Two identical sirens each emit a continuous tone of rest frequency
PHYSICS · Multi-part · AP · Physics 2: Algebra-based · Waves Sound and Physical Optics · Waves Sound and Physical Optics
Problem
Two identical sirens each emit a continuous tone of rest frequency 700Hz. Siren A is mounted on a sports car that passes a stationary roadside observer at t=0 with a speed of 30ms−1 and continues in a straight line at that constant speed. Four seconds later (t=4s), Siren B mounted on a delivery van passes the same observer, moving in the same direction at a constant speed of 20ms−1. Assume still air with sound speed vs=343ms−1.
Throughout the interval 0≤t≤6s, the two Doppler-shifted sounds reach the observer simultaneously and produce audible beats.
(a) For 0≤t≤4s, obtain an expression for the observed frequency from each siren as a function of time, measured at the roadside observer.
(b) Using the expressions from part (a), derive the time-dependent beat frequency fb(t) for the interval 0≤t≤4s.
(c) Repeat parts (a) and (b) for the interval 4≤t≤6s, during which both sirens are ahead of the observer and moving away.
(d) Without using calculus, estimate the average beat frequency heard by the observer over the entire 6s interval by evaluating fb(t) at t=0, t=3s, and t=6s and computing an arithmetic mean. Discuss whether this average is likely to overestimate or underestimate the true average value and explain why.