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Source: https://jmrplens.github.io/phonometry/reference/api/rooms/steady-field/

Steady-state sound field in a room: room constant, critical distance, level.

When a source of constant sound power runs in a room, the sound pressure level
at a receiver settles to a steady value made of two parts: the **direct field**
that falls with distance as any free-field source does, and the **reverberant
field** built up by the many wall reflections, which is (to the diffuse-field
approximation) the same everywhere in the room. This module gives the classical
statistical-acoustics relations between the source power, the room's absorption
and the received level (Bies, Hansen & Howard, *Engineering Noise Control* 5th
ed., 6.4; Kuttruff, *Room Acoustics* 6th ed., 5.6), the bridge between the sound
power of `phonometry.emission` and the received level indoors.

**Room constant** `R = S alpha_bar / (1 - alpha_bar)` (Bies Equation (6.44)),
with the total boundary area `S` and its area-weighted mean Sabine absorption
`alpha_bar` ([`phonometry.room.mean_absorption`](/phonometry/reference/api/rooms/reverberation-prediction/#mean_absorption)). `R` has units of
area and measures how much reverberant field a given power builds up: a live
room (small `alpha_bar`) has a small `R` and a loud reverberant field, a
dead room a large `R`.

**Steady-state level** (Bies Equation (6.43)):

    Lp = Lw + 10 log10( Q / (4 pi r^2) + 4 / R )   [ + 10 log10(rho c / 400) ]

with the source directivity factor `Q` (`= 1` omnidirectional, `2` on a
hard floor, ...), the distance `r` and the room constant `R`. The first
term inside the bracket is the direct field, the second the reverberant field.
The optional `10 log10(rho c / 400)` term corrects for a characteristic
impedance `rho c` differing from the reference 400 Pa s/m; it is about
`+0.14 dB` at 20 degC and is omitted by default (Bies notes the `~0.1 dB`
it contributes).

**Critical distance** `rc = sqrt(Q R / (16 pi))` is where the direct and
reverberant terms are equal (setting `Q / (4 pi r^2) = 4 / R` in Equation
(6.43)); closer than `rc` the direct field dominates, farther the reverberant
field does. Kuttruff's reverberation distance (Equation (5.44),
`rc = sqrt(A / 16 pi)` for `Q = 1`) uses the Sabine absorption area
`A = S alpha_bar` in place of the room constant `R = A / (1 - alpha_bar)`;
the two coincide for a small `alpha_bar` and differ by the factor
`1 - alpha_bar` otherwise. This module uses the room constant, so `rc` is
exactly the crossover of its own [`steady_state_spl`](/phonometry/reference/api/rooms/steady-field/#steady_state_spl).

**Schroeder frequency** `f_s = 2000 sqrt(T / V)` (Kuttruff Equation (3.44),
`V` in cubic metres, `T` in seconds) marks the boundary between the
modal low-frequency regime -- where discrete room modes rule and the diffuse
assumption of `R` and `rc` fails -- and the high-frequency regime of
overlapping modes where the statistical field of this module applies.

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

## critical_distance

```python
critical_distance(
    room_constant: ArrayLike,
    *,
    directivity: float = 1.0,
) -> np.ndarray | float
```

Critical (reverberation) distance `rc = sqrt(Q R / (16 pi))`.

The distance at which the direct and reverberant fields of
[`steady_state_spl`](/phonometry/reference/api/rooms/steady-field/#steady_state_spl) are equal (Bies Equation (6.43) crossover;
Kuttruff Equation (5.44) states the `Q = 1` form with the Sabine
absorption area `A = S alpha_bar` instead of the room constant `R`,
the two differing by `1 - alpha_bar`).

**Parameters**

| Name | Description |
| :--- | :--- |
| `room_constant` | Room constant `R`, m2 (scalar or per-band); from [`room_constant`](/phonometry/reference/api/rooms/steady-field/#room_constant). |
| `directivity` | Source directivity factor `Q` (`1` omnidirectional, `2` on one reflecting plane, `4` in an edge, `8` in a corner). |

**Returns:** The critical distance `rc`, m.

## room_constant

```python
room_constant(
    surface_area: float,
    mean_absorption: ArrayLike,
) -> np.ndarray | float
```

Room constant `R = S alpha_bar / (1 - alpha_bar)` (Bies Equation (6.44)).

**Parameters**

| Name | Description |
| :--- | :--- |
| `surface_area` | Total boundary area `S` of the room, m2. |
| `mean_absorption` | Area-weighted mean Sabine absorption `alpha_bar` in `(0, 1)` (scalar or per-band); e.g. from [`phonometry.room.mean_absorption`](/phonometry/reference/api/rooms/reverberation-prediction/#mean_absorption). |

**Returns:** The room constant `R`, m2; a float for a scalar input, otherwise a per-band array.

## schroeder_frequency

```python
schroeder_frequency(
    reverberation_time: ArrayLike,
    volume: float,
) -> np.ndarray | float
```

Schroeder frequency `f_s = 2000 sqrt(T / V)` (Kuttruff Equation (3.44)).

