Source: https://jmrplens.github.io/phonometry/buildings/rooms/sound-strength/
# Sound strength G (ISO 3382-1)
Reverberation time says what a hall does to energy over time. It says
nothing about how **loud** the hall is: two rooms with the same T30 can sit
20 dB apart at the same seat. Sound strength, $G$, is the quantity that
closes that gap, and it is one of only two measures in ISO 3382-1 Table A.1
that a calibrated source is needed for, the other being the late lateral
sound level, which is referred to the same free field. Everything else on
the
[room acoustic parameters page](https://jmrplens.github.io/phonometry/buildings/rooms/room-acoustics/)
(EDT, T20, T30, C50, C80, D50, $T_\mathrm{s}$) is a decay time, a ratio of
energies or a centre time read from one recording alone, and survives any
gain you put in front of it. $G$ does not.
*Only the first cell is this page: $G$ is the measure that needs a reference,
and the reference is the same source at 10 m with no room around it.*
## 1. What G is
ISO 3382-1:2009, Equation (A.1) defines it as the energy of the measured
impulse response against the energy the **same source** produces at 10 m in
a free field:
$$
G = 10 \lg \frac{\int_0^{\infty} p^2(t)\ \mathrm{d}t}{\int_0^{\infty} p_{10}^2(t)\ \mathrm{d}t}
= L_{pE} - L_{pE,10}\ \text{dB}.
$$
Both integrals are sound pressure exposure levels, Equations (A.2) and
(A.3), referred to $T_0 = 1$ s and $p_0 = 20$ µPa:
$$
L_{pE} = 10 \lg \left[ \frac{1}{T_0} \int_0^{\infty} \frac{p^2(t)}{p_0^2}\ \mathrm{d}t \right] \text{dB}.
$$
The reference quantities cancel in the ratio. They are there so that each
half is a level in its own right, which is what makes the three routes of
section 2 possible: you can obtain $L_{pE,10}$ without ever measuring
$p_{10}(t)$, and subtract it from a level you did measure.
```python
import numpy as np
from phonometry import room
fs = 48000
rng = np.random.default_rng(3382)
# The source at 10 m in a free field: one arrival, taken as the unit.
reference = np.zeros(int(0.2 * fs))
reference[480] = 1.0
# The same source heard at 20 m in the hall: half the direct pressure, and a
# reverberant tail carrying 2.8 times the free-field energy.
t = np.arange(int(2.0 * fs)) / fs
ir = rng.standard_normal(t.size) * np.exp(-3.0 * np.log(10.0) * t / 1.8)
ir *= np.sqrt(2.8 / np.sum(ir**2))
ir[480] += 0.5
res = room.sound_strength(ir, reference, fs)
print(res.frequencies.round(0)) # [ 126. 251. 501. 1000. 1995. 3981.]
print(res.strength.round(1)) # [5.9 4.1 4. 4.8 4.7 4.9] dB
res.plot() # G against the Table A.1 typical range (needs matplotlib)
```
**Time zero is the direct sound, not the start of the file.** The trigger of
A.3.4 is applied per band to both responses, so any propagation and system
delay in the recording is removed on both sides and cancels.
The *upper* limit is the one to watch. A.2.1 asks the integral to reach at
least the point where the decay curve has fallen 30 dB, and puts no limit on
how much further it may go, so read literally it would let $G$ grow with the
length of the tape: everything past the decay is noise floor, and $G$ has no
compensating denominator to absorb it the way C80 and D50 do. The integral
is therefore truncated where the fitted decay meets the noise floor and the
missing tail compensated with the fitted rate, which is the treatment 5.3.3
Equation (3) prints for this same integral and which
[`decay_curve`](https://jmrplens.github.io/phonometry/buildings/rooms/room-acoustics/) already gives
the same response. A synthetic response with no measurable noise floor is
integrated whole, so nothing moves for one.
