Source: https://jmrplens.github.io/phonometry/buildings/rooms/room-image-sources/ # Image sources and the steady-state room field (Kuttruff / Vorländer / Bies) Where [reverberation-time prediction](https://jmrplens.github.io/phonometry/buildings/rooms/reverberation-prediction/) gives a single statistical decay rate and [room-acoustics measurement](https://jmrplens.github.io/phonometry/buildings/rooms/room-acoustics/) analyses a *measured* impulse response, this page covers the classical *predictions* of the sound field in a rectangular room, all 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); and - the **normal modes** of the shoebox (`room_modes` and its parts): the eigenfrequencies below the Schroeder frequency, their axial / tangential / oblique classification and the Morse/Pierce mode count and modal density (Long *Architectural Acoustics* 2e, Chapter 8). 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), $S_n = S - 2 d\,\mathbf{n}$) 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), $g(t) = \sum_n A_n\,\delta(t - t_n)$). Image $i$ at distance $r_i$ from the receiver arrives at $t_i = r_i / c$ (Vorländer Equation (11.38)) with amplitude $$ A_i = \left[ \prod_\text{walls} R_\text{wall}^{\,\text{(reflections there)}} \right] \cdot e^{-m r_i / 2} / (4 \pi r_i) $$ combining the $1 / (4 \pi r_i)$ spherical spreading, the product of the wall **pressure reflection factors** $R = \sqrt{1 - \alpha}$ (Vorländer Equation (11.39); $|R|^2 = 1 - \alpha$ in energy, Kuttruff 4.1) each raised to the number of reflections that image made off that wall, and the air pressure attenuation $e^{-m r_i / 2}$ over the path (Kuttruff 4.1; $m$ the *intensity* attenuation constant, so intensity falls as $e^{-m r}$). 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 $\tfrac{2}{3}(2 i_0^3 + 3 i_0^2 + 4 i_0)$ audible images up to order $i_0$ (Kuttruff Equation (9.23), e.g. 1560 at order 10), and the temporal density of reflections grows as $\mathrm{d}N/\mathrm{d}t = 4 \pi c^3 t^2 / V$ (Kuttruff Equation (4.6)). The construction in one plan: every wall reflection is the straight-line sound of an image in a mirror room, so the geometry alone fixes each arrival. In the 7 × 5 m room below the direct sound lands at 10.7 ms and the four first-order lateral images in plan follow between 17.3 and 21.6 ms; the ceiling and floor images of the 3-D room are omitted from the drawing. ```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() ```

Every dot of the reflectogram is a mirror image at a definite place, and `.plot_geometry()` shows where: the plan view below draws the room, the source, the receiver and every image up to third order on the mirror-room grid, coloured by reflection order. *The lattice the reflectogram comes from, to scale: each image sits in a mirror room at the height plane of the source, and its distance to the receiver alone fixes the arrival time and the $1/(4\pi r)$ spreading of that reflection.*

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```python import matplotlib.pyplot as plt 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), 0.3, fs=16000, max_order=3) # One line: the image lattice in plan, coloured by reflection order. res.plot_geometry() plt.show() ```

### `image_source_rir()` parameters | Parameter | Type | Units | Range / default | Notes | | :--- | :--- | :--- | :--- | :--- | | `dimensions` | (float, float, float) | m | all > 0 | Room lengths `(Lx, Ly, Lz)` | | `source` / `receiver` | (float, float, float) | m | strictly inside the room | Positions `(x, y, z)` | | `absorption` | scalar / (6,) / (n,) / (6, n) | -- | `[0, 1]` | Uniform, per-wall, per-band, or per-wall per-band | | `fs` | int | Hz | > 0 | Sample rate | | `max_order` | int | -- | >= 0, default 20 | Reflection-order cut-off | | `speed_of_sound` | float | m/s | > 0, default 343 | Speed of sound `c` | | `air_attenuation` | float or (n,) | Np/m | >= 0, default 0 | Air power (intensity) attenuation coefficient `m` | | `duration` | float, optional | s | > 0 | RIR length (default: last image arrival) | | `frequencies` | (n,), optional | Hz | -- | Band centres labelling a per-band result | Returns an `ImageSourceResult` (`ir`, `fs`, `frequencies`, and the exact `times`/`distances`/`orders`/`amplitudes`/`image_positions` reflection table) with `.plot()`, `.plot_geometry()` and a `direct_time` property. `audible_image_count(order)` gives the shoebox image count and `reflection_density(t, volume)` the density $4\pi c^3 t^2 / V$. **Reproducing the statistical decay.