the crystal is given by a transverse plane wave with the wave-vector perpendicular to the surface normal (thickness-shear mode). There are neither compressional waves nor flexural contributions to the displacement pattern. There are no nodal lines in the plane of the resonator. All stresses are proportional to strain. Linear viscoelasticity holds. Piezoelectric stiffening may be ignored. Semi-infinite viscoelastic medium For a semi-infinite medium, one has η’ and η’’ are the real and the imaginary part of the viscosity, respectively. Zac = ρc =(G ρ)1/2 is the acoustic impedance of the medium. ρ is the density, c, the speed of sound, and G = i ωη is the shear modulus. For Newtonian liquids (η’ = const, η’’ = 0), Δf and Δ(w/2) are equal and opposite. They scale as the square root of the overtone order, n1/2. For viscoelastic liquids (η’ = η(ω), η’’≠ 0), the complex viscosity can be obtained as Importantly, the QCM only probes the region close to the crystal surface. The shear wave evanescently decays into the liquid. In water the penetration depth is about 250 nm at 5 MHz. Surface roughness, nano-bubbles at the surface, slip, and compressional waves can interfere with the measurement of viscosity. Also, the viscosity determined at MHz frequencies sometimes differs from the low-frequency viscosity. In this respect, torsional resonators (with a frequency around 100 kHz) are closer to application than thickness-shear resonators. Inertial loading (Sauerbrey equation) The frequency shift induced by a thin sample which is rigidly coupled to the crystal (such as a thin film), is described by the Sauerbrey equation. The stress is governed by inertia, which implies σ = -ω2u0mF, where u0 is the amplitude of oscillation and mF is the (average) mass per unit area. Inserting this result into the small-load-approximation one finds If the density of the film is known, one can convert from mass per unit area, mF, to thickness, dF. The thickness thus derived is also called the Sauerbrey thickness to show that it was derived by applying the Sauerbrey equation to the frequency shift. The shift in bandwidth is zero if the Sauerbrey equation holds. Checking for the bandwidth therefore amounts to checking the applicability of the Sauerbrey equation. The Sauerbrey equation was first derived by Günter Sauerbrey in 1959 and correlates changes in the oscillation frequency of a piezoelectric crystal with mass deposited on it. He simultaneously developed a method for measuring