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Vibrating and Oscillating String

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Find the total energy of a vibrating string of length L, fixed at both ends, oscillating in its n'th characteristic mode with and amplitude A.
The tension in the string is T and its total mass is M.
Hint - consider the integrated kinetic energy at the instant when the string is straight so that it has no stored potential energy over and above what it will have when not vibrating at all).

Calculate the total energy of vibration of the same string if it is vibrating in the following superposition of normal modes (see file for exact expression). You should be able to verify that it is the sum of the energies of the two modes taken separately.

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The 5-pages solution shows, step by step, how to set up and evaluate the required integrals in order to find the expresion for the kinetic energy and show how the superposition of the modes influence it.

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The solution is attached below in two files. the files are identical in content, only differ in format. The first is in MS Word format, while the other is in Adobe pdf format. Therefore you can choose the format that is most suitable to you.
We start with the relations between the wave's velocity v, the tension T, the length L and the string's linear density  (recalling that the density is the mass M divided by the length):

(1.1)
For a standing wave, the wavelength of the nth mode is given by:
(1.2)
Combining ...

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