In general, a tangential harmonic oscillation of amplitude u0 and frequency f applied to a wall at z = 0 creates a viscous wave of the form

where, f is the frequency, ρ0 is the static density, and μ is the dynamic viscosity. The viscous shear waves are therefore dispersive with wavelength Lv and the associated wavenumber kv

The viscous waves are highly damped since their amplitude decays exponentially with distance from the boundary (see Ref. 3). In fact, in just one wavelength, the amplitude decreases to about 1/500 of its value at the boundary. Therefore, the viscous boundary layer thickness can for most purposes be considered to be less than Lv. The length scale δv is the so-called viscous penetration depth or viscous boundary layer thickness.

Similarly, a harmonically oscillating temperature with amplitude T0 and frequency f at z = 0 creates a thermal wave of the form

where Cp is the (specific) heat capacity at constant pressure and k is the thermal conductivity. The wavelength and wavenumber are here

and a decay behavior similar to the viscous waves. The length scale δth is the thermal penetration depth.

The ratio of viscous wavelength to thermal wavelength is a nondimensional number related to the Prandtl number Pr, as

In air, this ratio is roughly 0.8, while in water, it is closer to 2.7. Thus, at least in these important cases, the viscous and thermal boundary layers are of the same order of magnitude. Therefore, if one effect is important for a particular geometry, so is probably the other.