Pressures inside coupled rooms subjected to sonic boom

PRESSURES INSIDE COUPLED ROOMS SUBJECTED TO SONIC BOOM

FEBRUARY, 1983

BY

A. J. M. BRESSERS

I

TECHNISCHE HOGESCHOOL DELFT

LUCHTVAART- EN RUIMTEVAARTIECHNIEK

BIBLIOTHEEK Kluyverweg 1 - DELFT

6 SEP. 1983

UTIAS TECHNICAL NOTE NO. 241 CN

ISSN

0082-5263

PRESSURES INSIDE COUP LED ROOMS SUBJECTED TO SONIC BOOM

by

A. J. M. Bressers

Acknowledgements

I would like to express my appreciation to my supervisor, Prof. I. I. Glass, for his assistance and to N. N. Wahba for much useful advice and helpful suggestions.

The efforts made by Dr. W. G. Richarz and Dr. J. J. Gottlieb to obtain the necessary equipment are very much appreciated.

Thanks are also due to D. R. Wilmut for his help while building the experi-mental facility, to

J.

L.

Bradbury and his staff for skilled judgement and fine machining of necessary parts, and to fellow students and friends, who helped me to enjoy my stay in Toronto.

The financial assistance received from the Canadian Ministry of Transport, Transportation Development Centre and the Natural Sciences and Engineering Research Council is acknowledged with thanks.

, ,

~

Summary

An

aeoustie system of two eoupled rooms subjeeted to sonie boom was studied numerieally and experimentally. The physieal dimensions of the aeoustie system are small eompared to the wavelength of a sonie-boom signal and justify a Helmholtz-resonator approach. Oseillation frequeneies and dampipg ratios were ealeulated for windowareas ranging from 0 to 0.6 m2 , room volumes ranging from about

7

m3 to 42 m3 and the average absorption coeffieient ranging from 0 to 0.25. Pressure responses were ealeulated for similar eonditions using sonie-boom input-signals with durations varying from 100 ms to 300 ms and various rise times between 0 and 0.25 times the duration.

The UTIAS Travelling-Wave Horn-Type Sonic-Boom Simulator was available for the experiments. A wall was built in an existing room, attaehed to the horn, in order to create two eonneeted rooms with volumes of

7.9

m3 and 23 m3 , respeetively. Experiments were conducted, using windowareas up to about 0.6

mF,

and input durations between 100 ms and 300 ms. The sonic-boom overpressures at the test rooms varied between about 100 N/mf and 300 N/m2. Very satisfactory agreement was achieved between analysis and experiments by using an average absorption eoeffieient of 0.25 for both rooms.

The amplitude ratio is the ratio of the maximum amplitude of the response to the maximum amplitude of the input signal. For low damping eonditions (am

=

0) the amplitude ratios of the two-room system may exceed the ampli-tude ratio of a single room lalmost 4 for a two-room configuration, almost 2.5 for a single room (Ref.

9)].

For realistic damping eonditions the amplitude ratios slightly exeeed unity for both the two-room system and the single room. In eonelusion, for realistic damping eonditions a system of two coupled rooms is no more prone to boom-indueed damage than a single room and it is unlikely that sonie boom will cause more damage than other transient loadings, such as, for example, the passage of traffic, thunder or door slamming.

'ti

TABLE OF CONTENTS PAGE

Acknowledgements i i Surrmary i i i Table of contents iv List of symbols v 1. 2. 3. 4. 5. 6. 7. INTRODUCTION 1

THEORETICAL BACKGROUND AND GOVERNING EQUATIONS 1

2.1 INTRODUCTION 1

2.2 HELMHOLTZ RESONATOR 2

2.3 MECHANICAL-ACOUSTICAL EQUATION: A.NPJ..DGY 4

2.4 THE TWO-ROOM CONFIGURATION 6

2.4.1 THE TWO-ROOM CONFIGURATION WITH THREE WINDOWS 6 2.4.2 THE TWO-ROOM CONFIGURATION WITH TWO WINDOWS 7

2.5 OSCILLATION FREQUENCIES 7

2.5.1 OSCILLATION FREQUENCIES FOR THE SYSTEM

WITH DAMPING 7

2.5.2 OSCILLATION FREQUENCIES FOR THE SYSTEM

WITHOUT DAMPING 8

OSCILLATION FREQUENCIES; NUMERICAL CALCULATIONS 8 3.1 EFFECT OF ABSORPTION COEFFICIENT ON

OSCILLATION FREQUENCIES 8 3.2 EFFECT OF AREA OF COUPLING WINDOW ON

OSCILLATION FREQUENCIES 9 3.3 EFFECT OF AREAS OF OUTGOING WINDOWS ON

OSCILLATION FREQUENCIES 11 3.4 EFFECT OF ROOM VOLUMES ON OSCILLATION FREQUENCIES 11

PRESSURE HISTORIES 13

4.1 INTRODUCTION 13

4.2 EFFECT OF VARIOUS PARAMETERS ON PRESSURE-TIME

HISTORIES 14

4.3 AMPLITUDE RATlOS 14

EXPERIMENTAL WORK 16

5.1 INTRODUCTION 16

5.2 ABSORPTION COEFFICIENTS 16

5.3 SIMULATED N-WA VES 17

5.4 EXPERIMENTAL PRESSURE HISTORIES 17

5.5 COMPARISON OF ANALYSIS AND EXPERIMENT 19

CONCWSIONS 20

REFERENCES 21

FIGURES

APPENDIX A: FREQUENCY SPECTRUM OF N-WAVES APPENDIX B: RADIATION IMPEDANCE

APPENDIX C: VISCOUS RESISTANCE

APPENDIX D: DAMPING RATlOS DUE TO ABSORPTION APPENDIX E: DERIVATIONS OF EQUATIONS

List of symbols

a speed of sound A windowarea

l>f reduced window area A reference window area

0

A amplitude ratio C compliance C

a,bde coefficients in systems of differential equations D damping parameter

D _o additional damping parameter for three window configuration ET total energy contained in N-wave

E~) energy content

F(t) externally applied force F(w) Fourier transform of F(t)

j N

I sound intensity k spring constant L orifice length

L effective orifice length e

m mass

m- reduced mass

M acoustic inertance p room pressure

p(t) externally applied pressure P normalized room pressure P maximum amplitude N-wave

o

Presonator pressure

r

PI normalized input pressure P

1,max maximum pressure response P

2,max maximum pressure response

in room in room

1 2 P (51) dimensionless Fourier transform of p(t) P R (n.) Real part of P (.n)

PI (n.) Imaginairy part of P (S\.)

