A sound source sends a sinusoldal sound wave of angular frequency 2300 rad/s and

Question

A sound source sends a sinusoldal sound wave of angular frequency 2300 rad/s and amplitude 14.0 nm through a tube of air. The internal radius of the tube is 1.50 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 he 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)0.610/7 rad, and (e) rad?

Explanation / Answer

R = 0.015 m

= 2300 rad/s
A= 14*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 = 1.52*10^-10 W = 0.152 nW

b) if there are two pipes with two different waves that are identical...the new power, P' = 2P
So, P'= 0.304 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''= 0.608 nW

d) with 0.61*pi as phase change, the new amplitude is (2*A*cos(/2))^2 = 1.32A^2

and P3= 1.32*P= 0.200 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.

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