Room acoustical optimization: Average values of room acoustical parameters as a function of room shape, absorption and scattering

Transcription

1 Buenos Aires 5 to 9 September, 26 PROCEEDINGS of the 22 nd International Congress on Acoustics Architectural Acoustics for Non-Performance Spaces: Paper ICA26-56 Room acoustical optimization: Average values of room acoustical parameters as a function of room shape, absorption and scattering Uwe M. Stephenson (a) (a) HafenCity University, Germany, post@umstephenson.de Abstract Numerical methods like ray tracing nowadays allow a comfortable computation of room acoustical parameters like reverberation time (RT), definition (D), lateral efficiency (LE) and strength (G). However, they do not deliver rules for an optimum room design. It is an old dream to have an inverse method that, for some given target parameters describing room acoustical qualities, delivers the (or one) optimum room shape and the distribution of absorbers and diffusors. Going only a small part of that way, this approach aims at to just estimate average values of some room acoustical parameters as a function of room volume and proportions, mean absorption values, and source-listener distance. By methods of geometric-statistical room acoustics, some relations are derived from average echograms and average time delays of first reflections and verified by ray tracing experiments. Concerning the important lateral reflections, non-diffuse sound fields have to be considered, so numerical methods are needed, however, may be restricted to 2D. The relationships between different shapes of ground plans (like rectangular, trapezoidal or circular, or different kinds of zick-zack side walls), scattering coefficients and lateral efficiency are investigated by sound particle simulations. Keywords: room acoustics, diffuse sound field, room acoustical parameters, average values

2 22 nd International Congress on Acoustics, ICA 26 Buenos Aires 5 to 9 September, 26 Average values of room acoustical parameters as a function of room shape, absorption and scattering. Introduction As argued in the abstract, numerical methods like ray tracing may deliver reliable values for room acoustical parameters like reverberation time (RT), definition (D), lateral efficiency (LE) and strength (G) [] considered here to be the most important. But they do not deliver rules for an optimization. Therefore, analytical estimates for average values of room acoustical parameters are desirable. Here, only general relationships are aimed at, valid for an average over all source and receiver positions (S and R). In this paper, some approaches are presented strictly based on geometric statistical room acoustics (GSRA). Often, sufficient speech intelligibility (described by a Speech Transmission Index STI) is important. Therefore, beside the well known formulae for the RT and the stationary level (related to G), valid for a diffuse sound field DSF [2], mainly a formula is derived for the average Definition (distinctness, German: 'Deutlichkeit') in a semi-diffuse sound field (SDSF [3]), as a function of the RT T and the source-receiver-distance (r= SRD): D(T, r). D is defined by the energy fraction received within the first 5ms in the energetic impulse response. For simplification, only the average quantities of standard reverberation theory for a diffuse sound field shall be used: and a) the average (surface weighted) absorption degree α of the room (physical parameter) b) the mean free path length (mfp) Λ as a geometric-statistical quantity. Hence, D(T, r) = D(V, S, α, r) (with V= room volume, S=total surface) To describe also the influence of room shape, for simplification a rectangular room is assumed. Preferring dimensionless quantities, the width/height proportion p = W/H and the length/height proportion q = L/H are introduced in the typical and most occurring range of p q 3. This is hardly a restriction as far as a diffuse sound field, hence diffuse reflections are assumed. Hence, D(V, S, α, r) = D(V, p, q, α, r) = D(T, r, V, p, q). As usual, the main dimensions of the room should be large compared with wavelength such that higher room modes occur. Thus, above the certain frequency (the Schroeder frequency [4]), sound incidence can be assumed to be practically incoherent, energies may be added ('sound particle model'). Statistical room acoustics is energetic acoustics. In the following, all acoustical quantities are meant frequency dependent. For the lateral efficiency, related to the impression of spaciousness -highly important for concert halls different from the promise in the abstract, it must be referred to the oral presentation and a future paper. By time reasons, this is only a preliminary paper. 2

