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sol5 - Problem 4.4.3(a The term-ut has a meaning of an...

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Problem 4.4.3 (a) The term - βu t has a meaning of an external force; u t is the velocity. β > 0 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 ) φ ( x ) into the equation ρ 0 u tt = T 0 u xx - βu t to get ρ 0 g ( t ) φ ( x ) = T 0 g ( t ) φ ( x ) - βg ( t ) φ ( x ) . We move the last term on the right to the left hand side of the equation, and we divide both sides by T 0 g ( t ) φ ( x ): (1) ρ 0 T 0 g ( t ) g ( t ) + β T 0 g ( t ) g ( t ) = φ ( x ) φ ( 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 - λ . Then (1) splits into two equations: (2) φ ( x ) + λφ ( x ) = 0 and (3) ρ 0 g ( t ) + βg ( t ) + λT 0 g ( t ) = 0 . The boundary conditions imply φ (0) = φ ( L ) = 0 , so the solutions of (3) are λ n = π L n 2 , φ n ( x ) = sin π L nx . To solve (2), with λ = λ n , we first solve the characteristic equation ρ 0 μ 2 + βμ + λ n T 0 = 0 : (4) μ 1 , 2 = - β ± β 2 - 4 ρ 0 T 0 λ n 2 ρ 0 = - β 2 ρ 0 ± n where (5) ω n = 4 π 2 ρ 0 T 0 n 2 L 2 - β 2 2 ρ 0 .
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