vibration.structural.junction_transmission

Bending-wave transmission coefficients for rigid plate junctions (Hopkins 2007, Sound Insulation, Section 5.2.1.3; Cremer et al. 1973; Craik 1981, 1996).

The wave approach models a plane bending wave that is incident on a rigid junction of thin plates at an angle theta and, assuming the junction beam is simply supported (pinned so it can rotate but not translate), produces only reflected and transmitted bending waves (no in-plane conversion). The resulting angle-resolved transmission coefficients are frequency independent, which is what makes them convenient closed-form building blocks for statistical-energy-analysis (SEA) and the EN 12354 flanking model. This module implements the rigid X, T, L and in-line junctions of two thin, homogeneous, isotropic plates.

Wave parameters (Hopkins Eqs 5.10 and 5.11, after Cremer et al. 1973). With plate i of thickness h_i, quasi-longitudinal wave speed cL_i, surface density rho_s,i (kg/m^2), bending stiffness per unit width B_i and critical frequency fc_i:

$$\chi =\frac{k_{B2}}{k_{B1}}={\left(\frac{{\rho }_{s2}{B}_1}{{\rho }_{s1}{B}_2}\right)}^{0.25}=\sqrt{\frac{h_1{c}_{L1}}{h_2{c}_{L2}}}=\sqrt{\frac{f_{c2}}{f_{c1}}}\text{\hspace{1em}(5.10)}$$ $$\psi =\frac{B_2{k}_{B2}^2}{B_1{k}_{B1}^2}=\frac{h_2{c}_{L2}{\rho }_{s2}}{h_1{c}_{L1}{\rho }_{s1}}=\frac{{\rho }_{s2}{f}_{c1}}{{\rho }_{s1}{f}_{c2}}\text{\hspace{1em}(5.11)}$$

chi is the ratio of bending wavenumbers (it fixes the total-internal- reflection cut-off ${\theta }_{co}=\arcsin \chi$) and psi is the ratio of the plates’ bending-moment mobilities.

Transmission around a corner (Hopkins Eq. 5.12, Craik 1981/1996). For an incident wave on plate 1, if $\chi \ge \sin \theta$,

$${\tau }_{12}(\theta )=\frac{0.5J_1{J}_2\psi \cos \theta \sqrt{{\chi }^2-{\sin }^2\theta }}{(J_2\psi {)}^2+{\chi }^2+J_2\psi \left[\sqrt{(1+{\sin }^2\theta )({\chi }^2+{\sin }^2\theta )}+\sqrt{(1-{\sin }^2\theta )({\chi }^2-{\sin }^2\theta )}\right]}$$

and ${\tau }_{12}(\theta )=0$ for $\chi <\sin \theta$ (no propagating transmitted wave beyond the cut-off angle).

Transmission across a straight section (Hopkins Eq. 5.13, Craik 1981/1996). Only the X-junction and T-junction (1) have an in-line (straight-through) section. If $\chi \ge \sin \theta$,

$${\tau }_{13}(\theta )=\frac{0.5{\chi }^2{\cos }^2\theta }{(J_3\psi {)}^2+{\chi }^2+J_3\psi \left[\sqrt{(1+{\sin }^2\theta )({\chi }^2+{\sin }^2\theta )}+\sqrt{(1-{\sin }^2\theta )({\chi }^2-{\sin }^2\theta )}\right]}$$

(the same denominator shape as Eq. 5.12), and for $\chi <\sin \theta$,

$${\tau }_{13}(\theta )=\frac{{\cos }^2\theta }{2+\frac{(J_3\psi {)}^2{C}^2}{{\chi }^4}+\frac{2J_3\psi C}{{\chi }^2}\sqrt{1+{\sin }^2\theta }}$$

with $C=\sqrt{{\chi }^2+{\sin }^2\theta }+\sqrt{{\sin }^2\theta -{\chi }^2}$.

Junction constants. J1, J2 set the corner coefficient and J3 the straight one:

=============== ==== ===== ===== Junction J1 J2 J3 =============== ==== ===== ===== X 1 1 1 T-junction (1) 2 0.5 0.5 T-junction (2) 2 2 — L 4 1 — =============== ==== ===== =====

For T-junction (1) plates 1 and 3 are identical; for T-junction (2) plates 2 and 4 are identical. The straight section is undefined for T-junction (2) and for the L-junction.

