Pressures inside a room subjected to sonic boom

Hay, 1977

PRESSURES INSIDE A ROOM SUBJECTED TO SONIC BOOM

by

TEeH \SCHE HOGESCHOOL DelfT

AART El! "'u'r~TEVAARTTECHNIEK LUCHT - '0 rt • VI. BIBUOTHEi:K

Kluyverwe 1 - DELFT

~. j\l~\

\91'

Nabi 1 N. Wahba

UTIAS Technical Note

No. 207

CN ISSN

0082-5263

PRESSURES INSmE A ROOM SUBJECTED TO SONIe BOOM

Subm:i:tted March,

1977

May, 1977

by

Nabi1 N. Wahba

UTIAS Technical Note No. 207

CN ISSN 0082-5263

..

Acknowledgements

The author wishes to e;x:press hls sincere thanks to his supervisors, Dr. 1. I. Glass and Dr. R. C. Tennyson, for their guidance and encouragement throughout the course of the project.

A special acknowledgement is due Dr. J. J. Gottlieb for his valuable

comments and ~scussions.

The help received from

Mr.

R. Gnoyke in carrying out experiments is

grea,tly appreciated.

The work was financially supported by the National Research Council of Canada and the Ministry of Transport, Transportation Development Centre.

s~

The pressure field inside a room subjected to sonic boom is investigated ~n order to study the problems of dynamic structural response and crack propagation • The data obtained ultimately may be useful for the assessment of structural damage caused by supersonic transport overflights.

The problem of a room with an open window exposed to sonic boom was considered to be analogous to that of a Helmholtz resonator which is externally excited. The damping of the system was related to such physical quantities as the geometry of the room, the size of the open window and the average absorption coefficient of the interior surfaces of the room.

An

analytical method based. on a Fourier transform was developed and used to study the effects of room volume, windowarea and type of N-wave on the pressure-time variations inside the room.

The UTIAS travelling-wave sonic boom simulator was used to produce the N-waves. The pressure-time profiles inside the room were measured as a function of window size and a fixed room volume for various N-waves. Very satisfactory agreement was obtained between analysis and experiment.

The room pressure , in terms of i ts profile and amplitude, was studied in a comprehensive manner. The results indicate that the maximum. pressure amplitude inside the room was larger than in the incident sonic boom. The room pressure reaches its greatest vçllue when the room volume, the window size and the duration of N-wave combine to cause resonance. If this occurs, the maximum room overpressure can be approximately twice the maximum overpressure of the N-wave with "zero" rise time. This value can even increase to 2.29 when the N-wave rise time increases to 1/10 of its duration.

A

significant reduction in the overall pressure inside the room was observed as a result of the absorption due to the walls.

~. 1. 2.

3.

4.

5·

6.

7.

CONTENTS

Acknowledgements Summary Notation

INTRODUCTION

THEORErICAL BACKGROUND

2 .1 Mass Element 2.2 Sources of Dissipation

GENERAL EQUATIONS

3.1

Dimensionless Parameters

NUMERICAL SOLUTION

4.1

Numerical Procedure

4.2

Convergence and Accuracy of Method

4.3

Results and Discussions

EXPERIMENTAL WORK

5.1

Measurement of Average Absorption Coefficient

(01n)

5.2

Simulated Sonic Booms

5.3

Results and Discussions

COMPARISON OF PREDICTED AND MEASUREDf:R'tSULTS

DISCUSSIONS AND CONCLUSIONS

REFERENCES

TABLE

FIGURES

APPENDIX A - EXPRESSIONS FOR END CORRECTION OF VARIOUS APERTURES

APPENDIX B - FREQUENCY SPECTRUM OF N-WAVES

ii iii v 1 1 2 2

3

4

6

8

8 9 11 12

13

13

14

15

18

a e J L m PCT) ;;: p/po

pen)

l'I(n) PR(n)

pet)

Piet) Notation Windowarea Ampli tude ratio .

Sound speed in air Window width

Window height

Constant defined by Eq. 18 Constant defined by Eq. 18 Orifice di~ter

Power contained in a certain frequency band relati ve to the total power of the N-wave defined by Eq.

B.7

Eccentricity of elliptic opening .

Thickness of orifice plate Effective length of opening Mass element defined by Eq. 1 Dimensionless incident pre~sure

Fourier transform of PCT) defined by Eq.

15

lmaginary part of

pen)

Real part of

p(

n)

Instantaneous room pressure Instantaneous incident pressure Maximum room pressure

Maximum averpressure of sonic boom steady state pressure

Resistance element due to radiation defined by Eq.

3

Resistance element due to viscosity defined by Eq.

4

r s T

=

t/T

Tf'

=

tf'h T = t

/T

r r t tf' t n

t

r V

w

Weu)

weu)

p T Notation Continued

Radius of' circular opening

Total area of' interior surf'aces of' room Dimensionle s s time

Dimensionless f'aJ.1 time

Dimensionless rise t~e

Time

Fal1 time of' sonic boom Natural period

Rise time of sonic boom Room volume

Instantaneous power

Power. spectrum of' sonic boom def'ined by Eq. B.5

Dimensionless power spectrum of' sonic boom def'ined by Eq.

B.6

Total power of' sonic boom def'ined by Eq.

B.4

Dimensionless natural f'requency def'ined by Eq. 14 Dimensionlel3s room pressure

Constant def'ined by Eq. 18 Aspect ratiq of' a rectapgle

Average absorption coef'f'icient of' room End correction 1ength

Average acoustic energy density'def'ined by Eq. 8

Displacement of' air in window Damping ratio def'ined by Eq. 14 Viscosity coef'f'icient of' air Density of' air

T

=

T/t

n w Q

=

WT Notation Continued Period ratio

Constant defined by Eq. 18

Frequency

Dimension1ess frequency

Cutoff frequency

Infinitesimal change in

n

1. INTRODUCTION

Figure 1 shows same experimental pressure histories which were obtained during a supersonic (SST) overflight outside and inside a specially constructed room-sized cubicle with a window (Ref. 1). Tests were conducted with the window closed and also partly opened. Thé outside pressure-history trace is indicated at the top. The inside pressure t·race, wi th the window

closed, is given in the middle. The bottom trace represents the conditions with the window partly opened so as to create a Helmholtz resonator. It is seen that the overpressure fluctuations for the Helmholtz resonator case has a higher overpressure amplitude than the exciting sonic-boom overpressure. The signal has the appearance of a damped sine wave, and i t persists for a longer period of time than the initial N-wave.

Vaidya (Refs. 2,3) provided a theory for predicting the ·sound field in a room generated by a sonic boom incident on an open window. Expressions for the pressure field were obtained by viewing the room as a terminated duet and then using a Green-function methode The concept of mode excitation distribution functions was formulated and used to match the boundary conditions. The method is only satisfactory to explain the behaviour in the first stage,

"the mult;iple-reflection stage", in the transient response of a room. When the field inside the room becomes fairly diffuse, the predicted response does not agree with the experimentally observed responses.

