Instead of drawing a 3-D cutaway of the soil and computing each reflector's arrival time from a closed-form hyperbola formula, this version numerically solves the scalar wave equation directly on a 2-D grid representing the survey's vertical cross-section (position × depth):
∂²u/∂t² = v(x,z)² · ∇²u
v(x,z) = c / √ε_r(x,z) (c ≈ 0.30 m/ns)
A short Ricker wavelet is injected at the surface at the antenna's position each time a trace is recorded. The leapfrog finite-difference update propagates the resulting wavefield outward in every direction; it slows down in the lossy soil (v = c/√ε_r), speeds back up to c inside the air-filled void (a real velocity contrast, not a drawn circle), and is forced to zero inside the buried pipe to mimic a perfect electrical conductor's total reflection. A thin absorbing sponge layer at the domain edges suppresses artificial wall reflections.
The amplitude sampled back at the source position over time is the radar trace — the radargram below is built column-by-column directly from this simulated field, one trace per antenna position, exactly like real 2-D GPR forward modelling software. The classic reflection hyperbola still appears — not because it was drawn, but because it falls straight out of the wave equation once you fire a pulse at many positions above a point-like reflector.
- The "Wave velocity", "two-way time" and "estimated depth" readouts above use the same closed-form formulas as a quick analytic cross-check — comparing them against the true hyperbola in the simulated radargram is itself a demonstration that the numerical method reproduces the known physics.
- A higher εr slows the wave, stretching the radargram vertically for the same physical depth, both in the analytic estimate and in the simulated field.
- Moving the antenna or pressing Auto-scan fires a fresh simulated pulse at the new position and bakes its recorded trace into the radargram.
Real-world relevance: this is essentially what finite-difference time-domain (FDTD) GPR forward-modelling codes do to generate synthetic radargrams used to plan and interpret real utility, rebar and void surveys.