Wave speeds of a solid

• Covered

  The three longitudinal wave speeds of a homogeneous isotropic solid, each with the inverse that returns Young’s modulus, and the product of thickness and critical frequency that materials tables print.

• Not covered

  Bending and shear waves, anisotropic plates, and any dependence on temperature or on how old the specimen is.

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.

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.

The first two are Hopkins Eqs. (2.20) and (2.21), the third is Norton and Karczub Eq. (1.225). At $\nu =0$ 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.

Building acoustics prints the plate speed. EN 12354-1 Table B.3 and 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.

Each speed has its inverse, and they are what let the books be compared:

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.

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:

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 $c_0=343$ m/s.

The constant matters. ISO 12354-1 writes the critical frequency with a rounded 1,8 in the denominator, which is $2\pi /\sqrt{12}=1,8138$ 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.

Typing the row yourself is the other half of the problem the three speeds solve. PUBLISHED_SOLIDS holds two hundred and sixty rows read from nine published tables, keyed by the table they came from, so a plasterboard partition or a concrete floor starts from a number somebody published rather than one you half remember. Every one of them is also on the published catalogues page, filterable, with the book and the page each row was read on.

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 of the section above, run on the row’s own speed, density and Poisson ratio, and the row says so: board.is_derived("youngs_modulus_pa") is True and board.derived["youngs_modulus_pa"] names the three cells it came from. Hopkins marks this board’s Poisson ratio as an estimate, so the text also says that the modulus rests on that estimate, beside a speed and a density whose basis the page does not state. A number this library computed is never stored as though a page had printed it.

Seven of the nine tables print a row named Steel, and they are not the same steel: Hopkins gives a plate speed of 5 270 m/s and a Poisson ratio of 0,28, Cremer a modulus of 210 GPa and 0,31, Bies 207 GPa for mild steel and 210 for one per cent carbon, Long a speed and a density and no modulus at all, Norton & Karczub 195 GPa in their appendix and nothing but a loss factor in their Table 6.1, and Vigran a modulus printed as a range. Mechel prints a steel too and calls it “Steel, cast steel”, so the lookup does not reach it: matching a material across books is the agreement work’s job and not the name’s. A flat "steel" would have to pick one of them without saying so, and picking silently between published values is the thing a catalogue exists not to do.

for row in solids.solids_named("Steel"):
    modulus = row.youngs_modulus_pa
    print(row.table, row.variant or "-", modulus and round(modulus / 1e9, 1))
# hopkins-2007-table-a2 - 199.6
# cremer-2005-table-4-3 - 210.0
# bies-2017-table-c1 mild 207.0
# bies-2017-table-c1 1% carbon 210.0
# long-2014-table-12-1 - None
# norton-karczub-2003-appendix-4a - 195.0
# norton-karczub-2003-table-6-1 - None
# vigran-2008-table-3-1 - None

Long and the loss-factor table print no modulus, and Vigran’s is a range, which is why the loop asks whether there is one before dividing. A book that gives fewer columns is the normal case here, not the exception.

Every row carries all three longitudinal speeds whenever its own cells allow it, so the comparison is between the books and not between two waves. Two of Hopkins’ rows, aircrete and brick, print a density only as a range, and nothing follows from a density the page declined to collapse: those two keep the plate speed the page printed and why_missing says why the other two are absent. Cremer’s own column is the bar speed, which his Eq. (3.32) defines as sqrt(E/rho): the Roman subscript in c_LII counts the directions along which the cross-sectional contraction is unconstrained, so II is a rod and not a plate. Against the pure longitudinal speed of an unbounded solid he puts the gap at 16 per cent for nu = 0,3, which is what the three functions above reproduce; the plate speed sits between them, 4,8 per cent above the bar.

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 the bulk speed would need a ratio he does not print, and a 0,3 would reproduce his table nicely and would be a number he did not give. Fourteen of his thirty-eight rows do not even get that far, because the page prints their density or their modulus as a range and nothing follows from a cell the book declined to collapse.

steel = solids.PUBLISHED_SOLIDS["mechel-2008-table-3/steel"]
print(steel.poisson_ratio, steel.plate_longitudinal_speed_m_s)
# None None
print(round(steel.bar_longitudinal_speed_m_s), steel.is_derived("bar_longitudinal_speed_m_s"))
# 5064 True

What crosses between Hopkins and Mechel with no assumption at all is h f_c, the product of a plate’s thickness and its critical frequency, which Hopkins prints as a column and Mechel as f_cr d. It needs no Poisson ratio, no speed of sound and no conversion, which makes it the cheapest cross-check there is between books that share no other column: for steel, both print 12,3 m Hz. Cremer prints no such column, so his rows carry a derived one and join that comparison only through the heading assumption Hopkins states.

Bies prints one column headed “Internal-in situ Loss factor”, and every cell in it holds two numbers rather than an interval. His footnote says which is which: use the upper limit for panels used in building construction and the lower for panels welded together in an enclosure. For steel they are 0,0001 and 0,01, so reading them as a range of the internal loss factor would be a factor of a hundred.

steel = solids.PUBLISHED_SOLIDS["bies-2017-table-c1/steel_mild"]
print(steel.loss_factor, steel.in_situ_loss_factor)
# 0.0001 0.01

His internal end is where the other books put the same number: Cremer measures steel’s flexural loss factor between 0,2 and 3 times 10⁻⁴, and Mechel prints 1 times 10⁻⁴.