Since the dimensions of the room are small in camparison with the wavelength· of sonic boom, th~ room can be justifiably represented by a

lumped acoustic element, particularly when the sound field in the room is a diffuse one, i.e., the pressure is nearly uniform everywhere. Tlilen, the motion of the medium in the system is analogous

to

.

a Hemlholtz resonator having lumped elements of mass, stiffness and resistancé. Due to the lack of a proper description of the damping of a system in terms of known para-meters, Vaidya (Refs. 2,3) and Lin (Ref. 4) chgse it as an arbitrary constant. The present work reveals that the daroping affects the pressure inside the room quali tati vely and quant i tati vely . The damping factor is frequency dependent and it can be expressed as a funC"tion of the physical quantities of the system.

2. THEORErICAL BACKGROUND

Consider a room with an open window exposed to sonic booms. The room is assumed as a rigid enclosure of volume V, communicating with the external atmosphere through the window wi th an opening of area A. Such an opening, even if it is small, can transmit an appreciable amount of sound. The properties of the sound field inside the room depend on the geometry of the window, the geometry and structure of the room and the characteristics

of the incident wave.

It is assumed that the low-frequency components of the sonic boom, which contain most of the acoustical energy (Ref. 2), dominate the indoor pressure profile and that their wave lengths are much longer than the

dimen-sions of the room. Consequently the room can be represented by a Helmholtz resonator which is externally excited.

2.1 Mass Element

When an acoustic wave propagates through the window a certain alIlount of air is induced to move forward and backward like a piston. This provides the mass element of the system. Let this piston possess a mass of

(1)

where p is the density of air and Le is certain effective length which is greater than the true aperture length L, because some of the air beyond the ends of the constriction is entrained through the constriction so that

Le

=

L + 25 (2)

where 5 is the end correction. In Ref.

4

this effective length was chosen to be L + o.8../Ä. Actually the end correction 5 is a function of the dimensions

and shape of the window (see Appendix A) . 2.2 Sources of Dissipation

The losses may be divided into two general categories: those associated with the wi.ndow and those due to the air and the boundaries of the room. At the opening, there is a radiation of sound into the surrounding air, which leads to a dissipatiofl of acoustic energy and thus provides a resistance element Rr. It is aSSUllled that the moving mass of air in the opening radiates sound intQ the surrounding air in the same manner as a simple source mounted in an infini te baffle, then the low frequency radia-tion resistance is given by (Ref. 5)

(lb/ sec)

where a is the sound speed and w/27T' is the frequency.

Another loss is associated wi th the viscous force due to the motion of the air through the window. This resistance is given by (Refs.

6,7)

(lb/ sec)

(4)

where ~ is the viscosity coefficient, L is the thickness of the orifice plate, d is the orifice diameter.

Losses in the medium itself may be divided into three basic types: viscous losses, heat-conductiori losses and losses associated with molecular exchanges of energy. This loss of energy appears as a reduction in pressure amplitude because the sound energy is prop ort i onal to the square of the

pressure amplitude. In general, the sound absorption in air may be neglected for frequencies below 1000 Hz (Ref.

5).

The damped vibrations of the walls will extract energy from the

sound field. After successive reflections the sound in the room may be assumed

to become diffuse, i.e., the average acoustic energy density

Co

(ft ·lb

/rt

3) is

then the same throughout the entire volume of the room and all <;tirections of propagation are equaJ.ly probable. It is necessary to examine how the diffuse field wiil be absorbed by the walls of the enclosure. The total absorption is found by im.ü.tiplying each individual area by its absorption coefficient and summing. The a;verage value of the absorption coefficient for all the surfaces of the room, am, may be expressed mathematically as

n

I a i si 1

a

_m

=

--=--

_S

(5)

where al' a2, ••. ,

Ctn

are the respective absorption coefficients of the different

materials of areas sl, s2, ... , sn forming the interior walls of the room as well as aI!y other absorbing surfaces • Then the rate at which energy is bEÜng absorbed

by these surfaces is (Ref. 8)

éa~

m

4

Sabine's theory (Ref. 8) of growth aI!d decay of sound energy in a room

says that: the rate of increase of energy within a room v "dê/dt is equal to

the rate of emission by source W minus the rate of absorption by the walls. The fundamental differential equation governing the growth of sound energy in a room is therefore

Ea~

_m

4

= W (6)

It shQuld be borne in mind that the absorption coefficients depend on the type of material and the art of affixing. They vary somewhatwi th the

frequency of the incident sound (Ref.

9).

It should be noted that Eq.

5

gives

fairly good results if the average absorption coefficient is less than about

0.2 and the absorbing materials are distributed uniformly over the room. If i t

is grea1;er thaI! 0.2 Eyring' s equation is used (Ref.

8).

It is similar to

Sabine's, differing only in that the absorption coefficient is replaced by a logarithmic function of the absorption coefficient. Then

3 •

GENERAL EQ,UATIONS

a

= -

.,en

m

1

-f'

L

s.a.)

~ J. 1 " . S

(5a)

If Pi(t) and p(t) are the instantaneous acoustic pressures impinging on the opening and in the room, re spe ctively, the resulting differential equation

for the inward disp1aeement l(t) of the air in the window is

where t is the time variab1e. The aeoustic energy densi ty

f,

may be expressed

as (Ref.

5)

(8)

. 'rhe instantaneous power supp1ied to the system, W, is equal to

Ap(dt/dt). Substituting the appropriate expressions for

e

end W into Eq.

6,

d~

_ V

~ ~m

dt -

~ dt +

8paA

p Apa

A substitution for the expressions of dg/dt and

d2~/dt2

into Eq. 7 gives

]

~

dt

(10)

Tl1e in! tial eonditions for t

= 0 are

-

~I

p(t=O) - 0 and dt

(t=O)

:=

0 (11)

The acoustic system is subjected to an N-wave of maximum overpressure

Po and total duration T wi th finite ris~ and fal1 times tr· and tf' re spe ct:Lve1y ,

as shown in Fig. 2. 'rhe pressure history of such

a.n

idealization N-wave is

pet)

=

p t/t

_o

_r

~ 0 3.1 D1mensionless Parameters for 0

<

t

<

t - r (12) for - JJO

< t

< 0 á.nd

T

< t

< ()()

It is convenient to rewrite Eq. 10 in dimensionless form by nOrmalizing the pressures and time

/ "

Eq. 10 then becomes

where

w

_n 2 p

=

p./p and

T

=

t/T ~ 0 f!Ci (R + R )

m r

v

8paA2

(13)

(lOa)

(14)

Wn and

S

are called the dimensionless natural frequency and damping ratio, respecti vely. For most practical cases the quant i ties f!Cim(Rr + Rv)

/8

paA2

and L/d are small compared with unity. Using Eqs. 1,

3

and 4 the damping ratio

~ can be expressed as a function of the physical quantities of the problem if the incoming wave is harmonic,

(14a)

When the incoming wave is in the form of a transient signal such as an N-wave, it may be analyzed in terms of its frequency spectrum as it will be shown later.

It should be noted that the radiation and viscous terms of the damping are proportional to the square and squar~ root of the frequency, respectively, as given in Eq. l4a.

The damping ratio is directly proportional to the total absorption of interior room surfaces. In usual building construction there is sufficient low-frequency absorption present in the form of fibrous plaster or other panels with air-space behind as well as joist floors and ceilings (Ref.

9).

In general, absorption coefficients are detennined experimentally. Their values at low

frequencies are different than their values at high frequencies (Ref.

9).

Sin ce the whole range of frequencies of present interest is of the order of 50 cycles/ sec (a..t most, as it is shown in Appendix B),

<Xm

may be assumed constant in this range without any appreciable error.