Ps pressure amplitude of steady sound source R resistance to motion

R acoustic resistance a

R _m mechanical resistance S absorbing surface

t time variabIe

t natural oscillation time n

t rise time of N-wave r

tf fall time of N-wave

T dimensionless time variabIe

T dimensionless natural oscillation time n

T dimensionless rise time r

T

f dimensionless fall time V room volume

V reference room volume

o

V

tot volume of total system

V adjusted speed in coupling window c

W

n dimensionless natural frequency W(w) energy spectrum

W(~) dimensionless energy spectrum

x

Z r 0< m

b

.n.

volume displacement radiation impedance

average absorptiom coefficient end correct ion

acoustic energy density damping ratio

damping ratio due to absorption viscosity coefficient

displacement of air in window applied power at resonator density of air

duration of input signal period ratio

radiation reactance angular frequency

dimensionless angular frequency cut-off frequency

pressure difference across window

Subscripts

1 related to room 1 or related to outgoing

2 related to room 2 or related to outgoing c related to coupling window

b related to basic oscillation frequency

window 1 window 2

i related to input signal

r related to radiation losses

- - --- --- - - -- - - - "

1. INTRODUCTION.

A transport aircraft moving at supersonic speed (SST) creates a shock-wave

system that develops into an overpressure signature resembling the capital

letter N. Such an N-wave, also commonly known as a sonic boom, is shown in Fig. 1. Typical durations of a sonic boom range from 100 ms to about 300 ms, mainly depending on the size of the aircraft, and the maximum overpressure

at ground level may reach a value of about 0.001 atm (100 Pa or 2 psf) for

steady-flight conditions of the aircraft in a standard atmosphere.

An extensive sonic-boom research program has been carried out at UTIAS over

a period of more than ten years under the supervision of Drs. I.I.Glass and H.S.Ribner, involving various projects such as prediction techniques

of sonic-boom phenomena (corridor width, effects or aircraft manoeuvers on

focussing of sonic-boom), the development of sonic-boom simulation facilities (portable simulator, loudspeaker-driven booth, travelling-wave horn) , and the effects of sonic boom on humans (heart-rate change, automobile-driver behaviour), animals (damage to cochleae of mice) , and structures (cracking of plaster panels) (see Refs. 1-12).

Plaster panels have been tested for boom-induced damage, by Leigh (Ref. 7), including the fatigue-life properties of the plaster material and its continual degradation due to transient loadings other than sonic·boom. It was found that it is unlikely that plaster in good repair will be

signifi-cantly damaged by repeated exposure to sonic booms generated by SST

over-flights. In order to study the effect on complete structures, field tests

had been conducted elsewhere to detect the pressure-time histories both outside and inside a room with a window subjected to sonic boom (Ref. 8). The pressure histories are shown in Fig. 2. A room with an open window behaves like a Helmholtz resonator, which may result in overpressures

inside the room with amplitudes that exceed those of the exciting signals.

Wahba (Ref. 9) provided a method to calculate the pressure-time history in a single room subjected to sonic-boom. He described the damping in terms

of known parameters, instead of using an arbitrary constant, as was

pre-viously done by Vaidya (Ref.l0) and Lin (Ref.ll). Experiments were then

conducted in the UTIAS Travelling-Wave Hom-Type Sonic-Boom Simulator (Ref.12), to which a full scale test room had been attached, and it was shown that the room can be described as a Helmholtz resonator.

The present study deals wi th the effect of the coupling of two rooms, while either one or both are subjected to sonic boom. Oscillation fre-quencies and damping effects of a two-room system are studied numerically and explicit analytical expressions are derived for the oscillation

frequencies of the system without damping . Pressure responses are

calcu-lated for various configurations and input signals and are compared with

experiments. Calculations are carried out to determine the amplitude

ratio in either one of the coupled rooms, i.e. the ratio of the maximum amplitude of the response to the maximum amplitude of the input signal.

2. THEORETICAL BACKGROUND AND GOVERNING EQUATIONS. 2.1 INTRODUCTION.

The air in a room with an open window can be acoustically excited by sound pulses from the outer environment, penetrating through the window. Consider two rooms coupled by a connecting window, and assume the walls of the rooms to be rigide In this case the sound fields in the rooms are

coupled to each other because of the connecting window. The sound field in the total system is determined by the volumes Vl and V2 of the rooms, the areas

total system is determined by the volumes V

l and V2 of the rooms, the areas Al and A

2 of the outgoing windows, and the area A of the coupling window (Fig. 3a), as weIl as by the characteristics of the inctdent sound pulses. Sonic-booms are used for acoustical excitation of the system and are characterized

by their maximum overpressure po. rise time t , fall time tf and total

dura-tionY (Fig.4). The energy content of an ideaf sonic-boom wlth zero rise time is represented as a function of frequency in appendix A. It is shown that most of the energy is contained in the low-frequency components of the Fourier

spectrum and is therefore carried by waves with large wavelengths. The

dimen-sions of the acoustic system under consideration are much smaller than these wave I engths, which justifies a lumped element approach.

2.2 HELMHOLTZ RESONATOR.

A Helmholtz resonator consists of a rigid enclosure of volume V, communicating with the outer environment through a window of area A (Fig. 3b). The air in the window is moving inward and outward much like a piston in an infinite baffle. The sound field in the resonator is uniform because the wavelengths of the sound waves are much longer than the dimensions of the resonator. The differential

equation governing the inward displacement ~ of the air in the window is

2. d~

m

~

= -

R --.2

+

A

(Pi -

Pr)

dt'

dt

I.

where m is the mass of air effectively moving in the window, R represents the

resistance to motion, and A(P.- P~ describes the force due to the pressure

difference across the window that is induced by the sound source. The mass of the air moving in the window is expressed by

m

=

fAL~

2.

wherep is the density of the air, A is the area of the window, and Le is the effective length of the window. The effective length is

le =

L

+

2

t

3.

consisting of the aperture length L of the window, and the end correction

b ,

representing air beyond the constriction of the window that is induced to move in the window. The factor 2 represents the engagement of air both inside and

outside the room. Expressions for

6

for different window shapes under low-frequency conditions are given in appendix B. Resistance to motion is due to

radiation and viscous effects. The low-frequency radiation resistance is given

by (appendix B) ~ \

'i"}

=

fW

A

J.

~~ 2n~ ~.

which is proportional to the square of the frequency. The radiation resistance represents energy losses because of radiation of sound into the surrounding

air. Viscous resistance is due to the viscous damping in the air that is forced back and forth through the constricting window. With~ the viscosity coefficient

of air, the viscous re si stance is proportional to the square root of the fre-quency (Ref.13, appendix C)

s.

