← [Documentation index](README.md)

# Image sources and the steady-state room field (Kuttruff / Vorländer / Bies)

Where [reverberation-time prediction](reverberation-prediction.md) gives a
single statistical decay rate and [room-acoustics measurement](room-acoustics.md)
analyses a *measured* impulse response, this page covers the two classical
*predictions* of the sound field in a rectangular room, both in
`phonometry.room`:

- the **image-source room impulse response** (`image_source_rir`): the
  deterministic early reflection pattern built by mirroring a point source in
  the six walls of a shoebox (Kuttruff *Room Acoustics* 4.1; Vorländer
  *Auralization* 11.4); and
- the **steady-state room field** (`steady_state_field` and its parts): the
  statistical direct-plus-reverberant sound pressure level a source of known
  power produces, with the room constant, critical distance and Schroeder
  frequency (Bies *Engineering Noise Control* 6.4; Kuttruff 5.6).

Together they bridge the sound power of `phonometry.emission` and the reverberation
prediction of `phonometry.room`: one gives the full RIR, the other the level
the same room settles to.

## 1. Image-source room impulse response

A rigid or absorbing rectangular room reflects a point source in its walls;
each reflection is exactly the free-field sound of a **mirror image** of the
source. Mirroring a coordinate in a wall (Vorländer Equation (11.36),
`S_n = S − 2 d n`) turns the source into a regular lattice of images, and the
room impulse response is the sum of the direct sound and one delayed,
attenuated impulse per image (Kuttruff Equations (4.4)–(4.5),
`g(t) = Σ_n A_n δ(t − t_n)`).

Image `i` at distance `r_i` from the receiver arrives at `t_i = r_i / c`
(Vorländer Equation (11.38)) with amplitude

```text
A_i = [ Π_walls R_wall^(reflections there) ] · exp(−m r_i / 2) / (4 π r_i)
```

combining the `1 / (4 π r_i)` spherical spreading, the product of the wall
**pressure reflection factors** `R = √(1 − α)` (Vorländer Equation (11.39);
`|R|² = 1 − α` in energy, Kuttruff 4.1) each raised to the number of
reflections that image made off that wall, and the air pressure attenuation
`exp(−m r_i / 2)` over the path (Kuttruff 4.1; `m` the *intensity* attenuation
constant, so intensity falls as `exp(−m r)`).

Along one axis the reflection count off the two walls of an image at lattice
index `n` and mirror parity `p` is `|n − p|` (wall at 0) and `|n|` (wall at
`L`), so the total reflection order is
`|2 n_x − p_x| + |2 n_y − p_y| + |2 n_z − p_z|` (Allen & Berkley 1979). A
shoebox has exactly `(2/3)(2 i₀³ + 3 i₀² + 4 i₀)` audible images up to order
`i₀` (Kuttruff Equation (9.23), e.g. 1560 at order 10), and the temporal
density of reflections grows as `dN/dt = 4 π c³ t² / V` (Kuttruff Equation
(4.6)).

```python
import numpy as np
from phonometry import room

# A 7 x 5 x 3 m room, source and receiver placed off-centre.
res = room.image_source_rir(
    dimensions=(7.0, 5.0, 3.0),
    source=(2.0, 1.6, 1.5),
    receiver=(5.2, 3.4, 1.7),
    absorption=0.12,            # uniform wall absorption
    fs=48000,
    max_order=12,
)

print(res.ir.shape)                          # (n_samples,) broadband RIR
print(round(res.direct_time * 1000, 2))      # direct-sound arrival, ms
print(res.times.size, room.audible_image_count(12) + 1)  # images + direct source

# Feed the synthetic RIR straight into the ISO 3382 decay analysis.
params = room.room_parameters(res.ir, res.fs, limits=None)
print(bool(params.t30_valid[0]))             # True: the decay window is usable
# T30 rises toward the Eyring estimate as max_order grows (see below); at a low
# order the specular tail is truncated, so treat this as an order-limited value.
print(round(float(params.t30[0]), 2))        # reverberation time, s
```

`image_source_rir` returns an `ImageSourceResult`. Its `ir` is the sampled
RIR (a 1D array broadband, or one row per octave band for per-band
absorption); the **exact** sub-sample reflection table is kept separately in
`times`, `distances`, `orders`, `amplitudes` and `image_positions`, so the
geometry stays exact regardless of the sample rate. `.plot()` draws the
reflectogram (reflection level in dB versus arrival time, coloured by order).

Pass per-band coefficients (a `(6, n_bands)` per-wall array, a per-band vector,
or a `frequencies` list) to synthesise one decay per octave band; pass a
length-6 vector to set each wall separately (order
`x0, xL, y0, yL, z0, zL`); and pass `air_attenuation` (the intensity
coefficient `m` from `air_attenuation_m`) to add the `exp(−m r / 2)` air loss.

![Image-source reflectogram: the synthetic room impulse response of a 7x5x3 m room as a cloud of reflections coloured by reflection order, decaying under the 1/r spreading envelope with the direct sound marked at order 0](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/image_source_reflectogram.webp)

**Reproducing the statistical decay.** The fitted initial decay slope of the
reverberant energy density of the synthetic RIR recovers the **Eyring**
reverberation time `T = −24 V ln 10 / (c S ln(1 − ᾱ))` (Kuttruff Equation
(5.23)): the mean reflection rate `c S / 4 V` equals
`(c/2)(1/Lx + 1/Ly + 1/Lz)`, so the specular field's initial decay rate is the
one that defines that `T`. The match is exact only in the near-cubic
limit; an elongated room sustains energy along its long axis, so its pure
*specular* decay runs slower than Eyring's diffuse-field estimate (the regime
the [Fitzroy and Arau-Puchades models](reverberation-prediction.md) correct).
The model captures specular reflections only, with no diffraction or diffuse
scattering, and is exact only for real, angle-independent wall reflection
factors (Kuttruff 4.1).