3 22 nd International Congress on Acoustics, ICA 26 Buenos Aires 5 to 9 September, The general assumptions of standard reverberation theory: the diffuse sound field The required diffuse sound field as defined by [5] consists of: A) Isotropy (intensity per solid angle j = di/dω is constant. B) Homogeneity: the energy density is constant, hence is simply the total room energy per volume: U = E/V, so the intensity, here interpreted as a scalar, I = jdω = c U (a) is spatially constant and is (with c= sound velocity) I = c E/V (b) So also the irradiation strength onto all surfaces is constant. It should be kept in mind that only under this condition the definition of an 'equivalent absorption area' A makes sense. The average (surface weighted) absorption degree α of the room, as the main average physical parameter of the room, is α = K i= α is i S (where the α i are the absorption degrees of the single walls.) = A/S () The surface qualities, required for a diffuse sound field are, strictly speaking: C) all parts of the surface must not be absorbing (-absorption); D) all parts of the surface must be totally diffusely reflecting, according the Lambert law. As this is utopic, it is usually formulated, for an approximately ('sufficiently') diffuse sound field, the average α should be 'small' ('<.3'). (Actually all single values must be small), the room proportions should not be 'too unequal' (< 3:), and an often underestimated condition at least parts of the surface (at least perpendicular to each main direction) should be scattering ('rough') [ISO ]. Fortunately, this is often the case.under the conditions of a diffuse sound field, the mean free path length ('a sound particle sees between two reflections') is [4] Λ = 4V S as the main average geometric quantity. 3. The reverberation time and the intensity level in a diffuse sound field To enable a distinction between subsequent reflections, the Eyring theory is preferred. Assuming a diffuse sound field, this is based on the model of a 'representative sound particle' which, in time intervals t = Λ/c (2), stepwise loses the energy fraction α at each reflection. This process starts with an energy E at t=. This convention is important in the following. Thus, the energy as a function of reflection order k follows the exponential function () 3

4 22 nd International Congress on Acoustics, ICA 26 Buenos Aires 5 to 9 September, 26 E(k) = E ( α) k = E ρ k = E e k α (3) where ρ = α is the energetic reflection degree and an 'absorption exponent' is introduced: α = ln( α) = ln(ρ) (4). The following considerations are important for the energy estimations in the context of the semidiffuse sound field: The decay of the total energy in the room is a continuous process, 'infinitely many sound particles' are emitted starting from t=, so also the decay E(t) starts at t=. The direct sound ' the. reflection' - arrives on average at t/3 (see 4..). But the average arrival time of reflections is not correlated to the average arrival time of the direct sound. So, at the time k t (not later), on average over all sound particles, really k = t = ct/λ ( k is now a real number) (5) t reflections have happened and (inserting equ. 5 in 3) the total energy in the room is decayed to with the Eyring time constant E(t) = E(k) = E ρ k = E e k α = E e cα Λ t = E e t τ (6) τ ey = Λ c α In reverberation theory, the RT is always proportional to mean path length and inversely proportional to a mean absorption coefficient. The common value of T T 6 (for the decay by 6dB) follows from the equation e T/τ = 6, hence T = 6 ln() τ. (8) The known equation finally follows with the value of the sound velocity c = 34m/s at 4 C: T ey = 6 ln() c 4V S α.63 (7) V S α (9) (the units s and m are omitted in the following). Sabine formula follows as an approximation expanding α = ln( α) = α + α2 2 + α3 3 α ( + α 2 ) α ( α 2 ) α α = α ρ α () and substituting S α by the equivalent absorption area A: T Sab =.63 V A (a) 3.. The intensity level in a diffuse sound field The steady state (diffuse field) intensity with a sound source of constant sound power P is closely related to the RT. It is the integral over the intensity of the energetic impulse response I(t) (as in equ. 6) after one emitted impulse of energy E : I stat = I() = P E I(t)dt Inserting equns. 6,7, I = ce /V and finally equ. yields I stat = cp V e t /τ dt = cpτ = P Λ = 4P = 4P V V α S α A (b) (c). 4

