15.82: a) , cos sin sin cos SW W S ωt kx ω ωA ωt, kx kA t y x y-= = ∂ ∂ ∂ ∂ and so the instantaneous power is ). 2 sin( ) 2 sin( 4 1 ) cos )(sin cos (sin SW SW 2 2 ωt kx ωk FA ωt ωt kx kx ωk FA P = = b) The average value of P is proportional to the average value of ), 2 sin( t ϖ and the average of the sine function is zero; .0 av = P c) The waveform is the solid line, and the power is the dashed line. At time 0and0 , 2 = = = P y ω π t and the graphs coincide. d) When the standing wave is at its maximum displacement at all points, all of the energy is potential, and is concentrated at the places where the slope is steepest (the nodes). When
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