## 2. Steady-state room field

When a source of constant sound power runs in a room, the sound pressure level
settles to the sum of a **direct field** that falls with distance and a
**reverberant field** that (to the diffuse approximation) is the same
everywhere. The **room constant**

```text
R = S ᾱ / (1 − ᾱ)                       (Bies Equation (6.44))
```

with total boundary area `S` and mean Sabine absorption `ᾱ` measures how much
reverberant field a given power builds up. The **steady-state level** is

```text
Lp = Lw + 10 log10( Q / (4 π r²) + 4 / R )   [ + 10 log10(ρc / 400) ]
                                               (Bies Equation (6.43))
```

with the source directivity factor `Q` (1 omnidirectional, 2 on a hard floor,
4 in an edge, 8 in a corner). The optional `10 log10(ρc / 400)` term
(about +0.14 dB at 20 °C) corrects for a characteristic impedance differing
from 400 Pa·s/m and is omitted by default. The **critical distance**

```text
rc = √( Q R / (16 π) )
```

is where the two fields are equal (the crossover of Equation (6.43)); closer
than `rc` the direct field dominates, farther the reverberant field does.
Kuttruff's reverberation distance (Equation (5.44), `rc = √(A / 16 π)` for
`Q = 1`) uses the Sabine absorption area `A = S ᾱ` instead of the room
constant `R = A / (1 − ᾱ)`; the two coincide for a small `ᾱ` and this module
uses `R`, so `rc` is exactly the crossover of its own `steady_state_spl`.

```python
from phonometry import room

# A 90 dB source in a 100 m^2 room with 20 % mean absorption.
field = room.steady_state_field(
    sound_power_level=90.0,
    surface_area=100.0,
    mean_absorption=0.2,
)
print(round(field.room_constant, 1))          # 25.0 m^2
print(round(field.critical_distance, 2))       # 0.71 m
field.plot()                                    # direct / reverberant / total vs distance

# The building blocks are exposed individually, too:
print(round(float(room.room_constant(100.0, 0.2)), 1))            # 25.0
print(round(float(room.critical_distance(25.0)), 3))              # 0.705
print(round(float(room.steady_state_spl(90.0, 5.0, 25.0)), 2))    # far-field level
```

The **Schroeder frequency**

```text
f_s = 2000 √(T / V)                       (Kuttruff Equation (3.44))
```

(`V` in m³, `T` in s) roughly marks the modal-to-diffuse transition, a
heuristic crossover rather than a sharp cutoff: well below it discrete room
modes dominate and the diffuse assumptions of `R` and `rc` grow unreliable,
well above it the modes overlap and the statistical field of this section
holds. In borderline rooms it is worth checking band by band.

```python
from phonometry import room
print(round(float(room.schroeder_frequency(1.0, 200.0)), 0))   # 141 Hz (V=200, T=1)
```

## Validation

The implementations are checked against the closed forms and the source texts'
own numeric anchors (see [CONFORMANCE.md](CONFORMANCE.md)):

- the direct-sound amplitude `1/(4 π r)` and delay `r / c` (exact geometry),
  the audible image count (Kuttruff Equation (9.23)) and the reflection
  density (Equation (4.6));
- the Eyring reverberation time recovered from the decay of the synthetic RIR
  in the near-cubic limit (documented ≈ 10 % tolerance), and an independent
  2D FDTD (`phonometry.simulation`) reproducing the rigid-wall echo delay and
  the uniform-damping `T60`;
- the room constant, the critical distance as the exact direct/reverberant
  crossover, the Schroeder frequency (Kuttruff's classroom example, `V = 200`,
  `T = 1` → 141 Hz) and the steady-state level (Bies Equation (6.43)).

## References

- Kuttruff, H. (2016). *Room acoustics* (6th ed.). CRC Press.
  [doi:10.1201/9781315372150](https://doi.org/10.1201/9781315372150).
  Section 4.1 (image sources, Equations (4.4)–(4.6)), Section 5.5–5.6
  (Eyring reverberation, reverberation distance, Equations (5.23), (5.44)),
  Section 3.6 (Schroeder frequency, Equation (3.44)) and Section 9.8 (audible
  image count, Equation (9.23)).
- Vorländer, M. (2020). *Auralization: Fundamentals of acoustics, modelling,
  simulation, algorithms and acoustic virtual reality* (2nd ed.). Springer.
  [doi:10.1007/978-3-030-51202-6](https://doi.org/10.1007/978-3-030-51202-6).
  Chapter 11 (the image-source / mirror-source model, Equations (11.36),
  (11.38), (11.39)).
- Allen, J. B., & Berkley, D. A. (1979). Image method for efficiently
  simulating small-room acoustics. *The Journal of the Acoustical Society of
  America*, 65(4), 943–950.
  [doi:10.1121/1.382599](https://doi.org/10.1121/1.382599).
  The reflection-count decomposition of the rectangular-room image lattice.
- Bies, D. A., Hansen, C. H., & Howard, C. Q. (2017). *Engineering noise
  control* (5th ed.). CRC Press.
  [doi:10.1201/9781351228152](https://doi.org/10.1201/9781351228152).
  Section 6.4 (steady-state response and the room constant, Equations
  (6.41)–(6.44)).
