A source moves at constant speed along a straight line past a stationary listener in still air. When approaching directly, its observed tone is 662.8931 Hz; when receding directly, it is 582.3204 Hz. Sound speed is 340 m s⁻¹. Treat these as the limiting observations far before and after closest approach. Determine the source speed and its emitted frequency. Explain why using the simple mean of the observed tones does not give the emitted frequency exactly.

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Source speed 22 m s⁻¹; emitted frequency 620 Hz. The arithmetic mean is slightly too large. For a moving source and stationary listener, wavefront spacing changes in the air. Write f_a = fc/(c − u) and f_r = fc/(c + u), using c for sound speed here. Eliminating f gives u/c = (f_a − f_r)/(f_a + f_r). Substituting the two measurements gives u = 22 m s⁻¹. Recover the emitted frequency using f = f_a(c − u)/c, or equivalently f = 2f_af_r/(f_a + f_r). The result is 620 Hz. The approach and recession formulas are not symmetric additive shifts: their denominators differ. The arithmetic mean equals f/[1 − (u/c)²], which is greater than f for any nonzero subsonic speed. The inferred source speed is below sound speed and the emitted frequency lies between the two measured frequencies. These checks confirm the signs and the intended subsonic model.

Infer a moving source speed from two tones · set 6

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