PROC
CEEDINGS of the23rd Internationa
al Congresss on Acousttics
13 Septembber 2019 in Aachen,AGerrmany
9 to 1

Acoustic perrformanc
ce of souundprooff ventilattion unitts installe
ed indw
welling waallsSoohei NISHIM
MURA1; Yuyya NISHIMU
URA1; Thu Lan
L NGUYEN
N2
1

National Institute of T
Technology Ku
umamoto Colllege, Japan
2

Shimaane Universityy, Japan.

ABSTRA
ACTThis paper describes thet design off the soundprooof ventilatio
on unit for insstallation insidde the wall of a house.
rcuube or ellipticc cylinder formm is reasonabble as the unitt shape. In thiss paper, at
For this ppurpose, the rectangular
first, the theoretical calculations
cofo the sound ppressure insidde the rectang
gular cube annd elliptic cyylinder arence frequencies of the higgher-order mode
mwave
carried oout by solvingg the wave equation. Nexxt, the resonan
componeent are calcullated for a recctangular cubbe and ellipticc cylinder of the same vollume. In ordeer to havemore souundproofing efficient, thee comparisonn of resonancce frequenciees of the higgher-order moode wave
between the rectangullar cube and ellipticecylindder is also disscussed.
Keywordds: Higher-ordder mode, Reesonance, Souund propagation

1. INT
TRODUCTION
A largge number off houses in deeveloping As ia countries area being affeected by roadd traffic noisee emission
from motorcycles whhich create freequent klaxonn sounds by reported (1,2,3). On the other hand, thet use ofmonly found in
i those counntries. However, these
ventilatioon holes for providing heealthy indoorr air is commventilatioon holes also provide direcct paths for trraffic noise too enter a resid
dence by repported (4). ThisT paper
describess the design ofo the soundproof ventilatiion unit for innstallation insside the wall oof a house as shown in
Fig.1. Foor this purposse, the rectanngular cube oor elliptic cylinder form iss reasonable aas the unit shhape. The
sound prressure propagating in the unit has a sttanding wavee and higher-oorder mode w
wave compon
nents. The
latter occcurs in the high-frequency
y range. The nnoise tends too increase in this frequenccy range due to higher
order moode resonance. In this pap
per, at first, tthe theoreticaal calculation
ns of the soun
und pressure insideitherectangullar cube and elliptic cylinnder are carrried out by solving the wavew
equationn. Next, the resonancerfrequencies of the hiigher-order modemwave ccomponent arre calculated for a rectanngular cube oro elliptic
oto have more soundp
proofing efficcient, the com
mparison of resonancercylinder of the same volume. In ordermwave bbetween the rectangular cube and elllliptic cylindeer is alsofrequencies of the hiigher-order modediscussedd.

Figure 1 – S
Soundproof Ventilation
VUnitU1nisimurra@kumamoto-nct.ac.jp
nishimuura@kumamotto-nct.ac.jp
2
lan@rikko.shimane-uu.ac.jp12932

2. MEETHOD OFF ANALYSIIS
2.1

Reectangular Cube

The mhown in Figg. 2. A crosss-section arrea Sw = a × b and leng
gth L ofmodel of annalysis is sh
rectanguular cube th at has an in
nput and ouutput at bothh side, the cross-section
cnal area of them are
S0 = a0 × b0 and S L = aL × bL , respectively.

Figur e 2 Model of analysis
Wavee equation inn terms of veelocity potenntial Φ is givengby⎛ ∂2∂2∂2 ⎞
1 ∂2+Φ=Φ+⎜ 222 ⎟c 2 ∂t 2
⎝ ∂x ∂y ∂z ⎠(1)wherre c is the soound velocitty. Let Φ = 2 φ exp( jωt ) ( j 2 = −1 , ω = k c , k :wave numbber) thenthe geneeral solution of Eq.(1) caan be given aas(φ = ( Ae μ z + Be− μ z ) ( C sin α x + D cos α x ) E sin s 2 − α 2 y + F cos s 22 − α 2 y)(2)Heree, j 2 = -1, A,A B, C, D, E and F aare arbitrary constants determinablede from the boundarybconditio ns, other sym
mbols are coonstants.
Let Vx = − ∂φ / ∂x , Vy = − ∂φ / ∂y and Vz = − ∂φ / ∂z are the velocity coomponent in the x, yW assume thhe walls of the cavity too be perfectlly rigid therrefore theand z diirections, resspectively. We
boundaryy conditionss may be exp
pressed as[1] at x = 0 , Vx = 0(3)[2] at x = a , Vx = 0(4)[3] at y = 0 , Vy = 0(5)[4] at y = b , Vy = 0(6)[5] at z = 0 , Vz = V0 F0 ( x, y)(7)[6] at z = l , Vz = VL FL ( x, y)(8)wher e V0 and VL are the driving
dveloccity at the innput and outtput, F0 ( x, y ) =1 at the input
iandF0 ( x, y) ==0 elsewhere, FL ( x, y) =1= at the outpput and FL ( x, y) =0 elsew
where.
Accoording to the above boundary conditiions, the sound pressure at the output
ut side becommes∞ ∞ ⎧⎫⎪ ⎡ ⎛ mπ x ⎞cosh ( μm, n L )1⎪⎛ nπ y ⎞ ⎤PL = j k Z w S w ∑∑ ⎨U 0 Dma , n +U L Dmb , n ⎬ ⎢cos ⎜⎟ cos ⎜⎟⎥ (9)μm , n sinh ( μm, n L )⎝ b ⎠⎦
m = 0 n = 0 ⎪ μ m , n sinh ( μm , n L )⎪⎭ ⎣ ⎝ a ⎠⎩2933

