Elastic solids

A plate transmits sound because it bends, and how it bends is decided by three numbers: a density, a Young’s modulus and a Poisson ratio. Fifteen public functions in this library ask for one or two of them, and the tables that print them, in Hopkins, in Cremer, in Mechel, in Bies, rarely print the same pair. One book gives a wave speed and no modulus. The next gives a modulus and no speed. A third gives the product of thickness and critical frequency and neither.

phonometry.solids is where the conversions between them live, so that they are written once instead of in each caller’s head.

Sixteen of the twenty-one packages are domains of application: you go to building because you are measuring a building, to underwater because you are working in water. Nobody goes to solids because they are measuring a solid. They go there because whatever they are measuring happens in one, or through one.

So it sits beside fluids in the transverse toolbox with filters, signals and metrology: any package may import it without an architecture edge, because a material is not a subject some domains have and others do not.

A solid carries a longitudinal wave differently depending on the shape it travels in, and the three speeds are not interchangeable.

A beam is free to contract sideways as it is compressed, so the lateral strains cost nothing and Poisson’s ratio does not appear at all: $c_{\mathrm{L},\mathrm{b}}=\sqrt{E/\rho }$.

A plate is held across its width. The material cannot get out of the way, which stiffens it by $1/(1-{\nu }^2)$: $c_{\mathrm{L},\mathrm{p}}=\sqrt{E/(\rho (1-{\nu }^2))}$. This is the one the building-acoustics tables print, and the one a critical frequency is computed from.

An unbounded solid holds the material on every side at once: $c_{\mathrm{L}}^{\prime }=\sqrt{E(1-\nu )/(\rho (1+\nu )(1-2\nu ))}$. This is the fastest of the three and the one a time-domain elastic solver integrates.

For the steel of Hopkins Table A2, with $\rho =7800$ kg/m³, $\nu =0,28$ and a plate speed of 5 270 m/s, the three are 5 059, 5 270 and 5 720 m/s. Reading one into another is a four to thirteen per cent error in whatever it feeds, and it is an easy mistake to make, because a table that prints one of them often calls it $c_{\mathrm{L}}$ with no qualifier at all.

A catalogue of materials is built out of whatever its sources happened to print. Hopkins Table A2 gives a plate speed, a density and a Poisson ratio, and no Young’s modulus anywhere. Cremer and Mechel give the modulus and no speed. Neither table can be checked against the other until one of them is converted, and that is what the six functions here are for: three speeds, three inverses, and a seventh for the column two of those books share.

• No shear or bending waves. The longitudinal family is what a materials table prints and what the conversions need. The bending wave that actually radiates from a wall is computed where it is used, in phonometry.vibration.structural and phonometry.building.prediction, from the bending stiffness of the plate rather than from a wave speed.

• No anisotropy. Every relation here is for a homogeneous isotropic solid with one modulus and one Poisson ratio. An orthotropic panel has two of each and two critical frequencies, and that is orthotropic_critical_frequencies in phonometry.building.

• No temperature or ageing. The constants are the ones a source printed for one specimen at one moment. Concrete stiffens for years and a polymer softens with the room; neither is modelled here, and a table that holds a range says so in its own entry.

• Wave speeds: the three longitudinal speeds and their inverses, which one a printed table holds, and the thickness-critical-frequency product that lets two books be checked against each other for free.
• Published catalogues: the two hundred and nineteen rows six books print, filterable by material and by table, each naming the page it was read on.