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Influence of room modes on low-frequency transients: Theoretical modeling and numerical predictions
Journal of Sound and Vibration 448 (2019) 19–33
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Influence of room modes on low-frequency transients: Theoretical modeling and numerical predictions Mirosław Meissner ∗ , Krzysztof Wi´sniewski Institute of Fundamental Technological Research, Polish Academy of Sciences, Pawi´ nskiego 5B, 02–106 Warsaw, Poland
article info
abstract
Article history: Received 24 September 2018 Revised 8 February 2019 Accepted 11 February 2019 Available online 14 February 2019 Handling Editor: D. Juve
In the low-frequency range, a sound reproduction in enclosures is strongly influenced by excited room modes. While the spectral impact of acoustic modes on a room response is well recognized, there is no sufficient knowledge on how these modes affects transients. In the paper this issue has been examined theoretically and numerically for a room excited by a tone burst by using a modal expansion method supported by a computer implementation. To quantify a temporal accuracy of a sound reproduction, the new metrics referred to as the tone burst reproduction error was introduced. The basis for determining this quantity was a deviation between the tone burst amplitude and the amplitude of a sound pressure computed via the Hilbert transform. A numerical simulation was performed for an irregularly shaped enclosure having a form of two-room coupled system. Calculation results have proved that a high inaccuracy of a tone burst reproduction occurs at receiving points corresponding to sharp dips in a distribution of the steady-state sound pressure level. This is because in these points an amplitude of transient sound is much bigger than a tone burst amplitude. It was discovered that strong narrow peaks in the tone burst reproduction error are located at centers of vortices in the active sound intensity vector field. An influence of a sound damping in a room on a reproduction of a tone burst was also examined and it was found that a substantial increase in a wall sound absorption does not significantly improves a tone burst reproduction because it does not eliminate sharp dips in a distribution of the steady-state sound pressure level. © 2019 Elsevier Ltd. All rights reserved.
Keywords: Room acoustics Room modes Transients Tone burst Discrete Hilbert transform Sound intensity vector field
1. Introduction The main aim of theoretical room acoustics is to study the transient and steady-state behaviors of a sound field in enclosures. Various methods exist for modeling acoustic characteristics of rooms and among them are statistical-acoustic methods, diffusion-equation model, geometrical acoustics, wave-based methods and modal expansion method. Statistical-acoustic methods are based on the diffuse sound field hypothesis that the acoustic energy is uniform in the field [1]. The diffusion-equation model is an extension of the statistical theory to spatially varying reverberant sound fields [2]. The geometrical acoustics is adequate for high sound frequencies and most systems for geometric modeling are based on the ray tracing method [3], the beam tracing algorithm [4] and the mirror source method [5]. The fact that geometric methods do not include wave phenomena such as interferences or diffraction is a considerable disadvantage. Therefore, the results obtained with such methods are inaccurate for enclosures with complex shapes and coupled-room systems.
∗ Corresponding author. E-mail address: [email protected] (M. Meissner).
https://doi.org/10.1016/j.jsv.2019.02.012 0022-460X/© 2019 Elsevier Ltd. All rights reserved.
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In contrast to geometric approaches, wave-based methods solve the wave equation after suitable discretization to describe a complete sound field in enclosures. The most common among these numerical techniques are the finite element method (FEM) [6], the boundary element method (BEM) [7] and the finite-difference time-domain (FDTD) method [8]. Alternatives to the FDTD technique are the pseudospectral time-domain (PSTD) method [9] and adaptive rectangular decomposition (ARD) which achieve a good accuracy with a much coarser spatial discretization [10]. All these wave-based methods provide different advantages and disadvantages depending on their complexity and computational load. However, the most important difference lies in their applicability: the frequency domain methods, such as FEM and BEM, typically provide results for steady-state situations, whereas the time domain models, such as FDTD, allow to predict transient room responses [11]. The modal expansion method is a technique used for vibration analysis of mechanical objects, thus, it has been applied in several branches of technical sciences. In low-frequency room acoustics, this method allows to express the sound field in a room as a linear combination of the acoustic room modes of pressure vibrations [12–14]. The modal expansion method proved to be particularly useful for systems of coupled rooms because it enables to identify such phenomena as a localization of modes [15], a modal degeneracy [16] and a formation of energy vortices in the sound intensity vector field [17]. In this paper, the modal expansion method is applied to study the impact of room modes on low-frequency transients. In the theoretical modeling of a transient sound, a convolution theorem and a modal representation of the room impulse response are used and the sound source is assumed to generate a tone burst. To assess the temporal quality of sound reproduction, a new metrics named the tone burst reproduction error is introduced. A theoretical analysis is accompanied by numerical simulations performed for a system of two coupled rooms. For this reason, a numerical implementation exploiting the high-accuracy FEM procedure is used for calculating the mode shape functions. A detailed analysis of simulation results are carried out and final conclusions regarding the influence of room modes on transient sound are formulated. 2. Modeling of low-frequency room acoustics In room acoustics, the low-frequency band is usually defined as the frequency range bounded from above by the Schroeder frequency. This frequency allows us to subdivide the acoustic behavior of rooms into two regions. Below the Schroeder frequency there is the modal region where the modal density is low and particular modes can be observed in the room response, whereas above this frequency there is the region of diffuse sound field where there is a strong overlap of room modes. As was shown in Ref. [18], the Schroeder frequency depends on the sound damping inside the room and is determined by
√
fs = c
6
𝛼S
,
(1)
where c denotes the sound speed, S is the surface area of room walls and 𝛼 is the average absorption coefficient of wall surfaces. In the modal region, room dimensions are comparable with the sound wavelength, and the most appropriate method for determining the interior sound field is modal expansion. According to this method, the pressure room response can be described as a superposition of individual responses of acoustic modes excited inside a room by a sound source, i.e., p(r, t ) =
∞ ∑
pm (t )Φm (r),
(2)
m=1
where r = (x, y, z) is the position coordinate of a receiver, the functions pm determine a temporal behavior of the sound pressure and Φm are the mode shape functions. These functions satisfy the orthonormal property in the room volume and each Φm is related to the corresponding modal angular frequency 𝜔m through the eigenvalue equation
𝛁2 Φm (r) +
( 𝜔 )2 m
c
Φm (r) = 0,
(3)
where 𝛁 is the nabla vector operator. In a theoretical model it is assumed that room walls are covered by a sound absorbing material with a purely real admittance which is independent of frequency. It means that the mass and stiffness of this material are neglected. This is equivalent to the damping of a sound wave with no phase change upon reflection from room walls. In this case, the following boundary condition must be satisfied by the sound pressure
𝛁p · n = −
𝜉 (rs ) 𝜕 p(rs , t) , c 𝜕t
(4)
where n is the unit vector normal to the walls, which is directed away from the room volume, 𝜉 is the specific conductance of a wall material and rs is a position coordinate on room walls. It is assumed that the value of 𝜉 is small because typical materials covering room walls are characterized by a small sound absorption in low-frequency range [19]. The procedure for finding the function pm relies on a suitable solution of the wave equation and using the method presented in Ref. [20], it can be shown that for small sound damping inside a room the function pm is a solution of the following differential equation
𝜕 2 pm 𝜕p + 2rm m + 𝜔2m pm = c2 q(r′ , t )Φm (r′ )d3 r′ , ∫V 𝜕 t2 𝜕t
(5)
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where q represents the volume source term in the wave equation and d3 r′ = dx′ dy′ dz′ is the volume element. In Eq. (5) the parameter rm is the modal damping factor given by rm =
c 𝜉 (r )Φ2 (r )ds. 2 ∫S s m s
(6)
A transient room response, which is most important, is the impulse response and it corresponds to the case of an impulsive temporal excitation of a room by a sound source located at a point. Thus, assuming that the volume source term in Eq. (5) has the form q(r′ , t) = 𝛿 (r′ − r0 )𝛿 (t − t0 ) one can obtain
𝜕 2 pm 𝜕p + 2rm m + 𝜔2m pm = c2 𝛿 (t − t0 )Φm (r0 ), (7) 𝜕 t2 𝜕t where r0 = (x0 , y0 , z0 ) is the position coordinate of a source point and t0 is a time of the impulse generation. A method for resolving Eq. (7) was described in detail in Ref. [21] and the obtained solution has the form c2 e−rm (t−t0 ) sin[Ωm (t − t0 )]Φm (r0 )
pm (t ) =
Ωm
√
,
(8)
2 𝜔2m − rm are the modal angular frequencies for damped vibrations. A substitution of Eq. (8) into Eq. (2) leads to
where Ωm =
the function of the form h(r0 , r, t ) = c2
∞ ∑ e−rm t sin(Ωm t )Φm (r0 )Φm (r) Ωm m=1
(9)
describing the room impulse response (RIR). Because of the causality condition, the RIR function h(r0 , r, t) is zero for t < 0. It satisfies also the reciprocity principle because the right side of Eq. (9) is a symmetric function of the source and receiver points coordinates. The room impulse response is very useful in room acoustics because a knowledge of the RIR function enables to predict the room response to any sound source. In fact, when a volume source in the wave equation is described by the source function q, the pressure response to this excitation can be found from the following expressions [19]. p(r, t ) =
∫V t
=
∫V ∫−∞
q(r′ , t ) ∗ h(r′ , r, t )d3 r′
(10)
q(r′ , 𝜏 )h(r′ , r, t − 𝜏 )d𝜏 d3 r′ ,
(11)
where V is the room volume and the symbol ∗ denotes a convolution operation. For example, the steady-state room response to a point source can be found assuming that in Eq. (11) the source function q takes the form q(r′ , 𝜏 ) = Q 𝛿 (r′ − r0 )ej𝜔𝜏 ,
(12)
where 𝜔 is the angular source √ frequency and the amplitude Q of a sound excitation is dependent on the source power W according to the formula Q = 8𝜋𝜌cW [22], where 𝜌 is the air density. Thus, after performing the volume and time integrations in Eq. (11), a formula for the steady-state complex pressure amplitude Pc is found to be as follows Pc (r) =
∞ ∑
(𝛼m + j𝛽m )Φm (r),
(13)
m=1
where the quantities 𝛼 m and 𝛽 m are given by
𝛼m =
Qc2 (𝜔2m − 𝜔2 )Φm (r0 ) 2 2 (𝜔2m − 𝜔2 )2 + 4rm 𝜔
𝛽m = −
2Qc2 rm 𝜔Φm (r0 )
,
2 2 (𝜔2m − 𝜔2 )2 + 4rm 𝜔
(14)
.