where2μm,n =2⎛ mπ ⎞ ⎛ nπ ⎞2
⎟ +⎜
⎟ −k⎜
⎝ a ⎠ ⎝ b ⎠(10)

Dma , n =⎡ ⎛ mπ a01 ⎞⎛ mπ a000 ⎞ ⎤ ⎡ ⎛ nπ b01 ⎞⎛ nπ b00 ⎞ ⎤sin ⎜⎟ ⎥ ⎢sin ⎜⎟⎟ − sin ⎜⎟ − sin ⎜⎢m n π S0 ⎣ ⎝ a ⎠⎝ a ⎠⎦ ⎣ ⎝ b ⎠⎝ b ⎠⎦⎥(11)

Dmb , n =⎡ ⎛ mπ aL1 ⎞⎛ mπ a L 0 ⎞ ⎤ ⎡ ⎛ nπ bL1 ⎞⎛ nπ bL 0 ⎞ ⎤sin ⎜⎟ − sin ⎜⎟ − sin ⎜⎟⎟ ⎥ ⎢sin ⎜⎢m n π SL ⎣ ⎝ a ⎠⎝ a ⎠⎦ ⎣ ⎝ b ⎠⎝ b ⎠⎦⎥(12)4242

U 0 and U L are the volumevveloocity of inp
put and output, respecctively. Z w = ρ c / Sw
he cavity, k: wave num
mber, c is th
he soundrepresennts the charaacteristic accoustic impeedance of thvelocity..
2.2

Ellliptical cyliinder
The model of annalysis is sho
own in Fig. 3. A sectionn area S w and length L of ellipticcal cavity
a S0 and S L , respectiively.
that has an input andd output at both side, thee sectional arrea of them areFigu
ure 3 Model aanalysis of ellipticalecylinder
mplete solutioon of wave equation
ewheen expressed
d in ellipticall coordinatess is by refereences (5).
The com∞φ = ( A0 exp( μ z ) + B0 exp(− μ z )) ∑ Cm CCem (ξ , s ) cem (η , s )           (13)m=0wher e cem (η , s) and Cem (ξ , s) are the Mathieu fu
unction and modified MMathieu fun
nction of
mth-ordeer, respectivvely. Other syymbols are cconstants.
Let Vx = − ∂φ / ∂x , V y = − ∂φ / ∂y and Vz = − ∂φ / ∂z be the velocity compponents in th
he x , yr
ns areand z directions, respectively.
The boundaary condition[1] at z = 0 , Vz = −∂φ / ∂z = V0 F0 (ξ ,η )(14)[2] at z = L , Vz = −∂φ / ∂z = VL FL (ξ ,η )(15)[3] at ξ = ξ w , Vξ = −∂φ / ∂ξ = 0where2934(16)⎧1 (0 ≤ ξ ≤ ξ02 )F0 (ξ ,η ) = ⎨⎩0 elsewhere(17)⎧1 (0 ≤ ξ ≤ ξ L 2 )FL (ξ ,η ) = ⎨elsewhere⎩0(18)

From the above boundary conditions φ can be determined and the sound pressure at the output
piston becomes
Pout = jk ρ cφ (ξ ,η , L)
∞ ∞ ⎛⎞cosh μm ,i L
1H m ,i − U LGm ,i ⎟⎟ Cem (ξ , sm ,i ) cem (η , sm ,i )= jkZ w ∑∑ ⎜⎜ U 0μm ,i sinh μm ,i Lμm ,i sinh μm ,i Lm =0 i =0 ⎝⎠(19)

where k is wave-number, Z w = ρ c / Sw , U 0 and U L are the volume velocity of the input andoutput section, other symbols are defined by