5 22 nd International Congress on Acoustics, ICA 26 Buenos Aires 5 to 9 September, 26 (This is the physical base of the level-equation L = L w + 6 lg(a) where L w is the sound power level and closely related to the strength G). (d) 4. The average definition in the diffuse and semi diffuse sound field The target parameter 'definition' D is defined by the energy fraction received within the first 5ms after arrival of the direct sound: with and D = t dir +t D t I(t)dt dir I(t)dt t dir (). t D = L /c = 5ms L = 7m (2a) (2b) and the arrival time of the direct sound t dir = r/c (3). To achieve dimensionless parameters, some lengths are related to the mean free path length as a characteristic parameter of the room. So it is introduced A relative SRD is defined by With this is e t dir τ = e Inserting equ. in equ.6 the intensity function reads This inserted in the definition equ. yields D DSF = e t D (The effect of time delay by t dir cancels.) l = L /Λ > (4) r = r Λ (5) r Λ α = (e α ) r` = ρ r` (6) I(t) = c E /V e t τ (7). With some of the above equations the following useful equalities can be derived e t D τ L Λ α = e τ (8). = (e α ) l` = ρ l = e.3s ln ()/T =.3s/T 2 T (9) With that, the wanted relationship between D and RT for the diffuse sound field is T D DSF = 2 = ρ l (2). As expected, for T, D and for T, D, and e.g. for T=s, D =.5, the recommended value for sufficient speech intelligibility (see fig. 3, dotted curve below). 5

6 22 nd International Congress on Acoustics, ICA 26 Buenos Aires 5 to 9 September, The average delay of first reflections The direct sound is followed by the reverberation beginning from the first reflections. This is described by the same exponential function as with the diffuse sound field (equ. 7), however beginning at the moment, on average the first reflections of first order arrive. This time delay has to be estimated carefully. Therefore, the mean path lengths from S to R over a reflection are considered (see fig. ). In a diffuse sound field, any straight line crossing an equally distributed source point S with an equally distributed orientation has the average length Λ. Now, equal distribution in space means also equal distribution on any straight line (not equal distribution e.g. of x-values). So, on average, S is in the middle of the line and the average path length (not the distance) to the surface is Λ/ 2. Also the probability for forward and backward direction on the line is equal. Choosing the shorter one, the average of the shorter distances is Λ/ 4. By symmetry reasons, the same considerations are valid from a reflection point P on the surface to a receiver R: as the reflection law is random and the local probability density of all P constant, even for every already specified R, always an P can be found from which a line goes out reaching R. So, the average total running distance of first reflections from S to R is Λ (and of k th order on average kλ which confirms the considerations in section 3), and the average shortest running distance of those is Λ/ 2. So, the wanted average arrival time of first reflections is t = t/2 = Λ/(2c) (2) Of course, the arrival time ranges of reflections of higher order overlap and are certainly wider than ± t/2, but for simplicity the range is assumed to be ± t/2; this is confirmed also by energetic considerations. S* Λ Λ/3 R* S P Λ/4 Λ/4 R Λ Figure : Path lengths in a room: the mean of the shorter paths from a source point S to a wall is a quarter of the mean free path length Λ ; the same from a reflection point to the receiver R; thus the mean shortest once reflected path from S to R is Λ/2. The average path length of the direct sound averaged over all source S* and receiver R* positions is Λ/3. Proof: As argued already, equal distribution in space means the same for any straight line through any S-R-combination. The total distance may be set to. The coordinates of S and R in that one dimension be x and y. Then their mean distance is y d = x y dx dy = { (x y)dx + (y x)dx} dy = y { y2 y y2 + y 2 y2 } dy = 2 {y 2 y + } dy = y3 y2 + y = /3. So the average arrival time of the direct sound is t/3. 6

