room.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 $R=S\bar{\alpha }/(1-\bar{\alpha })$ (Bies Equation (6.44)), with the total boundary area S and its area-weighted mean Sabine absorption alpha_bar (phonometry.room.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)):

$$L_p=L_W+10{\log }_{10}\left(\frac{Q}{4\pi r^2}+\frac{4}{R}\right)\left[+10{\log }_{10}\frac{\rho c}{400}\right]$$

with the source directivity factor Q ($Q=1$ omnidirectional, $Q=2$ 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. Dropping the direct term (distance=None) leaves $L_p=L_W+10{\log }_{10}(4/R)$, the reverberant field alone: the level far from the source, and the one a diffuse-field calculation such as a room-to-room transmission (phonometry.noise_control.room_to_room) works from. The optional $10{\log }_{10}(\rho c/400)$ 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 $r_{\mathrm{c}}=\sqrt{QR/(16\pi )}$ 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), $r_{\mathrm{c}}=\sqrt{A/(16\pi )}$ for $Q=1$) uses the Sabine absorption area $A=S\bar{\alpha }$ in place of the room constant $R=A/(1-\bar{\alpha })$; the two coincide for a small alpha_bar and differ by the factor $1-\bar{\alpha }$ otherwise. This module uses the room constant, so rc is exactly the crossover of its own steady_state_spl.

Sound power model. Both relations above assume the radiated power itself is a property of the machine. Norton & Karczub, Fundamentals of Noise and Vibration Analysis for Engineers 2nd ed., 4.6 (Table 4.5), point out that this is only one of three positions: a constant-power source radiates the same ${\Pi }_0$ wherever it stands, a constant-volume source is loaded by nearby reflecting boundaries and radiates ${\Pi }_{0}Q$ (up to +9 dB in a corner, the conservative upper bound a design estimate uses), and a constant-pressure source, a theoretical lower bound, radiates ${\Pi }_0/Q$. Real machines sit between the first two whenever the source is closer to the boundary than a wavelength. source_model selects which of the three SOURCE_POWER_MODELS is applied.

Schroeder frequency $f_{\mathrm{s}}=2000\sqrt{T/V}$ (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.

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critical_distance

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

Critical (reverberation) distance $r_{\mathrm{c}}=\sqrt{QR/(16\pi )}$.

The distance at which the direct and reverberant fields of steady_state_spl are equal (Bies Equation (6.43) crossover; Kuttruff Equation (5.44) states the $Q=1$ form with the Sabine absorption area $A=S\bar{\alpha }$ instead of the room constant R, the two differing by $1-\bar{\alpha }$).

Parameters

Name	Description
room_constant	Room constant R, m2 (scalar or per-band); from 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

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

Room constant $R=S\bar{\alpha }/(1-\bar{\alpha })$ (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.

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

schroeder_frequency

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

Schroeder frequency $f_{\mathrm{s}}=2000\sqrt{T/V}$ (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.

SOURCE_POWER_MODELS

Constant (dict).

SOURCE_POWER_MODELS = {'constant_power': 0.0, 'constant_volume': 1.0, 'constant_pressure': -1.0}

steady_state_field

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 $10{\log }_{10}(\rho c/400)$ term (None omits it).

Returns: A SteadyFieldResult.

steady_state_spl

steady_state_spl(
    sound_power_level: ArrayLike,
    distance: ArrayLike | None,
    room_constant: ArrayLike,
    *,
    directivity: float = 1.0,
    source_model: str = 'constant_power',
    characteristic_impedance: float | None = None,
) -> np.ndarray | float

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

$$L_p=L_W+10{\log }_{10}\left(\frac{Q}{4\pi r^2}+\frac{4}{R}\right)$$

plus the optional $10{\log }_{10}(\rho c/400)$ characteristic-impedance term. The bracket sums the direct field $Q/(4\pi r^2)$ and the (position-independent) reverberant field $4/R$. Passing distance=None drops the direct term and leaves the reverberant field alone, $L_p=L_W+10{\log }_{10}(4/R)$: the limit far from the source, and the level a diffuse-field calculation such as a room-to-room transmission uses.

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), or None for the reverberant field alone.
room_constant	Room constant R, m2 (scalar or per-band); from room_constant.
directivity	Source directivity factor Q (default 1; 2 on one reflecting plane, 4 in an edge, 8 in a corner).
source_model	Sound power model of Norton & Karczub 2e Table 4.5: "constant_power" (default, $\Pi ={\Pi }_0$, the position of the source does not change its radiated power), "constant_volume" ($\Pi ={\Pi }_{0}Q$, reflecting boundaries raise the radiated power by $10{\log }_{10}Q$; the conservative upper bound) or "constant_pressure" ($\Pi ={\Pi }_0/Q$, the theoretical lower bound). See SOURCE_POWER_MODELS.
characteristic_impedance	Air characteristic impedance rho c, Pa s/m. When given, the $10{\log }_{10}(\rho c/400)$ 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.

Raises

Exception	When
ValueError	If distance or room_constant is not positive and finite, or source_model is not one of SOURCE_POWER_MODELS.

SteadyFieldResult

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 $L_W+10{\log }_{10}(Q/(4\pi r^2))$ per distance, dB.
reverberant	Reverberant-field level $L_W+10{\log }_{10}(4/R)$, 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()

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.