Rays leave the Mach cone and bend in the layered atmosphere. Drag the view to orbit. Jet not to scale (shown about 100 times larger).
Boom-carpet half-width on the ground against Mach number (dashed = cutoff, dot = now)
Speed of sound against altitude. A ray turns back where sound is faster than the jet (right of the red line)
The science
Mach cone and rays. A source moving at speed V = M c(h) radiates a conical shock whose normals make an angle cos(phi) = 1/M with the flight direction, i.e. cone half-angle mu = asin(1/M). Each normal is a sound ray, tilted by an azimuth theta measured from straight down.
Ray equations. In a layered atmosphere with sound speed c(z) = sqrt(gamma R T(z)) (gamma = 1.4, R = 287.05 J/kg/K) the ray position and unit normal obey dr/dt = c k, dk/dt = -(dc/dz)(z_hat - (k.z_hat) k). The sim integrates them with fourth-order Runge-Kutta. No Snell invariant is assumed.
Cutoff. Along the flight track the ray keeps k_x = c(z)/V (Snell: trace speed V is constant). It can only reach the ground while c(z) < V at every level below the jet, so booms vanish for M < M_c = max c(z) / c(h). In the standard atmosphere M_c = c(0)/c(h): about 1.15 for jets above 11 km. Rays at larger azimuth turn back earlier, which gives the finite carpet width.
Atmosphere. Standard-atmosphere temperature (288.15 K at sea level, -6.5 K/km to 11 km, then 216.65 K) plus your uniform offset plus an exponential surface-layer anomaly exp(-z/1500 m).
Simplifications. No wind or shear, no humidity or absorption, no turbulence, no acceleration or manoeuvre focusing (the superboom), steady level flight, flat ground, and no overpressure (N-wave) amplitude is predicted. Only the geometry of where a boom can arrive. Real cutoff Mach numbers are quoted near 1.1 to 1.2 for cruise altitudes: treat this as a typical-value cross-check, not as a design tool.