Impedance, acoustic equations

The only catch in the use of equation 6.4 is that the acoustic impedance must be known. There are a number of cases where a compromise has to be made with accuracy due to incomplete or restricted knowledge of the material constants of the coating material ... 

The Sauerbrey equation assumes a rigid film with density and transverse velocity of the acoustic wave identical to those of the quartz crystal (equal acoustic impedance of the quartz and the over-layer). It also assumes that the deposit is uniform while the sensitivity of the QCM is non uniform across the radial direction of the resonant quartz crystal, with maximum sensitivity at the crystal centre [8]. 

This observation illustrates a general principle the total influence of any structure behind a boundary can be combined and represented as the acoustic input impedance of the boundary. Hence the input impedance at the interface between media 1 and 2 is defined by a rearrangement of Equation 5 as ... 

The subscripts 1 and 2 refer to the material the wave travels in and the material that is reflected by or transmitted into, respectively. These equations show that the maximum transmission of ultrasound occurs when the impedances and Z2 of the two materials are identical. The materials are then said to be acoustically matched. If the materials have very different impedances, then most of the US is reflected. The reflection and transmission of ultrasound at boundaries has important implications on the design of ultrasonic experiments and the interpretation of their results. In addition, measurements of the reflection coefficient are often used to calculate the impedance of a material. 

Ultrasound is reflected at the boundary of two media possessing different acoustic impedances. 99.99% of ultrasound is reflected at the air-water boundary when an ultrasound beam is incident upon it from either side. Hence occurrence of air bubbles should be minimized in the coupling medium in order to avoid ultrasound reflection. The reflection coefficient for various interfaces may be estimated from the acoustic impedances of two media forming the interface using equations described in Refs. f... 

Equation 2 can be rewritten in a way that Z x can be presented as a parallel arrangement of Co, the only genuine electrical parameter in Eq. 2 (formed by the two electrodes with quartz as dielectric), and a so-called motional impedance, Z Z = Co Z (Pig. 4a). Zm contains two elements in series. The first summand, Z q, includes only crystal parameters and describes the motional impedance of the quartz crystal as a fimction of frequency co = litf. The second summand expresses the transformation of the acoustic load, Zl, into the (electrical) motional load impedance, Z il- We therefore call the fraction in front of Zl transformation factor. Applying some assumptions reasonable in most sensor applications Z becomes ... 

Equation 17 illustrates why the acoustic impedance is of such tremendous importance in the physics of the QCM. The acoustic impedance governs the condition of stress continuity, and thereby the reflectivity at acoustic interfaces. 

Zq is the acoustic impedance of AT-cut quartz its value is 8.8 x 10 kg m s Strictly speaking, Zq is a complex quantity Z + iZ", where Z" accounts for internal friction. Zq is often considered to be real. When this happens, the fundamental frequency/f must also be a real number (see end of Sect. 2). The Sauerbrey equation fails to account for viscoelasticity and also, when applied in liquids, cannot distinguish between the adsorbed material itself and solvent trapped inside the adsorbed film. When a mass is derived by means of the Sauerbrey equation, the interpretation of this mass parameter is sometimes difficult. The terms Sauerbrey mass and Sauerbrey thickness are used in order to indicate that the respective parameters have been calculated by the simple Sauerbrey equation. 

Figure 6a shows the transmission hne representing a viscoelastic layer [64]. Every layer is represented by a T . The apphcation of the Kirchhoff laws to the Ts reproduces the wave equation and the continuity of stress and strain. The detailed proof is provided in [4]. To the left and to the right of the circuit are open interfaces (ports). These can be exposed to external shear waves. They can also be connected to the ports of neighboring layers (Fig. 6b). Alternatively, they may just be short-circuited, in case there is no stress acting on this surface (left-hand side in Fig. 6c). Finally, if the stress-speed ratio Zl (the load impedance, see below) of the sample is known, the port can be short-circuited across an element of the form AZl, where A is the active area (right-hand side in Fig. 6c). Figure 6c shows a viscoelastic layer which is also piezoelectric. This equivalent circuit was first derived by Mason [4,55]. We term it the Mason circuit. The capacitance, Co, is the electric capacitance between the electrodes. The port to the right-hand side of the transformer is the electrical port. The series resonance frequency is given by the condition that the impedance of the acoustic part (the stress-speed ratio, aju) be zero, where the acoustic part comprises all elements connected to the left-hand side of the transformer.

The variable k is called the scattering coefficient, (from wave scattering theory) and the equations are called scattering equations. They express the behavior of the wave equation at the boimdary between acoustic tube segments of different characteristic impedances, where part of the incoming wave is... 

Note that / can be an arbitrary function of its scalar argument and the above would satisfy the equations of linear acoustics. For instance, for propagation along the jc-direction in a Cartesian coordinate system, rightgoing pressure waves of the formp =f x — c t) produce a velocity of the form vi = ipi/ipQCo), whereas left-going pressure waves of the form pi = g(x + cof) lead to the velocity vi = —ipi/(poco). Here, i is the unit vector in the increasing x direction. This becomes an important distinction when the problem of reflection and transmission of a wave across an interface between two media is considered. It is also seen that the product of fluid density and sound speed, poCo, provides the proportionality between pressure and speed. This combination occurs frequently in acoustics and is known as the acoustic impedance. 

Consider a rectangular acoustic space occupying a volume V = abd zis shown in Fig. 1. The interior surface of the enclosure is assumed to be covered with absorptive materials for which the impedance characteristics are specified. Noise is generated in the acoustic enclosure through the vibration of the flexible portions of the side-walls, the partitions or the sound sources located in the interior. The perturbation pressure p within the enclosure satisfies the linearized acoustic wave equation... 

The sound pressure levels are given in Fig. 5 for an acoustic enclosure with small amount of absorption at the walls (Za — oo), and for the case where the walls are treated with porous materials for which the point impedance is given in equations (36)-(38). For this case, inputs to the stiffened panel are four point forces acting on the two intermediate stringers. The spectral densities of those four forces are assumed to be the same and equal to 0.S4 jHz over the frequency range 0... 

For planar layers, analjftic equations predicting A/ and AF from the model parameters (layer thickness, storage modulus, loss modulus) exist. Mathematical details of modeling have been provided in a recent book chapter. The models basically amount to a prediction of the acoustic reflectivity of the resonator t sample interface. For planar layers, this reflectivity is calculated in essentially the same way as in optics. The role of the refractive index is taken by the material s acoustic impedance, Zac =... 

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