Two things still need saying out loud, and `sound_strength` says them with
an `AuditoriumWarning`: a room response cut short of the 30 dB A.2.1 asks
for, and a response of either kind too short for its lowest band's filter
to ring down. The second catches the reference more often than the room: a
free-field window of 40 ms after the arrival is 0.04 dB light at 125 Hz, one
of 20 ms is 0.9 dB light and one of 10 ms is 13 dB light, and nothing about
the recording says so.
## 2. Three printed routes to the reference
An anechoic room 10 m across is rare, so A.2.1 prints three ways to obtain
$L_{pE,10}$ without one.
*The same source, measured three ways. The spread on the left is 0.0206 dB
and it is not a rounding error in the library: see section 3. The shaded
band on the right is what Table A.1 gives for the single number, the mean of
the 500 Hz and 1 kHz octaves, not for one band at a time.*
**Measure closer and correct.** Equations (A.4) and (A.8) apply the
inverse-square law from a distance $d \geq 3$ m:
$$
L_{pE,10} = L_{pE,d} + 20 \lg (d/10)\ \text{dB}.
$$
```python
from phonometry import room
print(round(room.free_field_reference_level(75.03, 5.0), 4)) # 69.0094
print(round(room.free_field_reference_level(83.4, 10.0), 4)) # 83.4 (the identity)
```
The note under (A.4) adds that the measurement is repeated around the
source and **energy**-averaged, so that the source's own directivity does
not decide the reference. `directivity_energy_average` is that mean:
```python
import numpy as np
from phonometry import room
bearings = np.arange(29) * 2.0 * np.pi / 29.0
levels = 80.0 + 20.0 * np.log10(np.abs(np.cos(bearings)) + 1e-12)
# The energy mean of a cosine pattern is exactly 10 lg(1/2) below its peak,
# whatever the bearing count; the arithmetic mean of the same decibels is not.
print(round(room.directivity_energy_average(levels), 4)) # 76.9897
print(round(float(np.mean(levels)), 1)) # 74.2 (wrong: arithmetic)
```
**Measure in a reverberation room.** Equation (A.5) converts a diffuse-field
reading into the free-field one through the room's absorption area, with
$S_0 = 1$ m²:
$$
L_{pE,10} = L_{pE} + 10 \lg (A/S_0) - 37\ \text{dB}.
$$
```python
from phonometry import room
# A = 0,16 V / T (Equation (A.6)): 200 m3 at 3,2 s gives exactly 10 m2.
print(round(room.reverberation_room_reference_level(80.0, 10.0), 4)) # 53.0
```
Watch the constant if you are reproducing a hand calculation. (A.6) prints
$A = 0{,}16\ V/T$, which is $24 \ln 10 / c_0$ at $c_0 = 345{,}4$ m/s, while
`room.sabine_absorption_area` defaults to 343 m/s and so to 0.1611. The
difference moves $10 \lg(A/S_0)$ by 0.030 dB; pass
`speed_of_sound=345.39` to get the printed constant back.
(A.5) also carries **no Waterhouse correction**, unlike the
reverberation-room sound power method of ISO 3741 that it otherwise
mirrors. The omitted $10 \lg(1 + S\lambda/8V)$ is worth over a decibel in
the 125 Hz band of a small room, above the 1 dB just-noticeable difference
Table A.1 gives $G$. That is a property of the printed method, and the
library reproduces the method rather than quietly improving it.
**Use the source's sound power level.** Equation (A.9) needs no free-field
measurement at all:
$$
G = L_p - L_W + 31\ \text{dB}.
$$
```python
from phonometry import room
print(round(room.sound_strength_from_power(80.0, 100.0), 4)) # 11.0
```
A.2.1 asks for $L_W$ to be measured to ISO 3741, which lives in
[Sound Power](https://jmrplens.github.io/phonometry/devices/emission/sound-power/).
