# Ultrasound wave propagation and parameters

Ultrasound waves are acoustic pressure waves that interact with propagation media. Ultrasound visualization normally is based on wave reflectance from regions with different acoustic properties. When wave meets different acoustic media, part of it transmits further while other reflects. This happens due to different acoustic impedance. The ration between reflected and transmitted energy purely depends on difference of acoustic impedance in both regions.

As you can see each material can be characterized by its acoustic impedance Z which is equal to ultrasound speed v and material density r.

Z = r∙ v;

Acoustic impedance units are MRayl [10-6 kg∙s-1∙m-2]. If we know acoustic impedances Z1 and Z2 of two regions crossed by ultrasound waves, we can calculate reflectance coefficient R:

$R = \frac{Z_{2}-Z_{1}}{Z_{2}+Z_{1}}$

And Transmit coefficient T:

$T = \frac{2Z_{2}}{Z_{2}+Z_{1}}$

As you can see if there is bigger difference between two acoustic impedances the more wave energy gets reflected. For instance air acoustic impedance is much lower than liquid or human tissue, this is why a water based gel is used to avoid air gap between measured media. Otherwise almost all ultrasound energy would be reflected due to large difference of acoustic impedances. Gel ensures that practically all ultrasound energy is transmitted to tissue.

When ultrasound wave is transmitted to different impedance area it refracts according to Snell law. According to it incident wave angle and refracted wave angle ration is proportional to speed ratio in both medias:

$\frac{sin\varphi}{sin\theta} = \frac{v_{1}}{v_{2}}$

These are very basics of wave propagation and applies to other wave phenomena like optics.

## Ultrasound attenuation

Attenuation is another important feature of wave propagation. In all cases wave traveling through media is attenuated. This is because media absorbs of energy which is converted to heat, so in its way ultrasound wave loses its energy and so amplitude drops. Mathematically attenuation can be modeled using exponential equation:

$P(x) = P_{0}e^{-ax}$

P0 – is the wave energy at entry point, x – is the point of interest; a – attenuation coefficient. Speaking of attenuation coefficient it isn’t constant value. It depends on frequency a = H(f) – the higher wave frequency, the higher attenuation coefficient value. If we take biological tissue, attenuation can be modeled with two mechanical components – elasticity and viscosity. Elasticity is modeled with spring, while viscosity with damper.

Both mechanical components together create viscoelasticity effect which works similar to car shock absorber. When shock frequency is low absorber dampens and practically copies the surfaces, but when car speed is high, absorber is stressed at high frequency and elasticity takes over the springy action. If you would touch shock absorber after intense driving on non-even surface you would see that it becomes hot. So similar situation is with ultrasound propagation. If high frequency ultrasound is transmitted, it gets attenuated more intensively due to viscosity of tissue. So practically in all ultrasound visualization devices there are time gain compensation feature installed which compensate the signal level over all depth of scan.

Next time we will continue with other features of ultrasound propagation.