(15)
Since the amplitude Pc is complex, a quantity suitable for a prediction of the steady-state room response is the real pressure amplitude P determined by absolute value of Pc , i.e.
√
P (r) =
Pc (r)Pc ∗ (r),
(16)
where an asterisk as a superscript denotes the complex conjugate. Thus, after inserting Eq. (13) into Eq. (16) one finds the formula 1∕ 2
]2 [ ∞ ]2 ⎫ ⎧[ ∞ ∑ ⎪ ∑ ⎪ 𝛼m Φm (r) + 𝛽m Φm (r) ⎬ P (r) = ⎨ ⎪ m=1 ⎪ m=1 ⎩ ⎭
.
(17)
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Eqs. (14), (15) and (17) indicate that in the low-frequency range, where individual modes are well separated from each other, the steady-state room response will contain peaks at frequencies of these modes. A magnitude of these peaks depends on the damping of energy in modes represented by the damping factors rm , thus, a smaller damping leads to more intense peaks. 3. Transients Transients have been tested using the tone burst excitation method. In the theoretical modeling, the usual form of the tone burst was assumed, i.e. the burst was created by multiplying a sinusoidal signal of angular frequency 𝜔 with a rectangular window whose length T corresponds to a specified number of full sine-wave cycles. It was postulated that 𝜔 belongs to a lowfrequency range and the burst duration is long enough to distinguish transients during a sound build-up and a sound decay. Thus, for the tone burst appearing at the time t = 0, the source function q in Eq. (11) is expressed by
{
Q 𝛿 (r′ − r0 ) sin(𝜔𝜏 ),
q( r , 𝜏 ) = ′
0 ≤ t ≤ T,
0,
t > T,
(18)
where the burst duration satisfies the condition T = 2𝜋 n∕𝜔, where n is a positive integer. The room response to the tone burst for the time t within the interval 0 ≤ t ≤ T can be found after inserting Eq. (18) into Eq. (11), integrating over the volume V and the time 𝜏 from zero to t. It gives the following formula for the sound pressure ∞ ∑
p(r, t ) = Im
[ ] (𝛼m + j𝛽m )Φm (r) ej𝜔t − Fm (t ) ,
(19)
m=1
where Im(·) is the imaginary part of a complex number and the function Fm is given by
[
]
r + j𝜔 Fm (t ) = e−rm t cos(Ωm t ) + m sin(Ωm t ) .
Ωm
(20)
Equation (19) describes the process of the sound build-up inside a room because in the bracket, the component which depends on the tone frequency, represents a steady-state response, whereas the component, which is a superposition of modal vibrations decaying exponentially with the time, constitutes a transient term. If the modal damping factors rm are sufficiently large and the time t is long enough, a steady-state will be reached inside the room. The transient response of the room at times t greater than the burst duration can be obtained by calculating the time integral in Eq. (11) from zero to T. This leads to the equation ∞ ∑
p(r, t ) = Im
[ ] (𝛼m + j𝛽m )Φm (r) Fm (t − T ) − Fm (t )
(21)
m=1
showing that when the tone burst is finished, the room response consists of two decaying transients arising at moments of turning on and turning off the sinusoidal signal. Of course, when the tone burst is long enough, the effect of the first transient on the room response will be small, so it can be omitted. To investigate how excited acoustic modes affect the reproduction of the tone burst, it should be assumed that the time interval of interest is from zero to the burst duration T. It means that the transient sound occurring just after starting the tone burst will only be taken into consideration. In this case the sound pressure is determined by Eq. (19) and this formula can be transformed into the form p(r, t ) = P (r) sin[𝜔t + 𝜙(r)] +
∞ ∑
Pm (r)e−rm t sin(Ωm t + 𝜙m )
(22)
m=1
after taking into account Eqs. (13)–(15) and (20). In the above equation P is the steady-state pressure amplitude from Eq. (17), the phase 𝜙 is given by
[ ∑∞
]
𝛽m Φm (r) , 𝛼 Φ (r) m=1 m m
m=1 𝜙(r) = arctan ∑∞
(23)
the quantities Pm are the pressure amplitudes for decaying modal vibrations Qc2 𝜔Φm (r)Φm (r0 )
Pm (r) =
Ωm
√[
2 2 (𝜔2m − 𝜔2 )2 + 4rm 𝜔
]
(24)
and the phases 𝜙m are determined by
(
𝜙m = arctan
2rm Ωm
2 𝜔2 − 𝜔2m + 2rm
)
.
(25)
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From an energetic point of view, an indicator which is adequate to quantify an accuracy of a tone burst reproduction is a deviation of the amplitude of the pressure p from the tone burst amplitude P. As shown by Eq. (22), this deviation depends on the time, thus, the following metric was proposed to evaluate the tone burst reproduction error
√
1 T ∫0
E(r) =
T
[
Pw (r, t ) − P (r) P (r)
]2 dt ,
(26)
where Pw is the amplitude of the sound pressure p from Eq. (22). Equation (26) indicates that the error E is in fact a root mean square value of the relative amplitude deviation in the time interval 0 ≤ t ≤ T and it is equal to zero when a reproduced sound is not influenced by room acoustics. For practical reasons, the amplitude Pw is best calculated using the Hilbert transform. In the continuous-time domain, this transform is defined as [23]. H[s(t )] =
∞
∞
s(𝜏 ) 1 s(t − 𝜏 ) d𝜏 = d𝜏, 𝜋 ∫−∞ t − 𝜏 𝜋 ∫−∞ 𝜏 1
(27)
where s(t) is a real-valued signal and the integral is considered as a Cauchy principal value because of the possible singularity at 𝜏 = t or 𝜏 = 0. The signal s(t) and its Hilbert transform H[s(t)] are related to each other in such a way that they together create the so-called analytic signal sa (t) defined as sa (t ) = s(t ) + jH[s(t )] = A(t )ej𝜓 (t) ,
(28)
where A(t) is the amplitude of the analytic signal A(t ) =
√
s2 (t ) + H2 [s(t )]
(29)
and 𝜓 (t) determines its phase
{
𝜓 (t) = arctan
H[s(t )] s(t )
}
.
(30)
In numerical simulations the pressure signal is of finite length and digitally sampled. This signal is represented as a sequence of numbers, in which the nth number in the sequence is denoted by s[n]. For convenience, it is assumed that n = 0, 1, … , N − 1. In the discrete-time domain, the Hilbert transform H is replaced by the discrete Hilbert transform Hd which for non-periodic signals is determined by Ref. [24]. N −1 ⎧ ∑ s[m] ⎪2 , ⎪ 𝜋 m=odd n − m Hd {x[n]} = ⎨ N −1 ⎪ 2 ∑ s[m] , ⎪𝜋 ⎩ m=even n − m
n even, (31) n odd.
The analytic discrete time signal sa [n] corresponding to s[n] is then given by sa [n] = s[n] + jHd {s[n]} = A[n]ej𝜓 [n] ,
(32)
where the envelope A[n] and the phase 𝜓 [n] are determined in the same way as A(t) and 𝜓 (t) in the continuous time case. 4. Numerical simulation 4.1. Room system under study Most theoretical studies of steady-state and transient room responses are limited to the rectangular room shape. However, in practice it is not uncommon to find that a room actually consists of several partial rooms which are coupled to each other. This fact was a motivation to carry out a numerical simulation for an enclosure having a form of a system of coupled rooms. The chosen room system had a relatively simple shape because it consisted of two rectangular subrooms of the same height which adhere to each other. The subrooms are acoustically coupled through a transparent area, so the exchange of acoustic energy takes place between them. The considered enclosure and the associated coordinate system are shown in Fig. 1. In the numerical simulation, the following enclosure dimensions were assumed: l1 = 6 m, l2 = 4 m, d1 = 5 m, d2 = 1 m, d3 = 3 m and h = 3 m. The room system was excited by a point source with the power W of 10−3 W located at the position: x0 = 1 m, y0 = 1 m, z0 = 1 m. It was assumed that wall surfaces are assigned equal and uniform sound absorption characterized by the random-incident absorption coefficient 𝛼 . In a computer program the coefficient 𝛼 was posited to be an input data, thus, its values were set in advance. Therefore, values of the specific wall conductance 𝜉 , needed to calculate the modal damping factors rm , were found by a numerical solution of the equation [22].
