How Ultrasound Sees Inside the Body: Echoes, Impedance, and the Physics of the Pulse

The one equation that makes an image: pulse-echo ranging

Strip a sonogram down and it is a stopwatch. The transducer emits a short pulse of ultrasound and then falls silent, listening. When an echo returns after a delay t, the depth of the reflector is

• d = c·t / 2

where c is the speed of sound in the medium and the factor of ½ accounts for the round trip. Clinical scanners hard-code c = 1540 m/s, an average over soft tissues. At that speed sound covers 1 cm in 6.5 μs one-way, so an echo from 10 cm deep arrives about 130 μs after transmission. The machine sends one pulse, records the full train of echoes along that line, then steers the beam a fraction of a degree and repeats. Stack a few hundred such A-lines side by side, map echo amplitude to brightness, and you have a two-dimensional B-mode (brightness-mode) slice — a fan of tissue rebuilt entirely from arrival times.

The pulse must be finished before its own echo returns from the shallowest structure, which sets the pulse repetition frequency: to image 20 cm deep you wait ≈ 260 μs per line, capping repetition near 4 kHz. That timing budget — not electronics — is why deep abdominal scans update more slowly than a shallow thyroid scan.

Acoustic impedance: why sound reflects at all

A wave only echoes where the medium's resistance to acoustic motion changes. That resistance is the acoustic impedance,

• Z = ρc, units Pa·s/m (rayl); tissues sit near 1.5 MRayl = 1.5×10⁶ rayl.

Impedance is the ratio of acoustic pressure to particle velocity, ρc — dense, stiff media have high Z. At a boundary between impedances Z₁ and Z₂ at normal incidence, the fraction of pressure amplitude reflected is

• r = (Z₂ − Z₁) / (Z₂ + Z₁),
• and the fraction of intensity reflected is R = r² = ((Z₂ − Z₁)/(Z₂ + Z₁))².

This is the acoustic twin of the Fresnel equations for light. Two crucial regimes fall out. Between fat (1.38 MRayl) and muscle (1.69 MRayl), R ≈ ((1.69−1.38)/(1.69+1.38))² ≈ 0.01 — only about 1% of the intensity bounces, so 99% penetrates to image deeper structures. But at a tissue–air interface Z drops by a factor of ~3000, giving R ≈ 0.999: essentially a perfect mirror. That is why gel is smeared on the probe (to banish the air gap) and why lungs and gas-filled bowel are acoustic blind spots. At tissue–bone the mismatch gives R ≈ 0.43, producing a bright surface and a dark acoustic shadow beyond it.

Frequency is a resolution-versus-depth bargain

Ultrasound is just sound above ~20 kHz; medical probes run 2–15 MHz. Frequency f, speed c, and wavelength λ obey c = fλ, so at 1540 m/s a 5 MHz pulse has λ = 1540/5×10⁶ ≈ 0.31 mm. Resolution splits into two kinds:

• Axial resolution (along the beam) is set by the spatial pulse length — roughly half of it. A pulse only a couple of cycles long at 5 MHz is ≈ 0.6 mm long, giving ~0.3 mm axial resolution. Shorter wavelength → shorter pulses → finer detail.
• Lateral resolution (across the beam) is the diffraction-limited beam width, θ ≈ λ/D for an aperture of diameter D — the same Abbe/Rayleigh limit as optics. Higher f narrows the beam.

So higher frequency always sharpens the picture. The catch is attenuation: sound is absorbed and scattered as it goes, and in soft tissue the loss runs at about 0.5 dB per centimeter per megahertz. A 10 MHz beam reaching 8 cm deep and back travels 16 cm, losing 0.5 × 10 × 16 = 80 dB — a factor of 10⁸ in intensity, drowned in noise. That is the fundamental trade: 5 MHz for a 20-cm-deep abdomen, 12–15 MHz for a 3-cm-deep thyroid or a superficial vessel. You cannot have both resolution and depth from one probe.

