Wave Equations and Continuity Conditions The Mathematical Approach

Wave Equations and Continuity Conditions The Mathematical Approach 

The following section describes the canonical mathematical way of finding the resonance frequencies of coated quartz crystals. The method is described in detail in [54]. The wave equations for the different layers and the boundary conditions at the interfaces between the layers form a homogeneous system of linear equations, which can only be solved if the determinant of the system is zero. Since the determinant depends on frequency, the zeros of the determinant lead to the resonance frequencies. 

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. 

Equations 15 and 17 constitute a homogeneous system of 2N + 2 linear equations. A nontrivial solution for the set of amplitudes only exists if the determinant of this equation system vanishes. The search for the zeros of the determinant as a function of frequency will in general be carried out numerically. The zeros define the resonance frequencies. Since, for a real material, the shear modulus always contains a dissipative component, G , the resonance frequencies are complex (where the imaginary part is the halfband-half-width, r). 

Let us consider a simple example. For the bare crystal in air, the number of layers is equal to 1 and the adjacent bulk media have vanishing impedance (Zo = 0, Z2 = 0). Here and in the following, we neglect the impedance of air 

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