Wave Propagation

Ultrasound is just
a shout in the dark.

Before any image exists, there is only a mechanical pressure wave moving through tissue. The wave equation governs everything — and the speed of sound in tissue is the fundamental clock the machine reads.

∂²u∂t² = c² ∇²u
D'Alembert Wave Equation
d = c · t2
Time-of-Flight Depth
Pressure Wave Through Tissue
Wave Properties
Transducer Frequency 7.5MHz
Higher frequency → better resolution, less depth penetration
Attenuation coefficient 0.5dB/cm/MHz
Bone ~22 · Muscle ~1.09 · Blood ~0.18 dB/cm/MHz
Depth of interest 4.0cm
Machine adjusts PRF and TGC to match selected depth
Speed of Sound
m/s
Wavelength (λ)
mm
Total Attenuation @ depth
dB
Round-trip time
μs
Speed of Sound in Biological Tissue
Fat
Speed1450 m/s
Attenuation0.63 dB/cm/MHz
Impedance Z1.38 MRayl
Blood
Speed1570 m/s
Attenuation0.18 dB/cm/MHz
Impedance Z1.61 MRayl
Muscle
Speed1580 m/s
Attenuation1.09 dB/cm/MHz
Impedance Z1.65 MRayl
Soft Tissue (avg)
Speed1540 m/s
Attenuation0.54 dB/cm/MHz
Impedance Z1.63 MRayl
Bone (cortical)
Speed3500 m/s
Attenuation22 dB/cm/MHz
Impedance Z7.8 MRayl
Air / Lung
Speed330 m/s
Attenuation>100 dB/cm/MHz
Impedance Z0.0004 MRayl
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1540 m/s: The Machine's Assumption

All ultrasound machines assume sound travels at 1540 m/s. When you're imaging through fat (1450 m/s), the machine slightly miscalculates depth — creating the "speed artifact" that shifts structures deeper than they are.

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Frequency-Resolution Tradeoff

Wavelength λ = c/f. A 15 MHz probe in soft tissue has λ ≈ 0.1 mm — near-capillary resolution. But attenuation scales with frequency: 15 MHz loses signal 30× faster per cm than 0.5 MHz.

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The "Echo Cave" Mental Model

You shout. You hear echoes from different walls at different times. The time delay tells you distance. The loudness tells you the reflectivity of the surface. Ultrasound is exactly this — just at 7 million Hz.

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Why Lung and Bone Are Difficult

Air has an acoustic impedance 4000× lower than tissue. Nearly 100% of the wave reflects at any air-tissue interface. Bone creates the opposite problem — nearly total reflection and extreme attenuation of what passes through.

Historical Origin
D'Alembert, Poisson, Fourier

Jean le Rond d'Alembert formulated the wave equation in 1747. Siméon Poisson extended it to 3D. These were pure mathematics — the medical application came 200 years later when Ian Donald first used ultrasound clinically in obstetrics (1950s). The physics waited for the technology to catch up.

Intracav Opportunity
Physics-Aware AI

Most AI ignores the physics it's built on. Intracav can incorporate speed-of-sound constraints and attenuation models directly into the inference layer — giving the system anatomical priors that pure pixel-based models lack.

Speed of sound → depth correction
Attenuation model → TGC guidance
Reflection coefficients → boundary detection
Tissue priors → vessel vs non-vessel