PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Correction of Sound Pressure Levels Calculated for Small Rooms
Using Diffuse-Field Theory
Kiyoshi MASUDA1; Hikari TANAKA2
1

Taisei Corporation, Japan

2

Taisei Corporation, Japan

ABSTRACTThe most frequently used method for predicting sound pressure levels arising in a room as a result oftransmitted sound or floor impact sound from adjacent rooms is an acoustic energy calculation assuming a
diffuse sound field. If the dimensions of the room under investigation are sufficiently large, pressure levels
can be predicted accurately by this method. However, especially at low frequencies, the wavelength of thesound may exceed the room dimensions, leading to reduced predictive accuracy. Although the Waterhousecorrection is typically used to correct for this error, its reliance on statistical energy considerations means thatit cannot take into account the response to the frequency range lower than the first-order mode and tofrequencies between adjacent lower-order modes. In this work, a new correction method is derived from therelationship between the sound pressure level calculated using wave theory and the sound pressure level
using diffuse-field theory in the frequency range from lower than the first-order mode to a low-order mode.
By comparing calculated results obtained using the new method with both actual measurements and FEM
calculations, its accuracy is examined.
Keywords: Correction, Diffuse-Field Theory, Small Room, Wave theory, Prediction

1. INTRODUCTION
It is important to be able to predict sound pressure levels arising in a room as a result of transmitted
sound or floor impact sound from adjacent rooms. This is typically done using an acoustic energy
calculation in which a diffuse sound field is assumed. This method gives accurate results if thedimensions of the room under investigation are sufficiently large. However, in smaller rooms the
wavelength of the sound may exceed the room dimensions, especially at low frequencies. This results
in predictions of lower accuracy.
Two specific problems must be faced when applying the energy calculation to low frequencies.
The first is the frequency characteristic of the amount of energy flowing into the sound field of theroom. It occurs because certain phenomena cannot be taken into consideration in the energy
calculation: resonance and antiresonance in the low-order mode of the sound field in the room, andperiodic compression and rarefaction of air in the room at a frequency lower than the first-ordermode frequency. The second problem is the deviation between the space-averaged squared soundpressure level of the entire room and that of the center area where measurement points are actuallyprovided. It occurs because of the sound pressure distribution in the room. The energy calculation
does not consider the distribution, because it assumes a diffuse sound field.
The Waterhouse correction [1] is typically used to correct for these problems [2]. This is acorrection applied to the space-averaged squared sound pressure in a room as calculated by energy
theory; it assumes that the energy density near the room boundary is higher than that near the centerof the room. As a method of correcting the sound pressure at any arbitrary point away from theboundary of the room, it is unable to fundamentally solve the first of the above problems while its
correction for the second problem may include a large error depending on the relationship between
the size of the room and the wavelength.
The purpose of this research is to derive a correction that approximates the wave phenomenon
12

kiyoshi.masuda@sakura.taisei.co.jp
tnkhkr00@pub.taisei.co.jp3411

separately for both of the above-mentioned problems. The authors propose a method that canapproximate the sound pressure level of a small room using diffuse sound field theory without theneed to invoke wave theory calculations. The accuracy of the method is examined by comparing its
calculation results with results of the wave theory calculations and laboratory experiments.

