periodic.

3. f is written as f( x) = p( x) + q( x) where p and q are also 2 L-periodic, with p an even function ( p( x) = p( -x)) and q an odd function ( q(- x) = - q( x)). Thus we have

   
   

   The initial conditions are given by

   
   

   Show that

   
   

   where = v ₀( x) and A is a constant.

4. These equalities are true for any x if we assume that u ₀ and V ₀ are extended to be an odd and an even function respectively, both 2 L-periodic. Infer from this result that

   
   

   and that this function is T-periodic in...