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**Unformatted text preview: **10.7: THE WAVE EQUATION: VIBRATIONS OF AN ELASTIC STRING KIAM HEONG KWA Consider an elastic string of length L which is tightly stretched be- tween two supports at the same horizontal level, so that it lies along the x-axis. Set the string in motion such that it vibrates in a vertical plane. If u ( x,t ) denotes the vertical displacement experienced by the string at the point x at time t , then it can be shown that u ( x,t ) is governed by the one-dimensional wave equation (0.1) a 2 u xx = u tt in the domain 0 < x < L for all t > 0, where the constant a 2 is given by a 2 = T ; T is the tension in the string and is the mass per unit length of the string material. It should be remarked that the applicability of the wave equation presupposes the absence of damping effects and the smallness of the amplitude of the motion. On the other hand, the fact that the two ends of the string are fixed is mathematically expressed as the boundary conditions (0.2) u (0 ,t ) = 0 , u ( L,t ) = 0 , t . Finally, when the string is set in motion, it acquires an initial position and an initial velocity at each of its spacial coordinate x . This gives rise to the initial conditions (0.3) u ( x, 0) = f ( x ) and u t ( x, 0) = g ( x ) , x L, where f ( x ) and g ( x ) are functions that depend only on the spacial coordinate x . Since the string is held fixed at its two ends, it is clearly necessary that (0.4) f (0) = f ( L ) = 0 and g (0) = g ( L ) = 0 ....

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