2. OUTLINE OF CORRECTION METHOD
A correction equation is defined as follows for approximating the space-averaged squared soundpressure level of the entire room of the actual wave field (𝐿𝑝.𝑤𝑎𝑣𝑒 ) from the space-averaged squaredsound pressure level obtained using the energy calculation based on diffuse sound field theory (𝐿𝑝.𝑒𝑛𝑒 ).
𝐿𝑝.𝑤𝑎𝑣𝑒 ≈ 𝐿̃𝑝.𝑤𝑎𝑣𝑒 = 𝐿𝑝.𝑒𝑛𝑒 +∆𝐿𝑠𝑓(1)
Here, 𝐿̃ 𝑝.𝑤𝑎𝑣𝑒 is the approximated 𝐿𝑝.𝑤𝑎𝑣𝑒 and ∆𝐿𝑠𝑓 is a correction factor for the frequency
characteristic of the amount of energy flowing into the sound field of the room that accounts forresonance and anti-resonance, in the low-order mode and periodic compression and rarefaction of air
in the room at a frequency lower than the first-order mode frequency.
In addition, taking into consideration that measurements are actually performed near the center ofa room, another correction equation is defined for approximating the space-averaged squared soundpressure level of the sound field within a central volume defined by a fixed distance from the boundaryof the room ( 𝐿𝑝,𝑎𝑟𝑒𝑎 ) from the space-averaged squared sound pressure level of the entire room of theactual sound field (𝐿𝑝.𝑤𝑎𝑣𝑒 ) as follows.
𝐿𝑝,𝑎𝑟𝑒𝑎 ≈ 𝐿̃𝑝,𝑎𝑟𝑒𝑎 = 𝐿̃𝑝.𝑤𝑎𝑣𝑒 − ∆𝐿𝑚𝑝
(2)
Here, 𝐿̃𝑝,𝑎𝑟𝑒𝑎 is the approximated 𝐿𝑝,𝑎𝑟𝑒𝑎 , and ∆𝐿𝑚𝑝 is the correction value for the soundpressure distribution in the room.
Therefore, the full equation for correcting the energy calculation result obtained using diffuse
sound field theory is as follows.
𝐿𝑝,𝑎𝑟𝑒𝑎 ≈ 𝐿̃𝑝,𝑎𝑟𝑒𝑎 = 𝐿𝑝.𝑒𝑛𝑒 +∆𝐿𝑠𝑓 − ∆𝐿𝑚𝑝
(3)
The corrected value will then reflect actually measured sound pressure levels. In the sections that
follow, the derivation of correction values ∆𝐿𝑠𝑓 and ∆𝐿𝑚𝑝 will be examined.

3. CORRECTION FOR LOW-ORDER MODE OF SOUND FIELD
3.1 Examination by One-Dimensional Sound Field
In order to explain in an easy-to-understand manner the characteristics of the discrepancies that
occur between actual wave phenomenon and the diffuse sound field theor y, we here consider aone-dimensional sound field and predict the sound pressure resulting from a floor impact as shown in
Fig. 1. A ceiling with a sound absorption coefficient of zero vibrates at velocity 𝑣 at all frequencies
and emits sound. The sound is reflected at the floor which has sound absorption 𝛼 at all frequencies.
The sound pressure 𝑝(𝑥) at any point 𝑥 can be calculated by the following equations:𝑝(𝑥) = 𝜌𝑐𝑣 [𝑒 −𝑗𝑘𝑥 +(𝑍 − 1)𝑒 −𝑗𝑘𝐻 𝑗𝑘𝑥(𝑍 − 1)𝑒 −𝑗𝑘𝐻
𝑒
]
⁄
[1
−
],
(𝑍 + 1)𝑒 𝑗𝑘𝐻(𝑍 + 1)𝑒 𝑗𝑘𝐻

2𝜋𝑓
4𝑍⁄𝜌𝑐
𝑘=
,𝛼 =
(𝑍
⁄
𝑐
𝜌𝑐 + 1)2(4)

Here, 𝜌 is the density of air, 𝑐 is the speed of sound, 𝐻 is the ceiling height, 𝑍 is the floor
impedance, and 𝑓is the frequency. 𝐿𝑝.𝑤𝑎𝑣𝑒 of the one-dimensional sound field is the space-averaged
sound pressure level of the entire sound field calculated using Eq. (4). On the other hand, 𝐿𝑝.𝑒𝑛𝑒 ofthe one-dimensional sound field using the energy calculation not considering phase is given by
Eq. (5).
(5)
𝐿𝑝.𝑒𝑛𝑒 = 10log10 (𝜌𝑐𝑣 2 ⁄𝛼̅), 𝛼̅ ≈ 𝛼⁄2
Figure 2 shows the calculated results of 𝐿𝑝.𝑤𝑎𝑣𝑒 and 𝐿𝑝.𝑒𝑛𝑒 when the ceiling height H is 2,650
mm, the average sound absorption coefficient 𝛼
̅ is 0.09, and the vibration velocity level of the slab
is -120 dB re 1m/s. According to Fig. 2, 𝐿𝑝.𝑒𝑛𝑒 is calculated to be smaller than 𝐿𝑝.𝑤𝑎𝑣𝑒 at the mode3412

