proportional to the bandwidth (although it should be according to the BvD circuit; see below). Also, in absolute terms, R1 – being an electrical quantity and not a frequency – is more severely affected by calibration problems than the bandwidth. Equivalent circuits Modeling of acoustic resonators often occurs with equivalent electrical circuits. Equivalent circuits are algebraically equivalent to the continuum mechanics description and to a description in terms of acoustic reflectivities. They provide for a graphical representation of the resonator's properties and their shifts upon loading. These representations are not just cartoons. They are tools to predict the shift of the resonance parameters in response to the addition of the load. Equivalent circuits build on the electromechanical analogy. In the same way as the current through a network of resistors can be predicted from their arrangement and the applied voltage, the displacement of a network of mechanical elements can be predicted from the topology of the network and the applied force. The electro-mechanical analogy maps forces onto voltages and speeds onto currents. The ratio of force and speed is termed "mechanical impedance". Note: Here, speed means the time derivative of a displacement, not the speed of sound. There also is an electro-acoustic analogy, within which stresses (rather than forces) are mapped onto voltages. In acoustics, forces are normalized to area. The ratio of stress and speed should not be called "acoustic impedance" (in analogy to the mechanical impedance) because this term is already in use for the material property Zac = ρc with ρ the density and c the speed of sound). The ratio of stress and speed at the crystal surface is called load impedance, ZL. Synonymous terms are "surface impedance" and "acoustic load." The load impedance is in general not equal to the material constant Zac = ρc = (Gρ)1/2. Only for propagating plane waves are the values of ZL and Zac the same. The electro-mechanical analogy provides for mechanical equivalents of a resistor, an inductance, and a capacitance, which are the dashpot (quantified by the drag coefficient, ξp), the point mass (quantified by the mass, mp), and the spring (quantified by the spring constant, κp). For a dashpot, the impedance by definition is Zm=F / (du/dt)=ξm with F the force and (du/dt) the speed). For a point mass undergoing oscillatory motion u(t) = u0 exp(iωt) we have Zm = iωmp. The spring obeys Zm =κp/(iω). Piezoelectric coupling