The radiation resistance R and the mass m are incorporated in the radiation _r

'"

impedance Z

=

R - i~m (Ref.14), (Z is considered in more detail in appendix B. )Considerrthersound field in the Fesonator, the sound intensi~ I is defined

as the power incident on a unit area. In the case of a uniform sound distribu-tion, the intensity I and the mean acoustic energy densityE of the sound field are related through the speed of sound a (Ref. 15)

b.

The absorption coefficient ~ is the fraction of the acoustic energy flux incident on the wall that is absorbed. The average absorption coefficient ~ is related

m

to

cc.

through

0(",

5 ;:

(f

~

clS

S

where the integration is over the surface area S. which is of importance here, Ol. may by considered

(Ref. 16). The rate of loss otnacoustic energy at

In the low-frequency range independent of frequency the surface is then

0(",

SI::

~

ot._rn

~

af.

i.

With a power TT(t) introduced into the resonator, the rate equation for the acous-tic energy

Ve

is or

A..(v!) :: nlt) -

J~

o(rnSa..t. c:l~ .,

V d{.

+ d~

The power supplied to the resonator is

where P" is the pressure in the resonator. The energy densi ty E depends on the pressure Pr according to (Ref. 1 5 )

~

This allows the rate

E. _

PI"

- 1. .ra~ equation to be expressed as 10.

".

\2.

which relates the pressure and the inward displacement to the properties of the resonator. Sound absorption in air need not be taken into account since it is negligible for frequencies below 1000 Hz (Ref. 16) and only small volumes are considered. Equations I and 12 are combined (Ref 9) into

+

_13.

+ where the dimensionless room pressure,

the dimensionless input pressure, the dimensionless time.

't'~[1

+-'Lf ,

v

.s

o(rn

(i~(

..

~

v)'

IS,

a

ft:4

A

•

A

~f

a 2

The latter two equations are functions of frequency through eguations

4

and!)

for R and R . However, for most practical cases

.sec""

(QI' " ~v)

'-<,

and

(~t'''~;) V vt\'\()(",S 9rQ,.Al

A,1 01\ "' &" Ai , so that the right hand tenns of equations ILt and IS'

can

8Ç

e

v

a

l

uat~d

with good approximation by inserting a relevant frequency.

The quantity W can then be referred to as the dimensionless natural frequency. n

) is the dimensionless damping ratio and can be approximated by

S

~

\

lN

~ ~VA

'

+

~~!M=.'~WVi...

oCmS

~

Lf.·

4rra~ Lt 2.a

f

Ak

Ib

VA

Radiation losses are represented by the first tenn and viscous losses by the second tenno Both loss effects originate at the window. The last tenn represents losses due to absorption, occurring at the walls. Radiation and viscous losses are only important when absorption is very small. For most practical cases the absorption is predominant. For later use define

r

ah

~

oCm

S

~

Le '

.}

~

\b

VA

as the contribution of absorption to the damping ratio. Note that )~S depends on the shape of the volume (appendix D). It can easily be shown from the homoge-neous fonn of equation 13 that a homogeneous solution of the fonn

- SW"

T :t

i

W" V

I -

ç

~

i

T

-

D·

r

~

L

n. ..

T &

Pho~

=

c..Or'\~~

•

e.

•

e..

=

<.on~~

11

e

..

e. I .

applies . The dimensionless oscillation frequency is lt:= \./" ~ and the damping is incorporated through the damping parameter D=

S_

W

.

Wahba (Ref. 9) provided an approximate means to solve equation 13 for the p~ssure-time history of a Helmholtz resonator subjected to sonic-boom. The results agree well with experiments and the same method is adopted in this work: to use a Fourier transfonn to model the input signal by harmonie components and to employ the differential equations to describe the two room system (seetions 2.4.1. and

2.4.2.). The equations are solved with the harmonie components as inputs. The frequency of each of the components is used in equations ~ and S to set a constant radiation resistance R and viscous resistance R along with that par-ticular input frequency. Eventuhlly all the responses to '{he indi vidual harmonie components are added together to compose the response of the system to the input signal. A critical examination of this method reveals that same objections can

be made to the way the frequency dependent effects are taken into account. A discussion of this matter is given in appendix F.

2.3 MECHANICAL-ACOUSTICAL EQUATION: J>.NP.JJ:X;Y.

Equation I for the inward displacement ~ of air in a window with area A is an applied version of the general machanicál equation for a mass-spring-dashpot system, given by

m represents the mass, Rm the resistance to motion and k the spring constant. F(t) is an external force. When considering the motion of air in the window of a resonator one would generally write

~

M

d

X ...

'R.Q

Q ...

-!...

X ::.

pl.e)

c;\ ~1. d~ C.

20.

X is defined as the volume displacement and related to ~ through the area of the window: X= A~ . p(t) is an externally applied pressure at the outside of the window and related to F(t) through P(t)=F(t)/A. Comparison of equation ,~ and equation

20

reveals for the acoustic inertance

m

M :;

'-"""'i""

A

for the acoustic resistance

Ra= 'Rm

A3.

and for the acoustic compliance 1 _

k.

G -

Al.

21.

2~.

2.l.

In order to understand the acoustic quantities, let us consider a Helmholtz resonator wi th volume V and assume an external pressure that creates a volume displacement, X, inwards into the resonator. This will result in an overpressure in the resonator according to

~

X

pt':::

f"

V

which is proportional to the relative

the resonator is (equation 20 )

volume change. The acoustic compliance of

and is spring

~

...!...:

.e.-c.

V

inversely proportional to the volume V.

constant k becomes ~

k :::

Al.. ::.

!"

~

A

C.

V

2.5. Its mechanical counterpart, the

2.b.

Equation

2b

shows clearly that a small volume corresponds to a large spring constant. The aperture length L of a window can be neglected with respect to the end correction (equation:3 ), which allows L to be approximated by L ~

'fA'

(Ref. 9). The mechanical mass then becomes, witheequation 2. e

showing that a small mass is related to a small window area.However, the acoustic inertance

M:::~=~=...L

Al.

A

A

'11-28. is large for a small windowarea. This is due to the fact that, given a certain pressure difference across the window, the acceleration of the gas particles has

óp

f

A

II~

Undamped natural frequencies are found from the homogeneous equations without

damping. The result for the mechanical case is the familiar

while for the acoustic case

or, wi th L

=

'VÀ

e

"": Irt'

~ ~A>f}' ~

a.

IrA

'M"C

f

VL

e

Yvr;-2.4 THE TWO-ROOM CONFIGURATION.

2.4.1 THE TWO-ROOM CONFIGURATION WITH THREE WINDOWS.

30.

When two rooms are coupled by a connecting window (Fig. 5a), equation

I

has to

be applied to each of the windows;

mi

a~~1

+

'RI

&

...

A.p.