[
𝛼 = 8𝜉 1 +
𝜉
1+𝜉
( )] 1 − 2𝜉 ln 1 +
𝜉
(33)
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Fig. 1. The room system under study containing two coupled rectangular subrooms.
describing the theoretically determined relationship between the absorption coefficient 𝛼 and the conductance 𝜉 . The minimum value of 𝛼 assumed in the numerical simulation was 0.05 and the enclosure having wall surfaces with such absorption coefficient will be termed in the forthcoming analysis as a hard-walled room. On the other hand, it was assumed that the maximum value of 𝛼 is 0.3. Using Eq. (33) it can be found that the conductance 𝜉 corresponding to such value of 𝛼 amounts to 0.051, approximately. This value of 𝜉 is sufficiently small to treated the considered enclosure as a lightly damped room.
Fig. 2. Shapes of the functions Ψ𝜈 for the mode number 𝜈 : (a) 56, (b) 67, (c) 76, (d) 88, (e) 104, (f) 116. Modal frequencies corresponding to these functions are: (a) 206.2 Hz, (b) 227.48 Hz, (c) 242.3 Hz, (d) 260.32 Hz, (e) 286.68 Hz, (f) 303.05 Hz.
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4.2. Mode shape functions As indicated by Eqs. (17), (22) and (24), a room response to the tone burst excitation depends on the mode shape functions
Φm which, on the other hand, are dependent on a room shape. Unfortunately, an irregular geometry of the considered room system causes that there is no analytical form of the functions Φm , and thus numerical methods are required. However, the fact, that in this system there is a weak sound damping, enables us to assume that a distribution of the functions Φm is well approximated by mode shape functions computed for perfectly rigid room walls. In this case the functions Φm are replaced by the double-indexed functions Φ𝜅𝜈 whose dependence on the coordinate z describe clearly defined cosine functions. Therefore, the expression for Φ𝜅𝜈 can be written as √ ⎧ 1 ∕ h Ψ ( x , y) , 𝜅 = 0, 𝜈 > 0, 𝜈 ⎪√ Φ𝜅𝜈 (r) = ⎨ 2∕V cos(𝜋𝜅 z∕h), (34) 𝜅 > 0, 𝜈 = 0, ⎪√2∕h cos(𝜋𝜅 z∕h)Ψ (x, y), 𝜅 > 0, 𝜈 > 0, 𝜈 ⎩ where 𝜅 and 𝜈 are non-negative integers and the mode shape functions Ψ𝜈 satisfy the orthonormal property over a room horizontal cross-section. To calculate the functions Ψ𝜈 a finite element method (FEM) with 105902 nodes was applied (a distance between adjacent nodes amounted to 2 cm). When the mode shape functions Ψ𝜈 were computed, the angular modal frequencies 𝜔𝜈 were calculated from the formula
[
𝜔𝜈 = c −
]1∕2
2
Ψ 𝛁 Ψ𝜈 ds ∫S0 𝜈
(35)
found directly from the following eigenvalue equation
𝛁2 Ψ𝜈 (r) +
( 𝜔 )2 𝜈
c
Ψ𝜈 (r) = 0,
(36)
√
where S0 is a surface of a room horizontal cross-section. Since Ψ0 = 1∕ S0 , the frequency 𝜔0 corresponding to this mode shape function is obviously equal to zero. Finally, the angular modal frequencies 𝜔𝜅𝜈 for the room system were calculated from the expression
𝜔𝜅𝜈 =
√ (
𝜋 c𝜅 )2 h
+ 𝜔2𝜈 ,
(37)
Fig. 3. (a) Distribution of the tone burst reproduction error E and (b) a mapped distribution of log(E) on the observation plane z = 1.6 m. The tone burst frequency of 140 Hz, the absorption coefficient 𝛼 of 0.05.
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Fig. 4. Time dependencies of (a) the sound pressure p and (b) the pressure level Lw at the point: x = 5.3 m, y = 0.64 m, z = 1.6 m, for the tone burst frequency of 140 Hz and the absorption coefficient 𝛼 of 0.05. The tone burst reproduction error E equal to 36.8.
where the indices 𝜅 and 𝜈 are not simultaneously equal to zero. 4.3. Analysis of simulation results By using the FEM, the mode shape functions Φ𝜅𝜈 for the first 500 room modes were calculated and the modal frequencies f𝜅𝜈 = 𝜔𝜅𝜈 ∕2𝜋 corresponding to these functions were in the range 17.5–308.5 Hz. This means that a minimum number of nodes per wavelength is about 56, thus, in this frequency range the utilized finite element procedure demonstrates a high accuracy. Examples of computed shapes of the functions Ψ𝜈 are plotted in Fig. 2 in the form of filled contour maps which are a twodimensional representation of three-dimensional data. These results imply that for some modes the acoustic energy can be concentrated inside the one of subrooms (Fig. 2(a) and (c)). This effect is called the localization of modes and it is observed in coupled-room systems and enclosures having an irregular geometry. To assess a temporal accuracy of a tone burst reproduction, a computer program has been developed to calculate the tone burst reproduction error E with the aid of Eqs. (21)–(26) and Eq. (31) describing the discrete Hilbert transform. In all calculations,
Fig. 5. Distribution of the steady-state sound pressure level L on the observation plane z = 1.6 m for the source frequency of 140 Hz and the absorption coefficient 𝛼 of 0.05.
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the burst duration T of 3 s was assumed. Such value of T was chosen to ensure that the transient sound occurring at the moment of a burst start disappears over the burst duration. A transient room response was predicted for the tone burst frequency of 140 Hz. Calculation of E were carried out for the observation plane located at the distance z = 1.6 m from the floor. 4.3.1. Hard-walled room Simulation results obtained for a hard-walled room are depicted in Fig. 3. Numerical data exhibit a distribution of E on the observation plane and also a mapped distribution of log(E) to better show areas where E achieves small values. The tone burst reproduction error, as seen in Fig. 3(b), is smaller than unity on a predominant part of the observation plane. However, a surprising thing is that at some points sharp peaks in E appear (Fig. 3(a)). Values of E in these peaks are clearly larger than unity and in the extreme case at the point: x = 2.3 m, y = 4.16 m, the error E reaches the value of 68.6. On the other hand, for the narrowest peak which occurs at the point: x = 5.3 m, y = 0.64 m, the error E has the value of 36.8. To explain the appearance of strong peaks in E, the transient behavior of a room response in a time interval larger than the burst duration T was examined by a visual inspection of temporal changes in the sound pressure p and the sound pressure level Lw given by
[
Lw (r, t ) = 20 log
]
Pw (r, t ) , p0
(38)
where p0 = 20 μPa is the reference pressure and Pw is the amplitude of the pressure p calculated via the discrete Hilbert transform. Simulation results in Fig. 4 illustrate time dependencies of p and Lw at the position of the narrowest peak in E. Fig. 4(a) proves that a high inaccuracy of the tone burst reproduction is caused by a transient sound with amplitude much bigger than the burst amplitude. Moreover, the onset and cut-off transients are replica of each other and this result is compatible with experimental observations of Bladier [25]. A difference between the transient sound amplitude and the burst amplitude reaches a value of over 40 dB (Fig. 4(b)). Since, as shown by Eq. (22), the burst amplitude is equal to the pressure amplitude P at steadystate, a large burst reproduction error is a result of a very small value of P. This fact is confirmed by Fig. 5 depicting a distribution of the steady-state sound pressure level L, L = 20 log(P∕p0 ), because peaks in E, visible in Fig. 3, have approximately the same
Fig. 6. Dependencies of (a) the tone burst reproduction error E, (b) the steady-state sound pressure level L on the coordinate y for x = 5.3 m and z = 1.6 m. The tone frequency of 140 Hz, the absorption coefficient 𝛼 of 0.05.