In-line junction (Hopkins Eq. 5.14, Cremer et al. 1973). Two collinear plates (a change of section). Only normal incidence is used; it is within 1 dB of the angular average when $\chi \ge 1$:

$${\tau }_{12}\approx {\tau }_{12}(0^{\circ })={\left[\frac{2(1+\chi )(1+\psi )\sqrt{\chi \psi }}{\chi (1+\psi {)}^2+2\psi (1+{\chi }^2)}\right]}^2\text{\hspace{1em}(5.14)}$$

Angular average (Hopkins Eq. 5.6). In a diffuse vibration field every angle of incidence is equally probable and the incident intensity carries a $\cos \theta$ obliquity factor, so the average transmission coefficient is:

$${\bar{\tau }}_{ij}={\int }_0^{\pi /2}{\tau }_{ij}(\theta )\cos \theta \mathrm{d}\theta \text{\hspace{1em}(5.6)}$$

(the $\cos \theta$ weight already normalises the average, since ${\int }_0^{\pi /2}\cos \theta \mathrm{d}\theta =1$).

Coupling loss factor (Hopkins Eq. 2.154). For a source plate i of area S_i, bending-wave group velocity cg_i and junction length L_ij:

$${\eta }_{ij}=\frac{c_{g,i}{L}_{ij}{\tau }_{ij}}{2{\pi }^{2}f{S}_i}\text{\hspace{1em}(2.154)}$$

Vibration reduction index (Hopkins Eq. 5.116). The wave-approach value of the EN 12354 junction descriptor, with fc_j the critical frequency of the receiving plate and the reference frequency $f_{\text{ref}}=1000$ Hz:

$$K_{ij}=10{\log }_{10}\left(\frac{1}{{\tau }_{ij}}\right)+5{\log }_{10}\left(\frac{f_{cj}}{f_{\text{ref}}}\right)\text{\hspace{1em}(5.116)}$$

Combined with the reciprocity relationship below (${\bar{\tau }}_{12}=\chi {\bar{\tau }}_{21}$ with $\chi =\sqrt{f_{c2}/f_{c1}}$) this form is symmetric, $K_{ij}=K_{ji}$, as EN 12354 and ISO 10848 require of the junction descriptor.

Reciprocity (Hopkins Eq. 5.7, the SEA consistency relationship). The angular averages of the two directions are linked by ${\bar{\tau }}_{ij}={\bar{\tau }}_{ji}\sqrt{h_i{c}_{Li}/(h_j{c}_{Lj})}={\bar{\tau }}_{ji}\sqrt{f_{cj}/f_{ci}}$, i.e. ${\bar{\tau }}_{12}=\chi {\bar{\tau }}_{21}$.

Two shortcuts for right-angle joints (Norton & Karczub 2003, Section 6.6.1). Alongside the angle-resolved wave approach above, the SEA literature uses a pair of closed forms that need no integration, which is what the experimental SEA of phonometry.vibration.structural.experimental_sea is normally compared against:

• right_angle_transmission_coefficient (Norton Eqs. 6.53 to 6.55, after Bies & Hamid and Cremer et al.). The normal-incidence coefficient of two plates at right angles is ${\tau }_{12}(0)=2({\psi }_N^{0.5}+{\psi }_N^{-0.5}{)}^{-2}$ with ${\psi }_N={\rho }_1{c}_{L1}^{1.5}{h}_1^{2.5}/({\rho }_2{c}_{L2}^{1.5}{h}_2^{2.5})$ (note this is a different psi from Hopkins Eq. 5.11 above), and the random incidence value follows from the empirical factor $2.754X/(1+3.24X)$ with $X=h_1/h_2$. Feeding it to coupling_loss_factor reproduces Norton Eq. (6.52), ${\eta }_{12}=2c_{B}L{\tau }_{12}/(\pi \omega S_1)$, identically: that equation and Hopkins Eq. (2.154) are the same expression once $c_g=2c_B$.
• point_connection_coupling_loss_factor (Norton Eq. 6.56, after Clarkson & Ranky). Plates joined at N discrete points (bolts, rivets, spot welds) rather than along a line. Use it when the bending wavelength is shorter than the joint length, and the line-junction route above when it is longer.