=

A, Pi

cil

~ c::U 31.

ml

cl~~~

+-

'R~ df~

-4-

A~ p~

e

k

_l

pi

d\:.~

cU:

.3.2.

rt\,

d

~f

<.

+

~,&

+

Ac.

(PI -

f>~

) ':

0

cÜ~ ol.~ '33.

and equation

Il

has to be applied to each of the rooms:

VI

~

+-

o<m,

SI PI ':

A

~

+

A, df<.

~

_f,a,

L Id~"-dl; 9y,c.,

colt

~4. V'L

~ ~p).

-4-

O<N\~ .s~ p~

':

A~ ~{~

A,

~

f~Q~

c:;n

8f~ C4₁ ~t eH 3$'.

In reality, of course, f,:'f~~f and a₁=a₂=a. These five equations (al ~o 3S)

are rewri tten (appendix

EJ

in'"'to three equations in the variables

i1::

PI/~ I

l\:.

P~,po cv\Q

V,::

~

.

i;)

the dimensionless pressures in rooms 1 and 2 and

an adJusted velocity ln the coupling window:

where

A

The coefficients are defined according to C bd where _{a, e} a

b d

e

nurnber of windows in system (2 or 3), nurnber of equations, 1 for P l' 2 for P 2' nurnber of variables, 1 for P l' 2 for P 2' order of time derivative.

3 for V , c 3 for V ,

c

The coefficients are given in appendix E. The system of differential equations is of fifth order since the equations in P

1 and P2 are of second order and the equation in V is of first order. A solution was obtained by numerical means using a Fourigr-transform methode

2.4 • .2 THE TWO-RooM CONFIGURATION WITH TWO WINDOWS.

When window 2 is closed (Fig. 5b), equation 32 is cancelled and A

2

=

0 in

equation lS ~ Equations 3' and 3~ through 15 can then be rewritten (appendix E) into two equations, in P

1 and P2:

•

C.~.I~I

dl>l •

C.~,I~;) 'j)~

=

j)l:

l

dT

~

31.

<'l.~«>

'PI. ...

<1.n2

c:ll~

..

(.~.nl

dl\.

+,

c~.),~o

i>l

~

0

~

en

This system of differential equations is of of second order. Again, a solution has been

fourth order since both equations are obtained by numeri cal means using a Fourier transform methode

2.5 OSCILlATION FREQUENCIES.

2.5.1 OSCILlATION FREQUENCIES FOR THE SYSTEM WITH DAMPING.

In the analytical solution to the problem of the Helmholtz resonator, the homo-geneous sloution was shown to be of the form

_,....

_...

-~~

T

j:~", ... V ( - r l

T

-

,bT :!.

"n.

T

i>ho,""

la

c.on:t

11 e., •

e.

•

c.o.~ ~

e.

..

e.

where D is defined as the damping parameter and.n. as the oscillation frequency. In the two roan si tuations, two oscillation frequencies and at least two damping parameters appear. They are calculated numerically fram the characteristic

equations,,. 9.S wUI be shown shortly. The damping ratio ') can be canputed fram D'

and...Il. :

D

) :: Y

l l.'

o ...

n

39.

D,..n. and

S

are dimensionless quant i ties. In order to find the oscillation

frequencies and damping parameters for either one of the systems of differential equations 3b and 37 , only the homogeneous equations are considered, with

solutions of the form

resulting in the characteristic equations. The characteristic equations for the situation with three windows is a polynanial in s of the fifth order, with one real root and two sets of canplex conjugate roots. The characteristic

equation for the system with two windows is a polynomial in s of fourth order with two sets of complex conjugate roots. The real parts of the roots are the

damping parameters, expressing the decay'of the response of the system with

time. The imaginairy parts represent the oscillation frequencies. The charac-teristic equations are found in appendix E. The smaller one of the oscillation frequencies will be called the basic oscillation frequency and denoted bYllb' since it appears to be most important in describing the behaviour of the

system. The higher oscillation frequency is denoted by ~ . The other variables are labelled accordingly: Db and) b along wi th ~, Dh and

Sb

along wi th ~. , The additional damping parameter in the case when A

=

0 wiTI be referred to as D . No analytical expressions exist for any of tfiese variables. However, someoillustrative analytical expressions do exist for the Helmholtz resonator

(equations 14 and Ib ) and also for the two'room configuration without damping

as will be shown in the next section. A dimensionless oscillation time T is n related to a frequency.n. through'

The period ratio

f

is defined as

-

\

.

st

,'INt"

1:"= ::. : : :

-T" 20rr 20'

_ r

-1:

Lio,

and compares the true oscillation time of thB system, t , with the duration

of the exci ting N-wave 1:'. The peri6d ratio is also labe:Cled according to Tb and f h' Equation 14 for the Helmhol tz resonator gi yes wi th S 0(1"1'1 (~\" t ~., ) ,

.;;;:..~;...;~;..:..;..« I I

afQ.A~

This relates equations 30 and 'tO.

2.5.2 OSCILLATION FREQUENCIES FOR THE SYSTEM WITHOUT DAMPING.

Analytical expressionsfor the oscillation frequencies of the system can be found

when no damping is involved. The oscillationfrequencies then are the natural frequencies of the system. The góverning equations follbw from the general equa-tions (3b and 31 ) by setting the terms that describe damping equal to zero. For three windows, A

2 # 0, the dim~nsionless natural frequencies are given by

Abh~~\ ~\I+~)

+

§.H~):!: JA~(I+~)+~(I+~)

_.2

YM:

(1+

f

+~ -~

\'

41

if~v\

AI

'12,

A~

VI

Ac.

V~

À).

V,'Il.

A"

~

VA.At"'}

and for two windows, A? =

°

,

..slb

h"

~

OC

JA:'(l.

+.1.)

±

\/(OC

+

fA;

(.l

+

-L)~ ~

_

I.ffÄA '

'6:

VI , 'I, V). Y

V,

VI

V

1 '}

V,

V"I,.

In both equations, the effective length Le is approximated by Le

=

~.

3. OSCILLATION FREQUENCIES. NUMERI CAL CALCULATIONS.

3.1 EFFECT OF ABSORPTION COEFFICIENT ON OSCILLATION FREQUENCIES.

Equation IS defines the damping parameter D = ~ W and the oscillation frequency

therefore equations 17 and

40

can be used to evaluate the approximations and

D:.

't' CÀ,. o(m ~

,b

V

St ::

1:' 0.. \/

A.

-r

'JL~

~3.

4lt.

which clearly show the dependence of D and~ on the absorption coefficient~.