frequency of the sound field, and larger than 𝐿𝑝.𝑤𝑎𝑣𝑒 at the antiresonance frequency. Also, at
frequencies lower than the first-order mode, the lower the frequency, the smaller the difference
between 𝐿𝑝.𝑤𝑎𝑣𝑒 and 𝐿𝑝.𝑒𝑛𝑒 . If the frequency is sufficiently high in a three-dimensional sound field,
and in the 1/3 octave band description, such a difference is considered to be almost eliminated ,because the mode density rises rapidly. However, in small rooms, the same tendency is expected tooccur at lower frequencies.
𝐿 : Vibration velocity level of ceiling slab

𝑣

)

-110

l(

-120

𝑝

-130

𝐿𝑝,𝑒𝑛𝑒

-140

(

-150

𝐻

at on

𝑝 𝑥

lo t

)

𝑥

-160

𝐿𝑝,𝑤𝑎𝑣𝑒

𝑝−

-170
-180

𝛼
Frequency (Hz)∆𝐿𝑠𝑓 𝑓𝑝𝑚 2 ⁄ 𝑝𝑒 22

𝑝𝑝 ⁄ 𝑝𝑒 2

𝐿 𝑝.𝑤𝑎𝑣𝑒-𝐿̃ 𝑝.𝑤𝑎𝑣𝑒 (dB)Figure 2 – Calculated results by wave and energy theory(H=2650mm)n

SPL Correction value (dB)Figure 1 – Calculation model1

o t. anFrequency (Hz)Frequency (Hz)Figure 3 – Sound level correction curve

Figure 4 – Difference between wave andenergy theory with correction

3.2 Correction Value ∆𝑳𝒔𝒇 for One-Dimensional Sound Field
Here, a correction value ∆𝐿𝑠𝑓 for the one-dimensional sound field that can approximate 𝐿𝑝.𝑤𝑎𝑣𝑒
from 𝐿𝑝.𝑒𝑛𝑒 without a wave theory calculation is derived.
The space-averaged squared sound pressure |𝑝𝑒 |2 is the physical quantity of 𝐿𝑝.𝑒𝑛𝑒 normalized
by the sound pressure 𝑝𝑚𝑎𝑥 which is the the physical quantity of 𝐿𝑝.𝑤𝑎𝑣𝑒 at the 𝑚th -orderresonance frequency 𝑓𝑚 and is approximated by the following equations:|𝑝𝑒 |2 =𝑏
2𝑓1−𝑐log 𝑒 (1 − 𝛼̅)tan−1, 𝑓𝑚 = 𝑚𝑐 ⁄2𝐻 , 𝑏 =𝑓1
𝑏6.27 × 𝐻(6)

On the other hand, the space average of the squared sound pressure |𝑝𝑚 |2 is the physical quantityof 𝐿𝑝.𝑤𝑎𝑣𝑒 normalized by 𝑝𝑚𝑎𝑥 and is approximated by the following equations:∞|𝑝𝑚 |2 = ∑ [1 + 4|𝑓 − 𝑓𝑚 |2 ⁄𝑏 2 ]−1
𝑚=1

(7)
2

Furthermore, the ratio of the space average of the squared sound pressure |𝑝𝑝 |first-order mode frequency of 𝐿𝑝.𝑤𝑎𝑣𝑒 to |𝑝𝑒 |2 is given by the following equation:

below the2(8)̅⁄(2𝜋𝑓𝐻)2|𝑝𝑝 | ⁄|𝑝𝑒 |2 = 𝑐2 𝛼
Therefore, the correction value ∆𝐿𝑠𝑓 for a one-dimensional sound field is obtained as below.
2∆𝐿𝑠𝑓 (𝑓) = 10log10 [|𝑝𝑚 |2 |𝑝𝑝 |+]|𝑝𝑒 |2 |𝑝𝑒 |2(9)2

The calculated values of ∆𝐿𝑠𝑓 , |𝑝𝑚 |2 ⁄|𝑝𝑒 |2 , and |𝑝𝑝 | ⁄|𝑝𝑒 |2 under calculation conditions

