Concept of wave motion

Question

!!2!! identical strings are used to generate waves under similar conditions. Ratio of average power for the waves is:

(a) !!16/25!!

(b) !!4/25!!

(c) !!16/5!!

(d) !!4/5!!

Solution:

Intensity = Power/Area.

Intensity, !!I = 1/2 rho V A^2(omega)^2!!

Power !!prop!! Intensity.

!!I prop A^2(omega)^2!!

!!=> P prop A^2(omega)^2!!

!!=> P_1 prop A_1^2 (omega_1)^2 and P_2 prop A_2^2 (omega_2)^2 !!

Therefore,

!!P_1/P_2 = (A_1^2(omega_1)^2)/(A_2^2(omega_2)^2)!!

!!omega = (2pi)/(Time period)!!

!!A_1 = 2, A_2 =5 and 2T_1 = T_2!!

!!=> P_1/P_2 = (A_1 T_2)^2/(A_2T_1)^2!!

!!=>P_1/P_2 = (2xx2T_1)^2/(5xxT_1)^2!!

!!=> P_1/P_2 = (4/5)^2= 16/25!!

Therefore, option (a) is correct.