The influence of~ on the oscillation frequencies and damping parameters in the two-room systeill is investigated numerically. The results are shown in Fig. 6 and Fig. 7. The damping parameters Db and Dh and the damping ratios ~ b and ~ h appear to be proportional to ~ , while the oscillation frequencies w_b andw h decrease with increasing ~ , aill expected, since the oscillations are slowed down with increased damping. D ~ the additional damping parameter when A

#

0 does not depend

on~

, but it aoes depend on frequency. However, within €he frequency

range of intereillt Do is of little importance. Both Do/Db and D /Dh will be less than 0.03 for ~ = 0.25 and an inserted frequency St. = 20. Rad~atlon and viscous losses are neglWcted in the calculations shown in Flgs.

6

artd

7.

3.2 EFFECT OF AREA OF COUPLING WINDOW ON OSCILLATION FREQUENCIES.

Consider the two individual rooms, labeled 1 and 2 (Fig. 5), each with its own period ratio

T' 1 and

1f? and damping ratio) 1 and ~ 2' When the rooms are coupled through an area A , tney lose their indivlduality and become part of a system of two rooms, des8ribed by the basic and higher period ratios ~b and Th and the basic and higher damping ratios ~ h and ~ h' lp b appears to be mainly detennined by the room with the lower indivioual period ratio, say for example room 2. In that case,

'r

~

:r

l' and room 2 is called the basic room. In the limi ting case of A ... 0, "b wifl approach f 2 and:t_h will approach f l' and similarly

~

b approaches ~2 aBd ~h approaches )1' Equations 'ft reduce for Ac = 0 to

't", ~

"0.

~

'IA,'

20'

V,

45.

t'.'

!9..

~

VA;'

1"

V),

4b.

Figure

8

shows th at Tb and especially Th increase as a function of Ac while both

~h and ~h decrease. The configuration under consideration is characterized in tfle figure. Volumes V1 and V2 are given in m3 , absorbing surfaces Sl and S2 in m2 • The average absor'Ption coefficients, ALPHAl and ALPHA2, are dimensionless. The area of ~he outgoing window in room 1 is denoted by AREA1 and given in m2 , while the areas .of the outgoing window in room 2 and of the coupling window are nonnalized with respect to AREA1 and denoted by NORMALIZED AREA 2 and NORMALIZED AREA C. The inserted frequency, FREQ, is given in rad/sec, and the duration of the input signa~ TDUR, is given in seconds. Oscillation

frequen-cies are represented in the figures in rad/sec, using ,.n.

W~=~

1:

The damping parameters Di are related to the damping ratios ~ i and the dimen-sionless oscillation frequencies Ai through

Lis.

Their behaviour can always be deduced fram the damping ratios and the oscilla-tion frequencies.

An interesting situation is created when the rooms have equal individual period

ratios,

ot

2 = ~ l' As can be deri ved fram equation ~ I wi th ~ ~

VA:

~ V~

~

V. V~ VO 'tI == 't CÄ.

~A:

t> 1

V

4!j.

rr

0 and ~.

i

=

-F

h for A = 0 and fb is not affected by A . The same tendency shows up in tRe numericalccalculations, where damping is iRcluded and

~1

=

~2'

This is shown in Fig. 9. Also, ~b appears to be almost independent of A , since there is hardly any change in the basic oscillation pattem. Note that Db ~ D •

Consider when Al

=

A

2

=

A and V1

=

V2

=

V . The basic period ratio wi12 be

given by equation lt~, butOequation So willobecome

Th ::

1:(4

~

~'fA:

..

2.

'fA;

I SI.

llf

Wo

If then Ao

=

0 there will be no basic oscillation pattem since tb 0, but

ii.

< 't'o.

~

1

'fA: '

51..

llf

V

o

The factor 2 in the term 2(;\ can be understood from equation 33 : since in this situation P1

=

-

P

2, the repèlling force on the air in the coupling window is

twice that on the air in the window of a Helmholtz resonator. This also clari-fies the nature of the higher oscillation frequency as the frequency for the excange interaction between the two rooms. If A = A equation ~ will again describe the basic period ratio for the system äs we~l as the individual period ratios for each of the rooms. But

ti,

=

~~3fA:'

53.

~n V_o

indicating that all three inertances pattem wi th frequency ~. When A

2 =

become, fram equation ~L

;r,::

~

~~.,

I l.rr V

I

are equally involved in the oscillation

0, the individual period ratios (A

=

0)

c

SLt .

Ss.

t

2 .{. 1:'1 in all situations and the limit for tb when A .... 0 will reach the value

:r

2

=

0 and ~ will approach t 1. A room is generally tRe basic room when i ts

outgoing winHow is closed. Similarly,

~

h approaches

~

1 for A - 0 and

5'

b - 1 , as Fig. 10 shows. S2

=

1 expresses, together withT

2

=

0 the noR-oscillatlng

behaviour of a completely closed room volume. When a certain amount of coupling is established, t _h increases more than fb as a function of Ac'

The damping ratios decrease wi th increasing A . The exchange between the two

rooms, as described by the higher oscillationCfrequency takes place through the coupling window. I t is therefore clear that 'r_h (and IN h) depends much

stronger on A than :eb (and W b). The behaviour or the damping ratios is

expected to b~ of simllar nature as described in equation

,b

for the Helm-holtz resonator.

In discussing equations 14 and IS for the Helmholtz resonator, it was mentioned

that any relevant frequency could be inserted to calculate approximate values

for the oscillation frequency and damping ratio. The same argument applies for

the two-room system as is shown in Figs. 11 and 12 for A

2

#

0 and in Figs. 13

and 14 for A

2

=

O. The inserted frequency is zero in Figs. 11 and 13. The

dam-ping ratios and damdam-ping parameters are zero, and the period ratios and oscil-lation frequencies agree exactly with equations ~I and~L. The inserted frequency is 50 rad/sec in Figs. 12 and 14. However, the deviation fram zero damping and the change in the oscillation frequencies are negligible.

Aremark can be made about the importance of the basic and higher oscillation

frequencies. As will turn out from the results in sect'ion 4, the system will mainly oscillate wi th frequency ~, wi th only small perturbations due to ~. The contribution ofSL

h is most apparent in the non-basic room, but almost

negligible in the baslc room. The basic oscillation pattern is therefore affecting the whole system, whereas the contribution of~h is apparent in

virtually one room only. Finally, the smaller the coupling area, the higher the

inertance it represents and the less coupling exists. The rooms virtually respond as individual rooms, each with its own oscillation frequency and dam-ping ratio. However, as the coupling increases, the rooms influence ~ach other and are tied together into one system that is mainly governed by ~ and ~ b.

3.3 EFFECT OF AREAS OF OUTGOING WINDOWS ON OSCILLATION FREQUENCIES.

The increase of the area of an outgoing window decreases the corresponding inertance. This will result in an increase in the oscil1ation frequencies and a decrease in the damping ratios, as is shown in Figs. 15 through 18.