3413

identical to those in Fig. 2 are shown in Fig. 3. The calculated ∆𝐿𝑠𝑓 is very similar to the difference
between 𝐿𝑝.𝑤𝑎𝑣𝑒 and 𝐿𝑝.𝑒𝑛𝑒 in Fig. 2, so it can be considered an appropriate correction value.
The difference between 𝐿𝑝.𝑤𝑎𝑣𝑒 and 𝐿̃ 𝑝.𝑤𝑎𝑣𝑒 as calculated by Eqs. (5) to (9) is shown in Fig. 4. Here,the ceiling height H is 2,400 mm and the average sound absorption coefficient 𝛼̅ is 0.09. It can be seenthat the difference is small in all frequency bands.
3.3 Examination by Three-Dimensional Sound Field
Frequency response analysis is performed using the finite element method (FEM), assuming thatthe room affected by the sound is under a vibrating ceiling concrete slab, as shown in Fig. 5. Threeceiling heights 𝐻 are considered: 2,400 mm, 2,650 mm, and 2,800 mm. The ceiling slab is modeledusing shell elements, while the sound field in the room is modeled with fluid elements. The bound ary
condition at the four edges of the ceiling slab is a simple support and the boundary conditions applied
to wall and floor surfaces are an impedance condition where the absorption coefficient for normal
incidence is 0.09.
A vibrational excitation force of 1 N in the vertical direction is input at the excitation point on theceiling slab for each frequency, and the sound pressure at the nodal points of fluid elements and thevibration velocity of the nodal points of shell elements are calculated. The mean squared soundpressure level of all nodal points of fluid elements is taken as 𝐿𝑝.𝑤𝑎𝑣𝑒 .
𝐿𝑝.𝑒𝑛𝑒 is calculated using Eq. (10).
𝐿𝑝.𝑒𝑛𝑒 = 𝐿̅ + 10log104𝑆𝑒𝑓𝑓 𝜎𝜌2 𝑐 2
𝑆𝑎 𝛼̅𝑝02(10)Here, 𝐿̅ is the surface-averaged squared vibration velocity level of all nodal points of shellelements in the sound radiation area. 𝑆𝑒𝑓𝑓 is the radiation area of the ceiling slab, 𝜎 is the acoustic
radiation efficiency of the ceiling slab (set to 1.0), 𝑆𝑎 is the surface area of the room affected by thesound, and 𝑝0 is the reference sound pressure (𝑝0 = 2 × 10−5 [Pa]). The calculated results of 𝐿𝑝.𝑤𝑎𝑣𝑒 ,𝐿𝑝.𝑒𝑛𝑒 and 𝐿̅ for the 2,650 mm ceiling height are shown in Fig. 6. In addition, the difference
between 𝐿𝑝.𝑤𝑎𝑣𝑒 and 𝐿𝑝.𝑒𝑛𝑒 is shown in Fig. 7 for all three ceiling heights.
The frequency dependence of the 𝐿𝑝.𝑤𝑎𝑣𝑒 and 𝐿𝑝.𝑒𝑛𝑒 difference is more complicated than in the
one-dimensional sound field. However, in the frequency range below the vertical first-order mode
frequency, the frequency characteristics are very similar to those in the one-dimensional sound field.
The reason for this seems to be that the vertical vibration mode is more easily excited than thehorizontal vibration mode because the slab occupies only the horizontal plane and the mode density
is low. At higher frequencies, the number of modes rapidly increases, so that it is close to the diffuse
sound field, and the deviation itself becomes smaller.