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Fig. 7. Time dependencies of (a) the sound pressure p and (b) the pressure level Lw at the point: x = 5.3 m, y = 2.5 m, z = 1.6 m, for the tone burst frequency of 140 Hz and the absorption coefficient 𝛼 of 0.05. The tone burst reproduction error E equal to 0.212.
positions as sharp dips in a distribution of L. This rule is more spectacularly confirmed by charts in Fig. 6 showing changes in the error E and the level L versus the y-coordinate for x = 5.3 m, z = 1.6 m, because the maximum value of E matches exactly the minimum value of L. It should be noted that in this dip the level L amounts to around 35 dB which means that in this case a value of L is as much as 60 dB lower than a maximum value of L. On the other hand, for the receiving position: x = 5.3 m, y = 2.5 m, z = 1.6 m, where the level L achieves a maximum, the tone burst reproduction error has a minimum value (E = 0.212). In this case, as seen in Fig. 7, a growth in the pressure amplitude of about 2 dB above the burst amplitude is observed in a sound build-up and a sound decay is deformed by high pressure fluctuations. These effects are characteristic for hard-walled rooms and they are due to a strong excitation of two room modes of slightly different frequencies. This causes the pressure amplitude to fluctuate with a frequency equal to half the difference between frequencies of these modes (the beating effect). A detailed analysis of calculation data allowed to discover that these are the (0, 28) and (0, 29) modes with frequencies 139.21 Hz and 141.06 Hz. An area of the observation plane, where the tone burst reproduction error E assumes smallest values, are displayed in Fig. 3(b) by white color and a global minimum of E equal to 0.099 is reached at the receiving point: x = 0.3 m, y = 0.6 m, located on this area. Simulation results showing time dependencies of the pressure p and the pressure level Lw at this position are depicted in Fig. 8. These data indicate that a small value of E is the result of very short sound build-up. This is surprising because in room with acoustically hard walls the duration of a sound build-up should be significantly longer. Since the onset and cut-off transients are very similar to each other, just after the end of the tone burst a rapid decrease in the sound pressure is observed. It produces a concave-shaped decay curve characterized by a fast early decay and a slow late decay rate. Such sound decays occur at receiving positions close to the source, where turning off the source causes an initial rapid drop of a sound pressure because of a lack of strong sound reflections. As shown by calculation data, a high inaccuracy of a tone burst reproduction is a result of an appearance of sharp dips in a distribution of the level L. However, an in-depth physical understanding of this phenomenon is possible after examining a structure of the active sound intensity vector field. The active sound intensity I is a product of the sound pressure and the inphase component of the particle velocity and it describes the flow of acoustic energy in a pure-tone sound field. According to Ref. [20], a modal expansion of a sound pressure used in this study (Eq. (2)) leads to the following expression for the active sound intensity
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29
Fig. 8. Time dependencies of (a) the sound pressure p and (b) the pressure level Lw at the point: x = 0.3 m, y = 0.6 m, z = 1.6 m, for the tone burst frequency of 140 Hz and the absorption coefficient 𝛼 of 0.05. The tone burst reproduction error E equal to 0.099.
I=
1
∞ ∑ ∞ ∑
2𝜌𝜔 m=1 n=1
(𝛼n 𝛽m − 𝛼m 𝛽n )Φm (r)𝛁Φn (r).
(39)
Fig. 9(a) shows the two-dimensional vector plot illustrating a distribution of the vector I on the observation plane for the source frequency of 140 Hz and the absorption coefficient 𝛼 of 0.05. As may be seen, the characteristic objects in the active intensity field are distinct energy vortices positioned irregularly within the room. This type of vortices forms such patterns of flow where sound energy gyrates around closed paths. The measure of an intensity of this energy flow is the magnitude of the vector 𝛁 × I. This vector can be determined by taking the curl of the right-hand side of Eq. (39). The result is
𝛁×I=
1
∞ ∑ ∞ ∑
2𝜌𝜔 m=1 n=1
(𝛼n 𝛽m − 𝛼m 𝛽n )𝛁Φm (r) × 𝛁Φn (r),
(40)
where the sign × denotes a vector product. As is evidenced by calculation data in Fig. 9(b), the magnitude of 𝛁 × I maps the intensity of vortices much better than a two-dimensional vector plot of I. This graph also highlights the fact that this quantity reaches a local maximum close to the vortex center. On the contrary, the active intensity I is close to zero at the vortex center because the steady-state sound pressure vanishes in this point [26]. This means that a high inaccuracy of a tone burst reproduction should appear at the receiving point which is located in the near vicinity of the vortex center. To verify this thesis, in Fig. 9(c) calculation results of the error E have been presented in such a way as to show only those areas on the observation plane for which the value of E is greater than or equal to 2. In this figure, these areas are indicated by a black color and the smaller this area, the narrower the peak in E. A comparison between simulation data in Fig. 9(a) and (c) leads to the conclusion that narrow peaks in E are always located at centers of vortices and this rule does not apply in the case of wider peaks. As stated above, a vortex of the active sound intensity is characterized by zero pressure at the center. Equation (2) shows that, from a theoretical point of view, a null pressure can be reached when all mode shape functions Φm possess a zero value at the vortex center. For considered coupled room system this condition is practically impossible to meet, therefore, the condition for the pressure minimum is such that the functions Φm for the dominant modes must be close to zero at the vortex center. In analyzed case, these are the (0, 28) and (0, 29) modes since they are strongly excited when the room is subjected to a tone burst
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Fig. 9. (a) The active sound intensity I and (b) a magnitude of 𝛁 × I on the observation plane z = 1.6 m for the source frequency of 140 Hz and the absorption coefficient 𝛼 of 0.05, (c) areas on the observation plane for which E ≥ 2 (black color) and lines indicating zeros of the mode shape functions Φ𝜅𝜈 for the modes: (0, 28) (solid lines) and (0, 29) (dashed lines). Modal frequencies corresponding to these functions are 139.21 Hz and 141.06 Hz.
with the frequency of 140 Hz. This feature is confirmed by calculation results in Fig. 9(c), because the centers of the vortices lie almost exactly on lines indicating zeros of the mode shape functions for these modes.
Fig. 10. A distribution of the tone burst reproduction error E on the observation plane z = 1.6 m for the tone frequency of 140 Hz and the absorption coefficient 𝛼 of 0.3.
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Fig. 11. Time dependencies of (a) the sound pressure p and (b) the pressure level Lw at the point: x = 5.54 m, y = 4.38 m, z = 1.6 m, for the tone burst frequency of 140 Hz and the absorption coefficient 𝛼 of 0.3. The tone burst reproduction error E equal to 4.09.
Fig. 12. Distribution of the steady-state sound pressure level L on the observation plane z = 1.6 m for the source frequency of 140 Hz and the absorption coefficient 𝛼 of 0.3.
4.3.2. Absorbing room The effect of absorbing properties of room walls on the tone burst reproduction error E has also been investigated. Fig. 10 illustrates a distribution of E for the same burst tone frequency as before but the wall absorption coefficient 𝛼 of 0.3. This figure clearly shows that an increase in 𝛼 to the value of 0.3 causes a substantial diminution of the error E because the maximum value of E is now only 4.09. This maximum is achieved at the position: x = 5.54 m, y = 4.38 m, on the observation plane and time dependencies of the sound pressure p and the pressure level Lw at this point are depicted in Fig. 11. As seen, a much better reproduction of the tone burst is associated with a significant shortening of the duration of a transient sound. There is, however, still a 30-dB difference between the amplitude of a transient sound and the tone burst amplitude (Fig. 11(b)). It results from the fact that an increase in 𝛼 to the value of 0.3 causes only a small reduction of the maximum value of the level L and a change in positions of points, where there are local minima of L, but does not entail a significant increase in L in these points (Fig. 12). For the wall absorption coefficient 𝛼 of 0.3, the tone burst reproduction error E minimizes at the position: x = 0.92 m, y = 0.74 m, on the observation plane. Calculation results of the pressure p and the pressure level Lw obtained in this case are depicted in Fig. 13. As can be seen, due to a significant sound damping on room walls, there is a meaningful reduction of the transient duration, resulting in a high-quality reproduction of the tone burst.
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Fig. 13. Time dependencies of (a) the sound pressure p and (b) the pressure level Lw at the point: x = 0.92 m, y = 0.74 m, z = 1.6 m, for the tone burst frequency of 140 Hz and the absorption coefficient 𝛼 of 0.3. The tone burst reproduction error E equal to 0.07.
5. Summary and conclusions A transient sound always appears when a source signal suddenly rises or falls, and its presence can have a large impact on a correct sound reproduction. In this study, a rapidly changing source signal was modeled by a tone burst and a numerical procedure was developed for calculating a room response in the time interval greater than the burst duration. In a theoretical description of a transient sound, the convolution theorem and a modal representation of the room impulse response were applied. A tone burst modeled by the sine wave multiplied by a rectangular window was used to excite room modes. To assess a temporal accuracy of a sound reproduction, a metric quantifying the tone burst reproduction error was introduced. The basis for determining this metric was a difference between the tone burst amplitude and the amplitude of a sound pressure computed by using the Hilbert transform. The proposed theoretical model can be applied for any room geometry, therefore, in a numerical simulation an enclosure having a form of a system of coupled rooms was considered. The chosen room system had a shape often encountered in practice because it consisted of two rectangular subrooms of the same height which adhere to each other. As was evidenced by numerical data, a high inaccuracy of a tone burst reproduction is a result of an appearance of sharp dips in a distribution of the steady-state sound pressure level because an amplitude of transient sound is much bigger than a tone burst amplitude at the receiving point where a dip is present. A structure of the active sound intensity vector field was also examined and it was found that strong narrow peaks in the tone burst reproduction error are located at centers of energy vortices in this field. As demonstrated, vortex centers lie almost exactly on the lines corresponding to zeros of mode shape functions for dominant modes. Simulation results have also shown that a substantial increase in a sound absorption does not significantly improves a tone burst reproduction. This is because this increase does not eliminate sharp dips in a distribution of the steady-state sound pressure level. Acknowledgments This work was financially supported by the National Science Center, Poland under the project no. 2016/21/B/ST8/02427.