Figures 15 and

16

deal with the same room-volume configuration. The value of A2, the area of the outgoing window of the basic room, is increased in

Fig. 15, causing wb to increase more than Wh, while Al' the area of the outgoing window of the non-basic room, is increased in Fig.

16,

in which case the effects on wb and Wh are of t,he same order. Figures 17 and 18 show the effect of Al for configurations with A2

=

0 and different room volumes, showing the importance of the room volumes for the behaviour of the oscillation frequencies. Note that room 2 is the basic room when A2 =0.

The different behaviour of the damping parameters Db and Dh in the various figures is very illustrative for their dependence on the damping ratios and the oscillation frequencies.

3.4 EFFECT OF ROOM VOLUMES ON OSCILLATION FREQUENCIES.

The role of the room volumes can best be illustrated for A

2 # 0 through the use

of equation ~I • With A

=

A

=

A

=

A

1r-__ ~2 __ ~c~~~o __ ========~

'rbh

=

~; ~A.·

Vi, •

-k

±

~.'

-

V,~.

•

V:'

sb.

This yields for V1 ~ V2 (Serl'eS _expanslon ' _wit h _V /

't'b

:::

~F

\1-

.. 1-

~

I

ln

V2.

Lt

Lf

V,

51-t'~ ~

~~Y&'

.

