Speed of sound in thin rods vs infinite solids

Question

This is a followup to this answer, which states that there are two different sound velocities in a solid depending on the geometry:

1. For 3-dimensional, infinite solids, in which the boundaries are much bigger than the wavelength, we have $$v = \sqrt{\frac{E(1-\nu)}{\rho(1+\nu)(1-2\nu)}}$$

2. In contrast, for thin rods, in which the wavelength is larger than the diameter, we have $$v = \sqrt{\frac {E}{\rho}}$$

But these are only for two extreme cases (wavelength significantly smaller than diameter and wavelength significantly bigger than diameter).

What about for non-thin rods with dimensions that are similar to the wavelength? For instance, if we start with a thin rod and keep increasing its diameter, then surely the speed of sound must continuously transition between the second equation and the first equation, but I can find no mention of this nor any theory that would describe these in-between speeds of sound. What am I missing?

Answer 1

Limiting cases such as $d\ll \lambda$ and $d \gg \lambda$, where $d$ is the transverse extent/boundary of the object and $\lambda$ is the wavelength of the acoustic wave, allow us to deduce analytical, closed-form results such as the ones you mentioned. Certainly, as you said, as we vary $d$ from much less than $\lambda$ to much greater than $\lambda$, the speed will generally interpolate between those two analytical formulae. But the intermediate result may not expressible in closed-form.