<!-- canonical: https://docs.jmrp.io/phonometry/reference/api/vibration/junction-transmission/ -->
Source: https://docs.jmrp.io/phonometry/reference/api/vibration/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`:

$$
\chi = \frac{k_{\mathrm{B}2}}{k_{\mathrm{B}1}} = \left( \frac{\rho_{\mathrm{s}2} B_1}{\rho_{\mathrm{s}1} B_2} \right)^{0.25} = \sqrt{\frac{h_1 c_{\mathrm{L}1}}{h_2 c_{\mathrm{L}2}}} = \sqrt{\frac{f_{\mathrm{c}2}}{f_{\mathrm{c}1}}} \tag{5.10}
$$

$$
\psi = \frac{B_2 k_{\mathrm{B}2}^2}{B_1 k_{\mathrm{B}1}^2} = \frac{h_2 c_{\mathrm{L}2} \rho_{\mathrm{s}2}}{h_1 c_{\mathrm{L}1} \rho_{\mathrm{s}1}} = \frac{\rho_{\mathrm{s}2} f_{\mathrm{c}1}}{\rho_{\mathrm{s}1} f_{\mathrm{c}2}} \tag{5.11}
$$

`chi` is the ratio of bending wavenumbers (it fixes the total-internal-
reflection cut-off $\theta_\mathrm{co} = \arcsin\chi$) 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 $\chi \ge \sin\theta$,

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

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
$\chi < \sin\theta$,

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

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 $\chi \ge 1$:

$$
\tau_{12} \approx \tau_{12}(0^\circ) = \left[ \frac{2 (1 + \chi)(1 + \psi) \sqrt{\chi \psi}} {\chi (1 + \psi)^2 + 2 \psi (1 + \chi^2)} \right]^2 \tag{5.14}
$$

**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:

$$
\bar{\tau}_{ij} = \int_0^{\pi/2} \tau_{ij}(\theta) \cos\theta \,\mathrm{d}\theta \tag{5.6}
$$

(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`:

$$
\eta_{ij} = \frac{c_{\mathrm{g},i} L_{ij} \tau_{ij}}{2 \pi^2 f S_i} \tag{2.154}
$$

**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:

$$
K_{ij} = 10 \log_{10}\!\left( \frac{1}{\tau_{ij}} \right) + 5 \log_{10}\!\left( \frac{f_{\mathrm{c}j}}{f_{\text{ref}}} \right) \tag{5.116}
$$

Combined with the reciprocity relationship below
($\bar{\tau}_{12} = \chi \bar{\tau}_{21}$ with
$\chi = \sqrt{f_{\mathrm{c}2} / f_{\mathrm{c}1}}$) 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
$\bar{\tau}_{ij} = \bar{\tau}_{ji} \sqrt{h_i c_{\mathrm{L}i} / (h_j c_{\mathrm{L}j})} = \bar{\tau}_{ji} \sqrt{f_{\mathrm{c}j} / f_{\mathrm{c}i}}$,
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`](/phonometry/reference/api/vibration/experimental-sea/) is normally
compared against:

* [`right_angle_transmission_coefficient`](/phonometry/reference/api/vibration/junction-transmission/#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
  $\psi_N = \rho_1 c_{\mathrm{L}1}^{1.5} h_1^{2.5} / (\rho_2 c_{\mathrm{L}2}^{1.5} h_2^{2.5})$ (note this is a *different* `psi` from
  Hopkins Eq. 5.11 above), and the random incidence value follows from the
  empirical factor $2.754 X / (1 + 3.24 X)$ with $X = h_1/h_2$.
  Feeding it to [`coupling_loss_factor`](/phonometry/reference/api/vibration/junction-transmission/#coupling_loss_factor) reproduces Norton Eq. (6.52),
  $\eta_{12} = 2 c_\mathrm{B} L \tau_{12} / (\pi \omega S_1)$, identically:
  that equation and Hopkins Eq. (2.154) are the same expression once
  $c_\mathrm{g} = 2 c_\mathrm{B}$.
* [`point_connection_coupling_loss_factor`](/phonometry/reference/api/vibration/junction-transmission/#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

```python
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

