Frequency dependence of the speed of sound in metallic rods

Abstract

The speed of sound waves in rods depends on the relationship between wavelength and rod dimensions. 
It differs from the speeds readily available in tables, and from what is often learned during most introductory courses on solid-state physics. Metallic rods with diameters in the centimetre range excited with sound waves of tens of kHz will behave as dispersive media.
Here, the speed of sound in metallic titanium rods of different lengths is measured, by using two different methodologies: 1) from the time of flight and 2) from the wavelength and frequency of standing waves that form in the rod. The latter allows analyzing the results in light of Pochhammer-Cree dispersion. The reflection coefficient is also determined both from time and from frequency response. 
Two off-the-shelf piezoelectric transducers, a function generator, an oscilloscope, and a lock-in amplifier were used.
We have used a low-frequency square wave (of tens of Hz) in the first case and a sine wave with frequencies that range from audible to ultrasound in the second case.
Experimental results show that the speed of sound decreases as the wavelength decreases. The Pochhammer-Chree dispersion equation was numerically solved to fit the experimental data that can be used to estimate both the Young modulus and the Poisson ratio. A practical empirical formula that allows the analysis of data without explicitly solving the Pochhammer-Chree equation is suggested.