Room Acoustics and Noise Control

Room Acoustics and Noise Control

Reverberation, Transmission Loss, NC and RC Ratings, Barriers and Outdoor Propagation

What the tool answers

Three questions an acoustical consultant or a mechanical engineer meets on most projects: is a room too reverberant and how much absorption fixes it; how much of the noise in one room reaches the next through the wall; and does the noise, from an HVAC system indoors or from equipment outdoors, meet the criterion for the space. The methods are those of the Department of Defense design manual UFC 3-450-01 (2022, the successor of Army TM 5-805-4 and NAVFAC DM-3.10) with the classical closed forms written out, so every number on the page can be followed by hand. The three sub-tools share one page; the select at the top shows the inputs of the one chosen.

Room reverberation

Sabine's absorption of a room in an octave band is A = Σ S_i α_i, the sum over its surfaces of area times absorption coefficient, plus the absorption of the people and furniture, which is tabulated per person or per chair (UFC Table C-4 part B). The mean coefficient is ᾱ = A/S and the room constant R = A/(1 − ᾱ) = S ᾱ/(1 − ᾱ) (UFC eq. C-3 calls the plain sum the room constant; the tool keeps the two apart). Coefficients of general building materials are tabulated at 125 to 4,000 Hz; the UFC extends them to the 63 Hz band at 70 % of the 125 Hz value and to the 8 kHz band at 80 % of the 4 kHz value, and the tool does the same, greying those two columns. The air absorbs too: the term 4mV, with m the energy attenuation constant of the air (from the ISO 9613-1 pure-tone absorption coefficient α in dB/m, m = α/4.343), matters above 2 kHz in large rooms.

The reverberation time is the time for the sound to fall 60 dB after the source stops:

Sabine: T₆₀ = 0.161 V/(A + 4mV) (V in m³, A in m²; 0.049 V/A in feet),

Eyring: T₆₀ = 0.161 V/(−S ln(1 − ᾱ) + 4mV),

Millington: T₆₀ = 0.161 V/(−Σ S_i ln(1 − α_i) + 4mV).

Sabine's form is the small-ᾱ limit of Eyring's and overestimates the time in absorbent rooms; Millington's form counts each surface's own −ln(1 − α_i) and is the right one when the absorption is concentrated on one surface, but a surface with α = 1 would make it zero, so coefficients are capped at 0.999. The Schroeder frequency f_S = 2000 √(T₆₀/V) (V in m³) marks the lower limit of the diffuse-field region in which these formulas, and the NC rating, make sense; below it the room's individual modes govern.

Treatment. For a target time T the absorption required is A_req = 0.161 V/T − 4mV and the amount to add is ΔA = A_req − A. Replacing a surface of coefficient α_old by a treatment of coefficient α_t over an area S_t adds S_t(α_t − α_old), so S_t = ΔA/(α_t − α_old). The tool sizes the area at the design band and tabulates every band: porous treatments are weak below 250 Hz, so a target met at 500 Hz is usually missed at 125 Hz unless a thick or resonant absorber is used. ANSI/ASA S12.60 limits classroom reverberation to 0.6 s (rooms up to 10,000 ft³) or 0.7 s (up to 20,000 ft³); offices and conference rooms are commonly designed for 0.5 to 0.8 s, lecture rooms 0.8 to 1.0 s, music rooms 1.5 s and more; the target is an entered value.

Sound field of a source. A source of sound power level L_w (dB re 1 pW) and directivity Q (1 in free space, 2 on a floor or wall, 4 at an edge, 8 in a corner) produces at distance r

L_p = L_w + 10 log₁₀(Q/(4πr²) + 4/R) (r in m, R in m²; add 10.3 dB when r is in ft and R in ft²).

UFC Appendix C reads the same quantity from its Figure C-1 as the "relative SPL" for a non-directional source; the figure's curves agree with the formula at Q = 1 within 1 dB (the tool's test reproduces Tables C-5 and C-6 to that tolerance). The direct and reverberant terms are equal at the critical distance r_c = √(Q R/(16π)); a receiver beyond it hears the room, and added absorption lowers its level by 10 log of the ratio of room constants (3 dB per doubling), while a receiver inside it hears the source and only distance, enclosure or a barrier helps. The band levels are A-weighted with the IEC 61672-1 weighting and rated against the NC curves (below) and the UFC Table 3-1 design goal of the space type.

