The effect of dissipation on the resistive admittance of an elastic medium

The effect of dissipation on the real part of the admittance of an elastic half-space is typically thought to be unimportant if the loss factor ζ of the elastic medium is small. However, dissipation induces losses in the near field of the source and, provided the size of the source is small enough, this phenomenon can be more important than elastic wave radiation. Such losses give rise to a fundamental limit in the quality factor of an oscillator attached to a substrate. Near field losses associated with strains in the elastic substrate can actually be larger than intrinsic losses in the oscillator itself if the internal friction of the substrate is larger than the internal friction of the oscillator. For a uniform stress applied to a disk of radius a, a monopole source, such phenomena become significant for k(L)a<ζ, while for higher order multipole sources of order l, near field losses are important for (k(L)a)(l+1)<ζ, a far less restrictive constraint.