8.1 Speed of Sound & the Range Equation

What Sets the Speed of Sound

Propagation speed (also called the speed of sound, symbol c) is the rate at which the compressional wave moves through a medium. It is a property of the medium alone. Changing the transducer frequency, increasing output power, or raising amplitude does nothing to how fast the pulse travels — a 3 MHz pulse and a 10 MHz pulse move through the same liver at exactly the same speed. This is one of the most frequently tested facts in the Image Formation domain: the sonographer cannot alter propagation speed in tissue.

Two physical properties of the medium determine speed:

• Stiffness (resistance to compression, related to the bulk modulus): the stiffer the medium, the more readily one compressed layer pushes on the next, so the disturbance travels faster. Stiffness and speed are directly related.
• Density (mass per unit volume): denser particles have more inertia and are harder to accelerate, so increased density slows propagation. Density and speed are inversely related.

In practice, differences in stiffness between biological media are far larger than differences in density, so stiffness dominates. That is why sound races through bone — extremely stiff, only moderately denser than soft tissue — at about 4080 m/s, while it crawls through air — neither stiff nor dense — at only about 330 m/s. Speed ranking to memorize: bone (fastest) > soft tissue > fat > air (slowest).

Speeds Worth Memorizing

Every ultrasound scanner is calibrated to the soft-tissue average of 1540 m/s, which can also be written as 1.54 mm/µs or 0.154 cm/µs. Note that room-temperature water is markedly slower — about 1480 m/s, the value exam tables use — and only approaches 1540 m/s when warmed toward 50 degrees C. Coupling gel works not because its speed matches tissue but because its acoustic impedance does, which is what removes the near-total reflection at a tissue-air interface.

The Range Equation: Turning Time into Depth

Pulse-echo imaging has no depth ruler inside the body. The machine knows only one thing with precision: time. It starts a clock when the pulse leaves the transducer and stops it when the echo returns. Depth is then computed with the range equation:

d = c × t ÷ 2

where d is the one-way distance to the reflector, c is 1540 m/s, and t is the go-return (round-trip) time. Division by two is essential — the measured time covers the trip down and the trip back, so the one-way depth is only half of what the raw time would suggest. Omitting the ÷2 is the classic exam distractor.

The 13-µs Rule

The single most useful number in clinical physics: for every 1 cm of reflector depth, the round-trip travel time is 13 µs. Derivation: a 1 cm depth means 2 cm of travel; 0.02 m ÷ 1540 m/s = 12.99 µs ≈ 13 µs. Worked examples:

• Echo returns after 65 µs → d = 1540 × 65 × 10⁻⁶ ÷ 2 = 0.050 m = 5 cm (65 ÷ 13 = 5)
• A structure displayed at 10 cm → round-trip time = 10 × 13 = 130 µs
• Echo returns after 104 µs → depth = 104 ÷ 13 = 8 cm

The range equation also sets the maximum imaging depth for a given pulse repetition frequency (PRF). The system must wait for the deepest echo before firing the next pulse, so maximum depth = c ÷ (2 × PRF). At a PRF of 5000 Hz the pulse repetition period is 200 µs, limiting unambiguous depth to 1540 × 200 × 10⁻⁶ ÷ 2 = 15.4 cm. Raising PRF for a fast Doppler study therefore sacrifices depth — an inverse relationship you will use daily.

When 1540 m/s Is Wrong: Speed Propagation Artifact

The scanner does not measure speed; it assumes 1540 m/s everywhere along the beam. If part of the path travels through a medium with a genuinely different speed, the go-return time no longer matches the assumed calibration and the reflector is placed at the wrong depth — misregistration, clinically called the speed propagation artifact (or speed error/speed displacement artifact).

The classic setting is a fatty beam path, because fat transmits sound at only ~1450 m/s. Consider a reflector truly 5 cm deep behind a thick layer of fat. The actual round-trip time is 0.10 m ÷ 1450 m/s ≈ 69 µs, but the machine computes depth as 1540 × 69 × 10⁻⁶ ÷ 2 ≈ 5.3 cm — the reflector is displayed too deep (farther from the transducer than it really is). The rule:

• Sound slower than 1540 m/s in the path (fat) → echo arrives late → reflector displayed too deep
• Sound faster than 1540 m/s in the path → echo arrives early → reflector displayed too shallow

This is why a diaphragm seen through a diffusely fatty liver can appear discontinuous or stepped at the boundary of the fatty region, and why measurements taken across large fatty or fluid-solid mixed paths carry a small systematic error. The artifact affects only depth placement; it does not change the brightness or shape of the reflector, and it cannot be corrected with gain or TGC — only recognized.