Lec3 - F y = τ sin θ τ sin θ τ(tan θ tan θ = τ ∂y...

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Unformatted text preview: F y = τ sin θ- τ sin θ τ (tan θ- tan θ ) = τ ∂y ∂x x + dx- ∂y ∂x x ≡ τ ∂ 2 y ∂x 2 dx in the infinitesimal limit. Let μ := M/L be linear mass density, The y –component of the force on the string element dx also equals F y = ma y = ( dm ) ∂ 2 y ∂t 2 = μdx 1 + dy dx 2 ∂ 2 y ∂t 2 = μdx + O ( θ 2 ) ∂ 2 y ∂t 2 ≈ ( μdx ) ∂ 2 y ∂t 2 where d 2 = dx 2 + dy 2 , or d = dx 1 + ( dy/dx ) 2 , and dm = μd , where d is the length of the string element (in the direction along the string). Then μdx ∂ 2 y ∂t 2 = τ ∂ 2 y ∂x 2 dx ∂ 2 y ∂t 2 = τ μ ∂ 2 y ∂x 2 ⇒ ∂ 2 y ∂t 2 = v 2 ∂ 2 y ∂x 2 Where v 2 = τ μ . This is the wave equation in one dimension! It is a second–order, linear, (hyperbolic) partial differential equation. Solutions : y ( x,t ) = g ( x + vt ) + f ( x- vt ) → any smooth function ( C 2 ) 7 Proof: (Here, a prime denotes differentiation w.r.t. argument.) ∂y ∂x = g + f ∂ 2 y ∂x 2 = g + f ∂y...
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Lec3 - F y = τ sin θ τ sin θ τ(tan θ tan θ = τ ∂y...

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