** The fitted initial decay slope of the reverberant energy density of the synthetic RIR recovers the **Eyring** reverberation time $T = -24 V \ln 10 / (c S \ln(1 - \bar\alpha))$ (Kuttruff Equation (5.23)): the mean reflection rate $c S / 4 V$ equals $\tfrac{c}{2}(1/L_x + 1/L_y + 1/L_z)$, 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/buildings/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** $$ R = S \bar\alpha / (1 - \bar\alpha) \quad \text{(Bies Equation (6.44))} $$ with total boundary area $S$ and mean Sabine absorption $\bar\alpha$ measures how much reverberant field a given power builds up. The **steady-state level** is $$ L_p = L_W + 10 \log_{10}\!\left( \frac{Q}{4 \pi r^2} + \frac{4}{R} \right) \left[ + 10 \log_{10}\frac{\rho c}{400} \right] \quad \text{(Bies Equation (6.43))} $$ with the source directivity factor $Q$ (1 omnidirectional, 2 on a hard floor, 4 in an edge, 8 in a corner). The optional $10 \log_{10}(\rho c / 400)$ 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** $$ r_\mathrm{c} = \sqrt{Q R / (16 \pi)} $$ is where the two fields are equal (the crossover of Equation (6.43)); closer than $r_\mathrm{c}$ 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 $A = S \bar\alpha$ instead of the room constant $R = A / (1 - \bar\alpha)$; the two coincide for a small $\bar\alpha$ and this module uses $R$, so $r_\mathrm{c}$ 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 $r_\mathrm{c} = 1.11$ 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() ```

### `steady_state_field()` parameters | Parameter | Type | Units | Range / default | Notes | | :--- | :--- | :--- | :--- | :--- | | `sound_power_level` | float | dB re 1 pW | -- | Source power level `Lw` | | `surface_area` | float | m2 | > 0 | Total boundary area `S` | | `mean_absorption` | float | -- | `(0, 1)` | Mean Sabine absorption `alpha_bar` | | `distances` | 1D array, optional | m | > 0 | Distance grid (default: `0.1 rc` to `10 rc`) | | `directivity` | float | -- | > 0, default 1 | Source directivity factor `Q` | | `characteristic_impedance` | float, optional | Pa*s/m | > 0 | Adds the `10 lg(rho c / 400)` term | Returns a `SteadyFieldResult` (`distances`, `direct`, `reverberant`, `total`, `critical_distance`, `room_constant`) with `.plot()`. The pieces `room_constant`, `critical_distance`, `schroeder_frequency` and `steady_state_spl` are also callable directly, and each accepts per-band arrays. `steady_state_spl` additionally takes `source_model`, one of `constant_power` (the default), `constant_volume` or `constant_pressure`: whether the mounting that raises `Q` also raises the radiated power is a modelling choice worth up to 18 dB for a corner source, and it should be stated in any report. The **Schroeder frequency** $$ f_\mathrm{s} = 2000 \sqrt{T / V} \quad \text{(Kuttruff Equation (3.44))} $$ ($V$ 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 $r_\mathrm{c}$ 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) ``` ## 3. Modes of a rectangular room Below the Schroeder frequency the statistical picture of section 2 breaks down and the room is a handful of discrete standing waves. For the rigid-walled shoebox the wave equation separates and the eigenfrequencies are exact: $$ f(n_x, n_y, n_z) = \frac{c_0}{2} \sqrt{\left(\frac{n_x}{l_x}\right)^2 + \left(\frac{n_y}{l_y}\right)^2 + \left(\frac{n_z}{l_z}\right)^2} \quad \text{(Long Equation (8.43))} $$ with non-negative integer orders counting the nodal planes on each axis. How many of the three orders are non-zero names the family: **axial** (one, a wave bouncing between one pair of walls, the