Source: https://jmrplens.github.io/phonometry/vibration/structural/junction-transmission/ # Bending-wave transmission at plate junctions (Cremer / Craik / Hopkins) When a bending wave travelling on a wall or floor reaches a rigid junction with another plate, part of its energy is reflected and part is transmitted into the connected plates. The **wave approach** of Cremer et al. (1973), tabulated by Craik (1981, 1996) and collected in Hopkins (2007, *Sound Insulation*, Section 5.2.1.3), gives the transmission coefficient in closed form for the four most common junctions of thin, homogeneous, isotropic plates: the **X**, **T**, **L** and **in-line** junctions. Modelling the junction as a simply supported (pinned) massless beam forces an incident bending wave to generate only reflected and transmitted *bending* waves, with no conversion to in-plane waves, and the resulting coefficients are **independent of frequency**. That is what makes them convenient closed-form inputs for statistical energy analysis (SEA) and for the EN 12354 flanking-transmission model, where they feed the coupling loss factor $\eta_{ij}$ and the vibration reduction index $K_{ij}$. The object itself first: a T-junction where a 140 mm concrete floor runs into a continuous 200 mm wall. A `junction_transmission` result retains its plates, so `res.plot_geometry()` draws the junction to scale with the incident bending wave marked. *Drawn to scale: the incident bending wave arrives along the 140 mm floor, and everything the closed-form coefficients describe happens where it meets the continuous 200 mm wall.*

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```python import matplotlib.pyplot as plt from phonometry import junction_transmission # A 140 mm concrete floor meeting a 200 mm wall at a T-junction. res = junction_transmission("T2", 0.14, 3500.0, 320.0, 0.2, 3500.0, 460.0) res.plot_geometry() plt.show() ```

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```python import matplotlib.pyplot as plt from phonometry import junction_transmission # X-junction between a 100 mm and a 200 mm concrete plate (cL = 3200 m/s). res = junction_transmission("X", 0.1, 3200.0, 240.0, 0.2, 3200.0, 480.0) res.plot() # tau(theta) for the corner and straight paths, with averages plt.show() ```

## 1. Wave parameters χ and ψ (Eqs 5.10, 5.11) With plate $i$ of thickness $h_i$, quasi-longitudinal wave speed $c_{\mathrm{L},i}$ and surface density $\rho_{\mathrm{s},i}$, the whole family of coefficients depends on just two dimensionless ratios (Cremer et al. 1973): $$ \chi = \frac{k_{\mathrm{B}2}}{k_{\mathrm{B}1}} = \left(\frac{\rho_{\mathrm{s}2} B_1}{\rho_{\mathrm{s}1} B_2}\right)^{1/4} = \sqrt{\frac{h_1 c_{\mathrm{L}1}}{h_2 c_{\mathrm{L}2}}} = \sqrt{\frac{f_{\mathrm{c}2}}{f_{\mathrm{c}1}}} $$ $$ \psi = \frac{B_2 k_{\mathrm{B}2}^2}{B_1 k_{\mathrm{B}1}^2} = \frac{h_2 c_{\mathrm{L}2} \rho_{\mathrm{s}2}}{h_1 c_{\mathrm{L}1} \rho_{\mathrm{s}1}} = \frac{\rho_{\mathrm{s}2} f_{\mathrm{c}1}}{\rho_{\mathrm{s}1} f_{\mathrm{c}2}} $$ $\chi$ is the ratio of the plates' bending wavenumbers (equivalently the square root of their critical-frequency ratio) and sets the total-internal-reflection cut-off angle $\theta_\text{co} = \arcsin\chi$; $\psi$ is the ratio of their bending-moment mobilities. For **identical plates** both are 1. ```python from phonometry import junction_wave_parameters chi, psi = junction_wave_parameters(0.1, 3200.0, 240.0, 0.2, 3200.0, 480.0) # -> (sqrt(0.5), 4.0) ``` ## 2. Transmission around a corner and across a straight section For an incident wave on plate 1, transmission **around the corner** (into the perpendicular plate 2) is $\tau_{12}(\theta)$ (Eq. 5.12), and transmission **across the straight section** (into the collinear plate 3, X- and T-junction (1) only) is $\tau_{13}(\theta)$ (Eq. 5.13): $$ \tau_{12}(\theta) = \frac{0.5\,J_1 J_2 \psi \cos\theta \sqrt{\chi^2 - \sin^2\theta}} {(J_2\psi)^2 + \chi^2 + J_2\psi\left(\sqrt{(1+\sin^2\theta)(\chi^2+\sin^2\theta)} + \sqrt{(1-\sin^2\theta)(\chi^2-\sin^2\theta)}\right)}, \qquad \chi \ge \sin\theta $$ $$ \tau_{12}(\theta) = 