Auto-generated from the source docstrings by scripts/generate_api_docs.py (make api-docs). Do not edit by hand.

angular_average_transmission_coefficient

angular_average_transmission_coefficient(
    chi: float,
    psi: float,
    junction: str = 'X',
    *,
    section: str = 'corner',
) -> float

Diffuse-field angular average of a transmission coefficient (Hopkins 5.6).

$\bar{\tau }={\int }_0^{\pi /2}\tau (\theta )\cos \theta \mathrm{d}\theta$, evaluated by adaptive quadrature.

Parameters

Name	Description
chi	Wave parameter chi (Eq. 5.10, > 0).
psi	Wave parameter psi (Eq. 5.11, > 0).
junction	"X", "T1", "T2" or "L".
section	"corner" (tau12, default) or "straight" (tau13; only for "X"/"T1").

Returns: The angular-average transmission coefficient tau_bar.

Raises

Exception	When
ValueError	for a non-positive chi/psi, an unknown junction or section, or a straight section that does not exist.

corner_transmission_coefficient

corner_transmission_coefficient(
    angle: ArrayLike,
    chi: float,
    psi: float,
    junction: str = 'X',
) -> NDArray[np.float64]

Transmission around a corner tau12(theta) (Hopkins Eq. 5.12).

Returns 0 for angles beyond the cut-off $\arcsin (\chi )$ (only reached when $\chi <1$).

Parameters

Name	Description
angle	Incidence angle theta, in radians (scalar or array, $0\le \theta \le \pi /2$).
chi	Wave parameter chi (Eq. 5.10, > 0).
psi	Wave parameter psi (Eq. 5.11, > 0).
junction	"X", "T1", "T2" or "L".

Returns: tau12(theta) (same shape as angle).

Raises

Exception	When
ValueError	for a non-positive chi/psi, an out-of-range angle or an unknown junction.

coupling_loss_factor

coupling_loss_factor(
    transmission_coefficient: ArrayLike,
    group_velocity: float,
    junction_length: float,
    frequency: ArrayLike,
    plate_area: float,
) -> NDArray[np.float64]

Coupling loss factor from a transmission coefficient (Hopkins Eq. 2.154).

${\eta }_{ij}=c_{g,i}{L}_{ij}{\tau }_{ij}/(2{\pi }^{2}f{S}_i)$ with the source-plate bending-wave group velocity cg_i, the junction length L_ij, the frequency f and the source-plate area S_i.

Parameters

Name	Description
transmission_coefficient	Angular-average tau_ij (scalar/array).
group_velocity	Source-plate bending-wave group velocity cg_i, in m/s (> 0). For a thin plate $c_g=2c_B$ with the bending phase speed cB (see phonometry.vibration.structural.point_mobility.plate_bending_wave_speed).
junction_length	Junction length L_ij, in m (> 0).
frequency	Frequency f, in hertz (scalar or array, > 0).
plate_area	Source-plate area S_i, in m^2 (> 0).

Returns: The coupling loss factor eta_ij (broadcast of the inputs).

Raises

Exception	When
ValueError	for a non-positive input.

inline_transmission_coefficient

inline_transmission_coefficient(chi: float, psi: float) -> float

Normal-incidence transmission across an in-line junction (Hopkins 5.14).

${\tau }_{12}=[2(1+\chi )(1+\psi )\sqrt{\chi \psi }/(\chi (1+\psi {)}^2+2\psi (1+{\chi }^2)){]}^2$ (Cremer et al. 1973). For identical plates ($\chi =\psi =1$) this is 1 (a continuous plate transmits fully).

Parameters

Name	Description
chi	Wave parameter chi (Eq. 5.10, > 0).
psi	Wave parameter psi (Eq. 5.11, > 0).

Returns: tau12(0 deg).

Raises

Exception	When
ValueError	for a non-positive chi/psi.

junction_transmission

junction_transmission(
    junction: str,
    thickness1: float,
    wave_speed1: float,
    surface_density1: float,
    thickness2: float,
    wave_speed2: float,
    surface_density2: float,
    *,
    angles_deg: ArrayLike | None = None,
) -> JunctionTransmissionResult

Bending-wave transmission of a rigid perpendicular plate junction.

