The decaying wave pulse on a string is modeled by: y(x,t) = A max e -bx sin(kx -

Question

The decaying wave pulse on a string is modeled by: y(x,t) = Amaxe-bxsin(kx - t) 1. derive an expression for the power delivered by this pulsefor any x>0, t>0. Answer only in terms of A, b, s, , k,. 2 derive an expression for the ratio of the power at any pointx1 to the powe at x2 where x>0. Answeronly in terms of A, b, x, , k, . 3.if A=9cm, b=0.5m-1,=60s-1,k=20m-1. Calculatethe distance traveled by the pulse before the power delivered hasdecreased by a factor of 20. I promise to rate you as lifesavers if youranswer is correct! The decaying wave pulse on a string is modeled by: y(x,t) = Amaxe-bxsin(kx - t) 1. derive an expression for the power delivered by this pulsefor any x>0, t>0. Answer only in terms of A, b, s, , k,. 2 derive an expression for the ratio of the power at any pointx1 to the powe at x2 where x>0. Answeronly in terms of A, b, x, , k, . 3.if A=9cm, b=0.5m-1,=60s-1,k=20m-1. Calculatethe distance traveled by the pulse before the power delivered hasdecreased by a factor of 20. I promise to rate you as lifesavers if youranswer is correct! I promise to rate you as lifesavers if youranswer is correct!

Explanation / Answer

(a) Power= P= F.v = dot product of force andvelocity. So, P = m(dv/dt).v = m[d(dx/dt)/dt] . (dx/dt) = m[d(-wAmaxe-bxcos(kx-wt)) / dt ] .(-wAmaxe-bxcos(kx-wt)) = m [-w2Amaxe-bxsin(kx-wt)] .(-wAmaxe-bxcos(kx-wt)) = [(mAmax2w2)/2]sin(2kx -2wt)e-2bx (b) P(1) / P(2) =[(mAmax2w2)/2]sin(2kx1- 2wt)e-2bx1 /[(mAmax2w2)/2]sin(2kx2- 2wt)e-2bx2 =e2b(x2-x1)sin(2kx1-2wt)/ sin(2kx2 - 2wt) (c) In this case, P(1)/P(2) = 20/1 Taking x1 = 0, x2 = x, t = /4w = 1/240s,    e2b(x) / sin(2kx + /2) =20      => ex / sin(40x +/2) = 20.      => ex =20*cos(40x)      Now, x>0, => ex > 1,=> cos(40x) > 0.05    You can obtain x=0.012 m by graphical method(preferable, also possible by numerical method)

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