Bat Doppler Shift: Wavefront-Timing Model

The 3D version of this sim computes the Doppler-shifted echo frequency by plugging bat and target velocities straight into the classical two-leg Doppler formula. This 2D companion instead simulates the actual sound field: every call the bat emits spawns a circular wavefront that expands outward at the true speed of sound through still air, oblivious to how fast the bat itself is moving. Because the bat keeps moving after each call, successive wavefronts are emitted from different points in space, so they arrive at the moth compressed together (if the bat is closing in) or stretched apart (if it's falling behind) — the moth's own position becomes a moving reflector that sends a fresh echo wavefront racing back to the bat. The bat's live "measured echo" readout comes from timestamping when those echo wavefronts physically arrive, not from evaluating an equation, and it reproduces the same closed-form Doppler result the 3D sim assumes — this time earned from geometry.