Element Boundary Conditions and Other Modeling Issues

, and a normal velocity, ${{v}_{z}}(x,y,t)$ , are produced over the face of the element and that over the remainder of the plane z = 0 outside the element either the velocity or pressure (or combinations of velocity and pressure) are specified. If the surface outside the element is a pressure-free surface, then we would have on that surface. If instead the element is imbedded in a rigid “baffle”, then we would have ${{v}_{z}}(x,y,t)=0$ on the baffle. Although both of these types of conditions are commonly used in modeling large, single element transducers, neither of these extreme conditions may hold for an element in an array since the array elements are often embedded in a surrounding matrix of material that has a different acoustic impedance from the piezoelectric element or has facing layers that have a different acoustic impedance from either the element or the surrounding fluid. Techniques such as finite elements can be used to model in detail these features of an array element but here we want to develop a much simpler model that can examine the effects of the boundary conditions surrounding an element (see Pesque and Fink [1] for a similar approach). Specifically, we will model the array element as embedded in a baffle having a finite specific acoustic impedance, ${{z}_{\text{b}}}$ , where on the baffle the condition

$p(x,y,t)/{{z}_{\text{b}}}+{{v}_{z}}(x,y,t)=0$

(14.1)

is satisfied (see Fig. 14.1). The case ${{z}_{\text{b}}}\to \infty$ then corresponds to the rigid baffle and the case ${{z}_{\text{b}}}\to 0$ models the pressure-free surface. Other finite values of the baffle impedance can then be used to model conditions that are in between these two extreme limits.

Fig. 14.1

A 1-D array element in a finite impedance baffle radiating into a fluid

In the 2-D model discussed in this section, we will assume that the length of the element is of length 2b over the interval [− b, b] in the x-direction. On the plane z = 0, we will specify the pressure and velocity fields as

$\frac{p\,(x,z=0,t)}{{{z}_{\text{b}}}}+{{v}_{z}}(x,z=0,t)=\left\{ \begin{array}{*{35}{l}} {{v}_{0}}(x,t) & -b\le x\le b\\ 0 & \text{otherwise}\\\end{array} \right.,$

(14.2)

which satisfies Eq. (14.1) on the surface outside the element and assumes that the pressure and velocity fields on the face of the element combine to generate a net non-zero driving term, ${{v}_{0}}(x,t)$ , having the dimensions of a velocity, but which we see from Eq. (14.2) is not the actual velocity on the face of the element. Taking the Fourier transform of Eq. (14.2) gives

$\frac{p\,(x,z=0,\omega )}{{{z}_{\text{b}}}}+{{v}_{z}}(x,z=0,\omega )=\left\{ \begin{array}{*{35}{l}} {{v}_{0}}(x,\omega ) & -b\le x\le b\\ 0 & \text{otherwise}\\\end{array} \right..$

(14.3)

To obtain solutions for the sound beam generated by this element, as done in Chap. 2, we will express the pressure field $p\,(x,z,\omega )$ in the form of an angular spectrum of plane waves . Specifically, we will write:

$p\,(x,z,\omega )=\frac{1}{2\text{ }\!\!\pi\!\!\text{ }}\int_{-\infty }^{+\infty }{\left[ P({{k}_{x}})/{{z}_{b}}+V({{k}_{x}}) \right]G({{k}_{x}})\exp \,(\text{i}{{k}_{x}}x+\text{i}{{k}_{z}}z)\,\text{d}{{k}_{x}}},$

(14.4)

where

${{k}_{z}}=\left\{ \begin{aligned} {\text{}}&\sqrt{{{k}^{2}}-k_{x}^{2}},\quad k\ge {{k}_{x}} \\& \text{i}\sqrt{k_{x}^{2}-{{k}^{2}}},\quad k<{{k}_{x}} \\ \end{aligned} \right.$

(14.5)

and $k=\omega /c$ is the wave number for pressure waves in the fluid. Since the right side of Eq. (14.4) is a superposition of plane waves and inhomogeneous waves , both of which are exact solutions of the Helmholtz equation , the pressure $p\,(x,z,\omega )$ in Eq. (14.4) will also be an exact solution to that equation. The “amplitude” terms $P({{k}_{x}}),\,V({{k}_{x}}),\,G({{k}_{x}})$ in Eq. (14.4) are at present undefined. The particular combination of these terms given in Eq. (14.4) was chosen simply to help satisfy the boundary conditions of Eq. (14.3), as we will now show. First, we note that from the equation of motion of the fluid in the z-direction, the pressure and the z-velocity must satisfy the differential relationship [Schmerr]

$-\frac{\partial p\,(x,z,t)}{\partial z}=\rho \frac{\partial {{v}_{z}}(x,z,t)}{\partial t}.$

(14.6)

Taking the Fourier transform of both sides of this equation and solving for ${{v}_{z}}(x,z,\omega )$ we find

${{v}_{z}}(x,z,\omega )=\frac{1}{\text{i}k{{z}_{\text{f}}}}\frac{\partial p(x,z,\omega )}{\partial z},$

(14.7)

where ${{z}_{\text{f}}}=\rho c$ is the specific impedance of the fluid.

