# Reference for the Temperature Dependent Speed of Sound in Common Solids

## Main Question or Discussion Point

Hi friends,

I need the average speed of sound in some common solids, such as Si, Cu and Al, over temperature ranges from 100K to 500K.

After 5 hours of surveying acoustic and physical properties handbooks and googling the web, what I find is almost nothing.

Does anybody know a reference which has clear tables of the average speed of sound (longitudinal or transverse) in common solids in the mentioned temperature range?

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The effect on densities of solids between those temperatures is small.

From wiki:

"A one percent expansion of volume typically requires a temperature increase on the order of thousands of degrees Celsius."

That is why you only find temp based charts for gasses. Just use a standard chart.

http://www.engineeringtoolbox.com/sound-speed-solids-d_713.html

I don't know a reference, but you might use V=$\sqrt{\frac{B}{\rho}}$ where B is the bulk modulus, and $\rho$ is the density. You might be able to find an equation for the bulk modulus as a function of temperature somewhere online, and the coefficients of expansion are certainly related to density as a function of temperature.

I googled for a few minutes and couldn't find anything better than that. Hope it helps.

I don't know a reference, but you might use V=$\sqrt{\frac{B}{\rho}}$ where B is the bulk modulus, and $\rho$ is the density. You might be able to find an equation for the bulk modulus as a function of temperature somewhere online, and the coefficients of expansion are certainly related to density as a function of temperature.

I googled for a few minutes and couldn't find anything better than that. Hope it helps.
Good Point. I will search for the temperature dependence of bulk modulus, hope to find something.

The effect on densities of solids between those temperatures is small.
Thank you, but what about bulk modulus?
Below 1% change in 100K to 500K is acceptable for me but 10% is not. are you sure that the variation of sound velocity, for all common solids, is under 1% in this temperature range?

Thank you, but what about bulk modulus?
Below 1% change in 100K to 500K is acceptable for me but 10% is not. are you sure that the variation of sound velocity, for all common solids, is under 1% in this temperature range?
I am not sure but since we are only talking 100's of degrees Celsius, not thousands I wouldn't expect so.

AlephZero
Homework Helper
Below 1% change in 100K to 500K is acceptable for me but 10% is not. are you sure that the variation of sound velocity, for all common solids, is under 1% in this temperature range?
You will certainly get changes of more than 1% for typical metals between 100 and 500K. See http://www.engineeringtoolbox.com/young-modulus-d_773.html (apologies for the stupid units on that web page!)

I'm not familiar with temperature-dependent properties of Si.

BTW, temperature dependent values of Young's modulus will be easier to find than Bulk Modulus. Since poisson's ratio is very unlikely to be temperature dependent, the two moduli are proportional to each other for isotropic materials.

The change in density from thermal expansion will be negligible compared with the change in the elastic moduli.

You will certainly get changes of more than 1% for typical metals between 100 and 500K. See http://www.engineeringtoolbox.com/young-modulus-d_773.html (apologies for the stupid units on that web page!)

BTW, temperature dependent values of Young's modulus will be easier to find than Bulk Modulus. Since poisson's ratio is very unlikely to be temperature dependent, the two moduli are proportional to each other for isotropic materials.

The change in density from thermal expansion will be negligible compared with the change in the elastic moduli.
Thanks for the excellent tip my brother,
I would certainly consider searching for the Young's Modulus temperature dependence.

To whom may be coming here via a Search Engine:

I finally used these two equations from Ref1:

V_Longitudinal=$\sqrt{\frac{3K+4G}{3\rho}}$

V_Transverse=$\sqrt{\frac{G}{\rho}}$

$\rho$: Density
K: Bulk modulus
G: Shear modulus

(In fact, some manipulations are made to obtain the above formulas, See Ref2)

I calculate my desired average velocity from V_L and V_T according to below equation, Ref3:

V_Average=$\frac{3}{\frac{2}{V_T}+\frac{1}{V_L}}$

Then I used the demo version of MPDB software, Ref4.
The demo version includes full free access to all of the temperature dependent properties, such as density, Shear modulus, Bulk modulus, Elastic Young's modulus and many other properties, for the elements of periodic table.

The variation is above 1% for many common solids.

References

Ref1: Gray, D.E., 1972. “American Institute of Physics Handbook”, 3rd ed., McGraw-Hill, New York - p 3-98

Ref2: http://en.wikipedia.org/wiki/Lam%C3%A9_parameters" [Broken]

Ref3: M. Holland, "Analysis of Lattice Thermal Conductivity", Physical Review, Vol.132,6(1963).

Ref4: MPDB (Material Property DataBase) software, http://www.jahm.com/pages/about_mpdb.html" [Broken]

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