For a light damping system such as

om

= 0, thè total tlamping is proportional to the square root of room volume. For simplicity the effective length, ~e' may be assumed proportional to the square root of the windowarea, A

(Ret;n

6).

Thell the damping due to radiation and viscosity are proportional to Al / and A-3! , respectively.

For higher values of

om,

damping is attributed to the absorption of the room rather than t9 radiatioi and viscosi ty. Then tn,e damping ratio, ~,

is porportional to V-1/2 and A-1

4.

Both Vaidya (Ref. 2) and Lin (Ref.

4)

did not give an e:lq)ression for the damping ratio

S,

as a function of the

physical quant i ties and chose i t arbi trari1y . Figure

3

shows computed pressure histories in a room normalized by the N-wave overpressure for ,different va1ues of the parameter ~. It can be seen that the general shape and the maximum value of the pressure inside the room is affected significant1y by the va1ue

of ~.

The initial conditions can be expressed in ,dimensionless form as:

X(T=O)

=

0 and : ' tT=O)

= 0

Equation 12 may be rewritten as:

P = T/T _r for 0

<

T

<

T

- r

(lla)

(12a)

o

for - 00

<

T

<

0 and 1

<

T

<

00

4 .

NUMERICAL SOL lJ.rION

The previous analysis shows that the system should be significant1y frequency dependent. When the incoming wave is in the form of a transient signal such as an N-wave, the spectrum of the sound field inside the room can be computed by using a Fourier transform which rep1aces the N-wave by a multi-tude of sinusoidal waves. The transform of PCT) is defined as

00

p(n)=

J

P(T) e-jar dT = PR(n) + jPI(n)

(15)

-00

where n

=

WT, !'R(n) and

Pr(n)

are real and imaginary parts of p( n), respec-tive1y, and j =

..r-ï

(see Appendix B). Then PCT) can be written as

00

PCT)

-00

Since PCT) is a real function, then (Ref. 10),

P(T)

=!

Real Part of [p(n) .

ej~]

•

~

TT

(16)

In this form P(T) appears as the SllID. of produets of the inf'initesimal

M

times

the value of the function of

u.

.

The general SQ1utiop of the second order different:Lal equation (Eq. 10a) with zero initial conditions is given by

X(T)

where c = -

X •

cos~ c [

e~~WnT{

Cc . cos[wn(l -

~2)T]

+ Cs +

X

cos(nT -

~)

J .

~

(17)

(18)

4.1 (1)

(2)

(4)

~.2 Numerical Procedure

Knowing the .dimensions of the window ~ i ts area A can be computed and the effecti ve length Le is o'Qtained as shown in Appendix A, and the mass element m is obtained from Eq. 1.

From the dimensions of the room and the absorption coefficients for t~e different materials forming the interior wal1s, f100r and cei1ing of the room as we11 .as any other absorbing surfaces, an average va1ue of the absorption coefficient am may be obtained using Eq. 5 or 5a. The volume V and tbe total surface area S also can be computed.

For a certain vaJ..ue of n, say

S'li,

the Fourier transform of the incomi~ transient wave pen) [pen) =

PR(.o)

+ j Pr(n)

1

is obtained as shown in -Appendix B. Using the values of resistance e1ements Rr and, Rv gi ven

by Eqs.3 and 4, re spe ctive1y , the '_{dimensionIess natural frequency Wn and}

the damping ratio ~ can be obtained from Eq. 14. Values of

X, cp,

Cc aJ),d Cs are given by Eq. 18.

step 3 is repeated for different vaJ.ues of n [+21

se .••

,ni, · ••

,On

=nc]. The infinitesimal M isequal to (ni+1 -

ili).

Finaliy , the pressure inside the room relàtive to the maximum overpressl,lre of the N-wave X(T), is

evaluated ~t different values of the time variab1e (Eq. 17). Convergence and ~ccuracy af Method

The accuracy of the numerical rethod depends on the vaJ,ue of nc and M. Much greater accuracy is obtained with a 1arger nc ~d a smaller

M.

The energy spectral density of a sonic boom signature was computed (Appendix ~) • Figure B1 shows that the first maximum is the dominaJlt one.. It corresponds to

.(2'. ~ 4 •. 16. Tbe area under a curve gi yes the weighted energy E. When the integra1 t~en over a dimension1ess frequency band n of 0 to 30, the weighted energy is approximate1y E ~ 0.94 of the total energy of the N-wave (Fig. Bl). It on1y exceeds this va!ue somewhat for N-waves with increasing rise times as shown in Fig. B3. Although short rise times are effective in causing start1e they have 1ittle effect on structural response. Here, the durations and overpressures are most important.

The procedure for calculating the pressure-time history for a specific room was repeated using different values of M (5, 3, 2, 1, 1/2 and 1/4). The results are shown in Tab1e 1. When NI. decreases, the accuracy is improyed but the time of computation increases. Sufficient accuracy is obtained when

M

is chosenas uni ty. Using these values of nc and Nl., incoming N-waves with rise times of 0.0, 0 005, 0.1 and 0.15 were calculated using Eq. 16. They are compared wi th ideal N-waves in Fig. 4. The computed N-WaV82' seem to have

longer rise-times than the ideal ones owing to disregarding frequency components beyond nc. The agreement between the computed and ideal N-waves is much better for longer rise-times.

To i11ustrate the degree of accuracy of a numerical method, it should be compared with an exact solution. Therefore, the computations were carried out on a system wi th a constant damping ratio ~, where the exact solution can be obtained. A comparison between the exact and numerical solut:i,ons using different values of ~ is given in Fig~ 5. The agreement is very good.

~~--

----

-

- - - -

- - - -

..

It can be summarized that good (and usually suf'ficient) accuracy is obtained for Uc and 6n of 30 and 1, respective1y.

4.3 ResUlts and Discussions

The pressure-time history inside a room is affected by the f0110wing factors:

(1) Characteristics of incident N-wave in terms of its overpressure Po, duration T, and rise-time tr' It is worth mentioning that the room

pressure is direct1y proportional to incident N-wave overpressure . (2) Open window in terms of its area A, and effective length Le, which is a

function of the dimensions and shape of the window.

(3) Room volume

v.

(4) Average coefficient· of absorption of room am'

Since the system is represented by a second order differential

e~uation, it is governed by two factors: the period ratio, i.e., the ratio of

the duration T, to the natural period of the system t n , and the damping ratio

S.

For most practical cases the quantity Bam(Rr + R_v

)/8paA

2 is smal1 compared with unity and the period ratio is given by

(19)

Figure 6 shows the pressure signature inside the room re1ati ve to the maximum overpressure of an ideal N-wave of duration T

=

200 ms as a function of the dimensionless time parameter T for a room of dimensions (13-1/2 ft x

10-5/12 ft x 7-11/12 ft) and an average absorption coefficient am

=

0.25. (These values agree wi th the characteristics of the room used in the experiments.) When a window of height b1 = 3 ft and width al = 1 ft was assumed, the corresponding natural period of the system tn was approximate1y 131 ms. It can be seen that the pressure signature has the general appearancé of a damped sine wave when 'R exceeds unity. 11' T is 1ess than unity there is a noticeab1e change in the pressure-time history for various period ratios as shown in Fig. 7, for light damping am

=

0 and ~

=

1, 2 and 3. Since the period ratio is an integer number the 1argest peak occurs at the instant of attainment of the maximum negative value of overpressure of N-wave, i.e., it exists at T

=

1. This peak decreases

slight1y in magnitude as the integer period ratio increases and i t is approxi-mate1y twice the maximum overpressure of the N-wave corresponding to period

ratio of unity. From Figs. 7 and

8,

it can be seen that the ear1y room pressure (T

<

_{1) has the appearance of a damped sine wave of duration t n , superimposed}

on the N-wave. Since the damping is proportiona1 to am the amplitude of the sine wave decreases as am increases and this exp1ains the existence of a kink in the ear1y pressure-time history for higher values of am (Fig.