Thus, using Eqs. (14.4) and (14.7), we can write the left-hand side of Eq. (14.3) as

$\begin{matrix} p\,(x,z,\omega )/{{z}_{\text{b}}}+{{v}_{z}}(x,z,\omega )=\frac{1}{2\text{ }\!\!\pi\!\!\text{ }}\int_{-\infty }^{+\infty }{\left[ P({{k}_{x}})/{{z}_{\text{b}}}+V({{k}_{x}}) \right]} \\ G({{k}_{x}})\left[ \frac{1}{{{z}_{\text{b}}}}+\frac{{{k}_{z}}}{k{{z}_{\text{f}}}} \right]\exp \left( \text{i}{{k}_{x}}x+\text{i}{{k}_{z}}z \right)\text{d}{{k}_{x}} \\ \end{matrix}.$

(14.8)

However, if we let

$G\,({{k}_{x}})=\frac{1}{\tfrac{1}{{{z}_{\text{b}}}}+\tfrac{{{k}_{z}}}{k{{z}_{\text{f}}}}}$

(14.9)

we see that on z = 0, Eq. (14.8) is in the form of an inverse spatial Fourier transform , i.e.

$p\,(x,z=0,\omega )/{{z}_{\text{b}}}+{{v}_{z}}(x,z=0,\omega )=\frac{1}{2\text{ }\!\!\pi\!\!\text{ }}\int_{-\infty }^{+\infty }{[P({{k}_{x}})/{{z}_{\text{b}}}+V({{k}_{x}})]}\exp \text{(i}{{k}_{x}}x)\text{d}{{k}_{x}},$

(14.10)

so that from an inverse spatial Fourier transform we have

$P({{k}_{x}})/{{z}_{\text{b}}}+V({{k}_{x}})=\int_{-\infty }^{+\infty }{\left[ p(x,z=0,\omega )/{{z}_{\text{b}}}+{{v}_{z}}(x,z=0,\omega ) \right]\exp \,(\text{i}{{k}_{x}}x)\text{d}x}.$

(14.11)

Equation (14.11) shows that the boundary conditions of Eq. (14.3) will be satisfied if we let

$P\,({{k}_{x}})/{{z}_{\text{b}}}+V\,({{k}_{x}})={{V}_{0}}\,({{k}_{x}}),$

(14.12)

where the term ${{V}_{0}}({{k}_{x}})$ is just the spatial Fourier transform of the right side of Eq. (14.3), i.e.

${{V}_{0}}({{k}_{x}})=\int_{-b}^{+b}{{{v}_{0}}({x}',\omega )\exp \,(-\text{i}{{k}_{x}}{x}')\,\text{d}{x}'},$

(14.13)

and from Eq. (14.11) it can be seen that $P({{k}_{x}})$ and $V({{k}_{x}})$ can be identified as the spatial Fourier transforms of the fields $p\,(x,z=0,\omega )$ and ${{v}_{z}}(x,z=0,\omega )$ , respectively, given by

$\begin{aligned}& P({{k}_{x}})=\int_{-\infty }^{+\infty }{p\,(x,z=0,\omega )\exp \,(-\text{i}{{k}_{x}}x)\text{d}x}, \\& V({{k}_{x}})=\int_{-\infty }^{+\infty }{{{v}_{z}}\,(x,z=0,\omega )\exp \,(-\text{i}{{k}_{x}}x)\text{d}x}. \\ \end{aligned}$

(14.14)

Collecting all these results we then can obtain the pressure wave field of Eq. (14.4) explicitly as

$p\,(x,z,\omega )=\frac{1}{2\text{ }\!\!\pi\!\!\text{ }}\int_{-\infty }^{+\infty }{{{V}_{0}}({{k}_{x}})\left[ \frac{{{z}_{\text{b}}}{{z}_{\text{f}}}}{{{z}_{\text{f}}}+{{z}_{\text{b}}}{{k}_{z}}/k} \right]}\exp \,(\text{i}{{k}_{x}}x+\text{i}{{k}_{z}}z)\,\text{d}{{k}_{x}}.$

(14.15)

Equation (14.15) is an angular plane wave spectrum representation for the wave field of a transducer element embedded in an infinite baffle of acoustic impedance, ${{z}_{\text{b}}}$ . If we also assume that the driving velocity term, ${{v}_{0}}(x,z=0,\omega )$ , is spatially uniform over the face of the transducer, i.e.

${{v}_{0}}(x,z=0,\omega )=\left\{ \begin{array}{*{35}{l}} {{v}_{0}}(\omega ) & -b\le x\le b\\ 0 & \text{otherwise}\\\end{array} \right.,$

(14.16)

then the inverse Fourier transform of this velocity term can be performed analytically, giving

${{V}_{0}}({{k}_{x}})=2b{{v}_{0}}(\omega )\frac{\sin \,({{k}_{x}}b)}{{{k}_{x}}b}=2b{{v}_{0}}(\omega )\operatorname{sinc}\,({{k}_{x}}b)$

(14.17)

in terms of the sinc function sinc (x) = sin(x)/x. This case is similar to a model of a single element piston transducer in a rigid baffle where the normal velocity is assumed to be uniform on the face of the transducer. Thus, we will also refer to our model of a element in a finite impedance baffle that satisfies Eq. (14.16) as a piston model .

Since on the face of the element we have

$p\,(x,z=0,\omega )/{{z}_{\text{b}}}+{{v}_{z}}(x,z=0,\omega ={{v}_{0}}(\omega ),$

(14.18)

if we integrate Eq. (14.18) over the element face we find

$\frac{1}{{{z}_{\text{b}}}(2b)}\int_{-b}^{+b}{p\,(x,\omega )\text{d}x}+\frac{1}{2b}\int_{-b}^{+b}{{{v}_{z}}(x,\omega )\text{d}x}={{v}_{0}}(\omega ),$

(14.19)

which can be written in terms of the force/unit length, ${{F}_{\text{t}}}(\omega )$ , acting on the element face and the average velocity in the z-direction, ${{\bar{v}}_{z}}(\omega )$ , on the face as