7 22 nd International Congress on Acoustics, ICA 26 Buenos Aires 5 to 9 September, The average intensity-time function of the semi-diffuse sound field In a semi-diffuse sound field, the contribution of the direct sound is handled separately according to the 'r²-distance law'. If an δ-impulse of energy E had been emitted from the source point S, the strength (intensity * time) at the receiver point R (at the time t dir = r/c) is J = E /(4π r²) (22) For the reverberation, beginning with the first reflections, still a diffuse sound field and hence an exponential decay of the location-independent sound energy density is assumed. So, the average intensity-time function of the semi-diffuse sound field is composed by the peak of the direct sound (equ. 22), followed - after an initial time delay gap (t t dir ) by the continuous exponential reverberation decay (see figure 2): I(t, r) = E /(4π r²) δ(t r/c) + [c E /V e t τ ]valid from t>t (23). As even in the largest rooms of consideration (V=2m³, and p=2, q=3) the mean free path length is no longer than 5m, even if r =, is r + 7m > Λ > Λ/2. So, t dir + t D is always > t = Λ/(2c) and t dir < t/2 < t dir + t D. intensity I t Dir r/λ Δt/2 /2 t Dir + t D (r + 7m)/Λ time t -> reflection order k -> Figure 2: Typical intensity-time function of the semi-diffuse sound field: on the left, at t dir = r/c, the peak of the direct sound (depicted with an arbitrary height), on the right, starting from t = Λ/(2c) = t/2, the exponential reverberation decay (full line, dashed on the left, the decay as included in the diffuse sound field). Shaded, the range as evaluated for the definition D: from the average arrival time of the reflections t to t D = 5ms. On another axis, the corresponding reflection numbers k = ct/λ Derivation of the average definition in the semi diffuse sound field Now, the definition D can be computed inserting the intensity-time function (equ. 23) into the definition equation. Thereby, the integral over the direct sound peak is E /(4π r²) and the nominator of equ. is t dir +t D I(t)dt = E t + c E τ (2τ) (e t dir 4π r² V e (tdir+t D ) τ ) (24) 7

8 22 nd International Congress on Acoustics, ICA 26 Buenos Aires 5 to 9 September, 26 where from now on τ τ Ey. The integral of the denominator is I(t)dt = E t + c E τ dir 4π r² V t (2τ) e (25) To express this as a function of the room's reflection degree, equns. 3,5 and t τ = α are used to derive and especially These and equ. 6, inserted in equns.24 and 25, yield e k t τ = (e α ) k = ρ k (26) t (2τ) e = ρ (27) D SDSF (r, τ, T, ρ, Λ, V) = 4π r² +cτ ( ρ ρr Λ 2 T ) V 4π r² +cτ V ρ (28) To avoid the second occurrence of τ, with τ = Λ (c α ), Λ = 4V S it is cτ = Λ = 4. This is inserted in equ. 28. The result is the definition D in the semi-diffuse sound field as a function of V Vα Sα the RT, the source receiver distance r and additionally some room parameters: D SDSF (r, T, ρ, α, Λ, S) = 4.4. The reverberation radius S α + ρ(r Λ /2 6π r² ρ ) 2 T S α (29) 6π r² ρ + Here, it seems suitable to introduce the term 'reverberation radius' r r. This is defined as the distance from a source point S of constant sound power P, at which (in a stationary field) the direct sound immission equals the reverberation, here assumed to begin after the time t = Λ/(2c). The same is valid for a once emitted δ-impulse of energy E. So, one can set: first = second term in equ. 25. With equ. 27 and cτ = 4 this leads to the formula V Sα r r = S α S α = 6π ρ 6π ( α) S 6π ( ρ ) = S 6π eα (3a,b) The approximation is based on that of equ.. (The known equation r r A/(6π) A is an 5 approximation only valid for small average absorption degrees; it must be wrong for α as it then should go to infinity: r r ). Equ.3b can also be derived by considering a grid of mirror image sources around the original source which approaches a diffuse sound field. From a balance of number and energy of these, the formula for the stationary intensity (equ.c) can be derived and, by a classification of mirror sources of different orders, also the reverberation radius. This confirms the above mentioned time limit of t for the arrival of first reflections of first order. The 2 first term inserted into equ. 29 yields ρ(r Λ /2 D SDSF (r, T, r r, Λ, ρ) = ) 2 T (r r /r)²+ (3) 8