Vî

~

1 -

-

..

~~~

~IT v~

4

'-i

V,

showing the Vi>"')V₂

for fb and

f

_h. On the other hand, when

s<6.

The larger volume (in this case V

1) is predominant in the determination of

f

b,

and a change in _ the larger volume mainly affects 'Eb' while a change in the smaller

volume affects ~ • These volume effects may not be as clear in the general

si tuation where

Rl

f.

A₂ :/:. A

f.

Al and damping is taken into account, but they do occur as is shown in Flg.

19.

A~, ~,A and Vi are fixed while V

2 is increased from V

2/V1 ~ 1/3 through to V2/V1 _ 2. c

When setting Al

= Ac

= Ao with

A₂

= 0, equation

~2 yields

lV

I/

"K

~

-2-+1

:t1l1+"'1·

VI

~~

~.

Series expansion wi th V/V 1 = 1-€. results in

-q, :

'r,,-~R'ïl,_ ~

..

Ü

-

:L)~'

l.TI

2.V

1

~

G'

'fS

V,

bo

.

'ti.:

'r"-1F.'

I

+

~

..

(~

+

'-1 )

'!.l

1

rr

lV).

rs

V'S'

VI

The same conclusions that follow equation 58 apply here. The difference between equations 1" and

b,

is due to the asyrrmetry of the system when A

2 = O. Figure 20 shows the effect of increasing volume, comparable to Fig. 19. Note that the increase in V

2 in the numerical calculations has been achieved by increasing the length of roan 2, thereby leaving width and height of both roans equal. All graphs show an increase in~)b' which may, be understood fram the increase

in the ratio (Si + S2)/VV₁ + _V2

=

St t/~Vt t with an increase in v~_(appendix D). ~h appears to have a minimum thä~ is r8ughly located around V27V1 = 0.8 for aIl configurations.

4. PRESSURE HISTORIES. 4.1 INTRODUCTION.

Once the oscillation frequencies are found, the systems of differential equations can be solved. Examples of pressure histories are given in Figs. 22 and 23. They show the pressure signatures relati ve to the maximum over-pressure of an N-wave of duration 't' = 200 msec and 't"= 144 msec as a function

of the dimensionless time parameter T for a system with dimensions V₁

=

8.20 m3 , V2

=

23.33

m3,

Al

=

0.2791 m2 and

Ac

=

_{0.2791 m2 while A2}

=

0.048 m2 and A2

=

0 m2 • The ave rage absorption coefficient~m

=

0.25 and the dimensionless rise time of the symmetrical N-wave is Tr

=

0.1. The numerical solution to the pressure histories is used to provide an approximate means to deal with the radiation and viscous losses. However, when these losses are set constant, the numerical procedure can be compared with an exact solution. Figs. 24 and 25 show that the agreement is very good.

A coupling window between two roans ties the roans together into one system and the oscillation frequencies deviate fram those of the individual rooms as discussed in section 3. Therefore, the pressure histories in the rooms will also deviate fram those of the individual roans. As Figs. 26 and 27

illustrate, the pressure history in a room starts drifting away fram its single roan reference when the room is coupled to another roan, and more so when the coupling area is increased and the oscillation patterns of the indivi-dual rooms are different.

The importance of the change of the oscillation frequencies due to the intro-duction of coupling and its influence on the pressure histories can best be illustrated under low damping conditions (~m

=

0): Figs. 28 and 29 show that coupling may create or destroy resonance conditions, as can be seen for each configuration by comparing the pressure responses in roan 1 (P1) with their single room reference (PN).

Fig. 30 makes clear that the pressure responses have the general appearance of (damped) sine waves superimposed on the N-wave and it also shows the rela-ti ve importance of ~ and ~ on the pressure responses in each of the rooms: the air in the basic roan oscillates mainly with frequencY~b' with only small perturbations due to~. This perturbation is most apparant ln Fig. 30a. The response in the non-basic roan also oscillates with frequency~, but in addition the contribution of~ is evident.

4.2 EFFECTS OF VARIOUS PARAMETERS ON THE PRESSURE-TIME HISTORIES.

The early part of the pressure responses depend very much on Tb. Figure 31a shows a situation for which the rise times of the pressure responses in the basic room decrease and the first positive peak amplitudes increase as Tb

increases. The rise times increase because the basic natural time of the system tn b = T/Tb decreases in comparison to the duration. The increase in the first'positive peak amplitudes is due to the fact that the rise time of the response matches better with the rise time of the input for higher Tb. This increase, however, is bound to reach a maximum and then decrease again, since for Tb approaching infinity the response will solely follow the input signal and the first positive peak amplitude will approach the value 1. It should be noted that the maximum overpressure amplitude is usually achieved under resonance conditions in the form of a negative overpressure at T ~ 1, i.e., when the e"lapsed time t is approximately equal to the duration of the input signal. Resonance conditions are created wh en room volumes and window areas are such that the basic natural oscillation time of the system matches the duration of the input signal. They are characterized by integer values of the basic period ratio Tb (see equation

40).

Maximum overpressure ampli-tudes and the conditions for which they are achieved will be considered in the next paragraph. Th increases with~. That explains the behaviour of nh in subsequent responses in Fig. 31b, which also shows that, as

Ob

increases, its influence fades away because the damping mechanisms become more effective.

Fig. 31a shows the development of a kink K in the early part of the pressure response when =t b exceeds uni ty and rJ.,.

=

0.25. Since damping is proportional

to ~ , i t will reduce the arnpli tudesmof the sine waves that are superimposed on thW N-wave. Only a kink is left in the early part of the response for higher

values of 0<. ,as is shown in Fig. 32.

m

Fig. 33 shows the effects of the absorption coefficient on the pressure

histories. The overpressures are reduced, especially in the latter part of the

response and the oscillations are slowed down when ~ is increased. Again,

the influence of,Sl. b quickly fades away for higher 0( m. An increase in any of the window areas, either couplirgor outgoing, resultsmin higher values for

~ and ~h and changes the pressure histories accordingly as illustrated in Figs. 34 and 35.

The effects of volume changes are shown in Figs. 36, 37 and 38. Depending on the configuration of the system the increase of a volume mainly affects

.n.

b (Fig. 36) or S'l. h (Fig. 37). In Fig. 38 the total volume is kept constant at

approximately 3T m3 • This value agrees wi th the experimental set-up. The

volume ratios V

1/V2 are 0.2, 1 and 5 respectively, and Al and A2 are not too far apart. Therefore, room 2 is the basic room in Fig. 38a and room 1 is the

basic room in Fig. 38c. Fig. 38b represents a situation where the period ratios

of the indi vidual rooms are almost equal and the room pressures are not very

much affected by the coupling, as can be se en fram the comparison of the pressure responses in both rooms wi th the response in room 1 if i t would not have been coupled to room 2, as indicated by PN.

4.3 AMPLITUDE RATIOS

The maxima of the room responses are of special interest. All responses ·are

pro-por~ion~l to the maximum oyerpressure Po of the input N-wave, so the amplitude rat10s Al

=

Pl,max/Po and A2

=

P2 max/po are characteristic for the maximum

res-ponses ~n the system. Fi~es 39' and

40

give same examples of the amplit.ude ratias

Al and A2 as a function of Tb for various configurations and different values of CXm· The amplitude ratios increase when CXm is decreased and are largest for CXm :: O.

Maximum values are ~hieved for approximateg integer values of ~b' characteri-zing resonance conditions. A comparison of Al and A₂ for a gi ven configuration shows differences in maximum values and in behaviour. The amplitude ratio of the basic room will exceed the amplitude ratio of the non-basic room because the am-plitude of the oscillation with frequencyIl is much larger than the one with frequency.st

h and the relati ve contribution

o~

~

in cornparison wi th

~

is large st in the basic room. Figs. 39a and b are clear examples of this situation. In bath cases room 2 is the basic room. For the configuration of Fig. 39c the period

rati~ of the individual rooms are almost equal (r

2 is only slightly smaller than Tl). Therefore both rooms oscillate wi th almost the same oscillation pattern (a simllar situation is shown in Fig. 38b for the pressure histories), and have virtually the same amplitude ratio profile. Fig. 40 shows amplitude ratios for the same configuration , but for different input characteristics. The dimension-less rise time of the $Yffiffietrical input signal is T = 0.0, 0.10 and 0.25 in Fig. 40a, b and c, respectively. An increase in theramplitude ratio aroundT

b

= 1

and a shift in the subsequent peaks wi th increasing rise time is visible.

Fig. 40c shows a maximum amplitude ratio as high as 3.79 for the given configura-tion,o<.

= 0

and T

= 0.25.

In Fig. 41, the maximum amplitude ratio of the basic room ismplotted ve~sus T for the configuration of Fig. 40, for various values

of~ • It shows that thermaximum amplitude ratio in the basic room increases with T u~ to T

= 0.25,

indicating higher induced room pressures for N-waves with

5. EXPERIMENTAL WORK. 5.1 INTRODUCTION.

Expertffients were conducted in order to verify the analysis for the pressure

response calculations. The UTIAS Travelling-Wave Hom-Type Sonic-Boom

Simulator (Ref.12) was used to generate the sonic-booms. The simulator consists

of a horizontal concrete pyramid which is 25 m long and has a 3 m square base. A flap-valve mechanism separates the hom from high-pressure air reservoirs. The facility is illustrated in Figs. 42 and 43. When the simulator is fired

a certain amount of air is released through the narrow throat of the

flap-valve into the hom. The flap-valve mechanism is designed to release air accor-ding to a parabolic mass-flow-rate profile, so that an N-wave type pressure profile will be generated (Fig. 44, Ref. 17).Typical duration times for N-waves

