NBPhO 2026

Difficulty medium

Prerequisites

• Doppler shift for a moving source, $f_o = f_e/(1 - v_{\parallel}/c)$
• Speed of sound in still air; wavefronts as straight-line travelers
• Reading a logarithmic frequency axis
• Mach number and Mach angle, $\sin\alpha = 1/M$

Learning objectives

• Derive the supersonic Doppler factor $|1 - M\cos\theta|$ and locate its zero at the Mach angle
• Recognise that for $M > 1$ the source-time-to-observer-time map $t(\tau)$ is non-monotonic, so each observer instant carries two arrivals
• Identify the sonic boom as the singular pile-up at $dt/d\tau = 0$ and explain its broadband character
• Extract $M$ from the asymptote ratio $f_+/f_- = (M+1)/(M-1)$ and recover $f_e$ as the harmonic mean $2f_+f_-/(f_++f_-)$
• Convert the boom delay $t_{\mathrm{cone}}$ into a lateral distance via $d = cMt_{\mathrm{cone}}/\sqrt{M^2-1}$

Watch out for

• Treating the upper and lower curves as two different emitted tones. They are one emitted tone heard via two different propagation paths; the same $M$ governs both, and they meet at infinite frequency at the boom.
• Mis-anchoring the spectrogram's time origin. The boom delay $t_{\mathrm{cone}}$ is measured from the moment the jet flies overhead (visual sighting at $t = 0$), not from the start of recording. The observer sees the jet several seconds before she hears it.
• Forgetting the absolute value in the Doppler factor. On the upstream branch ($\tau < \tau_*$, jet still ahead) one has $1 - M\cos\theta < 0$; the observed frequency is $f_e/|1 - M\cos\theta|$, not a signed quantity.
• Reading a single descending Doppler curve as a subsonic flyby. A subsonic flyby produces one curve with a smooth high-to-low crossover. The presence of two simultaneous descending curves and a leading silence is the signature of $M > 1$.
• Confusing the Mach angle $\alpha$ ($\sin\alpha = 1/M$) with the look angle $\theta$ at which the boom arrives. The boom does pile up on the Mach cone, where $\cos\theta = 1/M$, but $\alpha$ refers to the cone's half-opening, while $\theta$ is the source-to-observer line of sight.