8) .

•

A study of the effect of damping indicates a slight decrease of the first positive peak. However, a rather significant reduction of the peak (for T

>

1) is expected due to damping, especial1y for higher Values of T. As shown in Fig.

8,

_{for T :;:: 2, when a m increases from 0 to 0.1 the first peak} decreases only from 1.52 to 1.34 whi1e peak at T

=

1 decreases considerab1y from 1.96 to 1.29.

Figure

9

shows the early part o~ the pressure-time history ~or

CXm

=

0.25 and T

=

0'.76, 1.0, 1.52 and 2.0. As the period ratio increases the

dimensionless rise time of the pressure decreases while the ~irst peak increases

consideral>ly.

One interesting result is that increasing the ave rage coe~~icient

of absorption CXm causes a signi~icant reduction in theroom pressure and a

rapid decay 'of its latter part as shown in Fig. 10. The behaviour of the early

part ,is consistent with the previous discussion.

Figure 11 shows the pressure signature of a speci~ic room where

tn

=

131 ms and cxm

=

0.25 for various duration

(T

=

100, 200 and 300 ms). When

the period of the N-wave is greater than the natural period a kink develops in

the early part o~ room response as shown in Figs. ll(b) and (c).

The rise time of the N-wave has a small effect on the wave shape of

the room pressure as shown in Fig. 12. However, increasing the rise time o~

an N-wave tends to produce a higher amplitude and longer initial rise time o~

the room pressure • . I

The behaviour of the internal pressure is a~~ected by the dimensions

o~ the open window. A room o~ volume 116.2 ~3 and average coef~icient of

absorption 0.25 subjected to an N-wave o~ 200 ms duration was assumed. The

p:resstlre-time history for various windowarea (A = 1, 6 and 10 ft2 ) is shown

in Fig. 13. To make i t appli cable to an arbi trary window shape, the effecti ve

length was as~umed to be 0.96 ~ (Ref. 6). The period ratio is then

propor-tional to Al/4. It can be seen that the initial rise time increases with the reduction in the area of the opening due to the decrease in the period ratio.

Computations were also carried out for a windowarea

3

~2 wi th various aspect

ratio (bl/al

=

1, 10 and 20). The corresponding effective lengths are 1.64,

1.24 and 1.04 (Appendix A). When bl/al increases from 1 to 20 the increase in

the peak pressure wa.s within

5%

as shown in Fig. 14. When the room volume

increases the period ratio decreases (Eq. 19) and the initial rise time becomes

longer. The response decay is rapid for a smaller volume owing to the increase

in damping as shown in Fig. 15. It should be noted that as V, A or bl/al varies,

the change in shape of the early part o~ pressure signature is attributed to

change in the period ratio and damping ratio.

Since the largest amplitude of the room pressure, Pro, is important

in structural response, the amplitude ratio Ä

=

pm!po is chosen as a

character-istic parameter ~or comparing di~ferent system response. Figure 16 shows the

amplitude ratio as a function o~ period ratio for various values o~

CXm.

A

room volume of 1116.2 ft3 and a 3 ft x 1 ~ window were assumed in these

cal-culations. From these plots i t can be seen that large amplitudes can be achieved

for smaller values of

CXm.

When

CXm

is zero the amplitude ratio has a maximum

value roughly whenever the abscissa has an integer value wi th a minimum in

between. When cxm increases all o~ the peaks, with the exception o~ the ~irst,

disappear. This is because damping has the e~~ect of considerably reducing

the maxima o~ the latter part o~ the pressure, especially as the period ratio

T is increased. In other words, increasing damping has the effect of smoothing

out the peaks o~ the amplitude ratio. The asymptotic values of

Ä

are 1.47,

1.26 and 1.12 ~or CXm

=

0.1, 0.25 and O.~, respe ctively, ~or higher values o~

T.

A change in the period ratio can result fram a change in N-wave duration (Fig. 16), windowarea (Fig. 17) or room volume (Fig. 18). Comparing these figures it can be observed that the plots of the amplitude ratio as a function of the period ratio are almost· identical for am = 0, where the damping effect is small. On the other hand, for am = 0.25 there is a noticeable change in these plots due to a considerable effect of damping, which decreases as the windowarea or room volume increases and higher amplitude ratios result.

To study the effect of the rise time ratio Tr , the maximum amplitude Po and the total duration T of the N-wave were kept constant (i.e., the total

power is constant, Appendix B). For simplicity the fall and rise times were assumed to be equal. The period· ratio was chosen to be unity (resonance con-dition) , and the amplitude ratio was plotted as a function of Tr , as shown in Fig. 19. These results show that the maximum amplitude induced by N-waves with longer rise times is larger than that induced by N-waves with shorter

rise times. This effect iS more noticeable for smaller values of am than for higher values. This behaviour was observed by Lin (Ref. 4). However, he did not give an analytic reason because he thought that N-waves with small rise times contain more acoustic energy in their low-frequency components than thosewith large rise times. This statement is correct up to

n

= 7. At this value, the weighted energy parameter E has an approximate value of 0.75 for Tr

=

0.0, 0.05 and 0.1 as shown in Fig. B3. Beyond this value, E is larger for longer rise times. This means that N-waves wi th 10nger rise times contain more energy in their low frequency components than those with smaller rise

times and·larger amplitudes of room pressure are obtained for longer rise times.

5 . EXPERIMENTAL WORK

The illIAS Travelling-Wave Horn-Type Sonic Boom Simulator (Refs. 11, 12,13) was the facility used to generate sonic booms. The simulator consists essentially of a concrete horizontal pyramidal horn (80-ft long, 10-ft square base) and a sonic-boom generating device installed near the apex of the hom. A schematic diagram of the horn facility in the sonic-boom lahoratory is shown in Fig. 20.

The horn may be used in two modes: with a shock-tube driver or with an air mass-flow valve. Both devices release stored high-pressure air into the horn near the apex. This sudden discharge ·creates a travelling N-wave that propagates through the interior test section to the base of the pyramid. A porous-piston reflection-eliminator is installed there to cancel some of the

sonic boom signals which would be reflected from an open-ended horn. The mass-flow valve provides 10ng-duration (uP. to 1 s) N-waves, whereas the shock tube generates short-duration (up to 20 ms) N-waves. However these are more useful for startle effe cts than for structural testing. Therefore, the flap-type mass-flow valve was used in the present experiments to simulate sonic booms encountered in SST overflights.