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Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Influence of room modes on low-frequency transients: Theoretical modeling and numerical predictions Mirosław Meissner ∗ , Krzysztof Wi´sniewski Institute of Fundamental Technological Research, Polish Academy of Sciences, Pawi´ nskiego 5B, 02–106 Warsaw, Poland
article info
abstract
Article history: Received 24 September 2018 Revised 8 February 2019 Accepted 11 February 2019 Available online 14 February 2019 Handling Editor: D. Juve
In the low-frequency range, a sound reproduction in enclosures is strongly influenced by excited room modes. While the spectral impact of acoustic modes on a room response is well recognized, there is no sufficient knowledge on how these modes affects transients. In the paper this issue has been examined theoretically and numerically for a room excited by a tone burst by using a modal expansion method supported by a computer implementation. To quantify a temporal accuracy of a sound reproduction, the new metrics referred to as the tone burst reproduction error was introduced. The basis for determining this quantity was a deviation between the tone burst amplitude and the amplitude of a sound pressure computed via the Hilbert transform. A numerical simulation was performed for an irregularly shaped enclosure having a form of two-room coupled system. Calculation results have proved that a high inaccuracy of a tone burst reproduction occurs at receiving points corresponding to sharp dips in a distribution of the steady-state sound pressure level. This is because in these points an amplitude of transient sound is much bigger than a tone burst amplitude. It was discovered that strong narrow peaks in the tone burst reproduction error are located at centers of vortices in the active sound intensity vector field. An influence of a sound damping in a room on a reproduction of a tone burst was also examined and it was found that a substantial increase in a wall sound absorption does not significantly improves a tone burst reproduction because it does not eliminate sharp dips in a distribution of the steady-state sound pressure level. © 2019 Elsevier Ltd. All rights reserved.
Keywords: Room acoustics Room modes Transients Tone burst Discrete Hilbert transform Sound intensity vector field
1. Introduction The main aim of theoretical room acoustics is to study the transient and steady-state behaviors of a sound field in enclosures. Various methods exist for modeling acoustic characteristics of rooms and among them are statistical-acoustic methods, diffusion-equation model, geometrical acoustics, wave-based methods and modal expansion method. Statistical-acoustic methods are based on the diffuse sound field hypothesis that the acoustic energy is uniform in the field [1]. The diffusion-equation model is an extension of the statistical theory to spatially varying reverberant sound fields [2]. The geometrical acoustics is adequate for high sound frequencies and most systems for geometric modeling are based on the ray tracing method [3], the beam tracing algorithm [4] and the mirror source method [5]. The fact that geometric methods do not include wave phenomena such as interferences or diffraction is a considerable disadvantage. Therefore, the results obtained with such methods are inaccurate for enclosures with complex shapes and coupled-room systems.
∗ Corresponding author. E-mail address: [email protected] (M. Meissner).
https://doi.org/10.1016/j.jsv.2019.02.012 0022-460X/© 2019 Elsevier Ltd. All rights reserved.
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In contrast to geometric approaches, wave-based methods solve the wave equation after suitable discretization to describe a complete sound field in enclosures. The most common among these numerical techniques are the finite element method (FEM) [6], the boundary element method (BEM) [7] and the finite-difference time-domain (FDTD) method [8]. Alternatives to the FDTD technique are the pseudospectral time-domain (PSTD) method [9] and adaptive rectangular decomposition (ARD) which achieve a good accuracy with a much coarser spatial discretization [10]. All these wave-based methods provide different advantages and disadvantages depending on their complexity and computational load. However, the most important difference lies in their applicability: the frequency domain methods, such as FEM and BEM, typically provide results for steady-state situations, whereas the time domain models, such as FDTD, allow to predict transient room responses [11]. The modal expansion method is a technique used for vibration analysis of mechanical objects, thus, it has been applied in several branches of technical sciences. In low-frequency room acoustics, this method allows to express the sound field in a room as a linear combination of the acoustic room modes of pressure vibrations [12–14]. The modal expansion method proved to be particularly useful for systems of coupled rooms because it enables to identify such phenomena as a localization of modes [15], a modal degeneracy [16] and a formation of energy vortices in the sound intensity vector field [17]. In this paper, the modal expansion method is applied to study the impact of room modes on low-frequency transients. In the theoretical modeling of a transient sound, a convolution theorem and a modal representation of the room impulse response are used and the sound source is assumed to generate a tone burst. To assess the temporal quality of sound reproduction, a new metrics named the tone burst reproduction error is introduced. A theoretical analysis is accompanied by numerical simulations performed for a system of two coupled rooms. For this reason, a numerical implementation exploiting the high-accuracy FEM procedure is used for calculating the mode shape functions. A detailed analysis of simulation results are carried out and final conclusions regarding the influence of room modes on transient sound are formulated. 2. Modeling of low-frequency room acoustics In room acoustics, the low-frequency band is usually defined as the frequency range bounded from above by the Schroeder frequency. This frequency allows us to subdivide the acoustic behavior of rooms into two regions. Below the Schroeder frequency there is the modal region where the modal density is low and particular modes can be observed in the room response, whereas above this frequency there is the region of diffuse sound field where there is a strong overlap of room modes. As was shown in Ref. [18], the Schroeder frequency depends on the sound damping inside the room and is determined by
√
fs = c
6
𝛼S
,
(1)
where c denotes the sound speed, S is the surface area of room walls and 𝛼 is the average absorption coefficient of wall surfaces. In the modal region, room dimensions are comparable with the sound wavelength, and the most appropriate method for determining the interior sound field is modal expansion. According to this method, the pressure room response can be described as a superposition of individual responses of acoustic modes excited inside a room by a sound source, i.e., p(r, t ) =
∞ ∑
pm (t )Φm (r),
(2)
m=1
where r = (x, y, z) is the position coordinate of a receiver, the functions pm determine a temporal behavior of the sound pressure and Φm are the mode shape functions. These functions satisfy the orthonormal property in the room volume and each Φm is related to the corresponding modal angular frequency 𝜔m through the eigenvalue equation
𝛁2 Φm (r) +
( 𝜔 )2 m
c
Φm (r) = 0,
(3)
where 𝛁 is the nabla vector operator. In a theoretical model it is assumed that room walls are covered by a sound absorbing material with a purely real admittance which is independent of frequency. It means that the mass and stiffness of this material are neglected. This is equivalent to the damping of a sound wave with no phase change upon reflection from room walls. In this case, the following boundary condition must be satisfied by the sound pressure
𝛁p · n = −
𝜉 (rs ) 𝜕 p(rs , t) , c 𝜕t
(4)
where n is the unit vector normal to the walls, which is directed away from the room volume, 𝜉 is the specific conductance of a wall material and rs is a position coordinate on room walls. It is assumed that the value of 𝜉 is small because typical materials covering room walls are characterized by a small sound absorption in low-frequency range [19]. The procedure for finding the function pm relies on a suitable solution of the wave equation and using the method presented in Ref. [20], it can be shown that for small sound damping inside a room the function pm is a solution of the following differential equation
𝜕 2 pm 𝜕p + 2rm m + 𝜔2m pm = c2 q(r′ , t )Φm (r′ )d3 r′ , ∫V 𝜕 t2 𝜕t
(5)
M. Meissner and K. Wi´sniewski / Journal of Sound and Vibration 448 (2019) 19–33
21
where q represents the volume source term in the wave equation and d3 r′ = dx′ dy′ dz′ is the volume element. In Eq. (5) the parameter rm is the modal damping factor given by rm =
c 𝜉 (r )Φ2 (r )ds. 2 ∫S s m s
(6)
A transient room response, which is most important, is the impulse response and it corresponds to the case of an impulsive temporal excitation of a room by a sound source located at a point. Thus, assuming that the volume source term in Eq. (5) has the form q(r′ , t) = 𝛿 (r′ − r0 )𝛿 (t − t0 ) one can obtain
𝜕 2 pm 𝜕p + 2rm m + 𝜔2m pm = c2 𝛿 (t − t0 )Φm (r0 ), (7) 𝜕 t2 𝜕t where r0 = (x0 , y0 , z0 ) is the position coordinate of a source point and t0 is a time of the impulse generation. A method for resolving Eq. (7) was described in detail in Ref. [21] and the obtained solution has the form c2 e−rm (t−t0 ) sin[Ωm (t − t0 )]Φm (r0 )
pm (t ) =
Ωm
√
,
(8)
2 𝜔2m − rm are the modal angular frequencies for damped vibrations. A substitution of Eq. (8) into Eq. (2) leads to
where Ωm =
the function of the form h(r0 , r, t ) = c2
∞ ∑ e−rm t sin(Ωm t )Φm (r0 )Φm (r) Ωm m=1
(9)
describing the room impulse response (RIR). Because of the causality condition, the RIR function h(r0 , r, t) is zero for t < 0. It satisfies also the reciprocity principle because the right side of Eq. (9) is a symmetric function of the source and receiver points coordinates. The room impulse response is very useful in room acoustics because a knowledge of the RIR function enables to predict the room response to any sound source. In fact, when a volume source in the wave equation is described by the source function q, the pressure response to this excitation can be found from the following expressions [19]. p(r, t ) =
∫V t
=
∫V ∫−∞
q(r′ , t ) ∗ h(r′ , r, t )d3 r′
(10)
q(r′ , 𝜏 )h(r′ , r, t − 𝜏 )d𝜏 d3 r′ ,
(11)
where V is the room volume and the symbol ∗ denotes a convolution operation. For example, the steady-state room response to a point source can be found assuming that in Eq. (11) the source function q takes the form q(r′ , 𝜏 ) = Q 𝛿 (r′ − r0 )ej𝜔𝜏 ,
(12)
where 𝜔 is the angular source √ frequency and the amplitude Q of a sound excitation is dependent on the source power W according to the formula Q = 8𝜋𝜌cW [22], where 𝜌 is the air density. Thus, after performing the volume and time integrations in Eq. (11), a formula for the steady-state complex pressure amplitude Pc is found to be as follows Pc (r) =
∞ ∑
(𝛼m + j𝛽m )Φm (r),
(13)
m=1
where the quantities 𝛼 m and 𝛽 m are given by
𝛼m =
Qc2 (𝜔2m − 𝜔2 )Φm (r0 ) 2 2 (𝜔2m − 𝜔2 )2 + 4rm 𝜔
𝛽m = −
2Qc2 rm 𝜔Φm (r0 )
,
2 2 (𝜔2m − 𝜔2 )2 + 4rm 𝜔
(14)
.
