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# p09fl30 - Lecture 30 String I(11 Nov 09 review 9 Nov HW HJ...

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Lecture 30: String I (11 Nov 09) review 9 Nov HW HJ theory of oscillator A. Review – continuum string 1. FW Secs. 25 and 38; G Sec. 13.2 2. The Lagrangian density for transverse ( u ) vibrations of a 1D string with mass density σ and spatially varying tension τ ( x ) is L = 1 2 σ ( x )( ∂u ∂t ) 2 - 1 2 τ ( x )( ∂u ∂x ) 2 3. The Euler-Lagrange equations are obtained from ∂t L ( ∂u/∂t ) + ∂x L ( ∂u/∂x ) - L ∂u = 0 and for the string L are σ ( x ) 2 u ∂t 2 = ∂x τ ( x ) ∂u ∂x 4. Hamiltonian formulation G Sec. 13.4 (FW Sec. 45), using notation ˙ u = ∂u/∂t : π = L /∂ ˙ u ; H = π ˙ u - L here: π = σ ( x ) ∂u ∂t ; H = Z dx [ 1 2 σ ( x )( ∂u ∂t ) 2 + 1 2 τ ( x )( ∂u ∂x ) 2 ] 5. The case σ ( x ) = σ and τ ( x ) = τ (constants) gives rise to the elementary form of the wave equation with constant speed c 2 = τ/σ : 1 c 2 2 u ∂t 2 = 2 u ∂x 2 1

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6. Normal modes: separate time and space variables u ( x, t ) = ρ ( x ) cos( ωt + δ ) and define k 2 = ω 2 /c 2 . The equation for ρ ( x ) is d 2 ρ dx 2 + k 2 ρ = 0 7. Discrete allowed values of k
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p09fl30 - Lecture 30 String I(11 Nov 09 review 9 Nov HW HJ...

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