Beats and the Doppler effect

Beats are the slow rise and fall in loudness heard when two tones of nearly equal frequency overlap, at a rate equal to the difference of their frequencies; the Doppler effect is the shift in received frequency produced when source or observer moves through the medium. Use the supplied topic model Y_out = 500*(343+w)/(343-u), where Y_out is f RECEIVED (Hz). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. u is Source speed toward listener, measured in m s⁻¹. Keep w (Listener speed toward source) = 0 m s⁻¹ fixed. For this exercise use only 29.75 m s⁻¹ ≤ u ≤ 250.25 m s⁻¹ on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. Calculate Y_out at u = 78.26 m s⁻¹ and 109.13 m s⁻¹. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δu across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.

Beats are the slow rise and fall in loudness heard when two tones of nearly equal frequency overlap, at a rate equal to the difference of their frequencies; the Doppler effect is the shift in received frequency produced when source or observer moves through the medium. Use the supplied topic model Y_out = 500*(343+w)/(343-u), where Y_out is f RECEIVED (Hz). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. u is Source speed toward listener, measured in m s⁻¹. Keep w (Listener speed toward source) = 0 m s⁻¹ fixed. For this exercise use only 29.75 m s⁻¹ ≤ u ≤ 250.25 m s⁻¹ on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 959.4674 Hz, with a bounded reading uncertainty ±61.797 Hz. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown u by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.02205 m s⁻¹, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.