thus generated vary fram 0.1 to 1 seconde An N-wave will travel down the hom from the apex to the base where it hits the reflection eliminator. The

reflec-tion eliminator is a moving porous piston with an impedance matched to the rest of the system in order to minimize reflections from the base back into

the hom. The hom was connected to a full-scale test room of width 4.12 m, height 2.41 m and depth 3.17 m through a cut-out of 1.83 m x 3.66 m in one

of its side walls. The walls and ceiling of the test room were made out of

gypsum plaster, the floor was made out of wood (Ref. 9). In order to create two connected rooms a wall was built to make the one room into two. This wall

extends from the cut-out to the opposing wall so as to facilitate the use of

the cut-out for window construction in ei ther one or both of the rooms. A lightweight wooden door connects the two roorns and provides the frame for the

connecting window. The wall was constructed out of 4 cm x 9 cm wooden studs,

covered by drywall at both sides and i t creates a volume ratio of

approxima-tely 1:3 as is shown in Fig. 45.

5.2 ABSORPTION COEFFICIENTS.

If a sound source of arbitrary frequency is suddenly shut off at t 0, the pressure amplitude will decrease according to (equation

12),

p

=-0

Solving for the average absorption coefficien~am gives

o(m

=

bLf.

where P is the pressure amplitude before the source is shut off. The value for 0( Sat low frequencies is of main interest, as can be understood from

. the fr.wquency spectrum of N-waves (appendix A). Sound waves of particular frequencyare generated using a sine generator, amplifier and.45-cm diameter

speakers. A Bruel and Kjaer Sonic-Boom Condensor Microphone 4'Z41 with its

Bruel and Kjaer Carrier System 2631 measures the pressure responses. Figure 46 shows sorne examples of measured pressure decay traces. The upper traces show

the pressure signals as measured by the microphone. The lower traces show

the same signal, but filtered by a low-pass or a narrow-band filter set at

the frequency of interest. In general, the traces show a beat pattem., when

the source is shut off, rather than an exponential decay. The unfiltered

signal and the occurring beat pattem are a clear indication that sound waves of various frequencies are excited. The decay phenomena shown in various parts of the photographs in Fig. 46 will therefore be the re sul t of sound absorption as well as interference pattems and a straight forward use of

'

..

a mul ti tude of frequencies are exci ted may very weil re sul t fram the use of a steady sound source, possibly allowing time for an active response of the system, and the clear distinction with transient sonic-boom excitation should be noted. Final conclusions for the ave rage absorption coefficients for either one of the rooms can not be drawn fram these experiments. Wahba (Ref. 4) showed satisfactory agreement between analysis and èxperiments for a one room Helmhol tz resonator configuration using 0< = 0.25. This value for 0< will be adopted in the comparison of analysis wTth experiments for the two~ room system (section 5.5).

5.3 SIMUlATED N-WA VES •

Figure

47

shows three pressure traces of an N-wave of 125 ros as generated by

the UTIAS Travelling-Wave Hom-Type Sonic-Boom Simulator. Upper, middle and lower trace show the generated signal at 13.5, 18 and 22.5 m fram the apex of the hom. A number of peaks occur in addition to the expected N-wave. These perturbations are due to reflections fram the wal 1 of the building that enc10ses the end of the hom and the test room, as shown in

Fig. 42. Although the large doors in this wall are kept open during experi-ments, the N-wave will partially reflect off the wall and travel back up

the hom. When this reflected wave travels back up the hom, its amplitude will increase due to the decrease in the cross-sectional area of the hom and eventually the wave will again reflect off the valve and travel down the hom. The reflections fram the front and tail shocks can be considered

separately and their reflections can be identified. A general idea of the

appearance of a simulated N-wave and its reflections as a function of time for a fixed position is given in Fig. 48. In addition to the explained

feature~, some small amplitude oscillations may occur as the enclosing room responds acoustically in combination ~ith the large doors, much like a Helm-holtz resonator.

In order to adequately compare analytical and experimental pressure responses in the two rooms, it is necessary to provide the proper input signals for the numerical calculations. This is achieved by approximating the experimental input signals by an arbitrary number of straight lines. Typical examples of simulated N-waves and their approximations are given in appendix A.

5.4 EXPERIMENTAL PRESSURE HISTORIES.

Some examples of pressure recordings are shown in Fig. 49. Each photograph shows three traces. The upper trace represents the N-wave as generated by the sonic-boom simulator. This N-wave is the input signal to which the two room system responds. The middle trace shows the response in the smaller one of the two rooms. The lower trace shows the response in the larger room. Fram now on the snaller room will be cal led room 1, the larger room will be called room 2. The area of the outgoing window of room 1 will be caJ.led Al' the area of the outgoing window of room 2 will be called A

2• The coupling window will be referred to as A •

c

Some important features of these responses can be pointed out. The traces in Fig. 49a show the responses for a configuration in which room 2 is the basic room, i.e. room 2 mainly determines the oscillation pattem of the

total system. lts pressure response shows the least contribution due to~, and i t becomes clear fram the response in room 1, the non-basic room, tha~ the oscillation wi th frequency

.n.

h merely results in a ripple on the basic oscillation pattem as caused by st . The response in the non-basic room will therefore have a smaller rise

~ime

and the initial instant of response may be faster than for the basic room. It is possible to compare the instant

of the ini tial responses in the rooms to the instant of arri val of the front shock of the input signal, since all pressure recordings are triggered simul-taneously. The delay of the response in the basic room is more clear in Fig. 49b, which shows a resonance condition, wi th room 1 as the basic room. Fig. 49b indicates the importance of the maximum amplitudes of the positive and negative parts of the responses and the times at which they are achieved. A resonance condition is created when room volumes and windowareas are such that the basic natural oscillation time of the system matches the duration of the input signal. The maximum amplitude under resonance conditions is achieved in the fonn of a negati ve overpressure , occurring roughly around T = 1. The maximum amplitude in room 1 is largest because of the predominant importance of the basic oscillation pattern in the basic room. The develop-ment of a kink K in the responses is clear in Fig. 49c, resulting from the

fact that

iF

b> 1, as is already considered in Fig. 32.

Pressure responses at different positions in the rooms are shown in Fig. 50 for EL 100 ms input signal and in Fig. 51 for a 200 ms input signal. 'IherE:

appear to be slight differences, especially for the non-basic rocms. This is due to the oscillations wi th frequency..n.

h because the wavelength is in some cases of the same order as the dimensions of the rooms, in which case a Helm-holtz resonator approach becomes questionable. The differences are minor because the responses are mainly detennined by ~h' but they should be con-sidered in a comparison of the experimental and analytical re sul ts, as will be done in the next section. It should be noted that these differences will diminish when the dimensions of the system are increased, since the resulting decrease in.n.. h and the corresponding increase in the wavelength exceeds the increase in the dimensions of the system. The actual dimensions of the experi-ment al rooms are smaller than in most realistic si tuations, which makes the Helmholtz resonator approach an appropriate model to deal with for arealistic

two-room situation.

Figure

52

shows the responses of the individual rooms, with outgoing windows of 0.279 m2 for Fig. 52D and 0.0933 m2 for Fig. 52~. The coupling window was kept closed during these experiments. The responses in Fig. 52b are larger because the windowareas are larger and their inertances correspondingly

smaller, allowing more exchange between the rooms and the input signal. In addi tion, a resonance condition is created for room 1, as can be se en from the large negative overpressure and the oscillating latter part of the res-ponse. The responses in room 2 merely follow the input signal, since

i

b

<.

1. Figure

52

serves as a reference,to show the effect of coup}ing, as is

2done in Fig. 53. The area of the coupling window, A

=

0.093 m, 0.279 m, and

0.558 m2 in Figs. 53a, 53b and 53c, respectivgly. Room 1 has an outgoing

windowarea of A = 0.279 m2 , so its response is to be compared with the middle trace of tig. 52b. Room 2 has an outgoing wondowarea of A₂

=

0.093 m2 . lts response is to be compared with the lower trace in Fig. 52a. It can be seen that due to the introduction of coupling the resonance condition for room 1 is destroyed and .~~ ie amplitude of i ts response is decreased, whereas the amplitude of the response in room 2 is slightly increased and

the contribution of

.n..

h becomes much more apparent. The same effects are clear in Fig. 54, where A

2

=

0, and A

=

0.046 m2 , 0.186' m2 and 0.558 m2 , respectively. Again, the resonance cogdition is totally destroyed and the oscillation wi th frequency.St is more pronounced as the coupling area is increased. The amplitude of tRe response in room 1 is decreased whereas the amplitude of the response in room 2, clearly the basic room, is increased. Fig. 55a shows the responses when A and A

2 are 0.279 m

2 _{and A}

₌

_{0.558 m}2_• Room 2 is clearly the basic room anêt an oscillation due to.n.

b Is visible in the non-basic room. A decrease in Al to 0.093 m2 causes a Sllght decrease in the amplitudes of the responses, as observed in Fig. 55b. Any other change