As mentioned in 1.1.5, the following derivation is based on [1,21,22].
In the current thesis only the fundamental frequency of ultrasound imagers are used. Avoiding the use of harmonic imaging allows us to consider the homogenous linear wave equation (LWE) as a good approximation of the underlying physical processes of the examined systems.
However, it is important to note which assumptions and simplifications are made
to be aware of the limitations of this model. The theory of LWE are best fit for ultrasound pulses travelling in idealized – lossless and homogenous – media. Gen-erally, in soft tissues (but not in hard tissues, such as bone) high frequency shear waves can be neglected due to the high rate of absorption (a factor of 535 for each wavelength travelled [23] p. 87). Due to this simplification, ultrasound wave prop-agation can be described by a scalar pressure field. However, it should be noted that scalar waves are also affected by attenuation (on the order of 0.5 dB/cm/MHz in tissue, as described in the previous chapter, Section 1.1), which effect in the fol-lowing derivation is ignored for simplicity. For a discussion of various attenuation phenomena and its effect on wave propagation, please see [6] Chapter 4.
To derive the LWE, the three governing equations of acoustics first need to be derived, namely: I. State equation, II. Continuity equation, III. Force equation.
I. State equation
When a wave propagates through a medium, the local density will change with pressure. This relation is generally superlinear (growing faster than a linear rela-tionship); however, if the changes in density (ρ) and pressure (p) are small compared to the environmental set pointsρ0, p0, the relationship can be approximated as lin-ear:
ptotal=p0+p , p << p0 (2.3)
ρtotal =ρ0+ρ , ρ << ρ0 (2.4)
Expecting a linear relationship, the pressure (p) could be expressed as:
p=ερ (2.5)
whereε is a constant and could be expressed from ρ0 and the compressibility of the mediumκ or its inverse; the elastic modulus K:
ε= 1
κρ0 = K
ρ0 (2.6)
II. Continuity equation
The continuity equation is based on the principle of conservation of mass. Let us suppose we have a cubic control volume V with dimensions dx·dy·dz (see Figure 2.1).
Figure 2.1: Pressure wave acts on a cubic control volume. The pressure change will induce density change of the faces of the cube combined with particle motion. The figure is adopted from [22] .
The mass entering the control volume from face x eqauls:
˙
mx =ρtotal~uxdy·dz (2.7)
where ~u denotes ‘particle velocity’; which describes the local motion of small elements of the medium displaced by the pressure. We can use the Taylor series expansion for the mass entering the face atx+dx:
˙
mx+dx = ˙mx+ ∂m˙x
∂x dx+...∼=ρtotal~uxdydz+∂ρtotal~ux
∂x dx·dy·dz (2.8)
Applying this to all directions, the total mass gain per V could be written as:
We know also that the mass gain in V could be written as:
˙
mtotal = ˙ρtotaldxdydz = ˙ρtotaldV (2.10) Thus, comparing equations 2.9 and 2.10 the following could be written:
˙
ρtotaldV =−O·ρtotal~u dV (2.11)
From this (original) form of the continuity equation, using Equation 2.4, after simplification and linearisation the final form will be:
−ρ0O·~u= ∂ρ
∂t (2.12)
III. Force equation
For the investigation of the conservation of momentum we use Newton’s law. In the calculations the applied force can be derived from the acceleration~a of particles as follows:
d ~f =dm·~a=dm∂~u
∂t (2.13)
where dm is the mass of the control volume and equals ρtotaldV.
As can be seen in Figure 2.2, using the pressure, the net force acting in the x direction could be written as:
d ~fx =ptotal(x)dydz−ptotal(x+dx)dy·dz (2.14) Using Taylor series expansion, the force acting onx+dx face could be expanded as:
ptotal(x+dx) =ptotal(x) + ∂ptotal
∂x +... (2.15)
Thus, the net force acting in the xdirection could be rewritten as:
d ~fx =−∂ptotal
∂x dx·dy·dz (2.16)
Figure 2.2: Force acting in the x direction on the faces of the control volume. The figure is adopted from [22].
Applying Equation 2.16 for all directions, the net force acting on the whole control volume will be:
d ~f =−OptotaldV (2.17)
Combining Equation 2.13 knowing that dm = ρtotaldV and Equation 2.17 we get:
ρtotaldV ∂~u
∂t =−OptotaldV (2.18)
which can be further simplified, using equations 2.3 and 2.4, to the final form of the force equation:
ρ0∂~u
∂t =−Op (2.19)
The linear wave equation
To derive the linear wave equation, divergence of the right side of the force equation multiplied by−1 should be taken first. This will give us:
O2p=O·Op=−ρ0O· ∂~u
∂t = ∂
∂t(−ρ0O·~u) (2.20) Using the continuity equation (2.12), this could be rewritten as:
∂
∂t(−ρ0O·~u) = ∂
∂t
∂ρ
∂t = ∂2ρ
∂t2 (2.21)
Combining this with the state equation (Equation 2.5) arranged to ρ=p/εand comparing it with the standard form of the wave equation, we see that ε should equal the square of the speed of sound c2. Thus, we will get the final form of the homogenous LWE:
The acoustic impedance is defined as the ratio of pressure to particle velocity, in other words, it represents the resistance of the medium to acoustic wave propagation.
It can be derived from the force equation (Equation 2.19). After rearrangement and integration in time we get:
~
u=Z −Op
ρ0 dt (2.23)
In the special case of planar wave propagation, a simple constant specific to the material is obtained, termed the characteristic acoustic impedance, as it is charac-teristic of the material itself. Without loss of generality, let us suppose the wave propagates in the +x direction, so it can be written as p=A f(x−ct), where f(.) is an arbitrary function describing the shape of the wave. Knowing that the resting state density of the medium ρ0 is independent of space and time we can get:
~
So, acoustic impedance Z could be defined as:
Z = p
~
u =ρ0c (2.25)
where cis the longitudinal propagation speed of sound in the medium.