p09fl31 - Lecture 31: Strings II (13 Nov 09) Move Set VIII...

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Lecture 31: Strings II (13 Nov 09) Move Set VIII to Nov 23? A. Continuum string 1. 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 2. The Euler-Lagrange equation gives a “wave equation” ∂t L ( ∂u/∂t ) + ∂x L ( ∂u/∂x ) - L ∂u = 0 σ ( x ) 2 u ∂t 2 = ∂x τ ( x ) ∂u ∂x 3. σ ( x ) = σ ; τ ( x ) = τ (constants) “elementary” wave equation: 1 c 2 2 u ∂t 2 = 2 u ∂x 2 ; c 2 = τ/σ 4. 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 5. Discrete values of k (and hence of ω ) are set by boundary conditions on a finite length 0 < x < ‘ . Fixed ends give (normalized solutions) ρ n ( x ) = q 2 /‘ sin k n x ; k n = nπ/‘ 1
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6. The solution for general ( x,t ) can then be constructed from the initial position u ( x, 0), here using the fourier sine series. u
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p09fl31 - Lecture 31: Strings II (13 Nov 09) Move Set VIII...

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