Without loss of generality, consider waves propagating along the x-axis. The waves are linear, so that small strains are assumed.

where K and G are the bulk and shear moduli, respectively; and ηb and ηv are the bulk and shear viscosity coefficients, respectively.

The shear waves are solutions of the form: , where the y-polarization is assumed, again without loss of generality. The corresponding momentum balance equation is

In an infinite domain, the solution can be expressed for any specified angular frequency ω in the following complex-values form:

where U+ and U- are constant amplitudes, and

Note that for most materials, even for very large values of ω.

For each wave, the spatial attenuation is characterized by α(ω) which is often called the attenuation coefficient. It has a SI unit (1/m), but is often given using neper per meter (Np/m). Alternatively, the value can be given in decibel per meter (dB/m). Both Np and dB are relative units (dimensionless). Values given in Np or Np/m can be used directly in any formula using SI units. The conversion formula needs to be applied for values using dB:

The wave attenuation can be also specified as attenuation per wavelength λ, which is defined as

The above formula can be directly applied to the cases of pressure and shear waves. Using given values of the attenuation per wavelength γp and γs measured in Np at the corresponding references frequencies fp,ref and fs,ref, the shear and bulk viscosity are found as

For given values of attenuation coefficients αp and αs measured in Np/m at the corresponding references frequencies fp,ref and fs,ref the approximate formulas for the viscosity coefficients are

and . If the values of αp and αs are given in dB/m, the conversion factor ln(10)/20 needs to be inserted.