Consider an observer at rest in a frame where the ambient air is also at rest. Far away to the left, a speaker emits sound waves at angular frequency ω0; far away to the right, an identical speaker (emitting the same frequency waves in its rest frame) is moving at velocity u to the right. Let v denote the speed of sound in air. A: Assuming the amplitude of waves from each speaker is the same, we might expect that the (complex) amplitude of sound waves is given by (for all points x between the two speakers) u(x, t) = A(e^(ik(x−vt)) + e^(ik'(x+vt))). (1) Here A is an unimportant overall constant prefactor. Use the Doppler effect (if needed), and other physics of waves, to determine k and k' in terms of u, v and/or ω. B: If the observer is at x = 0, the amplitude of oscillations is given by u(0, t). Show that if u << v, then Re[u(0, t)] = 2A cos(ω1t) cos(ω2t) (2) where |ω2| << |ω1|. Find explicit formulas for ω1,2 in terms of ω0, u, v. cos(ω2t) varies extremely slowly in time.
Consider an observer at rest in a frame where the ambient air is also at rest. Far away to the left, a speaker emits sound waves at angular frequency ω0; far away to the right, an identical speaker (emitting the same frequency waves in its rest frame) is moving at velocity u to the right. Let v denote the speed of sound in air.
A: Assuming the amplitude of waves from each speaker is the same, we might expect that the (complex) amplitude of sound waves is given by (for all points x between the two speakers)
u(x, t) = A(e^(ik(x−vt)) + e^(ik'(x+vt))). (1)
Here A is an unimportant overall constant prefactor. Use the Doppler effect (if needed), and other
physics of waves, to determine k and k'
in terms of u, v and/or ω.
B: If the observer is at x = 0, the amplitude of oscillations is given by u(0, t). Show that if u << v, then
Re[u(0, t)] = 2A cos(ω1t) cos(ω2t) (2)
where |ω2| << |ω1|. Find explicit formulas for ω1,2 in terms of ω0, u, v. cos(ω2t) varies extremely slowly in time. This leads to beating – two waves interfering at slightly different frequencies will lead to a slow variation in the amplitude with time. Beating waves are particularly audible, which
makes music that is even slightly out of tune very unpleasant.
![B: If the observer is at a = 0, the amplitude of oscillations is given by u(0, t). Show that if u « v, then'
Reſu(0, t)] = 2A cos(wit) cos(wzt)
(2)
where Jw2| < lwi|. Find explicit formulas for w1,2 in terms of wo, u, v. cos(wzt) varies extremely
slowly in time. This leads to beating - two waves interfering at slightly different frequencies will
lead to a slow variation in the amplitude with time. Beating waves are particularly audible, which
makes music that is even slightly out of tune very unpleasant.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff547c719-17ef-4176-928f-d8eeaf29e9d1%2F689ac9b2-946e-4bef-ad8a-0d188a635466%2Fmtw6kb_processed.png&w=3840&q=75)
Step by step
Solved in 4 steps