Builds the angle-resolved corner (and, for X / T-junction (1), straight) transmission coefficients of Hopkins Eqs 5.12/5.13 and their diffuse-field angular averages (Eq. 5.6) from the two plates’ properties, together with the thin-plate critical frequencies $f_c=\sqrt{12}{c}_0^2/(2\pi h{c}_L)$ ($c_0=343$ m/s) used by the Eq. 5.116 vibration reduction index. For the in-line junction (normal incidence only) use inline_transmission_coefficient.

Parameters

Name	Description
junction	"X", "T1", "T2" or "L".
thickness1	Thickness h1 of the source plate, in m (> 0).
wave_speed1	Quasi-longitudinal wave speed cL1 of the source plate, in m/s (> 0).
surface_density1	Surface density rho_s1 of the source plate, in kg/m^2 (> 0).
thickness2	Thickness h2 of the receiving plate, in m (> 0).
wave_speed2	Quasi-longitudinal wave speed cL2 of the receiving plate, in m/s (> 0).
surface_density2	Surface density rho_s2 of the receiving plate, in kg/m^2 (> 0).
angles_deg	Incidence-angle grid in degrees (Default: 0 to 90 in 91 one-degree steps).

Returns: A JunctionTransmissionResult.

Raises

Exception	When
ValueError	for a non-positive input or an unknown junction.

junction_wave_parameters

junction_wave_parameters(
    thickness1: float,
    wave_speed1: float,
    surface_density1: float,
    thickness2: float,
    wave_speed2: float,
    surface_density2: float,
) -> tuple[float, float]

Wave parameters chi and psi of a plate pair (Hopkins 5.10/5.11).

$\chi =\sqrt{h_1{c}_{L1}/(h_2{c}_{L2})}$ (Eq. 5.10) and $\psi =(h_2{c}_{L2}{\rho }_{s2})/(h_1{c}_{L1}{\rho }_{s1})$ (Eq. 5.11), with plate 1 the plate carrying the incident wave.

Parameters

Name	Description
thickness1	Thickness h1 of plate 1, in m (> 0).
wave_speed1	Quasi-longitudinal wave speed cL1 of plate 1, in m/s (> 0).
surface_density1	Surface density rho_s1 of plate 1, in kg/m^2 (> 0).
thickness2	Thickness h2 of plate 2, in m (> 0).
wave_speed2	Quasi-longitudinal wave speed cL2 of plate 2, in m/s (> 0).
surface_density2	Surface density rho_s2 of plate 2, in kg/m^2 (> 0).

Returns: The pair (chi, psi).

Raises

Exception	When
ValueError	for a non-positive input.

JunctionTransmissionResult

JunctionTransmissionResult(
    junction: str,
    chi: float,
    psi: float,
    critical_frequency1: float,
    critical_frequency2: float,
    angles_deg: np.ndarray,
    corner: np.ndarray,
    straight: np.ndarray | None,
    corner_average: float,
    straight_average: float | None,
    thickness1: float | None = None,
    thickness2: float | None = None,
)

Bending-wave transmission across a rigid plate junction (Hopkins 5.2.1.3).

Attributes

Name	Description
junction	Junction type ("X", "T1", "T2" or "L").
chi	Wave parameter chi (Eq. 5.10).
psi	Wave parameter psi (Eq. 5.11).
critical_frequency1	Critical frequency fc_1 of the source plate, in hertz (thin plate, $c_0=343$ m/s).
critical_frequency2	Critical frequency fc_2 of the receiving plate, in hertz (thin plate, $c_0=343$ m/s).
angles_deg	Incidence-angle grid, in degrees.
corner	Corner transmission coefficient tau12(theta) on the grid.
straight	Straight-section coefficient tau13(theta) on the grid, or None when the junction has no straight section.
corner_average	Diffuse-field angular average tau_bar_12 (Eq. 5.6).
straight_average	Angular average tau_bar_13, or None.
thickness1	Plate 1 thickness, in metres, retained (with thickness2) so plot_geometry can draw the junction; appended after the original fields and None for hand-built results.
thickness2	Plate 2 thickness, in metres, or None.