Sound transmission

Single panel. A limp panel of surface mass m (kg/m²) obeys the field-incidence mass law TL = 20 log₁₀(m f) − 47 dB: 6 dB per doubling of mass or of frequency. A real plate bends, and at the coincidence (critical) frequency f_c = c²/(1.81 c_L h), with c_L = √(E/(ρ(1 − ν²))) the longitudinal wave speed and h the thickness, its bending wavelength matches the trace of the incident sound and the panel radiates freely: the TL dips, then recovers under damping control, TL = 20 log₁₀(m f) − 47 + 10 log₁₀(2η f/(π f_c)) (Cremer), where η is the total loss factor of the installed panel. The tool keeps the mass law up to f_c/2, draws a straight line on a logarithmic frequency axis from there to the damping-controlled value at f_c, and follows the damping-controlled form above. Sharp's plateau method, which caps the mass law by a tabulated material plateau, is the usual alternative; both are laboratory estimates that field installations miss by 4 to 5 dB (UFC C-5.1), and the UFC's own tables of measured values (Tables C-8 to C-22) are the better source where they apply. For gypsum board, 1/2 in at 50 lb/ft³ and 0.29 Msi gives f_c near 3,100 Hz; thicker, stiffer panels have lower f_c, which is why 1/4-in glass dips in the speech range.

Composite wall. Each element of the wall transmits the fraction τ_i = 10^−TL_i/10 of the incident power, and the composite transmission loss is TL_c = 10 log₁₀(Σ S_i/Σ S_i τ_i) (UFC eqs. C-8 and C-9). An opening has τ = 1: a gap of 1 % of the area limits any wall to about 20 dB, and a door of 20 dB in a 40 dB wall sets the result near the door's value plus the area ratio in decibels.

Noise reduction between rooms. The level in the receiving room is L_p2 = L_p1 − TL + 10 log₁₀(1/4 + S_w/R₂) (UFC eq. C-4), where the 1/4 is the field close to the wall and S_w/R₂ the reverberant build-up of the receiving room; the noise reduction NR = L_p1 − L_p2 = TL + C with C = −10 log₁₀(1/4 + S_w/R₂), tabulated in UFC Table C-7 (whose last cell, −29 at a ratio of 100, is a misprint for −20). The classical Sabine form NR = TL + 10 log₁₀(A₂/S_w) keeps only the reverberant term; the tool prints both, taking A₂ as R₂.

Sound Transmission Class. ASTM E413 fits a reference contour (−16, −13, −10, −7, −4, −1, 0, 1, 2, 3, 4, 4, 4, 4, 4, 4 dB relative to its 500 Hz value, at the sixteen third-octave bands from 125 to 4,000 Hz) to the measured TL: the contour is raised as far as the sum of the deficiencies (where the contour lies above the TL) stays at or below 32 dB and no single deficiency exceeds 8 dB; the STC is the contour's 500 Hz value. The tool applies the rule as mathematics; when only octave-band TL values are entered it interpolates them to the third-octave centres on a logarithmic frequency scale, which reproduces seven of the eight STC values printed with the UFC's octave tables (the eighth differs by 2). The STC is a speech-privacy rating weighted to 500 to 2,000 Hz; the UFC advises against it for mechanical-equipment noise, whose energy sits lower.

Noise criteria

NC curves. Beranek's 1957 Noise Criterion curves (NC-15 to NC-70, reproduced in UFC Figure 3-1) are octave-band limits from 63 Hz to 8 kHz shaped to equal loudness and speech interference. The tangent rating (UFC 3-2.5) is the highest curve the spectrum touches: the tool interpolates between the 5-dB curves in each band, reports the largest value with its band, and names the lowest standard curve the spectrum does not exceed. UFC Table 3-1 gives the design goals by space type (private offices NC 30, open-plan 40, classrooms 30, concert halls 20, gymnasiums 45); the NC method ignores the bands below 63 Hz and does not reward a balanced spectrum, which the RC method addresses.

RC Mark II (Blazier 1997, ASHRAE Applications chapter 49; the UFC works an example in its Appendix C-14). The RC number is the arithmetic average of the 500, 1,000 and 2,000 Hz levels, rounded to an integer; the reference curve falls 5 dB per octave through that value at 1,000 Hz and is flat between 16 and 31.5 Hz. The deviations of the spectrum from the curve are energy-averaged in three regions, LF (16 to 63 Hz), MF (125 to 500 Hz) and HF (1,000 to 4,000 Hz): spectral deviation factor = 10 log₁₀[(Σ 10^Δ/10)/3]. The quality assessment index QAI is the largest factor less the smallest. With QAI at or below 5 dB the spectrum is neutral (N); above it the region with the largest positive factor gives the descriptor, LF rumble, MF roar or HF hiss, and above 10 dB the sound is objectionable. Levels above 65 dB in the 16 to 63 Hz bands add the vibration flag (B), above 75 dB (A). The UFC example (64, 65, 64, 57, 47, 40, 35, 30, 23 dB from 16 Hz to 4 kHz) gives factors 6.6, 4.0 and −0.6, QAI 7.2 and the rating RC 35(LF).