0, \qquad \chi < \sin\theta $$ (no propagating transmitted wave beyond the cut-off). The junction constants $J_1$, $J_2$, $J_3$ select the geometry: | Junction | $J_1$ | $J_2$ | $J_3$ | |---|---|---|---| | X | 1 | 1 | 1 | | T-junction (1) | 2 | 0.5 | 0.5 | | T-junction (2) | 2 | 2 | — | | L | 4 | 1 | — | The straight section is undefined for the T-junction (2) and the L-junction. In the assumed symmetry the X-junction has plates 1 and 3 identical and plates 2 and 4 identical; T-junction (1) has plates 1 and 3 identical; T-junction (2) has plates 2 and 4 identical. ```python import numpy as np from phonometry import ( corner_transmission_coefficient, straight_transmission_coefficient, ) theta = np.radians(np.linspace(0.0, 90.0, 91)) tau12 = corner_transmission_coefficient(theta, chi, psi, "X") tau13 = straight_transmission_coefficient(theta, chi, psi, "X") ``` The clip below runs this experiment in the time domain with the library's 2D [elastic FDTD solver](https://jmrplens.github.io/phonometry/simulation/elastic-waves/): a bending packet on a 10 mm steel plate reaches an L-junction with an identical plate, and the corner splits it into the reflected and transmitted waves this section prices at $\tau_{12}(0°) = 0.5$. The fast in-plane precursor racing down the receiving plate is the mode conversion the pinned-junction model deliberately leaves out. [Watch the high-resolution video (WebM)](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_elastic_plate_junction.webm) ## 3. Diffuse-field angular average (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 $$ \bar{\tau}_{ij} = \int_0^{\pi/2} \tau_{ij}(\theta)\cos\theta\,\mathrm{d}\theta $$ (the $\cos\theta$ weight already normalises the average). For **identical plates** the algebra collapses to exact fractions that serve as the library's first-principles oracle: * X-junction corner and straight: $\tau_{ij}(\theta) = \cos^2\theta / 8$, so $\bar{\tau}_{ij} = 1/12$; * L-junction corner: $\tau_{ij}(\theta) = \cos^2\theta / 2$, so $\bar{\tau}_{ij} = 1/3$; * in-line junction: $\tau_{12}(0°) = 1$ (a continuous plate transmits fully). ```python from phonometry import angular_average_transmission_coefficient angular_average_transmission_coefficient(1.0, 1.0, "X", section="corner") # 1/12 angular_average_transmission_coefficient(1.0, 1.0, "L", section="corner") # 1/3 ``` The two directions obey the SEA consistency relationship (Eq. 5.7), $\bar{\tau}_{12} = \chi\,\bar{\tau}_{21}$, so only one direction needs to be computed. ## 4. Coupling loss factor and vibration reduction index The angular average is the bridge to the two junction descriptors used in SEA-based building models. The **coupling loss factor** (Eq. 2.154) for a source plate $i$ of area $S_i$, bending-wave group velocity $c_{\mathrm{g},i}$ and junction length $L_{ij}$ is $$ \eta_{ij} = \frac{c_{\mathrm{g},i} L_{ij} \tau_{ij}}{2\pi^2 f S_i} $$ and the wave-approach **vibration reduction index** (Eq. 5.116) is $$ K_{ij} = 10\log_{10}(1/\tau_{ij}) + 5\log_{10}(f_{\mathrm{c},j}/f_\text{ref}), \qquad f_\text{ref} = 1000\ \text{Hz} $$ with $f_{\mathrm{c},j}$ the critical frequency of the *receiving* plate. Combined with the Eq. 5.7 reciprocity this form is symmetric, $K_{ij} = K_{ji}$, as EN 12354 requires of the junction descriptor. For the identical 100 mm concrete X-junction ($f_\mathrm{c} \approx 203\ \text{Hz}$), $K_{ij} = 10\log_{10} 12 + 5\log_{10}(203/1000) \approx 7.3\ \text{dB}$. ```python from phonometry import (coupling_loss_factor, junction_transmission, wave_vibration_reduction_index) eta = coupling_loss_factor(1.0 / 12.0, group_velocity=200.0, junction_length=4.0, frequency=500.0, plate_area=10.0) res = junction_transmission("X", 0.1, 3200.0, 240.0, 0.1, 3200.0, 240.0) kij = wave_vibration_reduction_index(res.corner_average, res.critical_frequency2) # 7.33 dB kij = res.corner_reduction_index # the same, precomputed on the result res.plot() # tau(theta) for this junction's corner and straight paths (needs matplotlib) ``` $K_{ij}$ is also what ISO 10848 measures on a built junction: excite one element, read the