(15)
Since the amplitude Pc is complex, a quantity suitable for a prediction of the steady-state room response is the real pressure amplitude P determined by absolute value of Pc , i.e.
√
P (r) =
Pc (r)Pc ∗ (r),
(16)
where an asterisk as a superscript denotes the complex conjugate. Thus, after inserting Eq. (13) into Eq. (16) one finds the formula 1∕ 2
]2 [ ∞ ]2 ⎫ ⎧[ ∞ ∑ ⎪ ∑ ⎪ 𝛼m Φm (r) + 𝛽m Φm (r) ⎬ P (r) = ⎨ ⎪ m=1 ⎪ m=1 ⎩ ⎭
.
(17)
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Eqs. (14), (15) and (17) indicate that in the low-frequency range, where individual modes are well separated from each other, the steady-state room response will contain peaks at frequencies of these modes. A magnitude of these peaks depends on the damping of energy in modes represented by the damping factors rm , thus, a smaller damping leads to more intense peaks. 3. Transients Transients have been tested using the tone burst excitation method. In the theoretical modeling, the usual form of the tone burst was assumed, i.e. the burst was created by multiplying a sinusoidal signal of angular frequency 𝜔 with a rectangular window whose length T corresponds to a specified number of full sine-wave cycles. It was postulated that 𝜔 belongs to a lowfrequency range and the burst duration is long enough to distinguish transients during a sound build-up and a sound decay. Thus, for the tone burst appearing at the time t = 0, the source function q in Eq. (11) is expressed by
{
Q 𝛿 (r′ − r0 ) sin(𝜔𝜏 ),
q( r , 𝜏 ) = ′
0 ≤ t ≤ T,
0,
t > T,
(18)
where the burst duration satisfies the condition T = 2𝜋 n∕𝜔, where n is a positive integer. The room response to the tone burst for the time t within the interval 0 ≤ t ≤ T can be found after inserting Eq. (18) into Eq. (11), integrating over the volume V and the time 𝜏 from zero to t. It gives the following formula for the sound pressure ∞ ∑
p(r, t ) = Im
[ ] (𝛼m + j𝛽m )Φm (r) ej𝜔t − Fm (t ) ,
(19)
m=1
where Im(·) is the imaginary part of a complex number and the function Fm is given by
[
]
r + j𝜔 Fm (t ) = e−rm t cos(Ωm t ) + m sin(Ωm t ) .
Ωm
(20)
Equation (19) describes the process of the sound build-up inside a room because in the bracket, the component which depends on the tone frequency, represents a steady-state response, whereas the component, which is a superposition of modal vibrations decaying exponentially with the time, constitutes a transient term. If the modal damping factors rm are sufficiently large and the time t is long enough, a steady-state will be reached inside the room. The transient response of the room at times t greater than the burst duration can be obtained by calculating the time integral in Eq. (11) from zero to T. This leads to the equation ∞ ∑
p(r, t ) = Im
[ ] (𝛼m + j𝛽m )Φm (r) Fm (t − T ) − Fm (t )
(21)
m=1
showing that when the tone burst is finished, the room response consists of two decaying transients arising at moments of turning on and turning off the sinusoidal signal. Of course, when the tone burst is long enough, the effect of the first transient on the room response will be small, so it can be omitted. To investigate how excited acoustic modes affect the reproduction of the tone burst, it should be assumed that the time interval of interest is from zero to the burst duration T. It means that the transient sound occurring just after starting the tone burst will only be taken into consideration. In this case the sound pressure is determined by Eq. (19) and this formula can be transformed into the form p(r, t ) = P (r) sin[𝜔t + 𝜙(r)] +
∞ ∑
Pm (r)e−rm t sin(Ωm t + 𝜙m )
(22)
m=1
after taking into account Eqs. (13)–(15) and (20). In the above equation P is the steady-state pressure amplitude from Eq. (17), the phase 𝜙 is given by
[ ∑∞
]
𝛽m Φm (r) , 𝛼 Φ (r) m=1 m m
m=1 𝜙(r) = arctan ∑∞
(23)
the quantities Pm are the pressure amplitudes for decaying modal vibrations Qc2 𝜔Φm (r)Φm (r0 )
Pm (r) =
Ωm
√[
2 2 (𝜔2m − 𝜔2 )2 + 4rm 𝜔
]
(24)
and the phases 𝜙m are determined by
(
𝜙m = arctan
2rm Ωm
2 𝜔2 − 𝜔2m + 2rm
)
.
(25)
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From an energetic point of view, an indicator which is adequate to quantify an accuracy of a tone burst reproduction is a deviation of the amplitude of the pressure p from the tone burst amplitude P. As shown by Eq. (22), this deviation depends on the time, thus, the following metric was proposed to evaluate the tone burst reproduction error
√
1 T ∫0
E(r) =
T
[
Pw (r, t ) − P (r) P (r)
]2 dt ,
(26)
where Pw is the amplitude of the sound pressure p from Eq. (22). Equation (26) indicates that the error E is in fact a root mean square value of the relative amplitude deviation in the time interval 0 ≤ t ≤ T and it is equal to zero when a reproduced sound is not influenced by room acoustics. For practical reasons, the amplitude Pw is best calculated using the Hilbert transform. In the continuous-time domain, this transform is defined as [23]. H[s(t )] =
∞
∞
s(𝜏 ) 1 s(t − 𝜏 ) d𝜏 = d𝜏, 𝜋 ∫−∞ t − 𝜏 𝜋 ∫−∞ 𝜏 1
(27)
where s(t) is a real-valued signal and the integral is considered as a Cauchy principal value because of the possible singularity at 𝜏 = t or 𝜏 = 0. The signal s(t) and its Hilbert transform H[s(t)] are related to each other in such a way that they together create the so-called analytic signal sa (t) defined as sa (t ) = s(t ) + jH[s(t )] = A(t )ej𝜓 (t) ,
(28)
where A(t) is the amplitude of the analytic signal A(t ) =
√
s2 (t ) + H2 [s(t )]
(29)
and 𝜓 (t) determines its phase
{
𝜓 (t) = arctan
H[s(t )] s(t )
}
.
(30)
In numerical simulations the pressure signal is of finite length and digitally sampled. This signal is represented as a sequence of numbers, in which the nth number in the sequence is denoted by s[n]. For convenience, it is assumed that n = 0, 1, … , N − 1. In the discrete-time domain, the Hilbert transform H is replaced by the discrete Hilbert transform Hd which for non-periodic signals is determined by Ref. [24]. N −1 ⎧ ∑ s[m] ⎪2 , ⎪ 𝜋 m=odd n − m Hd {x[n]} = ⎨ N −1 ⎪ 2 ∑ s[m] , ⎪𝜋 ⎩ m=even n − m
n even, (31) n odd.
The analytic discrete time signal sa [n] corresponding to s[n] is then given by sa [n] = s[n] + jHd {s[n]} = A[n]ej𝜓 [n] ,
(32)
where the envelope A[n] and the phase 𝜓 [n] are determined in the same way as A(t) and 𝜓 (t) in the continuous time case. 4. Numerical simulation 4.1. Room system under study Most theoretical studies of steady-state and transient room responses are limited to the rectangular room shape. However, in practice it is not uncommon to find that a room actually consists of several partial rooms which are coupled to each other. This fact was a motivation to carry out a numerical simulation for an enclosure having a form of a system of coupled rooms. The chosen room system had a relatively simple shape because it consisted of two rectangular subrooms of the same height which adhere to each other. The subrooms are acoustically coupled through a transparent area, so the exchange of acoustic energy takes place between them. The considered enclosure and the associated coordinate system are shown in Fig. 1. In the numerical simulation, the following enclosure dimensions were assumed: l1 = 6 m, l2 = 4 m, d1 = 5 m, d2 = 1 m, d3 = 3 m and h = 3 m. The room system was excited by a point source with the power W of 10−3 W located at the position: x0 = 1 m, y0 = 1 m, z0 = 1 m. It was assumed that wall surfaces are assigned equal and uniform sound absorption characterized by the random-incident absorption coefficient 𝛼 . In a computer program the coefficient 𝛼 was posited to be an input data, thus, its values were set in advance. Therefore, values of the specific wall conductance 𝜉 , needed to calculate the modal damping factors rm , were found by a numerical solution of the equation [22].