loudest), **tangential** (two, grazing four walls, about 3 dB weaker) and **oblique** (three, involving all six, weaker still). ```python from phonometry import room modes = room.room_modes( (7.0, 5.0, 3.0), # lx, ly, lz in metres max_frequency=100.0, speed_of_sound=344.0, reverberation_time=0.8, # optional: carries the Schroeder frequency ) print(modes.orders[0], round(float(modes.frequencies[0]), 1)) # [1 0 0] 24.6 Hz print(modes.count_by_kind()) # {'axial': 7, 'tangential': 10, 'oblique': 4} print(round(modes.schroeder_frequency, 0)) # 175 Hz modes.plot() # mode ladder by family + modal density ``` Counting lattice points inside the positive octant of a sphere of radius $k = 2 \pi f / c_0$, with the half- and quarter-weight corrections for the points on the coordinate planes and axes, gives the smooth **integrated mode count** and its derivative the **modal density**: $$ \begin{aligned} N(f) &= \frac{4\pi}{3} V \left(\frac{f}{c_0}\right)^3 + \frac{\pi}{4} S \left(\frac{f}{c_0}\right)^2 + \frac{L}{8} \frac{f}{c_0} && \text{(Long Equation (8.45), after Morse and Pierce)} \\ \frac{\mathrm{d}N}{\mathrm{d}f} &= \frac{4\pi V f^2}{c_0^3} + \frac{\pi S f}{2 c_0^2} + \frac{L}{8 c_0} && \text{(Long Equation (8.46))} \end{aligned} $$ with the volume $V$, the total wall area $S$ and the sum $L$ of the twelve edge lengths. These are asymptotic estimates: below a few dozen modes the exact enumeration of `room_modes` is the honest answer, while high up they are accurate and much cheaper. ```python from phonometry import room room_dims = (7.0, 5.0, 3.0) print(round(float(room.room_mode_count(200.0, room_dims, speed_of_sound=344.0)), 1)) # 128.5, against 128 modes actually enumerated below 200 Hz print(round(float(room.room_modal_density(1000.0, room_dims, speed_of_sound=344.0)), 1)) # 34.3 modes/Hz ``` *The 7 x 5 x 3 m room of Long's Table 8.1 up to 200 Hz. Below about 60 Hz the axial modes stand alone and each is separately audible; by the 175 Hz Schroeder frequency ($T = 0.8$ s) the oblique family has filled in and the modal density has passed one mode per hertz, which is where the statistical field of section 2 takes over.* ### `room_modes()` parameters | Parameter | Type | Units | Range / default | Notes | | :--- | :--- | :--- | :--- | :--- | | `dimensions` | 3-tuple | m | > 0 | Room dimensions `(lx, ly, lz)` | | `max_frequency` | float | Hz | > 0, default 200 | Highest mode enumerated | | `speed_of_sound` | float | m/s | > 0, default 343 | `c0` | | `reverberation_time` | float, optional | s | > 0 | Adds the Schroeder frequency to the result | Returns a `RoomModesResult` (`orders`, `frequencies`, `kinds`, `dimensions`, `speed_of_sound`, `schroeder_frequency`, plus the `volume`, `surface_area` and `edge_length` properties and `count_by_kind()`) with `.plot()`. The pieces `room_mode_frequency`, `room_mode_count` and `room_modal_density` are callable directly. **Reading the ladder.** Two things matter more than any single frequency. Modes that coincide colour the sound: if the dimensions are low integer multiples of one another the energy coalesces into a few frequencies, which is why a cubic room is the worst possible listening room and why the ratio tables of Bolt and others exist. And modes crowd together as frequency rises, until they merge into the continuum the statistical model of section 2 assumes. The mode calculator assumes rigid walls and a rectangular plan: it gives the eigen*frequencies*, not the pressure amplitude at a listening position, and says nothing about damping, mode shape or a non-shoebox geometry. For those, mesh the room and run the [2D FDTD solver](https://jmrplens.github.io/phonometry/simulation/fdtd-simulation/). ## 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 $1/(4 \pi r)$ 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 $T_{60}$; - 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)); - the six modes Long prints for his 7 x 5 x 3 m room (Table 8.1: 24.6, 34.5, 42.4, 49.2, 57.4 and 60.1 Hz, reproduced within one unit of the last printed digit at his $c_0 = 344$ m/s) and the 34 modes per hertz he states for the same room at 1 kHz (Equation (8.46)), plus the mode count matched against the exact enumeration and the degeneracy of a cubic room. ## What this guide covers **Covered.