velocity level difference on both, and normalize by the junction length, so the closed-form value above has a direct experimental counterpart. The junction descriptor is a *design* quantity: sweeping the receiving plate's thickness shows how much a mass change at the junction buys. The corner paths stiffen quickly with a heavier receiving plate, while the straight (in-line) path of the X-junction rises fastest of all, since the perpendicular plates increasingly pin the junction line:

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import junction_transmission, wave_vibration_reduction_index # Concrete plates (cL = 3200 m/s, rho = 2400 kg/m3): plate 1 fixed at # 100 mm, plate 2 swept from 50 mm to 400 mm. h1, cl, rho = 0.1, 3200.0, 2400.0 ratios = np.linspace(0.5, 4.0, 36) kij_corner = [] for ratio in ratios: h2 = h1 * float(ratio) res = junction_transmission("X", h1, cl, rho * h1, h2, cl, rho * h2) kij_corner.append(res.corner_reduction_index) fig, ax = plt.subplots() ax.plot(ratios, kij_corner, label="X corner") ax.set_xlabel("Thickness ratio h2/h1") ax.set_ylabel("Vibration reduction index Kij [dB]") ax.set_title("Wave-approach junction Kij (Hopkins Eq. 5.116)") ax.legend() plt.show() ```

## 5. Worked example: feeding Kij into EN 12354 The number this page predicts is exactly what the EN 12354-1 flanking model consumes. Take the 100 mm / 200 mm concrete X-junction from the figure at the top: its corner path gives $K_{12} = 9.8\ \text{dB}$. Handing that to `flanking_element` in place of a tabulated Annex E value prices the junction's three flanking paths and their effect on the apparent rating: ```python from phonometry import building, junction_transmission # The 100 mm / 200 mm concrete X-junction of the opening figure: res = junction_transmission("X", 0.1, 3200.0, 240.0, 0.2, 3200.0, 480.0) k12 = res.corner_reduction_index # 9.8 dB (corner path) # Feed it to the EN 12354-1 simplified model as this junction's Kij: ff, df, fd = building.flanking_element( label="floor", r_flanking=49.0, r_separating=57.0, k_ff=k12, k_fd=k12, k_df=k12, separating_area=11.5, coupling_length=4.5) pred = building.predicted_airborne_insulation(r_direct=57.0, flanking_paths=[ff, df, fd]) print(round(pred.r_prime_w, 1)) # 55.4 (Rw 57 direct) print(pred.dominant.label, round(pred.dominant.fraction, 2)) # Dd 0.68 ``` One junction with a moderate $K_{ij}$ already trims 1.6 dB off the direct $R_\mathrm{w} = 57\ \text{dB}$; a full building repeats this for every junction, which is the [EN 12354 prediction guide](https://jmrplens.github.io/phonometry/buildings/design/insulation-prediction/). The measured, EN 12354 counterpart of $K_{ij}$ (from the direction-averaged velocity level difference) is the separate [laboratory flanking-transmission](https://jmrplens.github.io/phonometry/buildings/insulation/flanking-lab/) `vibration_reduction_index`; this guide is the closed-form *predicted* value from the wave approach. ## 6. Experimental SEA: coupling loss factors from measured energies Everything above is the **predictive** route: a coupling loss factor derived from a wave model of the junction. Real joints - welds, bolt rows, spot welds, adhesives - are not tractable that way, and the **experimental** route inverts the steady-state SEA power balance from *measured* subsystem energies instead (Norton & Karczub 2003, Sections 6.3.3 and 6.3.4). For two subsystems, $$ \Pi_1 = \omega\left[(\eta_1 + \eta_{12})\,E_1 - \eta_{21}\,E_2\right] \quad (6.10) $$ $$ 0 = \omega\left[(\eta_2 + \eta_{21})\,E_2 - \eta_{12}\,E_1\right] \quad (6.11) $$ with $E_i = M_i \langle v_i^2 \rangle$ the band energy of subsystem $i$ (its mass times the space- and time-averaged mean-square velocity). Drive subsystem 1 only, add the SEA consistency (reciprocity) relationship $n_1\eta_{12} = n_2\eta_{21}$ (Eq. 6.8), and both coupling loss factors follow from the two measured energies (Eq. 6.15): $$ \eta_{12} = \frac{\eta_2 E_2}{E_1 - E_2\,n_1/n_2}, \qquad \eta_{21} = \eta_{12}\,\frac{n_1}{n_2}, \qquad \Pi_\text{in} = \omega\,(\eta_1 E_1 + \eta_2 E_2) $$ The input power collapses to