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Chapter 2 Characterization of the Rec Technique 2.1 the Effects of Nonlinear Distortion on the Compression of Rec Echoes 2.1.1 Introduction
| Content Provider | Semantic Scholar |
|---|---|
| Abstract | In ultrasound, when the excitation waveform is low in pressure amplitude, wave propagation is described by the linear wave equation. In the development of the linear wave equation, the higher order terms in the general acoustic equations (equation of state, equation of continuity, Euler's equation of force) are negligible. The one-dimensional linear wave equation [20] is ∂ 2 ξ ∂t 2 = c 0 ∂ 2 ξ ∂x 2 (2.1) where c 0 is the small signal speed of sound, ξ is the particle displacement, and t is time. In cases where low pressure amplitudes are used, the eSNR may be relatively low, which in turn provides poor sensitivity. Therefore, larger pressure amplitudes may be necessary to improve the eSNR but at the expense of possible nonlinear effects of the medium that distort the pressure wave. To further understand the effects of nonlinearity, the development of the linear wave equation must be modified to include higher order terms. In the derivation of the linear wave equation the higher order terms from the Taylor series expansion of the equation of state were assumed to be negligible. For the non-linear equation of state the quadratic and linear terms of the Taylor series are used. Specifically, the nonlinear equation of state [21] is expressed as p = As + B 2 s 2 , (2.2) 9 where s is the condensation. Furthermore, A and B are defined as: A = ρ 0 (∂P ∂ρ) = ρ 0 c 2 0 (2.3) B = ρ 2 0 (∂ 2 P ∂ρ 2), (2.4) where ρ 0 is the ambient density, ρ is density, and P is the total pressure. Moreover, the one-dimensional nonlinear continuity equation [21] can be described by s = − ∂ξ ∂x (1 − ∂ξ ∂x), (2.5) where x is the spatial coordinate. If expression (2.5) was substituted into (2.2) and then only the linear and quadratic terms were kept, the resulting development can be obtained: p = −A ∂ξ ∂x + Aβ n (∂ξ ∂x) 2 , (2.6) where β n [22] is a parameter that quantifies the nonlinear properties of the medium and is referred to as the Beyer parameter or the coefficient of nonlinearity. The Beyer [22] parameter is given by β n = 1 + B 2A , (2.7) where the ratio B/A [20] is B/A = ρ 0 c 2 0 (∂ 2 P ∂ρ 2). … |
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| Alternate Webpage(s) | http://www.brl.illinois.edu/Downloads/sanchez/Chapter_2.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Chapter |
