Follows from fma applied to string elements y t x dx

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Unformatted text preview: ion x: 2 ∂y ∂ t2 Acceleration felt e.g. by an ant at position x. Segment of string from x to x+dx has mass: µdx Linear mass density of the string Wednesday, October 9, 2013 Derive wave eqn. Follows from F=ma, applied to string elements. y (t, x) τ dx x ∂y ∂y dFy = τ ( |x+dx − |x ) ∂x ∂x Tension (“F” in book). ∂2y dFy = (dm)ay = (µdx) 2 ∂t Linear mass density Wednesday, October 9, 2013 dy ∂2y dFy = τ 2 dx ∂x Equating gives the 1d wave eqn, with v= ￿ τ µ Ex: verify the units work, DA! Wave energy, power y (t, x) τ dx x dy Force exerted on string to the right, by the string to the left Method 1: ￿ (x, t) · ￿ (x, t) = Fy (x, t)vy (x, t) = −τ ∂ y ∂ y P (x, t) = F v ∂x ∂t Method 2: 1 k (x, t) = µ 2 ∂y ∂t 2 dE dx dE = = (k + u)v P= dt dx dt Wednesday, October 9, 2013 1 u(x, t) = τ 2 ∂y ∂x 2 energy densities Both methods give the same answer (using the wave eqn): Wave power, cont. ∂y ∂y ￿ P (x, t) = F (x, t) · ￿ (x, t) = Fy (x, t)vy (x, t) = −τ v ∂x ∂t y = A cos(kx − ω t) P = τ k ω A2 sin2 (kx − ω t) Pmax = Pave Wednes...
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This note was uploaded on 02/11/2014 for the course PHYSICS 2C taught by Professor Hicks during the Fall '09 term at UCSD.

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