the total dissipated power, as it must in the steady state: substituting Eq. (6.11) into Eq. (6.10) cancels the two coupling terms exactly, which is a free check on any measurement. `power_injection_clf` performs that inversion, with the modal densities of Norton Section 6.4.1 alongside it: `flat_plate_modal_density` (Eq. 6.25), `bar_modal_density` (6.23), `beam_modal_density` (6.24) and `cylindrical_shell_modal_density` (6.27 to 6.29, the Szechenyi approximations in three regimes about the `ring_frequency` of Eq. 6.26). The flat-plate expression $n(f) = S\sqrt{12}/(2 c_\mathrm{L} t)$ is the same quantity as EN 12354-4's $n = \pi S f_\mathrm{c} / c_0^2$ used by the [flanking model](https://jmrplens.github.io/phonometry/buildings/insulation/flanking-lab/), only parametrised by the plate itself rather than by its critical frequency. *Left: two ways of joining the same two plates. Right: the loss-factor budget inverted from a pair of measured velocities; the coupling stays an order of magnitude below the damping, which is the condition for a two-subsystem SEA model to be trustworthy.*

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```python import math import matplotlib.pyplot as plt import numpy as np from phonometry import ( coupling_loss_factor, cylindrical_shell_modal_density, flat_plate_modal_density, plate_bending_stiffness, point_connection_coupling_loss_factor, power_injection_clf, right_angle_transmission_coefficient, ) from phonometry.vibration.structural.point_mobility import plate_bending_wave_speed rho, nu, young = 2700.0, 0.33, 7.1e10 # aluminium cl = math.sqrt(young / (rho * (1.0 - nu**2))) # 5432 m/s (Eq. 6.25) bands = np.array([125.0, 250.0, 500.0, 1000.0, 2000.0]) # Predicted: a 3 mm x 2.5 m x 1.2 m plate meeting a 5.5 mm x 2.0 m x 1.2 m # plate at right angles along the 1.2 m edge, welded and then bolted. h1, h2, area1, length = 0.003, 0.0055, 2.5 * 1.2, 1.2 tau = right_angle_transmission_coefficient(h1, h2, density1=rho, density2=rho, wave_speed1=cl, wave_speed2=cl) cb = plate_bending_wave_speed(bands, plate_bending_stiffness(young, h1, nu), rho * h1) welded = [float(coupling_loss_factor(tau, 2.0 * c, length, f, area1)) for c, f in zip(cb, bands, strict=True)] bolted = point_connection_coupling_loss_factor( bands, 12, thickness1=h1, thickness2=h2, surface_density1=rho * h1, surface_density2=rho * h2, wave_speed1=cl, wave_speed2=cl, plate_area1=area1) # Measured: a 5 mm platform driven directly, a 3 mm cylinder driven only # through the joints; 27.2 and 13.2 mm/s in the 500 Hz octave. t_p, t_c, radius = 0.005, 0.003, 0.75 area_c = 2.0 * math.pi * radius * 2.0 area_p = 3.5 * 3.0 - math.pi * radius**2 sea = power_injection_clf( 500.0, rho * t_p * area_p * 0.0272**2, rho * t_c * area_c * 0.0132**2, 4.4e-3, 2.4e-3, flat_plate_modal_density(area_p, t_p, cl), float(cylindrical_shell_modal_density(500.0, area_c, t_c, radius, cl)[0])) print(f"{float(sea.coupling_loss_factor12[0]):.2e}") # 4.26e-04 print(f"{float(sea.coupling_loss_factor21[0]):.2e}") # 3.91e-04 print(round(float(sea.input_power[0]), 2)) # 1.31 W sea.plot() plt.show() ```

Two closed forms sit between the two routes and are what a measured value is normally compared against. `right_angle_transmission_coefficient` gives the wave transmission coefficient of a right-angle plate junction without any angular integration (Eqs. 6.53 to 6.55, after Bies & Hamid and Cremer et al.); feeding it to `coupling_loss_factor` reproduces Norton Eq. (6.52) identically, because that equation and Hopkins Eq. (2.154) are the same expression once $c_\mathrm{g} = 2 c_\mathrm{B}$. And `point_connection_coupling_loss_factor` covers plates joined at $N$ discrete points instead of along a line (Eq. 6.56). Which one applies is decided by the bending wavelength: use the point form when it is shorter than the joint length, and the line form when it is longer. The two differ in slope as well as in level, $1/f$ against $1/\sqrt{f}$, so bolting and welding are not interchangeable across the spectrum. When no independent value of the internal loss factors is available, the single-drive