The frequency above which room modes overlap (on average three
eigenfrequencies per resonance half-width) so the statistical, diffuse
field of this module applies; below it the sound field is ruled by discrete
modes and `R` / `rc` lose their meaning.

**Parameters**

| Name | Description |
| :--- | :--- |
| `reverberation_time` | Reverberation time `T`, s (scalar or per-band). |
| `volume` | Room volume `V`, m3. |

**Returns:** The Schroeder frequency, Hz.

## steady_state_field

```python
steady_state_field(
    sound_power_level: float,
    surface_area: float,
    mean_absorption: float,
    *,
    distances: ArrayLike | None = None,
    directivity: float = 1.0,
    characteristic_impedance: float | None = None,
) -> SteadyFieldResult
```

Steady-state SPL versus distance for one source in a room (Bies 6.4).

Builds the room constant from `surface_area` and `mean_absorption`
(Bies Equation (6.44)), then evaluates the direct, reverberant and combined
fields (Equation (6.43)) over a distance grid together with the critical
distance (crossover of the two fields).

**Parameters**

| Name | Description |
| :--- | :--- |
| `sound_power_level` | Source sound power level `Lw`, dB re 1 pW. |
| `surface_area` | Total boundary area `S`, m2. |
| `mean_absorption` | Mean Sabine absorption `alpha_bar` in `(0, 1)`. |
| `distances` | Distance grid `r`, m; default 30 points log-spaced from one tenth of the critical distance to ten times it. |
| `directivity` | Source directivity factor `Q` (default 1). |
| `characteristic_impedance` | Optional `rho c` for the Bies `10 log10(rho c / 400)` term (`None` omits it). |

**Returns:** A [`SteadyFieldResult`](/phonometry/reference/api/rooms/steady-field/#steadyfieldresult).

## steady_state_spl

```python
steady_state_spl(
    sound_power_level: ArrayLike,
    distance: ArrayLike,
    room_constant: ArrayLike,
    *,
    directivity: float = 1.0,
    characteristic_impedance: float | None = None,
) -> np.ndarray | float
```

Steady-state sound pressure level in a room (Bies Equation (6.43)).

`Lp = Lw + 10 log10( Q / (4 pi r^2) + 4 / R )` (plus the optional
`10 log10(rho c / 400)` characteristic-impedance term). The bracket sums
the direct field `Q / (4 pi r^2)` and the (position-independent)
reverberant field `4 / R`.

**Parameters**

| Name | Description |
| :--- | :--- |
| `sound_power_level` | Source sound power level `Lw`, dB re 1 pW (scalar or per-band); e.g. from `phonometry.emission`. |
| `distance` | Source-receiver distance `r`, m (scalar or array). |
| `room_constant` | Room constant `R`, m2 (scalar or per-band); from [`room_constant`](/phonometry/reference/api/rooms/steady-field/#room_constant). |
| `directivity` | Source directivity factor `Q` (default 1). |
| `characteristic_impedance` | Air characteristic impedance `rho c`, Pa s/m. When given, the `10 log10(rho c / 400)` term is added (about `+0.14 dB` at 20 degC where `rho c = 413`); `None` (default) omits it, matching the common textbook form. |

**Returns:** The steady-state SPL `Lp`, dB; a float for scalar inputs, otherwise an array broadcasting `sound_power_level`, `distance` and `room_constant`.

## SteadyFieldResult

```python
SteadyFieldResult(
    distances: np.ndarray,
    direct: np.ndarray,
    reverberant: np.ndarray,
    total: np.ndarray,
    critical_distance: float,
    room_constant: float,
    sound_power_level: float,
    directivity: float,
)
```

Steady-state SPL versus distance in a room, split direct / reverberant.

**Attributes**

| Name | Description |
| :--- | :--- |
| `distances` | Source-receiver distances `r`, m. |
| `direct` | Direct-field level `Lw + 10 log10(Q / (4 pi r^2))` per distance, dB. |
| `reverberant` | Reverberant-field level `Lw + 10 log10(4 / R)`, dB (constant across distance; broadcast to the distance grid). |
| `total` | Combined steady-state level (Bies Equation (6.43)), dB. |
| `critical_distance` | Critical distance `rc`, m, where direct equals reverberant. |
| `room_constant` | Room constant `R`, m2. |
| `sound_power_level` | Source sound power level `Lw`, dB re 1 pW. |
| `directivity` | Source directivity factor `Q`. |

### SteadyFieldResult.plot()

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

Plot direct, reverberant and total SPL against distance.

Marks the critical distance `rc` where the direct and reverberant
fields cross. Requires matplotlib (`pip install phonometry[plot]`);
returns the `Axes`.