formulae – acoustics vtan01€¦ · 0 = = =× ÷÷=-ø ö çç è æ ×=+ xnx xy y x xyxy n...
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Acoustics (VTAN01) HT2 (2017/2018)
01log110
)log()log(
logloglog
loglog)log(
0
==
×=
-=÷÷ø
öççè
æ
+=×
xnx
yxyx
yxyx
n
Formulae – Acoustics VTAN01
TABLE OF CONTENTS
I) Some logarithmic rules: 1
II) Fundamental acoustic definitions 1 II.1) Frequency bandwidths, weighted sound pressure level 3 II.2) One dimensional wave propagation 4
III) Room acoustics 5 III.1) Inside a room : reflection and transmission 5 III.2) Between two rooms: sound isolation and absorption 7
I) Some logarithmic rules:
yxy
x
yxyx
x
x
y
xx
xxy
-
+
=
×=
=
=
=Û=
101010
10101010
)10log(10log
log
II) Fundamental acoustic definitions Sound pressure - One-dimensional harmonic plane sound field:
- Effective value (RMS, Root Mean Square) for sound pressure in a point: NOTE: For a harmonic wave,
Sound pressure level SPL - Sound pressure level: , where pref = 2·10-5 Pa (pref : approximately the quietest sound a young undamaged human hearing can detect at 1000 Hz)
- Equivalent sound level:
÷øö
çèæ +-= jww x
ctptxp cosˆ),(
òD+
D=
tt
t
dttxpt
p0
0
),(1~ 2
÷÷ø
öççè
æ=÷
÷ø
öççè
æ= òò
TtL
T
refTeq dt
Tdt
ptp
TL p
0
10/)(
02
2
, 101log10)(1log10
÷÷ø
öççè
æ= 2
2~log10
refp p
pL
2/ˆ~ pp =
;2 fpw =lpw /)2(/ == ck;/ fc=l
;/1 Tf =
Acoustics (VTAN01) HT2 (2017/2018)
- Summing of two sound sources (if uncorrelated the last term vanishes):
SPL difference (dB) between 2 sources 0 1 2 3 4 5 6 7 8 9 10 Added dB to the highest SPL 3 2,54 2,12 1,76 1,46 1,19 0,97 0,79 0,64 0,51 0,41 - Summing of N uncorrelated sources:
- Phon curves:
Figure 1 – Phon curves
Sound intensity
- The sound energy P and sound intensity I is )()()()()(
)()()(tvtptI
SttI
tvtFt×=Þ
ïþ
ïýü
P=
×=P
- The sound power level and sound intensity level are calculated (in decibels) according to
÷÷ø
öççè
æ
PP
=Pref
L log10 and ÷÷ø
öççè
æ=
refI I
IL log10 , where P ref = 10-12 W and Iref = 10-12 W/m2.
Relation (function of the directivity factor Q) between Lp and L∏ : .4,3,2,1;4
log10 2 =÷øö
çèæ+=P QrQLL p p
òD+
D++=
tt
ttot dttptp
tppp
0
0
)()(2~~~21
22
21
2
÷ø
öçè
æ= å
=
N
n
Ltotp
npL1
10/,
,10log10
Acoustics (VTAN01) HT2 (2017/2018)
The time mean values are: ò ×=PT
dttvtpTS
0
)()( and ò ×=T
dttvtpT
I0
)()(1
- For a wave propagating in the positive x-direction, the previous integral yields: cpI r2~= ,
with c being the speed of sound in m/s and cZ r= the specific acoustic impedance.
cair =γP0
ρair (T = 0!)1+ Tair[
!C]2 ⋅273
"
#$
%
&'= 331.4 ⋅ 1+
Tair[!C]
2 ⋅273"
#$
%
&'
II.1) Frequency bandwidths, weighted sound pressure level II.1.a) Octave bands and third octave bands:
II.1.b) Weighed sound level:
Figure 2 - A-, B-, C- and D-weightings (10 Hz – 20 kHz).
