This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Problem 4.4.3 (a) The term u t has a meaning of an external force; u t is the velocity. &gt; 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 ....
View
Full
Document
 Summer '08
 Hassen

Click to edit the document details