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Unformatted text preview: F y = Ï„ sin Î¸ Ï„ sin Î¸ Ï„ (tan Î¸ tan Î¸ ) = Ï„ âˆ‚y âˆ‚x x + dx âˆ‚y âˆ‚x x â‰¡ Ï„ âˆ‚ 2 y âˆ‚x 2 dx in the infinitesimal limit. Let Î¼ := M/L be linear mass density, The y â€“component of the force on the string element dx also equals F y = ma y = ( dm ) âˆ‚ 2 y âˆ‚t 2 = Î¼dx 1 + dy dx 2 âˆ‚ 2 y âˆ‚t 2 = Î¼dx + O ( Î¸ 2 ) âˆ‚ 2 y âˆ‚t 2 â‰ˆ ( Î¼dx ) âˆ‚ 2 y âˆ‚t 2 where d 2 = dx 2 + dy 2 , or d = dx 1 + ( dy/dx ) 2 , and dm = Î¼d , where d is the length of the string element (in the direction along the string). Then Î¼dx âˆ‚ 2 y âˆ‚t 2 = Ï„ âˆ‚ 2 y âˆ‚x 2 dx âˆ‚ 2 y âˆ‚t 2 = Ï„ Î¼ âˆ‚ 2 y âˆ‚x 2 â‡’ âˆ‚ 2 y âˆ‚t 2 = v 2 âˆ‚ 2 y âˆ‚x 2 Where v 2 = Ï„ Î¼ . This is the wave equation in one dimension! It is a secondâ€“order, linear, (hyperbolic) partial differential equation. Solutions : y ( x,t ) = g ( x + vt ) + f ( x vt ) â†’ any smooth function ( C 2 ) 7 Proof: (Here, a prime denotes differentiation w.r.t. argument.) âˆ‚y âˆ‚x = g + f âˆ‚ 2 y âˆ‚x 2 = g + f âˆ‚y...
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 Spring '09
 LIORBURKO
 Physics, Force, Mass, FY, dx

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