Source: https://jmrplens.github.io/phonometry/guides/room-image-sources/ # Image sources and the steady-state room field (Kuttruff / Vorländer / Bies) Where [reverberation-time prediction](https://jmrplens.github.io/phonometry/reference/api/rooms/reverberation-prediction/) gives a single statistical decay rate and [room-acoustics measurement](https://jmrplens.github.io/phonometry/reference/api/rooms/room-acoustics/) 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 res.plot() # the reflectogram of the figure below # 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. ```python import numpy as np from phonometry import room freqs = [250.0, 500.0, 1000.0, 2000.0] alpha = np.array([[0.10, 0.15, 0.25, 0.40]] * 6) # (6 walls, 4 bands) res = room.image_source_rir((7.0, 5.0, 3.0), (2.0, 1.6, 1.5), (5.2, 3.4, 1.7), alpha, fs=48000, max_order=12, frequencies=freqs) print(res.ir.shape) # (4 bands, n_samples) print(np.round(np.sum(res.ir ** 2, axis=1), 4)) # more absorption -> less energy ``` 

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import room res = room.image_source_rir((7.0, 5.0, 3.0), (2.0, 1.6, 1.5), (5.2, 3.4, 1.7), 0.12, fs=48000, max_order=10) # One line: the reflectogram (level in dB re direct vs arrival time, by order). res.plot() plt.show() # By hand: scatter the reflection amplitudes coloured by order. t_ms = np.asarray(res.times) * 1e3 amp = np.asarray(res.amplitudes) level = 20 * np.log10(np.abs(amp) / np.max(np.abs(amp))) order = np.asarray(res.orders) fig, ax = plt.subplots() sc = ax.scatter(t_ms[order > 0], level[order > 0], c=order[order > 0], cmap="viridis", s=18) ax.stem([t_ms[order == 0][0]], [0.0]) # direct sound fig.colorbar(sc, label="Reflection order") ax.set_xlabel("Arrival time [ms]"); ax.set_ylabel("Level re direct [dB]") ax.set_xlim(0, 120); ax.set_ylim(-60, 5) plt.show() ```

**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](https://jmrplens.github.io/phonometry/reference/api/rooms/reverberation-prediction/) 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 ``` *A 90 dB re 1 pW source in a 12 x 8 x 4 m workshop with a mean absorption of 0.15: within $r_c = 1.11$ m moving away drops the level 6 dB per doubling; beyond it the reverberant plateau takes over and only absorption, not distance, lowers the level (Bies 5e, §6.4).*

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import room field = room.steady_state_field( sound_power_level=90.0, # Lw, dB re 1 pW surface_area=352.0, # a 12 x 8 x 4 m workshop mean_absorption=0.15, ) # One line: direct, reverberant and total levels with rc marked. field.plot() plt.show() # By hand, from the result's fields: fig, ax = plt.subplots() ax.semilogx(field.distances, field.direct, "--", label="Direct field") ax.semilogx(field.distances, field.reverberant, ":", label="Reverberant field") ax.semilogx(field.distances, field.total, label="Total") ax.axvline(field.critical_distance, ls="-.", label=f"rc = {field.critical_distance:.2f} m") ax.set_xlabel("Distance from source [m]") ax.set_ylabel("Sound pressure level [dB]") ax.legend() plt.show() ```

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](https://jmrplens.github.io/phonometry/reference/conformance/)): - 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)).