( )å += 10/)(10log10 weighingLweighed
nL
Acoustics (VTAN01) HT2 (2017/2018)
II.2) One dimensional wave propagation
The general form of the equation of motion in fluids and solid media (applying Newton’s 2nd law) is
tv
xp
¶¶
-=¶¶ r ; p: pressure, r: specific mass, n: fluid particle velocity
For fluids, the relation between pressure and particle velocity is
xvP
tp
¶¶
-=¶¶
0g ; g = adiabatic index (g air =1.4)
For solid media we have the corresponding relation between force and displacement
xuESF¶¶
-= ; E: modulus of elasticity, S: surface
II.2.a) Waves in fluid media
- For a longitudinal wave propagating in air (one dimensional propagation in positive x-direction) expressed with sound pressure and particle velocity, respectively, as the field variables are:
012
2
22
2
=¶¶×-
¶¶
tp
cxp
2
22
2
2
xvc
tv
¶¶×=
¶¶
where rg 0Pc = is the propagation speed for the pressure wave in the air, with P0 being the atmospheric pressure. The general solution to the wave equation (with p being the field variable) is
)/()/(),( cxtpcxtptxp ++-= -+
The harmonic solution to the wave equation in complex form is
)()( ˆˆ),( kxtikxti epeptxp +-
-+ += ww
For physical interpretation, take the real part of the result, where
+p̂ and -p̂ are the pressure amplitudes for
the waves propagating in positive and in negative direction, respectively. ω is the angular frequency and ck wlp == 2 is the wave number. This yields the equality lfc =
- Specific acoustic impedance is defined as vpZ º
- For a wave propagating in air (1-D propagation in positive x-direction) the acoustic impedance is
cvpZ r==+
+
Acoustics (VTAN01) HT2 (2017/2018)
II.2.b) Waves in solid media
-For longitudinal and waves (infinite medium) and quasi-longitudinal waves (finite medium), their wave equations read, respectively:
02
2
2
2
=¶¶×-
¶¶
tv
xvE xx r
0' 2
2
2
2
=¶¶×-
¶¶
tv
xvE xx r
The propagation speed for longitudinal waves is rEcL = , whereas for quasi-longitudinal waves it is
)1(' 2urr -== EEcqL , υ being the Poisson’s ratio of the material where the wave is propagating through. Particle displacement, particle velocity, strain or force can be used instead of pressure as a field variable in the wave equation.
- For shear waves the wave equation can be expressed using the transversal displacement w as:
02
2
2
2
=¶¶×-
¶¶
tw
Gxw r where
rGcsh = , G being the shear modulus of the material.
)1(2 u+=
EG
- For bending waves in beams and plates the wave equation in one dimension is
02
2
4
4
=¶¶
+¶¶
twS
xwB r
where
12
3bhEB = for a rectangular cross section.
The propagation speed depends on the frequency (dispersive waves), i.e.
4)(SB
kc f r
www ×==
III) Room acoustics
III.1) Inside a room: reflection and transmission
III.1.a) Standing waves, eigenfrequencies
At normal incidence on a hard surface the pressure function is
tiekxptxp w×= + )cos(ˆ2),(
and the particle velocity function
( ) )2/(sinˆ2),( pw -+ ×= tiekxvtxv
The Helmholtz equation: 0)()( 22
2
=+¶
¶ xpkxxp has, in the one dimensional case with two hard boundary
surfaces at x = 0 and x = L the solution
tfi nexLnBtxp pp 2cos),( ×÷
øö
çèæ=
Acoustics (VTAN01) HT2 (2017/2018)
where fn are the resonance frequencies Lnccf
nn 2
==l
For the three dimensional case with six hard boundary surfaces the eigenfrequencies are
222
,, 2÷øö
çèæ+÷÷
ø
öççè
æ+÷
øö
çèæ=
Hn
Bn
Lncf zyx
nnn zyx
nx, ny and nz being indexes indicating the order mode in each direction of the room (L, B and H).