A full- scale test room is linked to the horn interior by a 6""ft x 12-ft cutout. Eight aluminum I-bearns, 6-in deep with 3-l/2-in x 3/8-in flanges are milled so that one flange is 2-in wide. These bearns are mounted vertically

in the cutout at l6-in centrelines with the sma11 flange facing into the horn. Aluminum architectural channel sections, 2-in wide, l-in deep, 1/8-in thick, are placed horizontally at l6-in centrelines betweenthe I-beams. This

structurally-stiff network is t·akento simulate 2-in x 4-in wooden studs, with wODden cross-braces backed by a poured-concrete, concrete-bleek or brick wall (Ref. 14). The cut out is sealed with a 3/4-in thick wooden board to permit various sizes of window or other openings.

The inner dimensions of the room are as follows: height - 7-11/12

ft, width - 13-1/2 ft, and depth - 10-5/12 ft. The walls and the ceiling of this room are made of gypsum p;Laster 5/16-in thick on a metal screen lath. The outer surfaces of the walls and the ceiling are made of 3/8-in thick plywood nailed to the framing members (3-1/2 in x 1-3/8 in) for the walls and

(7-1/2 in x 1-1/2 in) for the ceiling. The distance between centrelines of two consecutive members is 16 in. Thérefóre, the total thickness of each wall and the cEüling are 4-3/16 in and 8-3/16 in, respeetivelY. The floor of the room is of wood-joist eonstruction and 5/8-in plywood on joists of

1-1/2-in.width and 7-3/8-in thickness. The room has a wooden lightweight door. The height and the width of the door are 6-2/3 ft and 2-1/2 ft, res-pectively. This door was kept closed throughout the experiments.

The microphone used in measuring the pressure was of thé.eondenser type, 1/2-indiam, Bruel and Kjaer No. 4147. It was connected with an adapter UA 0271 and microphone carrier system 2631. This microphone is sui table for low-frequency measurements. It was calibrated using a B

&

K 4220 Pistonphone. A Tektronix 535A oscilloscope and a Tektronix C-5 Polaroid camera wére used to record the pressure traces •. To· measure the average coeffieient of absorption an 18-in diam Goodman loudspeaker was used. An F34 IEC function generator was utilized to produce a.n harmonie voltage signal.

5.1 Measurement of Average Absorption Coefficient (CX_m)

It was shown in the previous analysis that the damping of the pressure inside the room is influenced by the absorption characteristics of the walls, ceiling and floor. The plywood panels, the plasterboards and the air space in the walls and ceiling have a high absorption in the low-frequeney range (Ref.

9).

The loss of energy due to absorption appears as a reduction in the pressure amplitude because the energy density is proportional to the square of i ts pressure amplitude. When a steady source of sound in the room is shut off at t

=

0, the decay of the pressure will follow an exponential decay (Ref. 8), such that

p

=

p e s

ro

at

m

8v

where Ps is the amplitude of the pressure in the steady oase.

(20)

The loudspeaker was used as a steady sound souree. It was dri ven by the funetion generator at various low frequencies (20 to 30 Hz). Below 20 Hz the amplitude of the pressure signal was too small to measure. Aft er the sound field had reached a steady state the souree was shut off. The

oscilloscope and tl1e Polaroid camera were used to record the pressure hi story. Figure 21 shows the decay af the pressure in the room at 22, 25 and 30 cycles/ sec. Then the average absorption eoefficient CXm was computed from the equation

8v (Pps )

cxm

= S at

tn

(21)

Accorcling to these measurements am was found to have an approximate value of 0.25 in this frequency range.

5.2 Simulated Sonic Booms

Figure 22 shows the oscilloscope pressure signature traces of the incident sonic booms of three different durations (100, 190 and 285 ms)

measured outside the window, i.e., at a clistance of 70 ft from the apex of the horn. The window was kept closed during these measurements. There are

weak pressure perturbations superinu?osed on the N-wave. These perturbations

were explained by Gottlieb (Ref. 15) as resulting from the fact that the base of the pyramidal hom and the reflection eliminator are enclosed in a room 22-1/2 ft x 25 ft, which includes the test room (Fig. 20) '. After the simulated sonic boom passes through the porous piston and leaves the large end of the horn it interacts with the room. Even though the large doors in the room

wall innnecliately behind the porous piston can be left' open during tests to

minimize the enclosing effects, the wave leaving the horn is partially reflected

from this wall and propagates back into the horn through the porous piston.

Furthermore, as the room air responds acoustically to the wave ~eaving the horn

much like a Helmholtz resonator, additional pressure perturbations follow the

reflected wave into the horn. These disturbances clisrupt a cOnu?lete simulation of the flow and the pressure conclitions inside the horn, by causing small

perturbations on the N-wave.

An

experiment al investigation was made to determine the rise time of

the, simulated sonic boom. The measured rise time was taken to be 1.25 times

the time for the overpressure to rise from 10% to 90% of its peaK value. The

rise time ranges from 4 to 8

ms

for N-wave duration ranging from 100 to 300

ms.

Many measurements at clifferent conclitions were done to assess the

effect of the clifferent factors on the pressure signature. The effect of the

reflection from the wall (10 ft from the position of the measurement) can be observed as a spike in the earlier part of the time history as shown in Fig.

23. A pressure signal obtained when the window of the test room was kept open

during the measurement is shown in Fig. 23b. COnu?aring it with Fig. 23a where

the window was closed, it can be observed that the latter part of the wave was

changed due to outward reracliation from the window. Figure 24 shows the pressure

signal of 190

ms

duration after it was electronically filtered by a low-pass

filter set at 40

Hz.

It can be concluded that the N-wave is followed by

pertur-bations of low-frequency content due to the response of the large room.

There-fore, the test room responds to the N-wave as well as the perturbations. The maximum amplitude of the perturbations is usually less than 35% of the N-wave

amplitude.

5.3 Results and Discussions

Sonic booms of the type shown in Fig. 22 were used as incident waves.

Five window sizes were used: 1 ft x 1 ft, 3 ft x 1 ft, 2 ft x 2 ft,

3

ft x 2 f t

and 5 ft x 2 ft. The pressures were measured at the centre of the room. Figure

25 shows a sample of the results. It illustrates the time history of the pressure for a 3-ft x l-ft opening subjected to sonic booms of 100, 190 and 285 ms duration. The behaviour of the early part agrees adequately with that predicted by the

analysis (see Fig. 11). Since the natural period of the system corresponcling

bf tbe roam pressure for incident waves of 190 and 285 rosduration wbere T

>

1. Howeve:v, tbe latter part of tbe response does not bave tbe appearance of a damped sine wave as predicted (Fig. 11) because tbe room responds . acous-tically to the. noted pressure perturbations following an N-wave. Since tbe

.perturbations of v:arious incident waves are different in sbape, tbe corresponding . là-tter parts of tberoom-pressure records are different.

Tbe analytical metbod reveals tbat tbe bebáviour of tbe room pressure is not affected by tbe orientation of tbe window about an axis -perpendicular to its plane. However, tbeaspect ratio of tbe window bl/al lias a sligbt effect

on tbe room pressure (Fig. 14). Tbe press\ll'e histories for various windows baving tbe same area are sllown in Fig. 26. Tbe purpose of tbese measurements was to investigate tbe effects of tbe aspect ratio (Fig. 26(a) and (b)) and tbe orientation of tbe window (Fig. 26(b) and (c)). Tbe cbanges in tbe pressure

signatures are not severe. Tbis means tbat tbe aspect ratio and tbe orientation of tbe window bas a negligible effect on tbe room pressure.

