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 getA ( x ) = A sin kxAlso A ( x ) = 0 at x = L; hence, A sin kL = 0or, 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 v_{1} 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