String Vibration Fixed Ends

In case of the vibrations of a string, both its ends at x = 0 and x = L may be permanently fixed; y (x = 0) = 0, and y (x = L) = 0at all t. That gives,

For all t, A ( x ) = 0 at x = 0, and therefore we have B = 0. Hence, we get

A ( x ) = A sin kx

Also A ( x ) = 0 at x = L; hence, A sin kL = 0

or, kn L = n π         n = 1, 2, .....

 

The above equation gives frequencies of various possible normal modes of a system at both ends. For the string v =   .

In terms of linear frequency, we write

 

For n = 1, the frequency    is called the fundamental; it is the minimum frequency for a normal mode. The higher modes are called harmonics, i.e. v_n = n v₁ represents nth harmonic. (The fundamental is first harmonic.)

The wavelength λn in nth mode is given by

   

That is, the nth accommodates exactly n/2 wavelength in length L of the medium.

The solution is now written as