```python
corner_transmission_coefficient(
    angle_rad: 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_rad` | 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

```python
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).

$\eta_{ij} = c_{\mathrm{g},i} L_{ij} \tau_{ij} / (2 \pi^2 f S_i)$ 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 $c_\mathrm{g} = 2 c_\mathrm{B}$ with the bending phase speed `cB` (see [`phonometry.vibration.structural.point_mobility.plate_bending_wave_speed`](/phonometry/reference/api/vibration/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, or a coefficient and a frequency whose shapes cannot be broadcast together. |

## inline_transmission_coefficient

```python
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

```python
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
$f_\mathrm{c} = \sqrt{12}\, c_0^2 / (2\pi h c_\mathrm{L})$ ($c_0 = 343$ m/s)
used by the Eq. 5.116 vibration reduction index. For the in-line junction
(normal incidence only) use [`inline_transmission_coefficient`](/phonometry/reference/api/vibration/junction-transmission/#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`](/phonometry/reference/api/vibration/junction-transmission/#junctiontransmissionresult).

**Raises**

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

## junction_wave_parameters

```python
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).

$\chi = \sqrt{h_1 c_{\mathrm{L}1} / (h_2 c_{\mathrm{L}2})}$ (Eq. 5.10) and
$\psi = (h_2 c_{\mathrm{L}2} \rho_{\mathrm{s}2}) / (h_1 c_{\mathrm{L}1} \rho_{\mathrm{s}1})$
(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

```python
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).

$K_{12} = 10 \log_{10}(1 / \bar{\tau}_{12}) + 5 \log_{10}(f_{\mathrm{c}2} / 1000)$
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()

```python
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()

```python
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

```python
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`](/phonometry/reference/api/vibration/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.

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | for an unknown junction type, or a thickness that is not positive and finite. |

## point_connection_coupling_loss_factor

```python
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:

$$
\eta_{12} = \frac{4 N h_1 c_{\mathrm{L}1}}{\sqrt{3}\, \omega S_1} \frac{A_1 A_2}{(A_1 + A_2)^2}, \qquad A_i = \rho_{\mathrm{s}i}^2 h_i^2 c_{\mathrm{L}i}^2 \tag{6.56}
$$

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`](/phonometry/reference/api/vibration/junction-transmission/#coupling_loss_factor) on
[`right_angle_transmission_coefficient`](/phonometry/reference/api/vibration/junction-transmission/#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

```python
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:

$$
\psi_N = \frac{\rho_1 c_{\mathrm{L}1}^{1.5} h_1^{2.5}} {\rho_2 c_{\mathrm{L}2}^{1.5} h_2^{2.5}} \tag{6.54}
$$

$$
\tau_{12}(0) = 2 \left( \sqrt{\psi_N} + \frac{1}{\sqrt{\psi_N}} \right)^{-2} \tag{6.53}
$$

$$
\tau_{12} = \tau_{12}(0)\, \frac{2.754 X}{1 + 3.24 X}, \qquad X = h_1/h_2 \tag{6.55}
$$

`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`](/phonometry/reference/api/vibration/junction-transmission/#coupling_loss_factor) with the source plate's group velocity
$c_\mathrm{g} = 2 c_\mathrm{B}$ to obtain Norton Eq. (6.52).

`rho` here is the **volume** density in kg/m^3, not the surface density
of [`junction_wave_parameters`](/phonometry/reference/api/vibration/junction-transmission/#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

```python
straight_transmission_coefficient(
    angle_rad: 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_rad` | 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

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

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

$K_{ij} = 10 \log_{10}(1 / \tau_{ij}) + 5 \log_{10}(f_{\mathrm{c}j} / f_{\text{ref}})$
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
$\bar{\tau}_{ij} = \bar{\tau}_{ji} \sqrt{f_{\mathrm{c}j} / f_{\mathrm{c}i}}$
(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`. |