## 3. Why the routes cannot agree exactly
The 31 dB of (A.9) is the spread of a point source over the sphere of
radius 10 m and the 37 dB of (A.5) is the diffuse-to-free-field ratio at
the same radius. Both are printed as whole decibels, and both roundings are
correct:
$$
10 \lg (4\pi \cdot 10^2) = 30{,}9921\ \text{dB}, \qquad
10 \lg (16\pi \cdot 10^2) = 37{,}0127\ \text{dB}.
$$
The printed integers differ by exactly 6 dB. The closed forms differ by
$10 \lg 4 = 6{,}0206$ dB. So the reverberation-room route and the
sound-power route describe the same physical situation and **cannot** agree
to better than 0.0206 dB, whatever a library does. That is 2 % of the 1 dB
just-noticeable difference of Table A.1, so it never matters in a hall; it
matters when a test suite compares the two routes, which is why the
[conformance report](https://jmrplens.github.io/phonometry/reference/conformance/) pins the gap
rather than tolerating it.
The library prints what the standard prints. A version that quietly used
30.9921 dB would disagree with every hand calculation done from the page.
Both closed forms also hold at a characteristic impedance of exactly
400 N·s·m⁻³, which is the value that makes the reference quantities
consistent: $p_0^2 S_0 / \rho c = (20\ \text{µPa})^2 / 400 = 1$ pW, the
reference sound power. Neither equation prints that caveat. Air at 20 °C
and 101.325 kPa is nearer 413 N·s·m⁻³, worth 0.14 dB, an order of
magnitude more than either rounding. The offsets are a convention of the
decibel scales, not a property of the air in the hall, and the library does
not make them follow the weather.
## 4. What G does in a hall
The right-hand panel above is the plot A.5 itself suggests drawing: "some
measures such as sound strength, G, tend to vary with the distance, and a
graphical plot of G as a function of source-receiver distance can be
useful". Near the source the direct field dominates and $G$ falls 6 dB per
doubling; past the critical distance the reverberant field takes over and
the curve flattens, at a level set by the room's absorption alone. Table A.1
gives −2 dB to +10 dB as the typical range in unoccupied halls up to
25 000 m³ (for the **single number**, the arithmetic mean of the 500 Hz and
1 kHz octave bands), which is what the "m" of $G_m$ marks and what
`res.plot()` draws across the shaded band.
```python
import numpy as np
from phonometry import room
volume, surface, reverberation = 15000.0, 3800.0, 2.0
area = float(room.sabine_absorption_area(volume, reverberation))
constant = float(room.room_constant(surface, area / surface))
level = room.steady_state_spl(100.0, [10.0, 20.0, 40.0], constant)
print(np.round(room.sound_strength_from_power(level, 100.0), 1)) # [5.8 4.9 4.6]
print(round(float(room.critical_distance(constant)), 1)) # 5.9 m
```
Show the code for this figure
```python
import matplotlib.pyplot as plt
import numpy as np
from phonometry import room
power_level = 100.0
absorption_area = 0.16 * 200.0 / 2.0
diffuse_level = power_level + 10.0 * np.log10(4.0 / absorption_area)
level_at_5m = power_level - 10.0 * np.log10(4.0 * np.pi * 25.0)
routes = [
("anechoic room, 5 m", float(room.free_field_reference_level(level_at_5m, 5.0))),
(
"reverberation room",
float(room.reverberation_room_reference_level(diffuse_level, absorption_area)),
),
("sound power level", power_level - room.SOUND_STRENGTH_POWER_OFFSET_DB),
]
exact = power_level - 10.0 * np.log10(4.0 * np.pi * 100.0)
fig, (left, right) = plt.subplots(1, 2, figsize=(11.8, 5.2))
for position, (label, value) in enumerate(routes[::-1]):
left.plot([exact, value], [position, position], color="grey", lw=1.2)
left.plot([value], [position], "o", markersize=11)
left.annotate(f"{value:.4f} dB", (value, position), textcoords="offset points",
xytext=(0, 13), ha="center")
left.axvline(exact, color="black", ls=":", lw=1.3)
left.set_yticks(range(len(routes)))
left.set_yticklabels([label for label, _ in routes[::-1]])
left.set_xlabel("Reference level at 10 m, $L_{pE,10}$ (dB)")
left.set_title("Three printed routes, one reference")
