solids.elastic

Three longitudinal wave speeds of a solid, and which one a table prints.

A solid carries more than one longitudinal wave, and which one a number belongs to depends on the shape the wave travels in, not on the material. The three are

and they are not interchangeable. Take the steel of Hopkins Table A2, rho = 7800 kg/m3, nu = 0.28 and a plate speed of 5 270 m/s, which is a Young’s modulus of 200 GPa: the beam speed is 5 059 m/s and the bulk speed is 5 720 m/s. Reading one into another is a four to thirteen per cent error in whatever it feeds, and the gap grows with Poisson’s ratio. The first two are Hopkins (2007) Eqs. (2.20) and (2.21), PDF page 144 (printed p. 117), which also records that the b and p subscripts are dropped later in that book and the plate value is then called cL with no qualifier at all. The third is Norton & Karczub (2003) 2e Eq. (1.225), PDF page 94 (printed p. 74), the wave velocity of a pure longitudinal wave, after Fahy.

Which one a printed table holds has to be read off its own page, and the answer is not always the one the column heading suggests. EN 12354-1 Table B.3, Hopkins Table A2 and Mechel Table 3 tabulate the plate value; a table of bulk wave speeds, which is what an elastic solver integrates, is a different table. This module exists so the conversion between them is written once, named after the shape it belongs to, and never done in a caller’s head.

Each speed has its inverse here as well, because a catalogue is built from whatever its sources happened to print: Hopkins prints the plate speed and no Young’s modulus, Cremer and Mechel print the modulus and no speed, and the two only meet through these six functions.

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beam_longitudinal_speed(
    youngs_modulus_pa: float,
    *,
    density_kg_m3: float,
) -> float

Quasi-longitudinal wave speed along a beam (Hopkins Eq. 2.20).

$c_{\mathrm{L},\mathrm{b}}=\sqrt{E/\rho }$. A beam is unconstrained on its sides, so the lateral strains cost nothing and Poisson’s ratio does not appear.

Parameters

Returns: The wave speed, in m/s.

Raises

bulk_longitudinal_speed(
    youngs_modulus_pa: float,
    *,
    density_kg_m3: float,
    poisson_ratio: float,
) -> float

Pure longitudinal wave speed in an unbounded solid (Norton Eq. 1.225).

$c_{\mathrm{L}}^{\prime }=\sqrt{B/\rho }$ with the longitudinal modulus $B=E(1-\nu )/((1+\nu )(1-2\nu ))$. Nothing is free to move sideways, which is why this is the fastest of the three and why it is the speed an elastic solver integrates.

Parameters

Returns: The wave speed, in m/s.

Raises

Constant (float).

DEFAULT_SPEED_OF_SOUND_M_S = 343.0

plate_longitudinal_speed(
    youngs_modulus_pa: float,
    *,
    density_kg_m3: float,
    poisson_ratio: float,
) -> float

Quasi-longitudinal wave speed along a plate (Hopkins Eq. 2.21).

$c_{\mathrm{L},\mathrm{p}}=\sqrt{E/(\rho (1-{\nu }^2))}$. The plate is constrained across its width, which stiffens it by $1/(1-{\nu }^2)$ against the beam of beam_longitudinal_speed.

This is the value the building-acoustics tables print, and the one the critical frequency of a wall or floor is computed from.

Parameters

Returns: The wave speed, in m/s.

Raises

thickness_critical_frequency_product(
    plate_speed_m_s: float,
    *,
    speed_of_sound: float = 343.0,
) -> float

The h f_c a materials table prints, from the plate wave speed.

which is the critical frequency of a thin plate multiplied by the thickness it was computed for, so the product is a property of the material alone. Hopkins Table A2, Long Table 9.1 and Mechel Table 3 each print this column, which makes it the cheapest cross-check there is between three books that otherwise share no column at all.

The constant here is the exact $2\pi /\sqrt{12}=1.8138$ of the thin-plate dispersion relation, which is what phonometry.vibration.coincidence_frequency computes and what reproduces the printed column: Hopkins states c0 = 343 m/s in the heading of Table A2 and prints 12.3 m Hz for steel, which this returns and the rounded 1.8 of ISO 12354-1 misses by 0.8 per cent (12.40). phonometry.building.critical_frequency keeps the rounded constant, because there it is the standard’s own arithmetic and not the material’s.

Parameters

Returns: The product h f_c, in Hz m.

Raises

youngs_modulus_from_beam_speed(
    longitudinal_speed_m_s: float,
    *,
    density_kg_m3: float,
) -> float

Young’s modulus from a beam wave speed, the inverse of Eq. 2.20.

$E=\rho c_{\mathrm{L},\mathrm{b}}^2$.

Parameters

Returns: Young’s modulus E, in pascals.

Raises

youngs_modulus_from_bulk_speed(
    longitudinal_speed_m_s: float,
    *,
    density_kg_m3: float,
    poisson_ratio: float,
) -> float

Young’s modulus from a bulk wave speed, the inverse of Eq. 1.225.

$E=\rho c_{\mathrm{L}}^{\prime 2}(1+\nu )(1-2\nu )/(1-\nu )$.

Parameters

Returns: Young’s modulus E, in pascals.

Raises

youngs_modulus_from_plate_speed(
    longitudinal_speed_m_s: float,
    *,
    density_kg_m3: float,
    poisson_ratio: float,
) -> float

Young’s modulus from a plate wave speed, the inverse of Eq. 2.21.

$E=\rho c_{\mathrm{L},\mathrm{p}}^2(1-{\nu }^2)$. This is what turns a table that prints cL and nu and no modulus, such as Hopkins Table A2, into one that can be compared with a table that prints the modulus.

Parameters

Returns: Young’s modulus E, in pascals.

Raises