9 Deutlichkeit 22 nd International Congress on Acoustics, ICA 26 Buenos Aires 5 to 9 September, 26 depending, beside T, additionally from the room's mean reflection degree ρ depending from T; the mean free path length Λ depends from the volume V and the surface S, r r from S and ρ. 4.5 Specialization to a rectangular room To use just one parameter for the absolute size of the room, V, it is convenient to compute S and Λ from V assuming, as mentioned, a rectangular room with the proportions p and q. Volume is height* width* length. So V = H W L = H³ p q (32) vice versa 3 H(V) = V/(pq) (33) The surface is S = 2 (HBW + HL + WL) = 2H² (p + q + pq) (34) with equ. 33: S(V) = (V/(pq)) 2/3 2(p + q + pq) (35) Finally, also the central quantity mean free path length Λ = 4V/S (equ.) can be expressed as a function of V, p and q. Introducing a factor n = Λ/H by it is As p q 3 it is 2 3 n 6 5 n = 2pq = 2 (p+q+pq) +/p+/q 3 Λ(V) = n H = n V/(pq), so varying less than by a factor of 2. (36) (37). In the following examples, the typical values p=2 and q=3 are assumed The average definition as a function of reverberation time and distance Now the functions 29 or 3 may be expressed with V, p and q as parameters: D(T, r, V, p, q). Definition D Deutlichkeit im semidiffusen Schallfeld als f(t), par. r, V=m³ p=2, q= Nachhallzeit [s] -> Reverberation time [s] Figure 3: Definition D in the semi-diffuse sound field as a function of the RT, with the source receiver distance r as parameter (according equ. 3 with 3 and 37). T in the usual range of.2 2.6s, r in the range up to a half mean free path length (.3m 3m) (V=m³, p=2, q=3); dashed curve below: definition in a diffuse sound field according equ. 2 r=.3m r=.6m r=.9m r=.2m r=.5m r=.8m r=2.m r=2.4m r=2.7m r=3m source-receiver-distance - 9

10 22 nd International Congress on Acoustics, ICA 26 Buenos Aires 5 to 9 September, 26 Figure 3 shows the definition D in the semi-diffuse sound field as a function of the RT, with the source receiver distance r as parameter. As with a diffuse sound field, for T, D. Also for r, D,. As also expected, in the far field r, definition tends to be the same as in the diffuse sound field: D SDSF D DSF. For T=s and r,, D =.5 the recommended value for sufficient speech intelligibility. Comparing the different curves, the same results e.g. at a finite distance of 2m and a T=2s 5. Lateral efficiency The lateral efficiency (LE) or better, lateral energy fraction (LEF) [] is defined as the fraction of energy arrived before 8ms delay time from the side compared with the total early energy: LEF = 8ms t I(t)cos²(φ)dt 5ms 8ms I(t)dt The side effect is measured by a factor of cos²(φ), where φ is the angle between the incident ray (sound) and the axis through the listener s ears if he looks to the source. The idea for an approximation is: in medium sized rooms, only lateral mirror image sources of first order need to be considered, if diffuse reflections, roughly only constantly behind the middle of the lateral walls while the average over all source points is made only over the front quarter of the room (the stage) and the averaging over all receivers only over the rest, the audience area. The relationships between proportions shape and LF still have to be investigated. 6. Conclusions Based on statistical room acoustics, some formulas have been derived to estimate averaged room acoustical parameters, especially definition as a function of source-receiver distance and RT. As usual, they are only valid for rooms with a sufficiently diffuse sound field (sufficient scattering, not too unequal geometry and absorption distributions). Although some relationships depend on the room proportions p and q, their influence on RT, G and D is small. However, it is expected that the influence of room proportions on lateral efficiency, and hence spaciousness, is considerable. This is especially important for concert halls. Systematic investigations are planned and are possibly presented orally at the conference and published in a future paper. (38). References [] ISO 3382: Acoustics Measurement of Room Acoustic Parameters; part : Performance Spaces; part 2: Reverberation times in common rooms. International Standards Organization, Geneva, Switzerland, 22-5 [2] ISO , Building acoustics Estimation of acoustic performance of buildings from the performance of elements Part 6: Sound absorption in enclosed spaces. CEN, Brussels, 23 [3] Cremer, L., Müller, H.; Principles and Applications of Room Acoustics, Vol. ; translated by T.J. Schultz; Applied Science Publishers, London, New York 978 [4] Kuttruff, H.: Room Acoustics, Taylor & Francis, 5th edition, 29 [5] Stephenson, U.M.: A rigorous definition of the term diffuse sound field and a discussion of different reverberation formulae; in: proc. of ICA Buenos Aires, Argentina, 26