)

-120

50

-130
-140

40

𝐿 𝑝,𝑒𝑛𝑒

30

-150

20

-160

𝐿 𝑝,𝑤𝑎𝑣𝑒

10
0

l(

-110

60

(

70

lo t

-100

level of ceiling slab)

-90

𝐿̅ : Vibration velocity

80

波動計算

at on

90

振動速度

エネルギ計算

-170
-180

10

100

Frequency (Hz)Figure 5 – Calculation model for FEM

Figure 6 – Calculated results by wave and energy theory(H=2650mm)

3.4 Correction Value ∆𝑳𝒔𝒇 for Three-Dimensional Sound Field
As shown above, the difference between 𝐿𝑝.𝑤𝑎𝑣𝑒 and 𝐿𝑝.𝑒𝑛𝑒 in the three-dimensional sound fieldis very similar that in the one-dimensional sound field up to around the frequency of the first-ordermode in the vertical direction. Therefore, the correction ∆𝐿𝑠𝑓 for one-dimensional sound field is
modified for the three-dimensional case as follows.
First, the space average of the squared sound pressure |𝑝𝑚 |2 is assumed to be as given by Eq. (7)up to the second-order mode. Then the influence of rapidly increasing higher modes is approximated3414

by adding a large sound absorption coefficient in the second-order mode. Equation (7) is thus modified
to obtain the following equation:
2−1∑ [1 + 4|𝑓 − 𝑓𝑚 |2 ⁄𝑏𝑚 2 ]|𝑝𝑚 |2=|𝑝𝑒 |2𝑚=1
2

1
2𝑓1 −1[ tan−1
] ,
𝑓1
𝑏𝑚

−1

{

𝑚𝑎𝑥 ( ∑ [1 + 4|𝑓 − 𝑓𝑚 |2 ⁄𝑏𝑚 2 ]
𝑚=1

𝑏𝑚 =

𝑓 < 𝑓21
2𝑓1 −1[ tan−1] , 1)
𝑓1𝑏𝑚

𝑓 ≥ 𝑓2(11)−𝑐log𝑒 (1 − 𝛼̅𝑚)

6.27 × 𝐻
Here, 𝛼̅𝑚 is the sound absorption coefficient at the frequency of the 𝑚th -order mode in thevertical direction. Equation (8) is then modified to obtain the following equation to fit thethree-dimensional sound field:
2

|𝑝𝑝 |
̅
𝑆𝑎 𝑐2 𝛼=2|𝑝𝑒 |4𝑆𝑒𝑓𝑓 (2𝜋𝑓𝐻)2

(12)

2

𝐿 𝑝.𝑤𝑎𝑣𝑒-𝐿 𝑝.𝑒𝑛𝑒 (dB)n

The values of ∆𝐿𝑠𝑓 , |𝑝𝑚 |2 ⁄|𝑝𝑒 |2 , and |𝑝𝑝 | ⁄|𝑝𝑒 |2 for Fig. 5 calculated using Eq. (11) andEq. (12) are shown in Fig. 8. Here, the height of the ceiling is 2,650 mm and the average soundabsorption coefficient is 0.073 for the first mode and 0.7 for the second mode.
The difference between 𝐿𝑝.𝑤𝑎𝑣𝑒 calculated using FEM and 𝐿̃𝑝.𝑤𝑎𝑣𝑒 calculated using Eqs. (1), (9),(10), (11) and (12) is shown in Fig. 9. In this three-dimensional calculation, it can be seen that thewave calculation result is well approximated only with the addition of the correction value ∆𝐿𝑠𝑓 tothe energy calculation result 𝐿𝑝.𝑒𝑛𝑒 .
H=2,800

H=2,650

H=2,400.

Frequency (Hz)Frequency (Hz)Frequency (Hz)SPL Correction value (dB)Figure 7 – Difference between wave and energy theory (3D)
∆𝐿𝑠𝑓 𝑓𝑝𝑚 2 ⁄ 𝑝𝑒 22

𝑝𝑝 ⁄ 𝑝𝑒 2

Frequency (Hz)𝐿 𝑝.𝑤𝑎𝑣𝑒-𝐿̃ 𝑝.𝑤𝑎𝑣𝑒 (dB)nFigure 8 – Sound level correction curve
H=2,400

H=2,650

H=2,800.