H m ,i2π

ξ 02020∫ ∫ Ce (ξ , s ) ce (η , s )(cosh 2ξ − cos 2η ) d ξ dη
= π ξ∫ ∫ Ce (ξ , s )ce (η , s )(cosh 2ξ − cos 2η ) d ξ dη0Gm ,imw

2m02π

ξ 01

02000m ,imm ,im ,i2mm ,i(20)∫ ∫ Ce (ξ , s ) ce (η , s )(cosh 2ξ − cos 2η ) d ξ dη
= π ξ∫ ∫ Ce (ξ , s )ce (η , s )(cosh 2ξ − cos 2η ) d ξ dη

μ m ,i =mw

1aw

2mm ,imm ,im ,i2mm ,iλm2 ,i − ( kaw )

(21)

2(22)

3. RESULTS AND DISCUSSION
In the case of elliptical cylinder, the first term of Eq. (9) represented the sound pressure component ofplane wave. PL is greatin the frequencies where the nominator sin kL becomes zero, namelyk L = ηπ ∴ f = ηc
( η = 1, 2, ..... )
2L

(23)

The frequency illustrated in Eq. (23) is called as the resonance frequency of plane wave. The second termof Eq. (9) represented the sound pressure component of higher-order mode. PL is great in the frequencies
where the nominator sinh μ m,i L becomes zero, namely

μ m ,i L = jηπ∴f m ,i =c2π aw

ηπ aw ⎞λm ,i 2 + ⎛⎜⎟⎝ L ⎠2(η = 0,1, ..... )(24)

Similarly, in the case of rectangular cube, resonance frequency of plan wave is the same as Eq. (23). Theresonance frequency of higher-order modes are given as following.
cf m,n =2π22⎛ mπ ⎞ ⎛ nπ ⎞ ⎛η π ⎞⎜⎟ +⎜⎟ +⎜⎟
⎝ a ⎠ ⎝ b ⎠ ⎝ L ⎠2(η = 0, 1, ..... )(25)

For the purpose of attaching SVHU inside the wall, the cavity can be fabricated by a rectangular cube orelliptical cylinder form. Be aware that the shape of the cavity has to be selected based on the fundamental
feature such as low resonance levels, less of resonance generation, high frequency of resonance in order to2935

avoid redducing a size ofo the unit duue to the use oof a thick sounnd absorbing material.
mFuurthermore, id
deally, theresonancce frequency shouldsbe outt of the rangee where humaan ears are moost sensitive.
The ccalculation ofo the first resonancerfrrequencies iss performed by Eq. (244) and Eq. (225) for arectanguular cube a nd two elliiptical cylinnders with the same volume.
v
Thee dimension
n of therectanguular cube is a=0.37m, b=0.17mb
an d L=0.6m. The dimension of the eelliptical cyylinder is
aw=0.188m, bw=0.111m and L=0..6m. The caalculation ressults are sho
own in Fig. 4. It is cleaar that anelliptica l cylinder is advantageou
us in terms oof less reson
nance generation.

FFigure4 The first resonaance frequenccies of higheer order modde.

4. COONCLUSIO
ONThis ppaper describbes the desig
gn of the sounndproof venttilation unit forf installatioon inside the wall of ahouse. T
The rectangulaar cube or ellliptic cylindeer form is reaasonable as th
he unit shapee for this purppose. The
sound prressure propagating in the unit has a sttanding wavee and higher-oorder mode w
wave compon
nents. The
noise tennds to increasse in the frequuency range due to higherr order modee resonance. A
At first, the thheoretical
calculatioons of the souund pressure inside the recctangular cub
be and ellipticc cylinder aree carried out byb solving
the wavee equation. Neext, the resonance frequenccies of the higgher-order moode wave com
mponent are calculated
cfor a recttangular cubee or elliptic cy
ylinder of the same volumee. In order to have more sooundproofingg efficient,
the compparison of ressonance frequuencies of thee higher-ordeer mode wavee between thee rectangular cube andelliptic ccylinder is alsso discussed leadl
to the coonclusion thaat using the ellliptical cylinnder in the unnit will bee
more souundproofing efficient.

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Outline of social survey
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a Community response tto road trafficc noise in Han
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5. N. W. McLachLan. Theory and application of Mathieu Function (Clarendon Press, Oxford, 1951).
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