borne approximation, scattering and diffraction

what is the borne approximation?

• set of assumptions that is used when mathematically modelling the scattering of an incident wave when passing through a heterogenous medium

  • sound speed heterogeneities: monopole point sources

  • density heterogeneities: dipole point sources producing wave parallel to the incident wave → little to no scattering perpendicular to incident wave

assumes that the:

1. scattered wave is much weaker (lower amplitude) as the incident wave

2. each wave is only scattered once

3. wave speed and absorption is homogenous throughout the medium

under what conditions are the assumptions in the borne approximation true?

true when the impedance missmatch of the target and the background is very low →

homogenous wave equation with contrast source terms originating from heterogenous medium

$$\frac{1}{\bar{c_0²}}\frac{\partial²}{\partial t²}-\frac{\partial²}{\partial x²}p=S_s(p)$$

where bar c_0 is the average sound speed

what is the total acoustic field composed of

$$total = small~scattered+incident$$

how do we obtain the born approximation from the total pressure field

take the integral equation for the green function, since the field is the sum of your source with the scattered terms:

$$p(x,t)=\int_0^{t}\!\int_{volume}^{}\!G\left(x-x^{\prime},t-t^{\prime}\right)\left(S\left(x^{\prime},t^{\prime},p\,\right)+source\right)dt\,dx$$

if we take the convolution of the green’s function and the source term and bring it out of the integrals it now becomes our borne approximation$p_o$ → we can then plug it into our scatter terms to

$$p(x,t)=p_{o}\left(x,t\right)+\int_0^{t}\!\int_{volume}^{}\!G\left(x-x^{\prime},t-t^{\prime}\right)S\left(x^{\prime},t^{\prime},p_0\right)dt\,dx$$

we can also write this as a heterogenous wave equation with contrast source terms

$$S_{s}\left(p_0\right)\left(x,t\right)=\left(\frac{1}{\bar{c_0^{2}}}-\frac{1}{c_0^2\left(x\right)}\right)\frac{\partial^2p_0}{\partial t^{2}}-\frac{\nabla\rho_0\left(x\right)\cdot p_0{}}{\rho_0\left(x\right)}$$

implicit vs an explicit solution

• explicit function → y variable is separated from the x variable on the other side → if we plug in a value for x we can find y

• implicit solution → y appears on both sides or a function of x and y is equal to 0 → we need to have prior information of y to find an exact solution for y at x

what are the types of scattering

0. molecular/absorption → energy is converted into heat

1. diffusive

2. diffractive

3. specular

4. moving → doppler from moving RBC

explain diffusive scattering

occurs when the scatterer is smaller than the wavelength

• scattering is frequency dependent

• produces a spherical wavefront

  • responsible for speckling → grainy appearance in ultrasound images

diffractive scattering

occurs when when scatterers have a size between 0.1-1mm

• produces a distorted spherical wavefront

what is specular scattering

occurs when the scatterer is much larger than the wavelength ex: organ surfaces

• frequency independent

• causes the waves to be reflected (specular reflection) appears as a bright white surface on US

what is a boundary

area where medium of one density meets a medium with another density

what do mediums of different densities have

different characteristic acoustic impedance (Z) $=\rho_o c_o$

impedance is always calculated normal to the boundary

what conditions must hold at the boundary

1. continuity of pressure: pressure on both sides should be the same so the boundary does not move: $p_i+p_r=p_t$

2. continuity of normal particle velocity: particle velocity must be the same on both sides (fluid must stay in contact): $u_i+u_r=u_t$

relationship between reflection coefficient and transmission coefficient

$$1+R=T$$

reflection coefficient

tells you the amplitude of a reflected wave compared to the incident wave

$R=\frac{p_{r}}{p_{i}}=\frac{z_2\cos\left(\theta_{i}\right)-z_1\cos\left(\theta_{t}\right)}{z_2\cos\left(\theta_{i}\right)+z_1\cos\left(\theta_{t}\right)}$ remember that you can replace z by $\rho_0c_o$

if:

• z2> z1 = wave is reflected back positively

• z2 < z1: wave is reflected flipped (-ve)

transmission coefficient

amplitude of the transmitted wave compared to the incident wave

$R=\frac{p_{t}}{p_{i}}=\frac{2z_2\cos\left(\theta_{i}\right)}{z_2\cos\left(\theta_{i}\right)+z_1\cos\left(\theta_{t}\right)}$ remember that you can replace z by $\rho_0c_o$

energy reflection coefficient

must follow the conservation of energy: Re + Te = 1

since $intensity=\frac{pressure^{2}}{z}$ $Re=R² ~~~and~~~Te=1-R^2$

why is coupling gel required

• large impedance missmatch between the PZT, air and the body(Water) → most sound is reflected

  • gel reduces this impedance missmatch

refraction and snell’s law

Refraction: change in wave direction when entering a medium with different sound speed

• High to low density: wave speeds up and bends away from the normal

• Low to high density: wave slows down and bends towards the normal

snell’s law: $\frac{sin(\theta_i)}{C_1}=\frac{sin(\theta_t)}{C_2}$

critical angle =$sin^{-1}(\frac{C_1}{C_2})$ it is when the refraction angle = 90 degrees

diffraction

based on hyugen’s principles:

• every point on the wavefront acts as a source for a secondary wavelet

• when these wavefronts pass through a small opening the wavelets spread out beyond the opening's edge and interfere with each other

• This causes the wave to bend a spread out

  • smaller the slit the greater the diffraction