[
𝛼 = 8𝜉 1 +
𝜉
1+𝜉
( )] 1 − 2𝜉 ln 1 +
𝜉
(33)
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Fig. 1. The room system under study containing two coupled rectangular subrooms.
describing the theoretically determined relationship between the absorption coefficient 𝛼 and the conductance 𝜉 . The minimum value of 𝛼 assumed in the numerical simulation was 0.05 and the enclosure having wall surfaces with such absorption coefficient will be termed in the forthcoming analysis as a hard-walled room. On the other hand, it was assumed that the maximum value of 𝛼 is 0.3. Using Eq. (33) it can be found that the conductance 𝜉 corresponding to such value of 𝛼 amounts to 0.051, approximately. This value of 𝜉 is sufficiently small to treated the considered enclosure as a lightly damped room.
Fig. 2. Shapes of the functions Ψ𝜈 for the mode number 𝜈 : (a) 56, (b) 67, (c) 76, (d) 88, (e) 104, (f) 116. Modal frequencies corresponding to these functions are: (a) 206.2 Hz, (b) 227.48 Hz, (c) 242.3 Hz, (d) 260.32 Hz, (e) 286.68 Hz, (f) 303.05 Hz.
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4.2. Mode shape functions As indicated by Eqs. (17), (22) and (24), a room response to the tone burst excitation depends on the mode shape functions
Φm which, on the other hand, are dependent on a room shape. Unfortunately, an irregular geometry of the considered room system causes that there is no analytical form of the functions Φm , and thus numerical methods are required. However, the fact, that in this system there is a weak sound damping, enables us to assume that a distribution of the functions Φm is well approximated by mode shape functions computed for perfectly rigid room walls. In this case the functions Φm are replaced by the double-indexed functions Φ𝜅𝜈 whose dependence on the coordinate z describe clearly defined cosine functions. Therefore, the expression for Φ𝜅𝜈 can be written as √ ⎧ 1 ∕ h Ψ ( x , y) , 𝜅 = 0, 𝜈 > 0, 𝜈 ⎪√ Φ𝜅𝜈 (r) = ⎨ 2∕V cos(𝜋𝜅 z∕h), (34) 𝜅 > 0, 𝜈 = 0, ⎪√2∕h cos(𝜋𝜅 z∕h)Ψ (x, y), 𝜅 > 0, 𝜈 > 0, 𝜈 ⎩ where 𝜅 and 𝜈 are non-negative integers and the mode shape functions Ψ𝜈 satisfy the orthonormal property over a room horizontal cross-section. To calculate the functions Ψ𝜈 a finite element method (FEM) with 105902 nodes was applied (a distance between adjacent nodes amounted to 2 cm). When the mode shape functions Ψ𝜈 were computed, the angular modal frequencies 𝜔𝜈 were calculated from the formula
[
𝜔𝜈 = c −
]1∕2
2
Ψ 𝛁 Ψ𝜈 ds ∫S0 𝜈
(35)
found directly from the following eigenvalue equation
𝛁2 Ψ𝜈 (r) +
( 𝜔 )2 𝜈
c
Ψ𝜈 (r) = 0,
(36)
√
where S0 is a surface of a room horizontal cross-section. Since Ψ0 = 1∕ S0 , the frequency 𝜔0 corresponding to this mode shape function is obviously equal to zero. Finally, the angular modal frequencies 𝜔𝜅𝜈 for the room system were calculated from the expression
𝜔𝜅𝜈 =
√ (
𝜋 c𝜅 )2 h
+ 𝜔2𝜈 ,
(37)
Fig. 3. (a) Distribution of the tone burst reproduction error E and (b) a mapped distribution of log(E) on the observation plane z = 1.6 m. The tone burst frequency of 140 Hz, the absorption coefficient 𝛼 of 0.05.
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Fig. 4. Time dependencies of (a) the sound pressure p and (b) the pressure level Lw at the point: x = 5.3 m, y = 0.64 m, z = 1.6 m, for the tone burst frequency of 140 Hz and the absorption coefficient 𝛼 of 0.05. The tone burst reproduction error E equal to 36.8.
where the indices 𝜅 and 𝜈 are not simultaneously equal to zero. 4.3. Analysis of simulation results By using the FEM, the mode shape functions Φ𝜅𝜈 for the first 500 room modes were calculated and the modal frequencies f𝜅𝜈 = 𝜔𝜅𝜈 ∕2𝜋 corresponding to these functions were in the range 17.5–308.5 Hz. This means that a minimum number of nodes per wavelength is about 56, thus, in this frequency range the utilized finite element procedure demonstrates a high accuracy. Examples of computed shapes of the functions Ψ𝜈 are plotted in Fig. 2 in the form of filled contour maps which are a twodimensional representation of three-dimensional data. These results imply that for some modes the acoustic energy can be concentrated inside the one of subrooms (Fig. 2(a) and (c)). This effect is called the localization of modes and it is observed in coupled-room systems and enclosures having an irregular geometry. To assess a temporal accuracy of a tone burst reproduction, a computer program has been developed to calculate the tone burst reproduction error E with the aid of Eqs. (21)–(26) and Eq. (31) describing the discrete Hilbert transform. In all calculations,
Fig. 5. Distribution of the steady-state sound pressure level L on the observation plane z = 1.6 m for the source frequency of 140 Hz and the absorption coefficient 𝛼 of 0.05.
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the burst duration T of 3 s was assumed. Such value of T was chosen to ensure that the transient sound occurring at the moment of a burst start disappears over the burst duration. A transient room response was predicted for the tone burst frequency of 140 Hz. Calculation of E were carried out for the observation plane located at the distance z = 1.6 m from the floor. 4.3.1. Hard-walled room Simulation results obtained for a hard-walled room are depicted in Fig. 3. Numerical data exhibit a distribution of E on the observation plane and also a mapped distribution of log(E) to better show areas where E achieves small values. The tone burst reproduction error, as seen in Fig. 3(b), is smaller than unity on a predominant part of the observation plane. However, a surprising thing is that at some points sharp peaks in E appear (Fig. 3(a)). Values of E in these peaks are clearly larger than unity and in the extreme case at the point: x = 2.3 m, y = 4.16 m, the error E reaches the value of 68.6. On the other hand, for the narrowest peak which occurs at the point: x = 5.3 m, y = 0.64 m, the error E has the value of 36.8. To explain the appearance of strong peaks in E, the transient behavior of a room response in a time interval larger than the burst duration T was examined by a visual inspection of temporal changes in the sound pressure p and the sound pressure level Lw given by
[
Lw (r, t ) = 20 log
]
Pw (r, t ) , p0
(38)
where p0 = 20 μPa is the reference pressure and Pw is the amplitude of the pressure p calculated via the discrete Hilbert transform. Simulation results in Fig. 4 illustrate time dependencies of p and Lw at the position of the narrowest peak in E. Fig. 4(a) proves that a high inaccuracy of the tone burst reproduction is caused by a transient sound with amplitude much bigger than the burst amplitude. Moreover, the onset and cut-off transients are replica of each other and this result is compatible with experimental observations of Bladier [25]. A difference between the transient sound amplitude and the burst amplitude reaches a value of over 40 dB (Fig. 4(b)). Since, as shown by Eq. (22), the burst amplitude is equal to the pressure amplitude P at steadystate, a large burst reproduction error is a result of a very small value of P. This fact is confirmed by Fig. 5 depicting a distribution of the steady-state sound pressure level L, L = 20 log(P∕p0 ), because peaks in E, visible in Fig. 3, have approximately the same
Fig. 6. Dependencies of (a) the tone burst reproduction error E, (b) the steady-state sound pressure level L on the coordinate y for x = 5.3 m and z = 1.6 m. The tone frequency of 140 Hz, the absorption coefficient 𝛼 of 0.05.
