---
title: "Bending-wave transmission at plate junctions"
description: "The wave-approach, frequency-independent bending-wave transmission coefficients for rigid X, T, L and in-line plate junctions (Cremer et al. 1973; Craik 1981/1996; Hopkins 2007, Section 5.2.1.3): the wave parameters chi and psi, the corner and straight-section coefficients tau12(theta) and tau13(theta), their diffuse-field angular average, and the derived SEA coupling loss factor and vibration reduction index Kij."
references:
  - type: book
    authors: ["Cremer, L.", "Heckl, M.", "Ungar, E. E."]
    year: 1973
    title: "Structure-borne sound: Structural vibrations and sound radiation at audio frequencies"
    edition: "1st ed."
    publisher: "Springer"
    doi: "10.1007/978-3-662-10118-6"
    note: "The original derivation of the wave parameters chi and psi (Eqs 5.10/5.11 in Hopkins) and the in-line normal-incidence transmission coefficient. ISBN 978-3-540-06002-4."
  - type: book
    authors: ["Craik, R. J. M."]
    year: 1996
    title: "Sound transmission through buildings using statistical energy analysis"
    publisher: "Gower"
    note: "The SEA treatment of building sound transmission with the tabulated bending-wave transmission coefficients for X, T, L and in-line junctions used here (Eqs 5.12/5.13). ISBN 978-0-566-07572-5."
  - type: book
    authors: ["Hopkins, C."]
    year: 2007
    title: "Sound insulation"
    publisher: "Butterworth-Heinemann"
    doi: "10.4324/9780080550473"
    note: "Section 5.2.1.3 collects the rigid X, T, L and in-line junction coefficients, the angular average (Eq. 5.6), the SEA consistency relationship (Eq. 5.7), the coupling loss factor (Eq. 2.154) and the wave-approach Kij (Eq. 5.116) implemented on this page. ISBN 978-0-7506-6526-1."
---

import ThemeImage from '../../../components/ThemeImage.astro';

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, 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
`etaij` and the vibration reduction index `Kij`.

<ThemeImage src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/junction_transmission.svg" alt="Transmission coefficient versus incidence angle for a rigid X-junction of a 100 mm and a 200 mm concrete plate, showing the corner coefficient tau12 and the straight-section coefficient tau13 falling from their normal-incidence values to zero at grazing incidence, with their diffuse-field angular averages marked as horizontal lines" width="82%" loading="eager" />

<details>
<summary>Show the code for this figure</summary>

```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()
```

</details>

## 1. Wave parameters chi and psi

With plate `i` of thickness `h_i`, quasi-longitudinal wave speed `cL_i` and
surface density `rho_s,i`, the whole family of coefficients depends on just two
dimensionless ratios (Cremer et al. 1973):

```
chi = kB2 / kB1 = (rho_s2 B1 / (rho_s1 B2))**0.25 = sqrt(h1 cL1 / (h2 cL2)) = sqrt(fc2 / fc1)
psi = B2 kB2**2 / (B1 kB1**2) = (h2 cL2 rho_s2) / (h1 cL1 rho_s1) = (rho_s2 fc1) / (rho_s1 fc2)
```

`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_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. Corner and straight-section coefficients

For an incident wave on plate 1, transmission **around the corner** (into the
perpendicular plate 2) is `tau12(theta)` (Eq. 5.12), and transmission **across
the straight section** (into the collinear plate 3, X- and T-junction (1) only)
is `tau13(theta)` (Eq. 5.13). The corner coefficient is zero beyond the cut-off,
`tau12(theta) = 0` for `chi < sin(theta)`. 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.

```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

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 `tau_bar_ij = integral 0..pi/2 of tau_ij(theta)
cos(theta) d(theta)` (Eq. 5.6). 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 `tau_bar_ij = 1/12`;
- L-junction corner: `tau_ij(theta) = cos**2(theta) / 2`, so `tau_bar_ij = 1/3`;
- in-line junction: `tau12(0 deg) = 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),
`tau_bar_12 = chi * tau_bar_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)
`etaij = cg_i L_ij tau_ij / (2 pi**2 f S_i)` and the wave-approach **vibration
reduction index** (Eq. 5.116)
`Kij = 10 lg(1 / tau_ij) + 5 lg(fc_j / f_ref)` with `f_ref = 1000 Hz` and
`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
```

The measured, EN 12354 counterpart of `Kij` (from the direction-averaged
velocity level difference) is the separate flanking-transmission
`vibration_reduction_index`; this page is the closed-form *predicted* value from
the wave approach.
