Lec3 - F y = sin - sin (tan - tan ) = y x x + dx- y x x 2 y...

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Unformatted text preview: F y = sin - sin (tan - tan ) = y x x + dx- y x x 2 y x 2 dx in the infinitesimal limit. Let := 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 = dx 1 + dy dx 2 2 y t 2 = dx + O ( 2 ) 2 y t 2 ( dx ) 2 y t 2 where d 2 = dx 2 + dy 2 , or d = dx 1 + ( dy/dx ) 2 , and dm = d , where d is the length of the string element (in the direction along the string). Then dx 2 y t 2 = 2 y x 2 dx 2 y t 2 = 2 y x 2 2 y t 2 = v 2 2 y x 2 Where v 2 = . This is the wave equation in one dimension! It is a secondorder, 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 = sin - sin (tan - tan ) = y x x + dx- y x x 2 y...

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