A transverse wave pulse on a stretched string of mechanical

A transverse wave pulse on a stretched string of mechanical impedance Z1 carries a total energy of 5.00 J. This string is joined seamlessly to a second string of impedance Z2. Both strings are under the same tension. When the 5.00-J pulse is incident on the junction point, the reflected pulse (which is not inverted) carries an energy of 3.2 J. The ratio Z2/Z1 of the mechanical impedances of the two strings is therefore (b) 9/41 (c) 1 (d) 41/9 (e) 9

Solution

characteristic impedance of a string is given by T/c ( where c is wave speed and T is tnesion in sthe string)
so from the given question
E1 = 5 J, Z1 is the inpedance
both the strings are under same tension
reflected pulse has E2 = 3.2 J

now energy on a string is given by
E = 0.5*mu*w^2*A^2

so for the original wave, let the amplitude be A
for the reflected wave, amplitude is Ar = (Z1 - Z2)A/(Z1 + Z2)

hence
0.5*mu*w^2*A^2 = 5
0.5*mu*w^2*(Z1 - Z2)^2A^2/(Z1 + Z2)^2 = 3.2 J

(Z1 + Z2)/(Z1 - Z2) = sqroot(5/3.2) = 1.25
taking componendo and dividendo
Z1/Z2 = (1.25 + 1)/(1.25 - 1) = 2.25/0.25 = 9
hence
Z1/Z2 = 9
Z2/Z1 = 1/9 option A