JunctionTransmissionResult.corner_reduction_index

property

Wave-approach K_12 of the corner path, in dB (Hopkins Eq. 5.116).

$K_{12}=10{\log }_{10}(1/{\bar{\tau }}_{12})+5{\log }_{10}(f_{c2}/1000)$ with the receiving plate’s critical frequency fc_2. The value is symmetric: building the reverse junction (plates swapped, and for a T-junction the matching constants T1 <-> T2) gives the same $K_{21}=K_{12}$.

JunctionTransmissionResult.plot()

JunctionTransmissionResult.plot(
    ax: Axes | None = None,
    *,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Plot tau(theta) versus incidence angle for this junction.

Requires matplotlib (pip install phonometry[plot]); returns the Axes.

JunctionTransmissionResult.plot_geometry()

JunctionTransmissionResult.plot_geometry(
    ax: Axes | None = None,
    *,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Draw the plate-junction cross-section to scale.

Requires matplotlib (pip install phonometry[plot]); returns the Axes.

Raises

Exception	When
ValueError	If the result does not retain its geometry.

plot_junction_geometry

plot_junction_geometry(
    junction: str,
    thickness1: float,
    thickness2: float,
    ax: Axes | None = None,
    *,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Draw a plate junction cross-section to scale.

Plate 1 runs horizontally; the perpendicular plate(s) of thickness 2 form the L, T or X. The incident bending wave arrives on plate 1 and the junction type follows junction_transmission.

Parameters

Name	Description
junction	"L", "T1", "T2" or "X".
thickness1	Plate 1 thickness, in metres.
thickness2	Plate 2 thickness, in metres.
ax	Existing axes, or None to create a figure.
language	Label language, "en" (default) or "es".
kwargs	Forwarded to the plate-1 rectangle.

Returns: The axes.

point_connection_coupling_loss_factor

point_connection_coupling_loss_factor(
    frequency: ArrayLike,
    n_connections: int,
    *,
    thickness1: float,
    thickness2: float,
    surface_density1: float,
    surface_density2: float,
    wave_speed1: float,
    wave_speed2: float,
    plate_area1: float,
) -> NDArray[np.float64]

Coupling loss factor of a point-connected plate pair (Norton Eq. 6.56).

Two homogeneous plates joined by N bolts, rivets or spot welds rather than along a continuous line:

$${\eta }_{12}=\frac{4N{h}_1{c}_{L1}}{\sqrt{3}\omega S_1}\frac{A_1{A}_2}{(A_1+A_2{)}^2},A_i={\rho }_{si}^2{h}_i^2{c}_{Li}^2\text{\hspace{1em}(6.56)}$$

with rho_si the surface density in kg/m^2, $\omega =2\pi f$ and S1 the source-plate area. Norton recommends it when the bending wavelength in the plates is shorter than the length of the connected edge, and the line junction (coupling_loss_factor on right_angle_transmission_coefficient) when it is longer. The result falls as $1/f$, unlike the line junction’s $1/\sqrt{f}$.

Note

Equation (6.56) is printed in the book with the (A1 + A2) bracket not squared, which is dimensionally inconsistent (the coupling loss factor would not be dimensionless). The squared form implemented here is the one that reproduces the book’s own answer to problem 6.13; see docs/ERRATA.md.

Parameters

Name	Description
frequency	Band centre frequency f, in hertz (scalar/array, >0).
n_connections	Number of point connections N (integer >= 1).
thickness1	Thickness h1 of the source plate, in m (> 0).
thickness2	Thickness h2 of the receiving plate, in m (> 0).
surface_density1	rho_s1, in kg/m^2 (> 0).
surface_density2	rho_s2, in kg/m^2 (> 0).
wave_speed1	Quasi-longitudinal wave speed cL1, in m/s (> 0).
wave_speed2	Quasi-longitudinal wave speed cL2, in m/s (> 0).
plate_area1	Source-plate area S1, in m^2 (> 0).

Returns: The coupling loss factor eta_12 per band.