Tbe room pressure traces for various windows are sbown in Fig. 27. Son;i.c boom of 190-ms duration was used as an incident wave. Tbe natural period of tbe system is weakly dependent on tbe windowarea (tn

a

A-l/4). As tbe windowarea increases from

3

to 10 ft 2 tbe natural period decreases from 131 to 98 ros. All pressure traces bave a similar appearance and a kink. is also observed in tbe early part because 1

<

T

<

2 for all windows. Since tbe same incident wave was used for all windows, tbe latter portions of tbe room

pressure traces are almost identical in sbape confirming tbat tbe room responds to tbe perturbations fOllowing tbe N-wave. It can also be seen tbat tbe rise time of tbe pressure decreases (from ·50 ros to 40 ms) as tbe windowarea increases

(froI!1 3 ft 2 to 10 ft2 ) due to tbe decrease in tbe natural period, as predicted. Figure 28 sbows tbe pressure signatures measured at tbree different positions of tbe room. It can be seen tbat tbe tbree signatures are almost identical. Tbis means tbat tbe room pressure is nearly uniform everywbere and bence tbe lumped-element approacb is a valid one. Tbis confirms tbat a room

subjected to sonic booms of sucb durations can be treated as a Helmboltz resonato:r .

Tbe room baving a 5-ft x 2~ft window, wbere tbe natural period of tbe system is 98 ms, was subjected to N-wave of 100-ms duration. Tbe purpose of tbis experiment was to investigate tbe bebaviour of tbe system at resonance condition (T ~ '_{t n). Figure 29a sbows tbe incident N-wave and Fig. 29b sbows}

tbe corresponding room pressure at resonance. Tbe maximum room pressure is approximately 1.5 times tbe maximum positive overpressure of N-wave and it occurs at t R: 100 ms equal to botb of tbe duration of N-wave and tbe natural period of tbe system, as expected. Wben tbe present room (wbere

a

_m

=

0.25) baving a

3-ft

xl-ft window is sUbjected to an N-wave witb zero rise time at resonance, tbe amplitude ratio is 1.17 (Fig. 16). It sbould be notedtbat increasing tbe rise time

or

an N-wave and decreasing tbe damping of tbe system due to tbe increase in tbe windowarea, tends to produce a higber amplitude. A comparison between tbe predicted and measured pressures at resonance is sbown in Fig. 3le.

6. COMPARISON OF PREDICTED AND MEASURED RESULTS

It is sbown in Fig. 22 tbat tbe incident sonic booms are a fair approximation to N-wa.ves, except for tbeir bigb frequency ripp1e and tbe

pressure perturbations following the N-wave caused by room confinement. For numerical work the time signature of sonic boom can be described by three straight lines (Fig. 2). The boom' s high-frequency content usually has a

negligible effect on the structure's response. The"duration and overpressure

of sonic boom are the most important. The pressure-time variation of sonic

booms are shown as solid lines in Fig. 30. They were approximated by dashed

lines as shown, keeping the duration and the overpressure the same. For sonic booms of 190 and 285 ms duration, the spike in the early part (at 20 ros),

caused by the reflection from the wall at 10 ft from the positiqn of

measure-ment, was neglected. For sonic-boom durations of 100 ms the negative over-pressure was taken equal to the positive one.

Using these approximate N-waves, room pressures were calculated numerically for various window sizes. Figures 31, 32 and 33 show comparisons between predicted and measured pressures. For most cases the analysis predicts a peak room overpressure that is wi thin lCY'/o higher or lower than th at measured. The predicted pressures have longer initial rise-times than those measured as a result of disregarding high-frequency components of sonic-booms in calcula-tions of the room pressure.

While a sonic boom of 190 or 285 ros duration is exciting the system, a kink is seen in predicted and measured pressures as shown in Figs. 32 and 33. The predicted time elapsed to develop the kink and room pressure at this

instant are in good agreement with those measured. Consequently, expressions

given by Eqs. l4a and 19 describe adequately the damping and period ratio of

the system because the way of developing a kink depends on the value of

S

(Fig. 8) and

T

(Fig. 9) . On the other hand, when the room is subjected to au

N-wave of 100-ms duration, predicted and measured pressures have the appearauce

of a damped sine wave (Fig. 31) and the kink does not exist because the period

ratio is less than unity. Figure 3le represents the acoustic resonance of the

room. The agreement between theory and experiment of the early part in terms

of amplitude, signature and phase is very satisfactory.

The measured latter portions of the pressure do not have the

appearance of a damped sine wave, as predicted. However, they have a similar appearance when the same incident wave was used for all windows confirming

that the room responds to the perturbations following the N-wave. Since the

perturbations contain acoustic energy added to the system, the predicted oscillations seem to die faster and have lower amplitudes than the experi-mental ones.

In general , reasonable agreement is obtained for most cases between theory and experiment in terms of signature farm, phase and amplitude.

7.

DISCUSSIONS AND CONCLUSIONS

The pressure in a room generated by a sonic boom incident on an

open window was investigated in some detail. It was shown that since the

low-frequency components of the sonic boom contain most of the acoustic energy and

when the dimensions of the room are small by comparison to the waveJ.engtb of

a sonic boom, the room acts as a Helmholtz resonator. The different types of energy dissipation associated with the window, the air, the structure and the absorbing surfaces of the room were taken into account. The losses have a

damping of' the system was related to the physical properties such as the rlDom volume V, windowarea A, and the average absorption coef'f'icient f'or all

surf'aces of' the room am, in order to establish the relative importance of' the signif'icant contributors .. It was f'ound that the damping increases as the absorption coef'f'i cient a m increases, and for higher values of am, the damping ratio

S

is proportional to V-l/2 and A-l/4. The average absorption coefficient of the present room was evaluated experimentally f'ram the sound decay in the room af'ter i t had been exci·ted by means of a loudspeaker.. It · was found that am had an approximate value of' 0.25.

Since the damping is f'requency dependent, an analytical method based on a Fourier transform was developed to f'ind the pressure histories of the room. The method is fairly general and can be applied to any frequency dependent system of second order and to any f'orm of transient signals. It also gives the spectrum of the room pressure which may be used for structural response. Using this method, i t was found that the f'requency components from

o

up to 30/27rT cycles/sec, where T is the total duration of' an N-wave, contains

0.94 of the total energy of an N-wave with zero rise time, and more than that for N-waves with finite rise times. Consequently, although rise time is of' importance in the study of startle effects on humans and animals , it is

unimportant f'or structural response. Therefore, it can be recommended that f'or structural response studies, consideration need be given only to frequency components of the N-wave below 30/27rT cycles/sec.

The room pressure (in terms of i ts signature , amplitude and ini tial rise time) was studied analytically and measured as a f'unction of window size and fixed room volume for various N-waves. The analytical method reveals that the behaviour of the room pressure is governed by the period ratio T (Êq. 19) and the damping ratio

S

(Eq. 14a). The pressure-time history has the appearance of a damped sine wave and decays rapidly as the damping increases. Due to

higher values of the damping (higher values of' am, smaller room volume or window area), there is a noticeable reduction in the room pressure . A kink is observed in the early part of the response (while sonic boom is exciting the system) when the period ratio exceeds unity. The initial room-response rise time has been observed to be longer f'or higher natural periods of' the system tm and f'or longer N-wave rise times.

