analysis of sound fields in rooms using

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Ele c t r on i c s and Communica tions i n Japan , Pa r t 3, Vol. 7 2 , No. 1 2 , 1989

Translated from D e n s h i Joho Tsushin Gakkai R o n b u n s h i , Vol. 7 2 - A , No. 1, J anuar y 1989, pp . 1-11

Analysis of Sound Fields in Rooms Using

Bergeron's Method

Hidemaro Shimoda, Member

I n s t i t u t e o f Techno logy, Sh imizu Corpo ra t ion , Tokyo , Japan 104

Norinobu Yoshida and Ic h i r o Fuk ai, Members

Facu l ty o f Eng inee r i ng , Hokkaido Un iv e rs i ty , Sapporo , Japan 060

SUMMARY

A t the des ign ing s t ag e of an aud i to r ium ,e . g . , a co n c er t h a l l o r t h e a t e r , i t i s impor-

t a n t t o p r e d ic t i t s a c o u s t i c f i e l d v i a a com-

p u t e r . A conven t iona l p red ic t ion me thod

b a se d on g e o m e t r i c a l a c o u s t i c t h e o r y g i v e s

o n l y a macroview of th e f i e l d , and an add i -

t i o n a l s c a l e model t e s t i s r e q u i r e d f o r a

f u r t h e r i n v e s t i g a t i o n . To o ve rc om e t h i s p r ob -

l e m , a l t h o u g h a t e c hn o l og y t o t r e a t t h e s ou nd

f i e l d i n a room a s a wave f i e l d has been con-

s i de re d , th er e have been few examples of

th ree -dimens iona l t ime- re sponse so lu t i on of

t h e s ound f i e l d i n a room because of t h e d i f -

f i c u l t y of h a n d l i n g a wi de a n a l y t i c a l r e g i o n ,

wide frequency band, and complex boundaryc o n d i t i on s . T h i s p ap er ex am in es t h e j u s t i f i -

c a t i o n of a p p l i c a t i o n of B e r g e r o n ' s m et hod t o

a th ree -dimens iona l sound f i e l d in a room.

T h i s p a p e r a l s o a n a l y z e s t h e c o rr e s po n de n ce

between a th ree -d imens iona l equ iva len t -c i r -

c u i t r e p r e s e n t a t i o n a nd t h e sound f i e l d , a nd

examines th e fundamen tal na tu re of a f r e e

sound f i e l d . Based on th e re su l t s , a model

of a s imple rec ta ngu la r room was consi der ed ,

an d i t s s t e a d y - s t a t e sou nd p r e s s u r e d i s t r i b u -

t ion (wh ich i s a f u n d a m e n t a l c h a r a c t e r i s t i c

of t h e f i e l d ) a nd s i m u l a t i o n of r e v e r b e r a t i o n

( t ra ns ie n t phenomenon) w e r e examined. A l l

t h e s u c c e s s f u l r e s u l t s show t h e u s e f u l n e s s o f

the proposed method.

1. I n t r o d u c t i o n

P r e d i c t i o n o f a s o u n d f i e l d i s i n d i s p e n -

s a b l e i n a c o u s t i c a l l y d e s i g n i n g a room s u ch

a s a c o n ce r t h a l l o r s t u d i o . W ith t h e r ec e n t

p r o g r e s s of a c o u s t i c i n s t r u m e n t s an d t h e i n -

c r ea s e of u s e r i n t e r e s t i n a c o u s ti c e f f e c t ,

t h e d e s i g n of rooms wi th more p re ci se a.cau.s-

t i c s p e c i f i ca t i o n s ( i . e . , a c o n ce r t h a l l w i t h

7 3

much b e t t e r a c o u s t i c s ) i s requ i red . The re -

fo re , ac ou s t ic sca le -mode l expe r imen ts and

c om pu te r s i m u l a t i o n s i n c r e a s e t h e i r i mp or -tan ce . The former i s a r e l i a b l e and e st a b -

l i s h e d me tho d f o r t h i s p ur p os e , a nd i s used

f o r d e s i g n of a l m o s t a l l m a j o r a u d i t o r i u m s .

