1. In the following, we shall concern ourselves with first-order changes in frequency arising from small material inhomogeneities. Equation (1) is the exact displacement for a homogeneous structure, but to this order it can be substituted for the true (distorted) dimensional resonance. This is because the difference between Eq. (1) and the true dimensional resonance will be small when the material perturbation is small, and any change in frequency will be second order in this difference, and hence can be neglected.
2. While losses can be included in the analysis, their contribution is often negligible. An estimate of their magnitude can be obtained through the general relation for loss effects on (amplitude) resonant frequencies Δf∕f∼Q−2, where Q−1 is the usual acoustic loss parameter.
3. Exact numerical solutions for the resonances of an inhomogeneous bar I have been carried out for a few low-order modes with single Fourier components and these show that for MnL∕MoL and ρnL∕ρoL∼10−2 (which describes many experimental situations including those in Sec. Ill) the errors in the Fourier coefficients obtained using Eq. (4) are less than 1 %.
4. Note that since Eq. (4) relates to the linear modulus and density, this technique can also respond to spatial variations in the cross-sectional area.
5. In using this procedure, however, it is important to correct the observed fn for dispersive effects in the rod. This may be done (to first order) by increasing the observed frequencies by the fractional amount 1+(πσnR∕2L)2, where σ is the bar radius. This correction is not necessary if fiducial state measurements are made, as noted later in this section.