Source: https://jmrplens.github.io/phonometry/solids/wave-speeds/ # Wave speeds of a solid ## The problem this solves You have a table. It gives you a wave speed of 5 270 m/s for steel, a density and a Poisson ratio, and no Young's modulus. Your function wants the modulus. Another table gives you 200 GPa and no speed, and the function you are feeding next wants the speed. A third gives you 12,3 m Hz and neither. All three describe the same steel. Getting from one to another takes one line of algebra, which is one line of algebra too many to be doing from memory, because the answer depends on something the table often does not say: whether its speed is the speed in a beam, in a plate, or in an unbounded solid. ## The three speeds ```python from phonometry import solids modulus, density, poisson = 2.0e11, 7800.0, 0.28 beam = solids.beam_longitudinal_speed(modulus, density_kg_m3=density) plate = solids.plate_longitudinal_speed( modulus, density_kg_m3=density, poisson_ratio=poisson ) bulk = solids.bulk_longitudinal_speed( modulus, density_kg_m3=density, poisson_ratio=poisson ) print(f"{beam:.0f} {plate:.0f} {bulk:.0f}") # 5064 5275 5725 ``` The order is always the same, and it is the order of how much the material is held: a beam can contract sideways freely, a plate is held across its width, and an unbounded solid is held on every side. $$ c_{\mathrm{L,b}} = \sqrt{\frac{E}{\rho}} \qquad c_{\mathrm{L,p}} = \sqrt{\frac{E}{\rho\,(1-\nu^2)}} \qquad c'_{\mathrm{L}} = \sqrt{\frac{E\,(1-\nu)}{\rho\,(1+\nu)(1-2\nu)}} $$ The first two are Hopkins Eqs. (2.20) and (2.21), the third is Norton and Karczub Eq. (1.225). At a Poisson ratio of zero the three collapse into one, which is the quickest way to remember that the only thing between them is how the lateral strain is paid for. ## Which one is in your table Building acoustics prints the **plate** speed. EN 12354-1 Table B.3, Hopkins Table A2 both tabulate it, and Hopkins says so in a footnote: the values can be used as estimates for beams or plates, which is an admission that the two differ and a statement that for these materials the difference is inside the spread of the table. A time-domain elastic solver integrates the **bulk** speed, because the material it discretises is unbounded at the scale of a cell. Reading a plate table into a solver is a documented trap: for the steel above the gap is eight and a half per cent, and for aluminium fifteen. A **beam** speed turns up where a bar was actually measured, which is where a speed-of-sound table in a general acoustics text usually gets its solids. ## Going back to the modulus Each speed has its inverse, and they are what let the books be compared: ```python from phonometry import solids # Hopkins Table A2, steel: a plate speed, a density and a Poisson ratio. modulus = solids.youngs_modulus_from_plate_speed( 5270.0, density_kg_m3=7800.0, poisson_ratio=0.28 ) print(f"{modulus / 1e9:.0f} GPa") # 200 GPa ``` Two hundred gigapascals, which is what a structural steel has, and the number Cremer and Mechel print in the column Hopkins leaves out. ## The column that checks the table Hopkins, Long and Mechel each print the product of thickness and critical frequency. It is a property of the material alone, because the thickness cancels, and so it is the cheapest cross-check there is between books that share no other column: ```python from phonometry import solids print(f"{solids.thickness_critical_frequency_product(5270.0):.2f}") # 12.31 ``` Hopkins Table A2 prints 12,3 m Hz for that steel and states in its heading that the column assumes a speed of sound of 343 m/s. $$ h f_\mathrm{c} = \frac{c_0^2 \sqrt{12}}{2 \pi\, c_{\mathrm{L,p}}} $$ The constant matters. ISO 12354-1 writes the critical frequency with a rounded 1,8 in the denominator, which is 2π/√12 = 1,8138 to two figures. The rounded form is the standard's own arithmetic and `phonometry.building.critical_frequency` keeps it, because a measurement that claims to follow the standard has to. Here the exact constant is the right one: it reproduces all twenty-five rows of Table A2, and the rounded one misses the steel row by 0,8 per cent. ## The table, and what it does not say `PUBLISHED_SOLIDS` holds two hundred and sixty rows read from nine published tables: Hopkins **Table A2**, twenty-five building materials; Cremer 3e **Table 4.3**, thirteen metals over fifteen rows; and Mechel **Table 3**, thirty-eight construction materials, plastics and metals. The key names the table, because three of them print a steel and they are not the same steel. Bies 5e **Table C.1** adds one hundred and five metals, building materials, woods, plastics and honeycomb panels, and its loss factor column is two quantities and not a range: the low end is the internal loss factor and the high end the one a panel has once it is installed. Norton & Karczub **Appendix 4 A** adds twenty-one solids with the bar and the bulk speed in separate columns, their **Table 6.1** eleven structural loss factors, and Vigran **Table 3.1** nine building materials. Every row is also on the [published catalogues](https://jmrplens.github.io/phonometry/reference/catalogues/#solids) page, filterable, with the book and the page each one was read on. ```python from phonometry import solids board = solids.PUBLISHED_SOLIDS["hopkins-2007-table-a2/plasterboard_natural_gypsum"] print(board.density_kg_m3, board.plate_longitudinal_speed_m_s, board.flexural_loss_factor) # 