Long prints a longitudinal speed without saying which of the three it is, and both he and Arau print a loss factor without saying which; Arau prints no speed at all. Long’s column is headed “Speed of Longitudinal Waves” and there is no modulus and no Poisson ratio beside it, so nothing on the page can settle whether it is the bar, the plate or the unbounded one.

steel = solids.PUBLISHED_SOLIDS["long-2014-table-12-1/steel"]
print(steel.longitudinal_speed_m_s, steel.bar_longitudinal_speed_m_s)
# 5050.0 None

Comparing it with the four tables that do say points at the bar speed: his steel, at 5 050 m/s, is below every steel plate speed in the catalogue, which run from 5 270 to 5 458, and his aluminium matches Bies’ printed bar speed to the digit. That is a hint and not a reading, and a guess put into a named field would then be compared against columns that were read, so the value stays where the page left it.

The same page prints a row of dots in nine of its speed cells and the words “Varies with frequency” in one of its loss factors, and the rows say so rather than being quietly empty.

block = solids.PUBLISHED_SOLIDS["long-2014-table-12-1/hollow_dense_concrete"]
print(block.loss_factor, block.why_missing("loss_factor"))
# None the table prints “Varies with frequency” where the number would be

A materials table is not a list of measurements, and this one says so. Most of Hopkins’ Poisson ratios and most of his loss factors carry a footnote whose whole text is “Estimate”: of the twenty-five rows, only aluminium, glass, mortar and steel print a Poisson ratio the book stands behind. The catalogue keeps that distinction, because a 0,2 read as a measurement will be trusted further than Hopkins trusts it. What a source claims for a cell is its basis, and basis_of reads it: "estimated" for a footnoted cell, and an empty string where the page says nothing about how the number was obtained.

print(board.basis_of("poisson_ratio"))
# estimated
print(board.basis_of("flexural_loss_factor") or "not stated")
# not stated

Three more things the pages say are not single numbers, and the row carries each of them rather than flattening it:

• A cell printed as an interval. Aircrete is 400 to 800 kg/m³ and brick 1 500 to 2 000, because that is what is known about a block. Those rows have density_kg_m3 of None and the interval in ranges, and why_missing says which.
• A bound rather than a value. Aluminium and steel print a loss factor as ≤, and so do Cremer’s brass, nickel and silver; all of them are in bounded_above as well as in ranges.
• A number the author rounded on purpose. Cremer prints ≈ 3 · 10⁻⁴ for gold, which is neither an estimate nor an interval. That is approximate.

aircrete = solids.PUBLISHED_SOLIDS["hopkins-2007-table-a2/aircrete"]
print(aircrete.density_kg_m3, aircrete.why_missing("density_kg_m3"))
# None the page prints 400 to 800 and no value

gold = solids.PUBLISHED_SOLIDS["cremer-2005-table-4-3/gold"]
print(gold.flexural_loss_factor, gold.is_approximate("flexural_loss_factor"))
# 0.0003 True

Where a page prints two specimens under one name, they are two rows. Cremer’s lead is chemically pure in one and antimonial in the other, and they share every column but the flexural loss factor, where they differ by a factor of fifty.

for key in ("lead_chemically_pure", "lead_antimonial"):
    row = solids.PUBLISHED_SOLIDS[f"cremer-2005-table-4-3/{key}"]
    print(row.variant, row.ranges["flexural_loss_factor"])
# chemically pure (0.05, 0.3)
# antimonial (0.001, 0.004)

Rows whose columns a book takes from someone else 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].

steel = solids.PUBLISHED_SOLIDS["hopkins-2007-table-a2/steel"]
print(steel.attributed_to["flexural_loss_factor"])
# Heckl, 1981

Two books printing different numbers for the same material is the catalogue working, not failing. check_solid_agreement.py puts them side by side and sorts the disagreements into the only three things they can be: a digit somebody typed wrong, a difference between the books themselves, or a defect in a page. It fails on the first and reports the other two, and the property it fails on is density.

Density is the one column a table cannot get very wrong without describing a different material. Across these nine tables every density the gate does not already accept agrees with the other books’ to within 8 per cent, while the moduli disagree by nineteen per cent on aluminium without anybody being wrong. So a density more than 8 per cent apart is a transcription error, anything under that passes without a word, and a modulus twice its neighbour is the literature.

The first full run found tin, where the books do not agree about a factor of twelve.

for row in solids.solids_named("Tin"):
    if row.youngs_modulus_pa is not None:
        print(row.table.split("-")[0], round(row.youngs_modulus_pa / 1e9, 1), row.density_kg_m3)
# cremer 4.4 7280.0
# mechel 4.4 7280.0
# bies 54.0 7300.0
# norton 45.0 7300.0

The densities agree to 0,3 per cent, so nobody has confused the material, and each book is consistent with itself: Cremer’s 4,4 GPa reproduces his own printed 780 m/s, his shear modulus and his transverse speed; Bies’ 54 reproduces his printed 2 720 m/s exactly; Norton & Karczub’s 45 reproduces their printed 2 500 m/s to within one per cent; and Mechel’s h f_c of 81 m Hz follows from 4,4 and not from 54. No page contradicts itself, so this is not an errata to register and the catalogue keeps all four readings.

The rest of the catalogue does say which way to look. A tin of 4,4 GPa carries sound at 780 m/s, which lands it among the soft plastics, below high-density polyethylene at 1 030 and below every lead row here, and those run from 1 180 to 1 257 m/s although lead is the softer metal and half again as dense. So the two books that agree are the two the other two hundred and fifty rows argue against, and the fourth book to print a tin sides with the rows, which is the whole reason nothing in this catalogue votes by majority: two books agreeing is not evidence, and when they share a mistake it is usually because they share an ancestor.

None of this is a specification. Block densities vary by manufacturer and boards vary by batch, which is the reason 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. A row is for reproducing a worked example, sanity-checking a measurement or getting an order of magnitude.

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 $1-{\nu }^2$ 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.