Other single numbers. The A- and C-weighted levels are the decibel sums of the weighted bands; the speech interference level SIL is the arithmetic average of the 500 to 4,000 Hz bands (UFC 3-4 relates it to telephone use and to the voice level needed at a distance). From a time history the equivalent level is L_eq = 10 log₁₀(Σ t_i 10^L_i/10/Σ t_i); the EPA Levels Document (1974) defines the day-night level L_dn as the 24-hour L_eq with 10 dB added to the night hours 22:00 to 07:00 and identifies L_dn 55 dB outdoors and 45 dB indoors as the residential levels protecting activity, and L_eq(24) 70 dB for hearing; L_den adds 5 dB to the evening 19:00 to 22:00.

Barriers. A barrier that breaks the line of sight diffracts the sound over its top; with the path length difference δ = √(S² + h²) − S + √(R² + h²) − R (UFC eq. C-14) the Fresnel number is N = 2δ/λ and Maekawa's insertion loss IL = 10 log₁₀(3 + 20N) dB, about 5 dB at grazing, rising 3 dB per doubling of N, limited to 24 dB by sound passing through and around any practical wall. UFC Table C-28 tabulates the same loss by band; the manual adds that the barrier needs a TL at least 10 dB above its insertion loss, must extend 3h beyond the line of sight at each end, must have no reflecting surfaces around it, and loses about 10 % of its effect per 1,000 ft of distance.

Outdoor propagation. Band by band, L_p = L_w + 10 log₁₀(Q/(4πd²)) − α d − A_excess − A_ground − IL_barrier − NR_wall. The spreading term at Q = 2 is the −10 log(2πd²) of UFC eq. C-10 (the manual writes d in feet with −10 dB for the ft² to m² conversion, 10.3 dB exactly). Air absorption is the ISO 9613-1 closed form at the entered temperature and humidity, or nine entered coefficients (the UFC reprints the SAE ARP 866A table for this term with permission; the two standards agree within a few tenths of a decibel per 1,000 ft up to 2 kHz and differ above, ISO giving about twice the SAE value at 8 kHz). The anomalous excess attenuation of UFC Table C-24 (0.3 to 4.0 dB per 1,000 ft) is the manual's allowance for small-scale wind and temperature effects in long-term averages. The ground effect is offered as the ISO 9613-2 general-method estimate A_gr = 4.8 − (2h_m/d)(17 + 300/d), an A-weighted value for soft ground applied from 500 Hz up, or as the UFC Table C-27 insertion loss of medium-dense woods; the FHWA Traffic Noise Model computes the ground reflection and the diffraction at every terrain point from the measured profile (De Jong's method with Delany-Bazley impedances and the Embleton flow resistivities) and is named here, not reproduced. The exterior wall of the neighbour reduces the spectrum by UFC Table C-29 (types A to G, from open windows to a 50 lb/ft² solid wall), and the indoor spectrum is rated against the NC goal. The default inputs reproduce the UFC cooling-tower example of Appendix C-8.6 (Tables C-30 and C-31): 520 ft, a penthouse barrier with δ = 0.23 ft, type B wall with open windows, compared with NC-25 for sleeping; the manual's Table C-30 header calls the wall "type B" and its text "type E", the values being column B.

Units and limits

Lengths and areas pass through the site dictionary (feet and square feet in USCS, metres in SI); sound levels, coefficients and times are plain numbers; the temperature is a plain number in °F (USCS) or °C (SI) and is not converted by the unit switch. The formulas are diffuse-field statistics: they hold above the Schroeder frequency, in rooms whose dimensions are of the same order, with absorption spread over the surfaces; a long corridor, a very absorbent room or a room with one absorbent surface facing a reflective one departs from them. Transmission-loss estimates are laboratory values; flanking paths, leaks and double-wall resonances (UFC C-5.5) are not modelled. Mechanical-equipment noise that reaches a room through its duct is the domain of the Duct Sizing tool, not of this one.