sol5 - Problem 4.4.3 (a) The term- u t has a meaning of an...

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Unformatted text preview: Problem 4.4.3 (a) The term- u t has a meaning of an external force; u t is the velocity. > means that the external force is directed opposite to the velocity, so it is a damping force indeed. (b) We substitute u ( x, t ) = g ( t ) ( x ) into the equation u tt = T u xx- u t to get g 00 ( t ) ( x ) = T g ( t ) 00 ( x )- g ( t ) ( x ) . We move the last term on the right to the left hand side of the equation, and we divide both sides by T g ( t ) ( x ): (1) T g 00 ( t ) g ( t ) + T g ( t ) g ( t ) = 00 ( x ) ( x ) . The expression on the left in the last equation does not depend on x , the expression on the right is independent of t , so both sides equal to a constant. We denote this constant by- . Then (1) splits into two equations: (2) 00 ( x ) + ( x ) = 0 and (3) g 00 ( t ) + g ( t ) + T g ( t ) = 0 . The boundary conditions imply (0) = ( L ) = 0 , so the solutions of (3) are n = L n 2 , n ( x ) = sin L nx ....
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sol5 - Problem 4.4.3 (a) The term- u t has a meaning of an...

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