Resonance Spectra and Their Parameters

This note discusses how the various aspects of a resonance spectrum are influenced by the complex material properties for the 6mm thickness extensional resonance mode. In the following, 'µ' means 'is controlled by'. It does not imply the functional dependency. For example, the functional dependence of f_p on c^D is f_p µ (Re(c^D))^½. 

a) Elastic Stiffness (c^D)

The real part of the elastic stiffness (c^D) controls the position of the peak (f_p), and the imaginary part of the elastic stiffness controls the breadth (Df_p) of the peak in resistance. 
 

b) Clamped Permitivity (e^S)

Above f_p, the baseline of B or B/w is controlled by the real part of the clamped permitivity. Similarly, the baseline of G or G/w is controlled by the imaginary part of the clamped permitivity above f_p. 

c) Thickness Coupling (k_t)

The difference between f_p and f_s is dictated by the real part of the coupling coefficient (k_t). The change in Df_s/f_s from the mechanical bandwidth Df_p/f_p is controlled by the imaginary part of the coupling coefficient. This means that if the mechanical Q of the sample is low, a large imaginary component of k_t is required to change Df_s/f_s significantly from the mechanical bandwidth Df_p/f_p. 

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