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A harmonic wave is travelling on string 1. At a junction with string 2 it is partly reflected and partly transmitted . The linear mass densityof the second string is four times that of the first string , and that the boundary between the two strings is atx = 0 . If the expression for the incident wave isy_(i) = A_(i) cos (k_(1) x - omega _(1) t). What are the expressions for the transmitted and the reflected waves in terms of A_(i),K_(1) and omega_(1)? (b) Show that the average power by the incident wave is equal to the sum of the average power carried by the transmitted and reflected waves . |
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Answer» Solution :Since ` v= sqrt (T//mu), T_(2) = T_(1) and mu_(2) = 4 mu_(1)` We have , `v_(2) = (v_(1))/(2)`(i) Since the FREQUENCY does not change , that is , `omega_(1) = omega_(2)`(ii) Also , because `k = ( omega //v)` , the numbers of the harmonic wavesin the strings are related by `k_(2) = (omega_(2))/( v_(2)) = (omega_(1))/( v_(1)//2) = 2 (omega_(1))/( v_(1)) = 2 k_(1)`(iii) The AMPLITUDES are `A _(t) = (( 2v_(2))/( v_(1) + v_(2))) A_(i)` `= [ (2 (v_(1)//2))/( v_(1) + ( v_(1)//2))] A_(i) = (2)/(3) A_(i)`(iv) and `A_(r) = (( v_(2) - v_(1))/( v_(1) + v_(2))) A_(i)` ` = [ ((v_(1)//2) - v_(1))/(v_(1) + (v_(1)//2))] A_(i) = (-A_(i))/(3)`(iv) Now with Eqs. (ii) and (iii) and (iv) , the transmitted wave can be written as `y_(t) = (2//3) A _(i) cos ( 2k_(1) x - omega t)` SIMILARLY the reflecteed wave can be expressed as ` y _(x) = - (A_(i))/(3) cos ( k_(1) x + omega _(1) t)` ` = (A_(i))/(3) cos ( k_(1) x + omega _(1) t + pi)` (B) The average power of a harmonic wave on a string is GIVEN by `P = (1)/(2) rho A^(2) omega^(2) mu v``(as rho s = mu)` Now `P_(i) = (1)/(2) omega_(1)^(2) A_(i)^(2) mu_(1) v_(1)`(vi) `P_(t) = (1)/(2) omega_(1)^(2) ((2)/(3) A_(i))^(2) ( 4 mu _(1)) (( v_(1))/(2)) = (4)/(9) omega_(1)^(2) A_(i)^(2) mu_(1) v_(1)`(vii) `P_(r) = (1)/(2) omega_(2)^(2) ( - (A_(i))/(3))^(2) ( mu_(1))(v_(1))= (1)/(18) omega_(1)^(2) A_(i)^(2) mu_(1) v_(1)`(viii) From Eqs. (vi),(vii) and (viii) , we can show that `P_(i) = P_(t) + P_(r)`. |
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