Frequency (Hz)Frequency (Hz)Frequency (Hz)Figure 9 – Difference between wave and energy theory with correction3415

3.5 Laboratory Experiment on ∆𝑳𝒔𝒇
In the laboratory outlined in Fig. 10, an excitation point (V3) on the ceiling concrete slab is excitedwith a constant force. The vibration velocity level of the slab and the sound pressure level in the room
are measured at large numbers on grid points (on a pitch of approximately 280 mm).
𝐿𝑝.𝑤𝑎𝑣𝑒 is the space-averaged squared sound pressure level of all measurement points in the room’s
sound field. 𝐿𝑝.𝑒𝑛𝑒 is calculated using Eq. (10). 𝐿̅ is the surface-averaged squared vibration
velocity level of all measurement points on the ceiling slab. The correction value ∆𝐿𝑠𝑓 is calculatedusing Eqs. (9), (11) and (12), where 𝛼̅1 is 0.3 and 𝛼̅2 is 0.7.
The difference between measured 𝐿𝑝.𝑤𝑎𝑣𝑒 and approximated 𝐿̃𝑝.𝑤𝑎𝑣𝑒 calculated using Eqs. (1),(9), (10), (11) and (12) is shown in Fig. 11, and the difference between 𝐿𝑝.𝑤𝑎𝑣𝑒 and 𝐿̃𝑝.𝑤𝑎𝑣𝑒 isshown in Fig. 12. An improvement is seen in the 1/3 octave band, especially below 100 Hz.
Center volume
Measurement point (Sound)
Measurement point (Vibration)
Excitation point
V1

V2

V3

V4

V5

Drywall
2,920 mm4,200 mm
Ceiling slab plan

Center volume

2,650 mm3,810 mm5,110 mm

t=230 mm2,920 mm

Room plan

Room sectionn
𝐿 𝑝.𝑤𝑎𝑣𝑒-𝐿 𝑝.𝑒𝑛𝑒 (dB)ñ
𝐿 𝑝.𝑤𝑎𝑣𝑒-𝐿 𝑝.𝑤𝑎𝑣𝑒 (dB)Figure 10 – Experimental condition

Frequency (Hz)Figure 11 – Difference between measurement
and energy calculation

Frequency (Hz)Figure 12 – Difference between measurement
and energy calculation with correction

4. CORRECTION FOR SOUND PRESSURE DISTRIBUTION
4.1 Examination by One-Dimensional Sound Field
The variation in sound pressure distribution with frequency in the one-dimensional sound field isshown in Fig. 13. At frequencies lower than the first-order mode (A), the sound pressure in the soundfield is constant everywhere, as in the inner space of an air spring. At the frequency of the first-ordermode (B), the sound pressure increases at the boundary and decreases in the center volume of thesound field.
Therefore, in frequency range (A), since there is almost no variation over the entire sound field, it
can be considered that there is almost no difference between the average value in the center volume
and the average value of the entire sound field. However, at mode frequencies such as (B), the lower
the mode, the larger the difference between the average value in the center volume and the averagevalue of the entire sound field.

3416

)
(

1 mode2𝑛 modeag

𝐿𝑝,𝑒𝑛𝑒n g −a

B

A

A

B
𝑝𝑥

2

𝑝𝑥

𝐻

𝐻

on t.

a

𝐿𝑝,𝑤𝑎𝑣𝑒

2

Frequency (Hz)𝐿 𝑝.𝑤𝑎𝑣𝑒-𝐿 𝑝.𝑎𝑟𝑒𝑎 (dB)nFigure 13 –Squared sound pressure distribution in one-dimensional sound field

Frequency (Hz)Figure 14 – Difference between 𝐿𝑝.𝑤𝑎𝑣𝑒 and 𝐿𝑝,𝑎𝑟𝑒𝑎
4.2 Examination by Three-Dimensional Sound Field
In the FEM calculation model shown in Fig. 5, the mean squared sound pressure level of only thenodal points in the center volume 500 mm away from the ceiling, wall and floor respectively is taken
as 𝐿𝑝,𝑎𝑟𝑒𝑎 . The difference between 𝐿𝑝.𝑤𝑎𝑣𝑒 and 𝐿𝑝,𝑎𝑟𝑒𝑎 is shown in Fig. 14. Here the three ceilingheights of 2,400 mm, 2,650 mm and 2,800 mm are considered as before. The value of theWaterhouse correction at 𝐻 = 2,650 mm is also shown for reference.
The Waterhouse correction takes a larger value as the frequency decreases, but the calculateddifference starts to decrease near the first-order modal frequency in the vertical direction, and below
31.5 Hz becomes almost zero. The frequency range below 31.5 Hz corresponds to frequency range (A)in Fig. 13.
4.3 Correction Value ∆𝑳𝒎𝒑 for Three-Dimensional Sound Field
Based on the discussion that follows, correction values for the sound pressure distribution ∆𝐿𝑚𝑝
are defined as the following equations:0𝑓 < 𝑓1 ⁄2(2𝑓)⁄⁄
10log10
𝑓1
𝑓1 2 ≤ 𝑓 < 𝑓1
∆𝐿𝑚𝑝 = {𝑚𝑖𝑛( , 10log10 (1 + 𝑆𝜆⁄8𝑉 + 𝐿𝜆2 ⁄ 2𝜋𝑉 ))𝑓 ≥ 𝑓1(13)