volume, surface, reverberation = 15000.0, 3800.0, 2.0
area = float(room.sabine_absorption_area(volume, reverberation))
constant = float(room.room_constant(surface, area / surface))
distance = np.linspace(3.0, 45.0, 400)
strength = room.sound_strength_from_power(
room.steady_state_spl(power_level, distance, constant), power_level
)
direct = room.sound_strength_from_power(
power_level + 10.0 * np.log10(1.0 / (4.0 * np.pi * distance**2)), power_level
)
right.axhspan(-2.0, 10.0, color="#2ca02c", alpha=0.15)
right.plot(distance, strength, lw=2.2, label="$G$")
right.plot(distance, direct, ls="--", lw=1.5, label="direct sound alone")
right.axvline(float(room.critical_distance(constant)), color="grey", ls=":", lw=1.4)
right.set_xlim(3.0, 45.0)
right.set_ylim(-4.0, 14.0)
right.set_xlabel("Source-receiver distance (m)")
right.set_ylabel("Sound strength $G$ (dB)")
right.legend()
plt.tight_layout()
plt.show()
```
Classical theory flattens the curve completely. Real halls do not: Barron
and Lee measured a reverberant level that keeps falling with distance,
because the sound arriving at a far seat has taken longer to get there and
has been absorbed on the way. The curve above is the classical one, and it
reads as the optimistic bound on a real measurement.
## 5. The calibration that has to be shared
$G$ is a difference of two absolute levels, so it only survives a gain that
is applied to **both** recordings. `sound_strength` computes both exposure
levels from the responses it is given, so:
- a common calibration factor on the pair cancels exactly, and the
[conformance report](https://jmrplens.github.io/phonometry/reference/conformance/) has a row holding
it to that;
- a factor on one of them does not cancel, and shows up as a straight
offset in every band of $G$;
- when the reference arrives as a *level* rather than as a response, the
hall recording carries the calibration alone and must be an absolute one.
A bare NumPy array is read as pascals. A
[`Signal`](https://jmrplens.github.io/phonometry/io/audio-files/) brings its own calibration factor and
it is applied before the integral.
## 6. Where this sits
$G$ belongs to the group of Table A.1 quantities the library measures from
an impulse response. The decay times and energy ratios are in
[Room Acoustic Parameters](https://jmrplens.github.io/phonometry/buildings/rooms/room-acoustics/);
acquiring the response itself, with the source and microphone positions
ISO 3382-1 asks for, is in
[Measuring the Room Impulse Response](https://jmrplens.github.io/phonometry/buildings/rooms/room-impulse-response/).
## See also
- [Room Acoustic Parameters](https://jmrplens.github.io/phonometry/buildings/rooms/room-acoustics/): the decay times and energy
ratios read from the same impulse response.
- [Measuring the Room Impulse Response](https://jmrplens.github.io/phonometry/buildings/rooms/room-impulse-response/): the
ISO 18233 acquisition of the responses this page compares.
- [Sound Power](https://jmrplens.github.io/phonometry/devices/emission/sound-power/): the $L_W$ that
Equation (A.9) subtracts, measured to ISO 3741.
- [Absorption and Reverberation Time](https://jmrplens.github.io/phonometry/buildings/rooms/enclosed-space-absorption/): the
absorption area Equation (A.5) converts a diffuse-field reading through.
- API reference: [`room.auditorium`](https://jmrplens.github.io/phonometry/reference/api/rooms/auditorium/).
## Standards
ISO 3382-1:2009, Annex A (informative), A.2.1: the sound strength G of
Equation (A.1), the sound pressure exposure levels of Equations (A.2) and
(A.3), and the three routes to the free-field reference at 10 m of
Equations (A.4) to (A.9). Annex A is informative: it fixes the definition
everyone quotes, and nothing about G is normative in ISO 3382-1. Validated
against closed forms and the standard's own printed offsets in the
[conformance report](https://jmrplens.github.io/phonometry/reference/conformance/); the two defects the annex carries
are in the [errata register](https://jmrplens.github.io/phonometry/reference/errata/).