inversion is underdetermined, and the classical **power-injection method** drives each subsystem in turn while measuring both energies each time. That gives four equations for $\eta_1$, $\eta_2$, $\eta_{12}$ and $\eta_{21}$ with no prior assumption at all, which `power_injection_matrix` solves band by band. Reciprocity then becomes a *check* on the measurement rather than an input: `modal_density_ratio` compares the measured $\eta_{21}/\eta_{12}$ against the $n_1/n_2$ computed from the geometry, and a large disagreement means the subsystem boundaries were drawn in the wrong place. ## What this guide covers **Covered.** The frequency-independent, rigid-junction transmission coefficients of Cremer, Heckl & Ungar (1973), tabulated by Craik (1981/1996) and collected in Hopkins (2007, Section 5.2.1.3), for X, T, L and in-line junctions of thin, homogeneous, isotropic plates: the wave parameters $\chi$/$\psi$, the corner and straight-section coefficients $\tau_{12}(\theta)$/$\tau_{13}(\theta)$, their diffuse-field angular average, the SEA coupling loss factor and the wave-approach vibration reduction index $K_{ij}$, through `junction_wave_parameters`, `corner_transmission_coefficient`, `straight_transmission_coefficient`, `angular_average_transmission_coefficient`, `inline_transmission_coefficient`, `coupling_loss_factor`, `wave_vibration_reduction_index` and `junction_transmission`. Also the experimental route of Norton & Karczub Chapter 6: the two-subsystem power balance and its in-situ inversion (`power_injection_clf`, `power_injection_matrix`), the modal densities of bars, beams, flat plates and thin-walled cylindrical shells with the `ring_frequency` they are split about, and the two closed forms between the routes (`right_angle_transmission_coefficient`, `point_connection_coupling_loss_factor`). **Not covered.** The predicted $K_{ij}$ is a closed-form idealisation of a rigid, simply supported junction, not a measurement: the empirical $K_{ij}$ from a direction-averaged velocity level difference is ISO 10848's `vibration_reduction_index`, in [Laboratory Flanking Transmission](https://jmrplens.github.io/phonometry/buildings/insulation/flanking-lab/). The straight-section coefficient $\tau_{13}$ is undefined for the T-junction (2) and L-junction geometries, which have no collinear third plate, so only the corner path applies there. The tabulated coefficients assume a *symmetric* junction — opposite plates identical — and there is no closed form here for one that is not. On the experimental side the inversion is written for **two** subsystems only, and `PowerInjectionResult.modal_density_ratio` is a consistency check the reader makes, not one the library automates. ## See also - [Predicting Sound Insulation (EN 12354)](https://jmrplens.github.io/phonometry/buildings/design/insulation-prediction/): the flanking model that consumes $K_{ij}$ junction by junction. - [Laboratory Flanking Transmission (ISO 10848)](https://jmrplens.github.io/phonometry/buildings/insulation/flanking-lab/): the measurement of $K_{ij}$ from velocity level differences, the empirical counterpart of this page's closed forms. - [Structure-borne sound power of equipment (EN 15657)](https://jmrplens.github.io/phonometry/buildings/design/structure-borne-power/): the source power that these junction transmissions carry through a building. - [Mechanical mobility and the FRF family (ISO 7626-1)](https://jmrplens.github.io/phonometry/vibration/structural/mechanical-mobility/): the plate mobilities behind the wave parameters. ## References - Cremer, L., Heckl, M., & Ungar, E. E. (1973). *Structure-borne sound* (1st ed.). Springer. - Craik, R. J. M. (1996). *Sound transmission through buildings using statistical energy analysis*. Gower. - Hopkins, C. (2007). *Sound insulation* (Section 5.2.1.3). Butterworth-Heinemann. - Norton, M. P., & Karczub, D. G. (2003). *Fundamentals of noise and vibration analysis for engineers* (2nd ed., Chapter 6). Cambridge University Press. See the [bibliography](https://jmrplens.github.io/phonometry/reference/bibliography/) for full entries.