First, based on the above results, set 𝑓1 ⁄2 as the frequency at which the influence of the pressure
field is large, and then increase by 1 dB every 1/3 octave from that point up to the frequency of thefirst-order mode. Above the first-order mode frequency, the correction basically follows theWaterhouse rule, but a maximum value is set at 3 dB. In equation (13), 𝑆 is the surface area of theroom, 𝜆 is the wavelength, 𝑉 is the volume of the room, and 𝐿 is the entire perimeter of the room.
The calculation results of ∆𝐿𝑚𝑝 for 𝐻 = 2,400 mm in Fig. 5 and the difference between 𝐿𝑝.𝑤𝑎𝑣𝑒
and 𝐿𝑝,𝑎𝑟𝑒𝑎 are shown in Fig. 15. It can be seen that the correction value closely approximates thesound pressure distribution in the room.
4.4 Laboratory Experiment on ∆𝑳𝒎𝒑
In the laboratory results shown in Fig.10, the mean squared sound pressure level at only the

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𝐿 𝑝.𝑤𝑎𝑣𝑒-𝐿 𝑝.𝑎𝑟𝑒𝑎 (dB)measurement points in the center volume 500 mm away from the ceiling, wall and floor respectively istaken as 𝐿𝑝,𝑎𝑟𝑒𝑎 . Calculated ∆𝐿𝑚𝑝 values for the laboratory measurements and the difference
between 𝐿𝑝.𝑤𝑎𝑣𝑒 and 𝐿𝑝,𝑎𝑟𝑒𝑎 are shown in Fig. 16. In the frequency range below 80 Hz, thecorrespondence between ∆𝐿𝑚𝑝 and the measured difference is very good. At 100 Hz or more, thecorrection value ∆𝐿𝑚𝑝 is rather smaller, but the difference is at most about 1.5 dB, so accuracy is
sufficient as a correction value.

∆𝐿 𝑚𝑝

.(1 )

Frequency (Hz)n
𝐿 𝑝.𝑤𝑎𝑣𝑒-𝐿 𝑝.𝑎𝑟𝑒𝑎 (dB)Figure 15 – Correction value and difference between 𝐿𝑝.𝑤𝑎𝑣𝑒 and 𝐿𝑝,𝑎𝑟𝑒𝑎

∆𝐿 𝑚𝑝

.(1 )

Frequency (Hz)Figure 16– Actual difference between entire room average and center volume average in real room

5. CONCLUSIONS
A small-room correction method is proposed for the conventional diffuse sound field energy
calculation, in which it is assumed that frequencies are low. The proposed correction includes two
terms. One is for the influence of low order modes and periodic compression and rarefaction of air in
the room at a frequency lower than the first-order mode frequency, and the other is for the influenceof sound pressure distribution. The correction method well approximates the space -averaged squaresound pressure level in simulated and measured sound fields without the need to invoke wave theory
calculations.
Future work is to examine the correspondence between the correction and measurements in moresound fields with the aim of making the correction method more accurate.

REFERENCES
1. Waterhouse, R. V., “Interference patterns in reverberant sound fields.,” J. Acoust.Soc. Am., 1955.
2. ISO 12354-1:2017, Building acoustics – Estimation of performance of buildings from the performanceof elements – Part 1: Airborne sound insulation between rooms

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