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Unformatted text preview: 12.3 1D Wave Equation: Separation of Variables, Use of Fourier Series In the previous section, we have derived the one-dimensional wave equation ( 60 ), which governs small transverse vibrations of an elastic string of length L > 0. This PDE is given in := (0 , L ) (0 , ) R 2 , and according to our Definitions 1 and 2 , it is a second-order, linear and homogeneous partial differential equation. In particular, Theorem 1 applies to the solutions of ( 60 ). We complete the one-dimensional wave equation ( 60 ) by boundary and initial conditions, so that we obtain the following (well-posed) initial-boundary value problem: 2 u t 2 = c 2 2 u x 2 , x (0 , L ) , t > , c := radicalBigg T , (61) u (0 , t ) = u ( L, t ) = 0 , t > , (62) u ( x, 0) = f ( x ) , u t ( x, 0) = g ( x ) , x [0 , L ] . (63) For compatibility of initial and boundary conditions, we require that f (0) = f ( L ) = 0. The initial conditions ( 63 ) specify the initial deflection, f ( x ), and the initial velocity, g ( x ). The method presented here to solve the initial-boundary value problem ( 61 )( 63 ) consists of three steps: 1. separation of variables ( product method ): write the unknown function as a product of functions with fewer variables. From the PDE, derive separate differential equations for each factor. 2. Find solutions of these differential equations which satisfy the auxiliary conditions. 3. Using the superposition principle (Thm. 1 ), combine these solutions in a series ( squiggleright Fourier series) and determine coefficients from the data. Separation of Variables We are looking for solutions u negationslash 0 of the one- dimensional wave equation ( 61 ) of the form u ( x, t ) = F ( x ) G ( t ) , x (0 , L ) , t > . (64) 16 If we insert ( 64 ) into ( 61 ), we obtain F G = c 2 F G, x (0 , L ) , t > . (65) where denotes time derivatives and where denotes spatial derivatives. Di- vision by c 2 FG leads to G c 2 G = F F , x (0 , L ) , t > . (66) The left-hand side of ( 66 ) depends only on t , whereas the right-hand side of ( 66 ) depends only on x , thus the variables have been separated. We conclude that both sides must be equal to a constant, the separation constant k R . Then we obtain the following set of separate ODEs: F = kF, x (0 , L ) , (67) G = c 2 kG, t > . (68) Remark: A PDE which can be broken down into a set of separate equations of lower dimensionality by a method of separating variables is called a separable PDE....
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