Consider the 1D wave equation for a string of length l (lower case L) and with a

Question

Consider the 1D wave equation for a string of length l (lower case L) and with a wave speed c. Suppose the left endpoint of the string is fixed at a height of 1, while the right endpoint is attached to a mechanical device in such a way that a slope of zero is maintained at all times. the string has initial shape given by f(x) = cos(2*pi*x), 0 < x < l, and is released with initial velocity g(x) = 1, 0 <x< l.

a) Set up the initial-boundary value problem modeling this scenario.
b) Set up and solve the equilibrium state problem for (a).
c) Set up and solve the transient problem for (a).
d) Solve the problem in (a).

Explanation / Answer

We consider a string of length l with ends fixed, and rest state coinciding with x-axis. The string ispluckedintooscillation. Let u (x,t)bethepositionof the string attime t. Assumptions: 1. Small oscillations, i.e. the displacement u (x,t)is small compared to the length l. (a) Pointsmovevertically. Ingeneral,wedon’tknowthatpointsonthe string move vertically. By assuming the oscillations are small, we assume the points move vertically. (b) Slope of tangent to the string is small everywhere, i.e. |ux (x,t)|«1, so stretching of the string is negligible R l p (c) arc length a (t)= 1+u2dx ?l. 0 x 2. String is perfectly flexible (it bends). This implies the tension is in the tangent direction and the horizontal tension is constant, or else there would be a preferred direction of motion for the string. Consider an element of the string between x and x +?x. Let T (x,t)betension and ? (x,t)be the angle wrt the horizontal x-axis. Note that ?u tan? (x,t)= slope of tangent at (x,t) in ux-plane = (x,t). (1) Newton’sSecondLaw(F = ma)states that ?2u F =(??x) (2) ?t2 where ? isthelineardensityofthe string(ML-1)and?x isthelengthof the segment. The force comes from the tension in the string only -we ignore any external forces such asgravity. Thehorizontal tensionisconstant,andhenceitisthevertical tension thatmovesthe string vertically(obvious). Balancing theforcesinthehorizontaldirectiongives T (x +?x,t)cos? (x +?x,t)= T (x,t)cos? (x,t)= t = const (3) where t is the constant horizontal tension. Balancing the forces in the vertical direction yields

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