860.0 1490.0 0.0141 print(round(board.youngs_modulus_pa / 1e9, 2)) # 1.74 ``` That modulus is not in the table: it is the inverse above run on the row's own cells, and `is_derived` and `derived` say so, `derived` down to the Poisson ratio Hopkins marks as an estimate, which the modulus rests on. A number this library computed is never stored as though a page had printed it. `solids_named("Steel")` gives the steel of four books, from Hopkins' 199,6 GPa at a Poisson ratio of 0,28 to Cremer's 210 GPa at 0,31 and Long's row with no modulus at all, and every row carries all three longitudinal speeds whenever its own cells allow it, so the comparison is between the books rather than between two waves. Two Hopkins rows, aircrete and brick, print a density only as a range and nothing follows from it, so they keep the plate speed the page printed and `why_missing` says why the others are absent. Cremer's own column is the bar speed: his Eq. (3.32) defines `c_LII` as `sqrt(E/rho)`, and the Roman subscript counts the directions along which the cross-sectional contraction is unconstrained, so `II` is a rod. A page gets as far as its own columns allow and no further. Mechel prints a modulus and no Poisson ratio, so a row of his gets as far as the bar speed and stops: the plate and bulk speeds would need a ratio he does not print, and assuming one would put a number in his mouth. Fourteen of his thirty-eight rows do not get that far either, because their density or their modulus is printed as a range. What crosses between Hopkins and Mechel without any assumption at all is `h f_c`, which Hopkins prints as a column and Mechel as `f_cr d`: for steel both print 12,3 m Hz, to the digit. Cremer prints no such column, so his rows carry a derived one. Two of the nine tables qualify nothing. Long prints a "Speed of Longitudinal Waves" with no modulus and no Poisson ratio beside it, so nothing on the page says which of the three it is and the value goes in `longitudinal_speed_m_s` rather than in a named one. Comparing it with the tables that do say points at the bar speed, because his steel, at 5 050 m/s, is below every steel plate speed in the catalogue, but that is a hint and not a reading. Nine of his speed cells print a row of dots and one of his loss factors prints "Varies with frequency", and the rows say so. Arau prints the critical frequency of a one centimetre plate instead of the `h f_c` product, which is a change of unit; on that column he and Hopkins agree inside four per cent for aluminium, brick and glass and are nineteen per cent apart on steel. A materials table is not a list of measurements, and these say so. Most of Hopkins' Poisson ratios and loss factors carry a footnote whose whole text is "Estimate", and only aluminium, glass, mortar and steel print a Poisson ratio the book stands behind; each footnoted cell holds `"estimated"` in `basis`, which `basis_of(field)` reads, and a cell whose page says nothing about how the number was obtained answers with an empty string. A cell printed as an interval is in `ranges` and the field is `None`, with `why_missing` saying which; a cell printed as a bound is in `bounded_above` too; a cell the author rounded on purpose, like Cremer's `≈ 3 · 10⁻⁴` for gold, is in `approximate`. Where a page prints two specimens under one name, as Cremer does for lead and for copper, they are two rows and `variant` says which is which. Rows whose columns a book takes from another author keep the credit, and the bracketed numbers Cremer prints are resolved to their authors from his own reference list rather than left as `[4.19]`. Two books printing different numbers for the same material is the catalogue working. `check_solid_agreement.py` puts them side by side and fails only when two densities are more than 8 per cent apart, which is the one column a table cannot get very wrong without describing a different material: every density the gate does not already accept agrees with the other books' to within that, while the moduli disagree by nineteen per cent on aluminium without anybody being wrong, and those it reports and never fails on. Its first full run found tin, where Cremer and Mechel print 4,4 GPa, Bies 54 and Norton & Karczub 45, while the four densities agree to 0,3 per cent. Each book is consistent with itself, so no page contradicts itself and this is not an errata; but a tin of 4,4 GPa would carry sound at 780 m/s, among the soft plastics and below every lead row here at 1 180 to 1 257 m/s, so the two books that agree are the ones the rest of the catalogue argues against, and the fourth book to print a tin sides with the rest. That is why it keeps every book's reading and nothing in it votes by majority. None of this is a specification. Block densities vary by manufacturer, which is why the pages print ranges where they do, and Cremer says in as many words that the loss factor of a metal cannot be considered a material constant. ## What the functions refuse A Poisson ratio of 0,5 describes an incompressible material, and an incompressible material has no pure longitudinal wave to have a speed, so `bulk_longitudinal_speed` raises rather than dividing by zero. The plate form is looser, since it only needs one minus the square of the ratio to stay positive, but it refuses a ratio of 1 for the same reason. Both messages name the ratio they were given and the term that would have vanished.