The transducer: a quartz crystal that hears and shouts

The heart of the probe is a slab of piezoelectric ceramic — historically lead zirconate titanate (PZT), now often a single-crystal relaxor. Piezoelectricity, discovered by Jacques and Pierre Curie in 1880, is bidirectional: a voltage across the crystal strains it, and a mechanical strain generates a voltage. So the same element transmits (a voltage spike rings it like a bell) and receives (returning pressure waves squeeze out a millivolt signal). The crystal is cut so its fundamental thickness-mode resonance lands at the design frequency: a half-wavelength thick, t = λ_crystal/2, giving a natural ring at the operating MHz.

Two engineering tricks are decisive. A backing layer heavily damps the crystal so it stops ringing after a cycle or two — a long, resonant "ping" would smear axial resolution. And a matching layer a quarter-wavelength thick, with impedance near the geometric mean √(Z_crystal·Z_tissue), acts as an acoustic anti-reflection coating so energy actually crosses into the body instead of bouncing off the crystal face. Modern probes pack 128 to 512 elements in a line; firing them with staggered nanosecond delays lets the array electronically steer and focus the beam — a phased array — with no moving parts.

Doppler mode: turning frequency shifts into blood-flow color

Beyond anatomy, ultrasound measures motion via the Doppler effect. Sound scattering off red blood cells moving at velocity v returns with a shifted frequency. For a beam at angle θ to the flow, the round-trip shift is

• Δf = 2 f₀ v cosθ / c,

the factor of 2 again from the two-way path. Take f₀ = 5 MHz, v = 0.5 m/s (a healthy carotid), c = 1540 m/s, θ = 0: Δf = 2 × 5×10⁶ × 0.5 / 1540 ≈ 3.2 kHz — conveniently in the audible band, which is why old Dopplers literally played the heartbeat as sound. Colour-Doppler scanners compute Δf at every pixel and paint flow toward the probe red and away blue. Note the cosθ: at θ = 90° there is no measurable shift, so sonographers deliberately angle the probe. Pulsed Doppler also faces an aliasing limit — the Nyquist theorem caps the measurable shift at half the pulse repetition frequency, so very fast jets "wrap around" and mis-color unless the scale is raised.

Scattering, speckle, and the limits of the picture

Big smooth boundaries produce specular echoes, but most of what fills an organ is diffuse scattering off structures smaller than a wavelength — the Rayleigh regime, where scattered power scales as f⁴, the same law that reddens sunsets. The interference of countless tiny scatterers produces speckle: the grainy, shimmering texture inside a liver or muscle. Speckle is not noise about anatomy; it is a coherent interference pattern, and its statistics carry real tissue information, though it also blurs fine detail.

Several artifacts follow directly from the physics:

• Acoustic shadowing — behind a gallstone or bone, so much is reflected/absorbed that structures beyond go dark.
• Enhancement — behind a fluid-filled cyst, which barely attenuates, tissue appears abnormally bright.
• Reverberation — sound bouncing between two strong reflectors creates evenly spaced false echoes.
• Speed error — the machine assumes 1540 m/s, but fat is ~1450 m/s, so a reflector seen through fat is placed slightly too deep.

Safety, too, is set by physics. Ultrasound is non-ionizing (photon-free), but it deposits energy: the thermal index and mechanical index (MI, the peak rarefactional pressure divided by √f) bound heating and the risk of cavitation — the violent collapse of microbubbles that, pushed further, powers lithotripsy and sonoluminescence. Diagnostic scanners are held to MI ≤ 1.9 precisely to stay below that threshold.

Acoustic properties of tissues and the reflections at their interfaces (soft-tissue speed ≈ 1540 m/s assumed by the scanner)

Medium	Density ρ (kg/m³)	Speed c (m/s)	Impedance Z (MRayl)	Echo at soft-tissue boundary
Air (in lung/bowel)	1.2	330	0.0004	≈ 100% reflected — opaque
Fat	950	1450	1.38	weak echo (~1% intensity)
Blood / soft tissue	1060	1540	1.63	reference
Muscle	1070	1580	1.69	faint echo, texture
Bone	1900	4080	7.75	≈ 43% reflected — bright, shadows behind