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Diffraction effects from cylindrical transducers in a piezo-electric medium of hexagonal symmetry (class C6v(6mm))

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Abstract

The subject of investigation of the present paper is the integral equation formulation of diffraction problems associated with the two-dimensional wave motion generated by cylindrical transducers in a piezo-electric medium of hexagonal symmetry (class C6v (6mm)). It is noted that as to their elastic and piezo-electric properties poled piezo-electric ceramics fall into the same class of symmetry as the hexagonal crystals. In a two-dimensional wave motion the geometrical configuration as well as the field quantities involved are independent of one of the Cartesian coordinates. Now, using this property the relevant integral equations are derived from their three-dimensional counterparts obtained in an earlier paper. The axis of six-fold symmetry (or poling axis for ceramics) is chosen along the direction of cylindricity of the geometrical configuration. Then, two mutually uncoupled kinds of wave motion exist: viz. (a) waves with particle displacement in the basal plane (perpendicular to the axis of six-fold symmetry) and (b) waves with particle displacement along the axis of six-fold symmetry. For the two kinds of wave motion the auxiliary field quantities (Green's functions) that occur in the integral representations, are determined explicitly for an unbounded medium. When considering the piezo-electric wave motion generated by a number of perfectly conducting electrodes, also the boundary conditions associated with the two types of wave motion uncouple and hence a complete separation as to the diffraction problems to be considered takes place.

The configurations to be investigated in more detail are: the transmitting transducer, the receiving transducer and the piezo-electric delay line. In these configurations the wave motions that occur have their particle displacements along the axis of hexagonal symmetry, while the electrodes are taken to be of vanishing thickness. The directional properties of the transmitting transducer are described by its directivity which follows from the distribution of the electric surface charge on the electrodes of the transducer. This surface charge density is obtained by solving a system of integral equations. The properties of a receiving transducer are described by its absorption cross-section in relation to an incident plane wave. Through an appropriate reciprocity relation the absorption cross-section is directly expressed in terms of the directivity of the same transducer in the transmitting situation. The performance of the piezo-electric delay line is characterized by its impedance matrix. The elements of the impedance matrix are determined with the aid of an appropriate system of integral equations.

Numerical results are presented for transducers embedded in the piezo-electric ceramic PZT-4 and consisting of respectively two plane parallel electrodes, one plane electrode together with one parabolic electrode, and one plane electrode together with one flat-roof electrode. Furthermore, numerical results are presented for planar transducers and for delay lines consisting of two transducers with plane parallel electrodes embedded in the piezo-electric ceramic PZT-4.

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Abbreviations

c i, j; p, y :

stiffness tensor at constant electric field

c i, j; p, y :

piezo-electrically stiffened elastic constant

D i :

electric flux density

\(D(\hat x_\wp )\) :

directivity of a transmitting transducer

E i :

electric field vector

e i, j; y :

piezo-electric tensor

f i :

volume density of mechanical body forces

i:

imaginary unit

I n :

the net electric current per unit length in the x 3-direction fed into the n-th electrode

k :

wave number for piezo-electric waves

K EM :

electro-mechanical coupling coefficient

K S, PE :

shear piezo-electric coupling coefficient

n β :

unit vector in the direction of the outward normal to a closed contour in the x 1, x 2-plane

P a :

time averaged power per unit length in the x 3-direction absorbed by the receiving transducer

R :

real part of Z

s i :

displacement vector

S β :

complex electro-mechanical power flow density

t :

time

U :

electric scalar potential

V n :

electric potential of the n-th electrode

v i :

particle velocity

X :

—imaginary part of Z

x :

x 1 i 1+x 2 i 2=position vector in the x 1, x 2-plane

x β :

Cartesian coordinate in x 1, x 2-plane

Z :

impedance per unit length in the x 3-direction

δ :

delta function

δ i, γ :

permittivity tensor at constant strain

ρ e :

electric volume charge density

ρ m :

mass density of the piezo-electric medium

σ a :

absorption cross-section per unit length in the x 3-direction of receiving transducer

σ e :

electric surface charge density

τ i, j :

stress tensor

ω :

circular frequency

∂β :

partial derivative with respect to x β

∮ C :

integral along a closed contour ℒ

∫ C :

Cauchy principal value of ∫ C

i, j, k, p :

Latin subscripts running through the values 1, 2 and 3

β, γ, κ :

Greek subscripts running through the values 1 and 2

GE :

denotes the auxiliary fields generated by an electric line charge

GM :

denotes the auxiliary fields generated by a line source of mechanical body force

References

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Authors and Affiliations

  1. Dept. of Electrical Engineering, Division of Electromagnetic Research, Delft University of Technology, Delft, The Netherlands

    G. De Jong

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  1. G. De Jong
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De Jong, G. Diffraction effects from cylindrical transducers in a piezo-electric medium of hexagonal symmetry (class C6v(6mm)). Appl. Sci. Res. 27, 169–218 (1973). https://doi.org/10.1007/BF00382486

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  • DOI: https://doi.org/10.1007/BF00382486

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