Raises

Exception	When
ValueError	for a non-positive or non-integer input.

right_angle_transmission_coefficient

right_angle_transmission_coefficient(
    thickness1: float,
    thickness2: float,
    *,
    density1: float,
    density2: float,
    wave_speed1: float,
    wave_speed2: float,
    incidence: str = 'random',
) -> float

Right-angle plate junction tau12 (Norton Eqs. 6.53 to 6.55).

The closed form used throughout the SEA literature for two flat plates coupled at right angles along a line, with no angular integration:

$${\psi }_N=\frac{{\rho }_1{c}_{L1}^{1.5}{h}_1^{2.5}}{{\rho }_2{c}_{L2}^{1.5}{h}_2^{2.5}}\text{\hspace{1em}(6.54)}$$ $${\tau }_{12}(0)=2{\left(\sqrt{{\psi }_N}+\frac{1}{\sqrt{{\psi }_N}}\right)}^{-2}\text{\hspace{1em}(6.53)}$$ $${\tau }_{12}={\tau }_{12}(0)\frac{2.754X}{1+3.24X},X=h_1/h_2\text{\hspace{1em}(6.55)}$$

tau12(0) is symmetric in the two plates (swapping them inverts psi_N, which the expression does not see); the random-incidence factor is not, so tau12 and tau21 differ. Pass the result to coupling_loss_factor with the source plate’s group velocity $c_g=2c_B$ to obtain Norton Eq. (6.52).

rho here is the volume density in kg/m^3, not the surface density of junction_wave_parameters.

Parameters

Name	Description
thickness1	Thickness h1 of the source plate, in m (> 0).
thickness2	Thickness h2 of the receiving plate, in m (> 0).
density1	Density rho1 of the source plate, in kg/m^3 (> 0).
density2	Density rho2 of the receiving plate, in kg/m^3 (> 0).
wave_speed1	Quasi-longitudinal wave speed cL1, in m/s (> 0).
wave_speed2	Quasi-longitudinal wave speed cL2, in m/s (> 0).
incidence	"random" (Default, Eq. 6.55) or "normal" (Eq. 6.53 alone).

Returns: The transmission coefficient tau12.

Raises

Exception	When
ValueError	for a non-positive input or an unknown incidence.

straight_transmission_coefficient

straight_transmission_coefficient(
    angle: ArrayLike,
    chi: float,
    psi: float,
    junction: str = 'X',
) -> NDArray[np.float64]

Transmission across a straight section tau13(theta) (Hopkins 5.13).

Defined only for the X-junction and T-junction (1); both incidence regimes $\chi \ge \sin \theta$ and $\chi <\sin \theta$ are covered.

Parameters

Name	Description
angle	Incidence angle theta, in radians (scalar or array, $0\le \theta \le \pi /2$).
chi	Wave parameter chi (Eq. 5.10, > 0).
psi	Wave parameter psi (Eq. 5.11, > 0).
junction	"X" or "T1" (the only junctions with a straight section).

Returns: tau13(theta) (same shape as angle).

Raises

Exception	When
ValueError	for a non-positive chi/psi, an out-of-range angle, or a junction without a straight section.

wave_vibration_reduction_index

wave_vibration_reduction_index(
    transmission_coefficient: ArrayLike,
    critical_frequency_receiver: float,
) -> NDArray[np.float64]

Vibration reduction index from a transmission coefficient (Hopkins 5.116).

$K_{ij}=10{\log }_{10}(1/{\tau }_{ij})+5{\log }_{10}(f_{cj}/f_{\text{ref}})$ with fc_j the critical frequency of the receiving plate and the reference frequency $f_{\text{ref}}=1000$ Hz. Because the angular-average transmission coefficients satisfy the reciprocity relationship ${\bar{\tau }}_{ij}={\bar{\tau }}_{ji}\sqrt{f_{cj}/f_{ci}}$ (Eq. 5.7), this form is symmetric: $K_{ij}=K_{ji}$.

Parameters

Name	Description
transmission_coefficient	tau_ij (scalar or array, > 0).
critical_frequency_receiver	Critical frequency fc_j of the receiving plate, in hertz (> 0).

Returns: The vibration reduction index K_ij, in dB.

Raises

Exception	When
ValueError	for a non-positive tau or fc_j.