** Kuttruff's image-source construction (§4.1), the Eyring formula used for the near-cubic check and the Schroeder frequency of §3.6; Vorländer's mirror-source model with its reflection-factor and delay expressions (Chapter 11); and the Allen & Berkley reflection-order decomposition — all in `image_source_rir`. The Bies §6.4 steady-state field (room constant, directivity $Q$, critical distance) in `steady_state_field`, `room_constant`, `critical_distance`, `steady_state_spl` and `schroeder_frequency`. Long's Chapter 8 rectangular-room eigenfrequencies (Eq. 8.43) with their axial/tangential/oblique classification and the Morse/Pierce mode count and modal density (Eqs. 8.45, 8.46) in `room_modes`, `room_mode_frequency`, `room_mode_count` and `room_modal_density`. **Not covered.** Specular reflection only: no diffraction, no diffuse scattering. An elongated room therefore decays more slowly than the Eyring estimate the model is checked against — the anisotropy the Fitzroy and Arau-Puchades models of [reverberation-time prediction](https://jmrplens.github.io/phonometry/buildings/rooms/reverberation-prediction/) exist to correct. Kuttruff's reverberation distance, which uses the Sabine absorption area rather than the room constant, is quoted for comparison and not implemented: `steady_state_field` and `critical_distance` are always the Bies room-constant form. The mode calculator assumes rigid walls and a rectangular plan and returns eigen*frequencies* only — no amplitude at a listening position, no damping, no mode shape, no other geometry; for those, mesh the room and run the [2D FDTD solver](https://jmrplens.github.io/phonometry/simulation/fdtd-simulation/). Bolt's preferred dimension ratios are discussed, not tabulated. ## See also - [Room Acoustics](https://jmrplens.github.io/phonometry/buildings/rooms/room-acoustics/): the *measured* impulse response (ISO 18233) and the ISO 3382 parameters the synthetic RIR feeds. - [Reverberation-time prediction (Sabine, Eyring, Arau)](https://jmrplens.github.io/phonometry/buildings/rooms/reverberation-prediction/): the statistical decay rate the image-source model reproduces, and the anisotropy models beyond it. - [Sound absorption in enclosed spaces (EN 12354-6)](https://jmrplens.github.io/phonometry/buildings/rooms/enclosed-space-absorption/): the equivalent absorption area behind the mean absorption `alpha_bar`. - [Sound Power](https://jmrplens.github.io/phonometry/devices/emission/sound-power/): the `Lw` that drives the steady-state level. - [2D FDTD wave simulation](https://jmrplens.github.io/phonometry/simulation/fdtd-simulation/): the independent wave solver used to cross-check the rigid-wall echo and the decay. - [Theory: Rooms and buildings](https://jmrplens.github.io/phonometry/reference/theory/rooms-buildings/): the image-lattice and steady-state derivations. - [Conformance report](https://github.com/jmrplens/phonometry/blob/main/docs/CONFORMANCE.md): the closed forms and worked anchors these implementations are validated against. - API reference: [`room.image_source`](https://jmrplens.github.io/phonometry/reference/api/rooms/image-source/) and [`room.steady_field`](https://jmrplens.github.io/phonometry/reference/api/rooms/steady-field/). - Theory: [Image-source room impulse response](../../reference/theory/rooms-buildings.md#image-source-room-impulse-response-kuttruff-41-vorländer-11): the geometrical model behind the image method and its relation to the statistical one. ## 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)). - Long, M. (2014). *Architectural acoustics* (2nd ed.). Academic Press. [doi:10.1016/C2012-0-03257-5](https://doi.org/10.1016/C2012-0-03257-5). Chapter 8 (the rectangular-room eigenfrequencies of Equation (8.43) and Table 8.1, and the Morse/Pierce mode count and modal density of Equations (8.45) and (8.46)).