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Fig. 7. Time dependencies of (a) the sound pressure p and (b) the pressure level Lw at the point: x = 5.3 m, y = 2.5 m, z = 1.6 m, for the tone burst frequency of 140 Hz and the absorption coefficient 𝛼 of 0.05. The tone burst reproduction error E equal to 0.212.
positions as sharp dips in a distribution of L. This rule is more spectacularly confirmed by charts in Fig. 6 showing changes in the error E and the level L versus the y-coordinate for x = 5.3 m, z = 1.6 m, because the maximum value of E matches exactly the minimum value of L. It should be noted that in this dip the level L amounts to around 35 dB which means that in this case a value of L is as much as 60 dB lower than a maximum value of L. On the other hand, for the receiving position: x = 5.3 m, y = 2.5 m, z = 1.6 m, where the level L achieves a maximum, the tone burst reproduction error has a minimum value (E = 0.212). In this case, as seen in Fig. 7, a growth in the pressure amplitude of about 2 dB above the burst amplitude is observed in a sound build-up and a sound decay is deformed by high pressure fluctuations. These effects are characteristic for hard-walled rooms and they are due to a strong excitation of two room modes of slightly different frequencies. This causes the pressure amplitude to fluctuate with a frequency equal to half the difference between frequencies of these modes (the beating effect). A detailed analysis of calculation data allowed to discover that these are the (0, 28) and (0, 29) modes with frequencies 139.21 Hz and 141.06 Hz. An area of the observation plane, where the tone burst reproduction error E assumes smallest values, are displayed in Fig. 3(b) by white color and a global minimum of E equal to 0.099 is reached at the receiving point: x = 0.3 m, y = 0.6 m, located on this area. Simulation results showing time dependencies of the pressure p and the pressure level Lw at this position are depicted in Fig. 8. These data indicate that a small value of E is the result of very short sound build-up. This is surprising because in room with acoustically hard walls the duration of a sound build-up should be significantly longer. Since the onset and cut-off transients are very similar to each other, just after the end of the tone burst a rapid decrease in the sound pressure is observed. It produces a concave-shaped decay curve characterized by a fast early decay and a slow late decay rate. Such sound decays occur at receiving positions close to the source, where turning off the source causes an initial rapid drop of a sound pressure because of a lack of strong sound reflections. As shown by calculation data, a high inaccuracy of a tone burst reproduction is a result of an appearance of sharp dips in a distribution of the level L. However, an in-depth physical understanding of this phenomenon is possible after examining a structure of the active sound intensity vector field. The active sound intensity I is a product of the sound pressure and the inphase component of the particle velocity and it describes the flow of acoustic energy in a pure-tone sound field. According to Ref. [20], a modal expansion of a sound pressure used in this study (Eq. (2)) leads to the following expression for the active sound intensity
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Fig. 8. Time dependencies of (a) the sound pressure p and (b) the pressure level Lw at the point: x = 0.3 m, y = 0.6 m, z = 1.6 m, for the tone burst frequency of 140 Hz and the absorption coefficient 𝛼 of 0.05. The tone burst reproduction error E equal to 0.099.
I=
1
∞ ∑ ∞ ∑
2𝜌𝜔 m=1 n=1
(𝛼n 𝛽m − 𝛼m 𝛽n )Φm (r)𝛁Φn (r).
(39)
Fig. 9(a) shows the two-dimensional vector plot illustrating a distribution of the vector I on the observation plane for the source frequency of 140 Hz and the absorption coefficient 𝛼 of 0.05. As may be seen, the characteristic objects in the active intensity field are distinct energy vortices positioned irregularly within the room. This type of vortices forms such patterns of flow where sound energy gyrates around closed paths. The measure of an intensity of this energy flow is the magnitude of the vector 𝛁 × I. This vector can be determined by taking the curl of the right-hand side of Eq. (39). The result is
𝛁×I=
1
∞ ∑ ∞ ∑
2𝜌𝜔 m=1 n=1
(𝛼n 𝛽m − 𝛼m 𝛽n )𝛁Φm (r) × 𝛁Φn (r),
(40)
where the sign × denotes a vector product. As is evidenced by calculation data in Fig. 9(b), the magnitude of 𝛁 × I maps the intensity of vortices much better than a two-dimensional vector plot of I. This graph also highlights the fact that this quantity reaches a local maximum close to the vortex center. On the contrary, the active intensity I is close to zero at the vortex center because the steady-state sound pressure vanishes in this point [26]. This means that a high inaccuracy of a tone burst reproduction should appear at the receiving point which is located in the near vicinity of the vortex center. To verify this thesis, in Fig. 9(c) calculation results of the error E have been presented in such a way as to show only those areas on the observation plane for which the value of E is greater than or equal to 2. In this figure, these areas are indicated by a black color and the smaller this area, the narrower the peak in E. A comparison between simulation data in Fig. 9(a) and (c) leads to the conclusion that narrow peaks in E are always located at centers of vortices and this rule does not apply in the case of wider peaks. As stated above, a vortex of the active sound intensity is characterized by zero pressure at the center. Equation (2) shows that, from a theoretical point of view, a null pressure can be reached when all mode shape functions Φm possess a zero value at the vortex center. For considered coupled room system this condition is practically impossible to meet, therefore, the condition for the pressure minimum is such that the functions Φm for the dominant modes must be close to zero at the vortex center. In analyzed case, these are the (0, 28) and (0, 29) modes since they are strongly excited when the room is subjected to a tone burst
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Fig. 9. (a) The active sound intensity I and (b) a magnitude of 𝛁 × I on the observation plane z = 1.6 m for the source frequency of 140 Hz and the absorption coefficient 𝛼 of 0.05, (c) areas on the observation plane for which E ≥ 2 (black color) and lines indicating zeros of the mode shape functions Φ𝜅𝜈 for the modes: (0, 28) (solid lines) and (0, 29) (dashed lines). Modal frequencies corresponding to these functions are 139.21 Hz and 141.06 Hz.
with the frequency of 140 Hz. This feature is confirmed by calculation results in Fig. 9(c), because the centers of the vortices lie almost exactly on lines indicating zeros of the mode shape functions for these modes.
Fig. 10. A distribution of the tone burst reproduction error E on the observation plane z = 1.6 m for the tone frequency of 140 Hz and the absorption coefficient 𝛼 of 0.3.
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Fig. 11. Time dependencies of (a) the sound pressure p and (b) the pressure level Lw at the point: x = 5.54 m, y = 4.38 m, z = 1.6 m, for the tone burst frequency of 140 Hz and the absorption coefficient 𝛼 of 0.3. The tone burst reproduction error E equal to 4.09.
Fig. 12. Distribution of the steady-state sound pressure level L on the observation plane z = 1.6 m for the source frequency of 140 Hz and the absorption coefficient 𝛼 of 0.3.
4.3.2. Absorbing room The effect of absorbing properties of room walls on the tone burst reproduction error E has also been investigated. Fig. 10 illustrates a distribution of E for the same burst tone frequency as before but the wall absorption coefficient 𝛼 of 0.3. This figure clearly shows that an increase in 𝛼 to the value of 0.3 causes a substantial diminution of the error E because the maximum value of E is now only 4.09. This maximum is achieved at the position: x = 5.54 m, y = 4.38 m, on the observation plane and time dependencies of the sound pressure p and the pressure level Lw at this point are depicted in Fig. 11. As seen, a much better reproduction of the tone burst is associated with a significant shortening of the duration of a transient sound. There is, however, still a 30-dB difference between the amplitude of a transient sound and the tone burst amplitude (Fig. 11(b)). It results from the fact that an increase in 𝛼 to the value of 0.3 causes only a small reduction of the maximum value of the level L and a change in positions of points, where there are local minima of L, but does not entail a significant increase in L in these points (Fig. 12). For the wall absorption coefficient 𝛼 of 0.3, the tone burst reproduction error E minimizes at the position: x = 0.92 m, y = 0.74 m, on the observation plane. Calculation results of the pressure p and the pressure level Lw obtained in this case are depicted in Fig. 13. As can be seen, due to a significant sound damping on room walls, there is a meaningful reduction of the transient duration, resulting in a high-quality reproduction of the tone burst.
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Fig. 13. Time dependencies of (a) the sound pressure p and (b) the pressure level Lw at the point: x = 0.92 m, y = 0.74 m, z = 1.6 m, for the tone burst frequency of 140 Hz and the absorption coefficient 𝛼 of 0.3. The tone burst reproduction error E equal to 0.07.
5. Summary and conclusions A transient sound always appears when a source signal suddenly rises or falls, and its presence can have a large impact on a correct sound reproduction. In this study, a rapidly changing source signal was modeled by a tone burst and a numerical procedure was developed for calculating a room response in the time interval greater than the burst duration. In a theoretical description of a transient sound, the convolution theorem and a modal representation of the room impulse response were applied. A tone burst modeled by the sine wave multiplied by a rectangular window was used to excite room modes. To assess a temporal accuracy of a sound reproduction, a metric quantifying the tone burst reproduction error was introduced. The basis for determining this metric was a difference between the tone burst amplitude and the amplitude of a sound pressure computed by using the Hilbert transform. The proposed theoretical model can be applied for any room geometry, therefore, in a numerical simulation an enclosure having a form of a system of coupled rooms was considered. The chosen room system had a shape often encountered in practice because it consisted of two rectangular subrooms of the same height which adhere to each other. As was evidenced by numerical data, a high inaccuracy of a tone burst reproduction is a result of an appearance of sharp dips in a distribution of the steady-state sound pressure level because an amplitude of transient sound is much bigger than a tone burst amplitude at the receiving point where a dip is present. A structure of the active sound intensity vector field was also examined and it was found that strong narrow peaks in the tone burst reproduction error are located at centers of energy vortices in this field. As demonstrated, vortex centers lie almost exactly on the lines corresponding to zeros of mode shape functions for dominant modes. Simulation results have also shown that a substantial increase in a sound absorption does not significantly improves a tone burst reproduction. This is because this increase does not eliminate sharp dips in a distribution of the steady-state sound pressure level. Acknowledgments This work was financially supported by the National Science Center, Poland under the project no. 2016/21/B/ST8/02427.
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