In transmission from one medium with wave impedance Z1 = ρ1c1 to another medium with wave impedance Z2 = ρ 2c2 the transmission factor t and reflection factor r are given, respectively, as:
12
2
1122
22 22ˆˆ
ZZZ
ccc
ppti
t
+=
+==
rrr
12
12
1122
1122
ˆˆ
ZZZZ
cccc
ppri
r
+-
=+-
==rrrr
while the transmission coefficient t and reflection coefficient r is
( ) ( )212
122
1122
1122 44ZZZZ
cccc
II
i
t
+=
+
×==
rrrr
t ( ) ( )212
212
21122
21122
ZZZZ
cccc
II
i
r
+
-=
+
-==
rr
rrr
III.1.b) Eigenfrequencies types:
Axial modes 1D Tangential modes 2D Oblique modes 3D Figure 3 – Related geometry of eigenfrequencies
Source: Sengpielaudio
III.1.c) Reverberation time
Reverberation time can be estimated by using Sabine’s formula. Sabine based his empirical formula on the sound decay of 60 dB (1/1000 SPL) after the abruptly end of a test tone (shot, shooting, etc.).
𝑇"#(𝑓) = 0,16𝑉
𝐴(𝑓)
with: V: volume of the room in cubic meters A: effective absorption area of the room in square meter: 𝐴(𝑓) = ∝0(1). 𝑆04
056
with Si : surface of every absorption element αi(f): absorption coefficient of each element
Acoustics (VTAN01) HT2 (2017/2018)
III.2) Between two rooms: sound isolation and absorption
III.2.a) Airborne sound insulation
- Sound reduction index R:
- Measurement of sound reduction index R of a wall of surface S:
- Spectrum adaptation term: Li given in Fig.6
-Traffic spectrum adaptation term: Li given in Fig.6
- Combined R of a wall of surface S made by different elements (Si):
- Slot leakage:
- The mass law for a single leaf wall:
- The mass law for a double leaf wall:
- Coincidence frequency (critical frequency):
III.2.b) Impact sound insulation
- Measurement of step sound level L:
- Spectrum adaptation term:
NOTE: A prime (´) in the parameters R/L and its related ones means measurements performed in situ, whereas the same letter without the prime refers to measurements performed in the laboratory.
÷÷ø
öççè
æ+-=
)(log10)()()(
fASfLfLfR recsend
÷øö
çèæ+=10)(log10)()( fAfLfL recn
÷øö
çèæ=÷÷
ø
öççè
æPP
=t1log10log10
t
iR
÷÷ø
öççè
æ÷ø
öçè
æ-= å -
i
fRi
iSS
fR 10/)(1101log10)(
÷øö
çèæ +×-= -
SSfR sfR 10/)(10log10)(
hKBmcfc /
2
20 =
¢¢=
p
002log40)(
cmffRrp ¢¢
»
002log20)(
cmffRrp ¢¢
»
wnfiLn
l LC ,
2500
50
10/)(,250050, 1510log10 --÷
ø
öçè
æ= å-
wi
RiLi RC -÷ø
öçè
æ-= å
=
--
19
1
10/)(315050 10log10
wi
RiLitr RC -÷÷
ø
öççè
æ-= å - 10/)(10log10
hm r=''
)1(121 2
3
2 uu -=
-=
EhIEB
Acoustics (VTAN01) HT2 (2017/2018)
III.2.c) ISO-REFERENCE CURVES Airborne sound isolation
Measurement of sound reduction index:
Figure 4 – Reference curve for air borne sound insulation (ISO 717-1)
Impact sound isolation
Measurement of step sound level Ln:
Figure 5 – Reference curve for step sound insulation (ISO 717-2)
÷øö
çèæ+=10)(log10)()( fAfLfL recn
÷÷ø
öççè
æ+-=
)(log10)()()(
fASfLfLfR recsend
Acoustics (VTAN01) HT2 (2017/2018)
Figure 6 – Spectra for calculating the spectrum adaptation terms for different frequency spectra (ISO 717-1)
Acoustics (VTAN01) HT2 (2017/2018)