Echo Hunter

Ultrasound imaging is one equation used twice. The machine knows the time t at which an echo came back and the speed v of sound in the material, so the interface that made it sits at d = v t / 2 — the halving is there because the pulse made the trip twice. While the gate falls down the block it is showing exactly that: at scope time t the gate is at depth v t / 2, and a spike appears the instant the gate reaches something.

How big that spike is comes from acoustic impedance, Z = ρ c. At a boundary between materials the fraction of intensity sent back is α = (Z₂ − Z₁)²/(Z₁ + Z₂)², which is about 1% for a fat–soft-tissue boundary and about 43% for soft tissue against bone. The scope plots pressure amplitude rather than intensity, so the heights go as √α, then fade with depth: the beam is absorbed on the way down and on the way back, and the time-gain compensation in this machine leaves 0.5 dB/cm uncorrected. That is why the same interface returns a visibly shorter spike the deeper it sits — and why nearly all the sound is lost at a gas pocket, where α ≈ 1 and everything behind it goes dark.

The later levels change what the block is made of, and that is the whole point. Fat carries sound at 1450 m/s and cortical bone at 4080, so an echo at 90 µs is 65 mm deep in fat and 184 mm deep in bone. Real scanners assume 1540 m/s everywhere and are therefore quietly wrong wherever the tissue is not average — a known source of measurement error that sonographers correct for by hand. The width of the transmit pulse sets the other limit you meet here: two interfaces closer together than about half a pulse length merge into one bump, which is axial resolution, and the only cure is a shorter, higher-frequency pulse that in turn cannot travel as deep.