Problem Set

NBPhO 2026

8. Jet Sound Teo Kai Wen, Jaan Kalda 8 pts

Waves · Doppler effect, Sound propagation, Supersonic flow

Decode a supersonic flyby spectrogram to extract the Mach number and closest-approach distance from the Doppler asymptotes and the sonic-boom delay.

Problem by Teo Kai Wen, Jaan Kalda.

A fighter jet flies past a ground observer along a straight horizontal line at constant supersonic velocity, passing at closest-approach distance dd. The jet’s Mach number is M=v/c>1M = v/c > 1, where vv is the speed of the jet and cc is the speed of sound. The air is at rest. The observer records the sound as a spectrogram (intensity colour-coded as a function of frequency and time; black means silence), shown below.

Graph: Spectrogram with time on the horizontal axis (0 to 25 s) and frequency on the vertical logarithmic axis (100 to 3000 Hz). Two bright curves descend over time, sweeping from higher to lower frequencies, characteristic of a Doppler-shifted supersonic flyby.

i) (2 points) Explain, qualitatively, the features seen in the spectrogram.

ii) (3 points) From the spectrogram, find the Mach number MM.

iii) (3 points) Find the closest-approach distance dd given that c=340 m⋅s−1c = 340\,\text{m}\cdot\text{s}^{-1}.