Question

sinusoidal wave of angular frequency 1205 rad/s and amplitude 3.20 mm is sent along a cord with linear density 2.10 g/m and tension 1233 N. (a) What is the average rate at which energy is transported by the wave to the opposite end of the cord? (b) If, simultaneously, an identical wave travels along an adjacent, identical cord, what is the total average rate at which energy is transported to the opposite ends of the two cords by the waves? If, instead, those two waves are sent along the same cord simultaneously, what is the total average rate at which they transport energy when their phase difference is (c) 0, (d) 0.6? rad, and (e)? rad?

I figured everything out except part d

Answer 1

A sound source sends a sinusoidal sound wave of angular frequency 3000 rad/s and amplitude 12.0 nm through a tube of air. The internal radius of the tube is 2.00 cm. (a) What is the average rate at which energy (the sum of the kinetic and potential energies) is transported to the opposite end of the tube? (b) If, simultaneously, an identical wave travels along an adjacent, identical tube, what is the total average rate at which energy is transported to the opposite ends of the two tubes by the waves? If, instead, those two waves are sent along the same tube simultaneously, what is the total average rate at which they transport energy when their phase difference is (c) 0, (d) 0.40? rad, and (e) ? rad?

[SOLUTION]-

R=.02m radius.
?=3000 rad/s
A= 12*10^-9 m amplitude
?= 1.21 kg/m^3 air density
v = 343 m/s air speed
So, power = Intensity*area and intensity= 0.5?*v*?^2*A^2 and area = pi*R^2
a) P= 0.5?*v*?^2*A^2*pi*R^2
and i get 3.38E-10 W which is same as .338 nW
b) if there are two pipes with two different waves that are identical...the new power, P' = 2P
So, P'= .676 nW
c) if there is 0 phase change, the waves are completely constructive and the amplitude is the only thing that changes (it doubles) and since amplitude is squared, P''= 4P
P''= 1.35 nW
d) with 2/5*pi as phase change, the new amplitude is (2*A*cos(?/2))^2 which i get 2.62A^2 and
P3= 2.62P= .885 nW
e) if the waves have a pi difference, they will be completely destructive, so the resulting amplitude = zero. (also cos(?/2) will be cos(pi/2) which is zero!). So P4= 0 W