Lecture-20-03-27-08

Lecture-20-03-27-08 - Lecture-20 Energy Transported by...

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1 Lecture-20 Energy Transported by Waves in Rope Kinetic energy in a small piece of rope, dx in length and μ dx in mass, at time t is Or, power in form of kinetic energy per unit time is Averaging over wavelength, Since KE and PE switch back and forth, total power transported is given by Or, [] 2 2 ) cos( ) ( 2 1 ) ( 2 1 t kx A dx dK t y dx dK ω = = v A dt dK dt dV dt dK ave ave 2 2 2 1 2 = = + v A dt dK dx kx v A dx t x K dt dK ave ave 2 2 00 2 2 2 4 1 ) ( cos 2 1 ) 0 , ( 1 λ λλ = = = = ∫∫ dt dx t kx A dt t x dK ) ( cos 2 1 ) , ( 2 2 2 = v A P >= < 2 2 2 1 dx y x
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2 2 2 2 2 t D x D F T = μ Wave Equation for Rope Assume: - small amplitude; - vertical displacement; - at constant tension. Equation of motion for vertical displacement (D) of rope element dx with mass dx is Here D is a function of both position x and time t. Since angles are small, sin θ tan slope of rope. Then Taking Δ x to the differential limit, we arrive at This equation holds true for many different waves. 2 2 1 2 ) ( sin sin t D x F F ma F T T y y Δ = = 2 2 2 2 1 2 ) ( t D x slope F t D x slope slope F T T = Δ Δ Δ =
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3 ] ) ( 2 1 cos[ ] ) ( 2 1 cos[ 2 2 1 2 1 t t D D o ω + = Amplitude (a) (b) Beats
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Lecture-20-03-27-08 - Lecture-20 Energy Transported by...

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