It was desirable to determine an upper limit to the amplitude ratio

Ä (i. e ., the greatest room overpressure relati ve to the maximum overpressure of the N-wave). Theref'ore a m was assumed zero (light damping system). Then the amplitude ratio has peaks at

T

= 1, 2, 3, •.. It has a maximum value of' approximately 2 at the first peak. This value increases f'or longer N-wave rise times and reaches 2.29 and 2.44 when Tr = 0.1 and 0.2, respectively. Increasing am tends to produce lower ampli tude ratios and the damping becomes more ef'f'ective, especially for smaller room volumes and window areas. It was also observed that the room pressure changes only slightly wi th a change in aspect ratio of the window.

The room volu2;re, window size and the duration of' the N-wave must be

_such as to give a period ratio of' unity to cause resonance. For the present room where am

=

0.25 the ampl±tude ratio at resonance is 1.5 for an N-wave of 100-ms duration and a window of' 5 f't x 2 ft.

The agreement between predicted and measured pressures in terms of signature, phase and amplitude was satisfactory except f'or latter portions of

the pressure history owing to the perturbations following the N-wave caused by room confinement and the outward acoustic reradiation from the window. Measurements show that the room pressure is nearly uniform everywhere. Hence treating a room subjected to sonic booms as a HelIDholtz resonator is avalid approach.

A mathematical model of a wallof the room is currently being

investigated to find its natural frequency and strain produced by the pressures inside the room. strain was measured by a strain gauge placed at the centre of the rear wall. From a study of the data obtained it was found that the strain has a larger value for short-duration N-waves

(T

= 100 ms). It may be inter-preted that the pressures inside the room resulting from N-waves of 100 ms

duration cause higher loadings as the low-frequencies of the N-waves are nearer to the natural frequency of the wall than those resulting from longer duration N-waves. Consequently, for structural-response and crack-prQpagation problems, N-waves of 100 ms duration will be used as incident waves and a window of

5

ft x 2 f t is recommended in order to achieve aresonanee condition for both the room as an acoustic system and the wall as a structural one.

1. Hubbar.d, H. H. Mayes, W. H. 2 • Vaidya, P. G. 3.Vaidya, P. G.

4.

Lin,

s.

5. Kins1er, L. E. Fr~y, A. R.

6.

Ingard., U. 7. Ingard,

U.

Ising, H. 8 .• Hunter, J. L. 9·

Parkin, P.

H. I!UlIIlllreys, H.

R.

10. Jaeger, .J. C. 11. Glass, I. I-Ribner, H.

S.

GottliE;lb, J. J. 12. C&I'others,

R.

13· Gott1ie'Q, J. J. 14. Leigh,

B. R.

Tennyson,

R.

C. Glass, I-

r.

REFERENCES

Sonic Boom Effects on l'eop1e and Structures. NASA AP-~47 (Ed. A. R. Seebass), 1967.

The Acoustic Response of Rooms w~th ~en Windows to Airborne Sounds. J ~ Sound Vib., Vol. 25, No.

4,

1972,pp. 505-5~2.

The Transmission of Sonic Boom Signals into Rooms through Open Windows. J. Sound Vib., Vol.

25,

No. 4, 1972, pp. 533-559. .

Some-Boom Analog for Investigating Indoor Ácoustical Waves. J. ,Acoust. Soc. Amer., Vol ~ 49, No. 5, 1971, pp. 1386-1:392; also UTIAS Technical Note No. 158, 1970.

Fundamentals of Acoustics. Chapters 8, 9 and 14, John Wi1ey

&

Sons, 1962.

On the Theory and De~ign of Acoustic Resonators. J. Acoust. Soc. Amer., Vol. 25, No. 6, 1953, pp. 1037-1052.

Acoustic Nonline ari ty of an Orifice. J. Acoust. Soc •. ,AlDer., Vol. 42, No. 1, 1967, pp. 6-17. Acoustics. Chapter 9, Prentice-Hall Inc., Englewood Cliffs, N.J., 1962.

Acoustics Noise and Buildings. Chapters 2 and 9, Faber and Faber Ltd., London, 1969.'

An Introduction to Applied Mathematics . Çhapter 11, Oxford University Press, 1951.

Canadian Sonic-Boom Simulation Facili ties. CAS I Jour., Vol. 18, No. 8, 1972, pp. 235-246.

Initial Calibration and Physiological Response Data for the Travelling-Wave Sonic-;Boom Sj,mulator. lJrIAS Technical Note No. 180, 1972.

Sonic Boom Research

at

Ul'IAS. CASI Jow;., Vol. 20, No. 5, 1974, pp. 199-222.

Age~ Plaster P~els Subjected to Sonic Booms. CASI Jour., Vol. 21, No. 9, 1975, pp. 352-360.

15. Gott1ieb,

J.

J. 16. Ray1eigh, L. 17. Swenso, G. W. Johnson, W. E. 18. Morfey, C. L. 19· YoUIlg, J. R. 20. Howes, W. L. 21. Onc1ey, P. B. Dunn, D. O.

Simulation of' Trave1llng Sonic Boom in a Pyrami..dal Horn. Ph.D. Thesis, University of Toronto; a1so UTIAS Report No. 196, 1974 •

. The Theory of Sound. Vol. 2, Section 306, MacMillan & Co., London, 1945.

Radiation Impedance of' a Rigid Square Piston in an Infini te Baff1e. J. Acoust. Soc. Amer., Vol. 24, No. 1, 1952, p •. 84.

. Acoustic Properties of Openings at Low Frequencies. J. Sound Vib., Vol.

9,

No. 3, 1969, pp. 357-366. Energy Spectral Density of' Sonic Booms. J. AcouSt. Soc. Amer., Vol. 40, No. 2, 1966, pp. 496-498. Far Field Spectrum of the Sanic Boom. J. Acoust. Soc . .A:J:iY:r., Vol. 41, No. 3, 1967·, pp. 716-717. Frequency Spe ctrum of' N-Waves with Finite Rise Time. J. Acoust. Soc. Amer., Vol. 43, No. 4, 1968, pp. 889-890.

TABLE 1

CONVERGENCE OF NUMERICAL METHOD

Dimensionless Room Pressure X(T) using different

M

"

-

3

2

V :::;

1116~.2" ft

,ex

= 0, A = 3

ft

and

T

= 200 ms

m

~

5

3

2

-

.,

1

0·5

...

• 25

1.42

1.3655

1.3193

1.2939

1.29

.

.50

-0.il48

-0.1526

-0.1494

-0.1437

-0.1421

.75

-1.2368

-1.1270

-1.0858

-1.0687

-1.0672

1.00

-0.0533

-0.0561

-0.0552

-0.0545

-0.0544

1.25

-0.1878

-0.3019

-0.2597

-0.2219

-0.2149

1.50

1.7717

0.3510

0·3307

0.3419

0.3455

1.75

-0.4045

-0.2219

-0.2854

-0.2834

:'0.2788

2

"

.00

-0-.0931

0.0689

0.0391

0.0505

0.0576

0.25

1.2895

-0.1416

-1.0675

-0.0544

-0.2138

0.3461

-0.2779

0.0586

Internel Mike

~xternol

Mike Po pst

l

I-

0.1

Sec-t

O.83~

.#" _

t

---~---L

External O.19~ _ _

f

_{Internal- Window Closed}

-

'"'""""

- L

1.15~

~ ~ ~.

