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# p09fl19 - Lecture 19 Continuum string(16 Oct 09 Mid-term...

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Lecture 19: Continuum string (16 Oct 09) Mid-term exam, take-home, handed out in class today, due Monday 19 Oct. at 5 PM at 5338 Chamberlin or LB mailbox. No class on Monday. A. continuum approx – discrete chain 1. Chain of spacing a , N atoms in repeat unit (or in chain with specified ends). “Shortest wavelength” is λ = a , longest is λ = Na . 2. Wave-number is defined by k = 2 π/λ , long wavelength, smooth varia- tion has ka << 1. 3. Chain with successive displacements having relative phase exp( iq ); q ka ; x coordinate x j = ja 4. The equations of motion for the chain were m ¨ η j = - K [2 η j - η j +1 - η j - 1 ] with phased displacements η j exp( ijq ) having ω ( q ) = q 4 K/m sin( q/ 2). 5. Go to a notation η j ξ ( x j , t ) and use finite differences for the second derivative: ξ ( x j +1 , t ) + ξ ( x j - 1 , t ) - 2 ξ ( x j , t ) ( δx ) 2 2 ξ/∂x 2 6. Then the equations of motion become, with density σ = m/a and tension τ = Ka σ 2 ξ ∂t 2 = τ 2 ξ ∂x 2 7. The “smooth variation” of ξ corresponds to q << 1 or ω ( q ) q q K/m = k q τ/σ.

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p09fl19 - Lecture 19 Continuum string(16 Oct 09 Mid-term...

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