Source: https://jmrplens.github.io/phonometry/reference/api/vibration/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 `ηij` and the vibration reduction index `Kij`.

Show the code for this figure

```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 `cL_i` and surface density `ρs_i`, the whole family of coefficients depends on just two dimensionless ratios (Cremer et al. 1973): ``` χ = kB2 / kB1 = (ρs2 B1 / (ρs1 B2))**0.25 = sqrt(h1 cL1 / (h2 cL2)) = sqrt(fc2 / fc1) ψ = B2 kB2**2 / (B1 kB1**2) = (h2 cL2 ρs2) / (h1 cL1 ρs1) = (ρs2 fc1) / (ρs1 fc2) ``` `χ` 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 `θco = arcsin(χ)`; `ψ` 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 `τ12(θ)` (Eq. 5.12), and transmission **across the straight section** (into the collinear plate 3, X- and T-junction (1) only) is `τ13(θ)` (Eq. 5.13): ``` 0.5 J1 J2 ψ cos θ sqrt(χ² − sin²θ) τ12(θ) = ─────────────────────────────────────────────────── , χ ≥ sin θ (J2 ψ)² + χ² + J2 ψ ( sqrt((1+sin²θ)(χ²+sin²θ)) + sqrt((1−sin²θ)(χ²−sin²θ)) ) τ12(θ) = 0 , χ < sin θ (no propagating transmitted wave beyond the cut-off) ``` The junction constants `J1`, `J2`, `J3` select the geometry: | Junction | `J1` | `J2` | `J3` | |---|---|---|---| | 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") ``` ## 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 θ` obliquity factor, so the average transmission coefficient is ``` τ̄ij = ∫₀^{π/2} τij(θ) cos θ dθ ``` (the `cos θ` 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: `τij(θ) = cos²θ / 8`, so `τ̄ij = 1/12`; * L-junction corner: `τij(θ) = cos²θ / 2`, so `τ̄ij = 1/3`; * in-line junction: `τ12(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), `τ̄12 = χ · τ̄21`, 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 `cg_i` and junction length `L_ij` is ``` ηij = cg_i L_ij τij / (2 π² f S_i) ``` and the wave-approach **vibration reduction index** (Eq. 5.116) is ``` Kij = 10 lg(1 / τij) + 5 lg(fc_j / f_ref), f_ref = 1000 Hz ``` with `fc_j` the critical frequency of the *receiving* plate. Combined with the Eq. 5.7 reciprocity this form is symmetric, `Kij = Kji`, as EN 12354 requires of the junction descriptor. For the identical 100 mm concrete X-junction (`fc ≈ 203 Hz`), `Kij = 10 lg 12 + 5 lg(203 / 1000) ≈ 7.3 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) ``` 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:

Show the code for this figure

```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 `K12 = 9.8` 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 `Kij` already trims 1.6 dB off the direct `Rw = 57 dB`; a full building repeats this for every junction, which is the [EN 12354 prediction guide](https://jmrplens.github.io/phonometry/guides/insulation-prediction/). The measured, EN 12354 counterpart of `Kij` (from the direction-averaged velocity level difference) is the separate [flanking-transmission](https://jmrplens.github.io/phonometry/guides/insulation-field/) `vibration_reduction_index`; this guide is the closed-form *predicted* value from the wave approach. ## See also - [Predicting Sound Insulation (EN 12354)](https://jmrplens.github.io/phonometry/guides/insulation-prediction/): the flanking model that consumes `Kij` junction by junction. - [Laboratory Insulation Measurement](https://jmrplens.github.io/phonometry/guides/insulation-lab/): the ISO 10848 measurement of `Kij` 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/reference/api/building/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/reference/api/vibration/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. See the [bibliography](https://jmrplens.github.io/phonometry/reference/bibliography/) for full entries.