--,---~ 'C7

'C7

t

lnternal- Window Open

FIG. 1 INrERNAL ROOM PRESSURE-TD1E HISTORIES DUE TO SONIC BOOMS FOR BCY.rH WINDOW-CLOSED AND WINDOW-OPENED CONDrI'IONS (HEF. 1)

P(

t)

o~~

__________

~~

____________

~

____

~

t

T

X

,~A

X

-c.,-'"{

..

...,., )C!trn< 7'

T

oV

,\:> <

-I,

_V

a) ~

=

0.01

I _

~

T

-I _{b) ~}

₌

_0.08 3 4

T

-I

_c)

_C

₌

_0.15

FIG.

3

PRESSURE HISTORY INSIDE A ROOM USING

ANALYSIS OF REF.

4

V

=

1116.2

ft

3 ,

A

=

3

ft2, T

=

195

ros,

L

_e

=

0.8

~,

X

=

p/p , ₀

T

=

t/T

pi

°l

-.,.."""

1/

c=-....

T

.11 -...

-IL

a) Tr=O.O

PI

fI

_"""-

1

-r==::::.

~

,

"

T

-Il

b) T

_r

=

0.05

PI 1

T

-IL

c)Tr=o.l

PI

1

T

_I L d)

Tr = 0.15

FIG.

4

N-WAVES PREDICTED BY NUMERICAL METHOD

T = t

/T,

P = p. /p , Ideal

r r ~ 0

XI -I a)

t

= 0.05

-I

b)

t

= 0.1

XI

3

T

-I c)

t

= 0.15

FIG.

5

COMPARISON BErWEEN TEE EXACT SOLUTION .ANI) TEE NUMERICAL MErHOD

X

I

-I

V

=

1116.2 ft3, A

=

3 ft2, T

=

200

ms,

X

=

p/po' T

=

t/T,

- - Exact Solution, --- Numerical Method

2 3

T

FIG.

6

COMPtJI'ED PRESSURE FIELD INSmE ROOM

V = 1116.2 ft3, Window 3 ft x 1 ft, <Xm = 0.25, T

=

200

ms,

X = pip , T =

t/T

x

2

N-Wave

0

T

\

,

\

/

\ I

-I

\J

-2

FIG.

7

EFFECT OF INTEGER PERIon RATIO

T

ON OVERPRESSURE RESPONSE

SIGNATURES

CT

<

1)

X

2 N-Wove 1

T

-I -2

FIG.

8

EFFEm' OF ABSORPI'ION COEFFICIENT

exm

ON OVERPRESSURE RESPONSE SIGNATURES

(T< 1)

t

=

131

ms,

T

=

2,

a

m

=

0.0,

-

• -

ex

_m

=

0.1, ----

ex

_m

=

0.25,

... a

=

0

4

x

N-Wove

À.,

,

"

"

'

\~"\"

""""

"

"

\

\ ... \ , \ '.

'\

\ '\ \

\.

\

\.

\ .... , I . . . \~ \ , O~.... \ 4 . '. 1

T

-I ,

FIG.

9

EFFECT OF PERIOD RATIO

T

ON OVERPRESSURE RESPONSE SIGNATURES (T

< 1)

t

=

131

ms ,ex

=

0.25, - T

=

2.0,

n

m

- - • -

; =

1. 52, ;

=

1. 0,

X

x

"

₃

•

T

-I~

\

/

Or

-a) 0m= 0.0 \. 1 / 2 ~3

X

t

_"

-I. a) 1"= 100 ms, T = 0.76

X

3

T

-I~

"""

b) Om=O.l "'I \. 1 _{"--"" 2}

X T

--... . / T -I b) 1" = 200 ms, 'r= 1.52

--

I

•

2 3

X

T -I~ c) Om = 0.25

or

"

I I ~

--~

I

X •

~

IJ

V

3

T

C) T= 300 ms, f' = 2.28 I I

..

2 3

T

_{FIG. 11 EFFECT OF DURATION}

T

ON OVERPRESSURE

-I~

RESPONSE SIGNATURES

d) 0m=Q4

FIG. 10 EFFECT OF am ON OVERPRESSURE RESPONSE

V :::: 1116.2

ft

3,

Window

3

ft

,

x

1

ft,

SIGNATURES

am :::: 0.25

V :::: 1116.2

ft3,

Window

3 ft

x 1 ft,

x

2 3

T

-\ a) Tr= 0.0

x

2 3

T

b) Tr

=

0.1

x

I

L

\

I " d""'?"> ' ~

o

(

I

'":7

2 3 -I

T

c) Tr=0.2

FIG. 12 EFFECT OF RISE TIME OF N-WAVE ON

OVER-PRESSURE RESPONSE

V

=

1116.2

ft3,

Window

3

ft

x 1

ft,

om

=

0.25,

=

200 ros

x

, I . . . " , I I . .

ol'"

(

r ... ~::::;;::>" 3

T

-I a)

A= 1ft

2,

r=1.I1

2 3

T

-( b) A

=

6 ft2,

f

=

I. 74

x

3

T

-I c)

A= 10

ft

2,

f

=

1.97

FIG. l3 EFFECT OF WINDOW AREA A ON OVERPRESSURE

RESPONSE SIGNAXURES

x

x

2 :3

T

0) bi /01

=

1.

f

=

1.47 3 T -I b) bi/ai

=

10,

T

=

1.69 :3

T

c) bi /01 = 20. f'= 1.78

FIG.

14

EFFECT OF HEIGHT/WlJ>TH RA!rIO OF WINDOW

ON OVERPRESSURE RESPONSE SIGNATURES

V = 1116.2 ft3, A = 3 ft2,

a

m

=

0.25, T = 200

ms

x

2 :3

T

-I a) V= lOOO ft3 , T = 1.61 X 1

Or

)

\

r

I "

, _ I C - : : : = - " > c: .. :3

T

-I

b) V

=

2000 ft

3.

f

=

1.14

X 1 I \. I

er

"I

Ii

Ot"'

\

r

,...

>

3

-I c) V = 3000 ft 3, T"= Q93

FIG. 15 EFFECT OF ROOM VOLUME V ON OVERPRESSURE RESPONSE

A

2

o~~

__________

~

____________

~

____________

~

____________

~

4

r

FIG. 16 AMPLITUDE RATIO AS A FUNCTION OF FERIOD RATIO AND ABSORPI'ION COEFFICIENT 2 V :;:: 1116.2 :ft3, Window 3 ft x 1 ft, tn :;:: 131 ms, A E amplitude,

T

EPeriod Ratio 0~~---~---~~---~3----~ T I I . .

o

0.66 10.53 46.55

ft

2

Window Area A

FIG. 17 AMPLITUDE RATIO AS A FUNCTION OF WINDOW AREA

A

00 2587 647 ₂₈₈ I

_ft

3 •

Room Volume V

FIG.

18

AMPLITUDE RMIO AS A FUNarION OF ROOM VOLUME

Window

3

ft x 1 ft, T = 200 ms 3

A

am=O.

0.25

o

al

Q2 Q3

T,

FIG.

19

AMPLITUDE RMIO AS A FUNCTION OF RISE TIME OF N-WAVE