However , t h i s method has some disa dvan tage s:

a l a r g e model ( a t l e a s t on e- te nt h t h e a c t u a l

s i z e ) ; complex c o n d i t i o n s f o r i t s s p e c i f i c a -

t i o n s ( e . g . , m ater i a l s , sound-measurements

inc lu d ing u l t r a so n i c range , and sound med ium) ;

and cons ide rab le t i m e and expense [l].

A compu te r s imu la t ion i s a powerfu l a l -

t e r n a t i v e t o t h e s c a l e- m od e l e xp e ri m en t w hi ch

r e q u i r e s t i m e , e x pe n se a nd e x p e r i m e n t a l t e c h -

no logy . However , a t p re se n t , t h i s method ap -p l i e d t o p r a c t i c a l d e si gn i s based on geomet-

r i c a l a c o us t i cs [ 2 ] . This method i s used

ve ry wide ly because o f ea s ine ss o f fo rmula -

t i o n a nd h an d l i n g , s i n c e a s o u n d f i e l d i s

t r ea te d wi th a mac roview of g e o m e t r i c a l c h a r -

a c t e r i s t i c s of i n c i d e n t and r e f l e c t e d sound

r a y s [ 3 , 41. Th is method i s l i m i t e d t o a

low-f requency range s in ce t he wave p rope r ty

of sound i s n e g l e c t e d. I n t h e c a s e w he re

t h i s l i m i t a t i o n c a nn ot b e a cc e p t e d , a n an a l y-

s i s i n which t he approximated wave pr ope r ty

i s added [ 5 ] , and a method of g eom etr ica l

aco us t ic s imu la t i on wi th the wave -mot ion

pro per ty [ 6] have been proposed . However ,

t h e r e a r e l i m i t s t o improvements of geometr i -ca l -acous t ic me thods , and a new method based

on wave theory i s d e s i r e d .

Seve ra l an a l y t ic a l methods o f a sound-

wave f i e l d have been p roposed :

od [ 7 ] f o r m u l a t e s i t by u s i n g t h e K i r ch o f f -

H el mh ol tz i n t e g r a t i o n e q u a t i o n a s t h e b ou n-

d a r y - i n t eg r a t i o n e q u a t i o n , a nd t h i s h a s be en

a p pl i ed t o c a l c u l a t e a t r a n s i e n t s oun d f i e l d .

Some nume r ica l -ana ly t ic a l me thods fo r a t h r e e -

d imens io na l sound f i e l d have been p roposed ,

T e r a i ' s m e t h -

1SSN1042-0967/89/0012-0073$7.50/0

o 1990 Sc r i p t a T e c hn i ca , I n c .



(x,y, z tAd)

(x , y tAd,z)

(x,~,z+Ad/2)1 / % k d / 2 , z)

Fig. 1. Three- di mensi onal equi val entci rcui t for sound f i el d. (a) Equi va-l ent ci rcui t and (b) coordi nat es def i -

ni t i on.

i . e. , the f i ni t e el ement met hod [ 8 ] , and theboundary el ement method [9]. Al though t reat-ment of a sound f i el d as a wave f i el d compl i -cat es the whol e process, thei r i mport ance and

usef ul ness are i ncreasi ng w t h the recent ad-vance of hi gh-speed and l arge- capaci t y compu-ters. However , some of t hese numeri cal - ana-l yti cal methods requi re a hi gh- degree ofmathemati cal process, and unsol ved probl emsremai n i n t reatment of the actual boundarycondi t i ons.

Thi s paper descri bes f ormul ati on ofBergeron' s method anal ysi s of t he sound f i el di n a room Bergeron' s method i s a three-di -mensi onal numeri cal anal ysi s method whi ch

has general appl i cat i ons such as an el ect ro-magnet i c f i el d and an el ast i c wave anal ysi s[ l o , 111. The feat ure of thi s method i s thata sound f i el d i s represent ed by a three- di -mensi onal equi val ent - ci rcui t net work consi st -i ng of one- di mensi onal l y di st r i but ed- const antt ransmssi on l i nes and t hei r nodes so that aphenomenon i s anal yzed successi vel y on theci rcui t net work. Anal yti cal methods usi ng anequi val ent ci rcui t have been examned [12] ,but that onl y i s to i nt roduce di f f erent i al equa-t i ons f or a wave mot i on f i el d, and, f i nal l y,a deference method i s used f or the anal ysi s.By cont rast , the proposed method deduces di -rectl y node-equati ons whi ch represent thepropagat i on character i st i cs of the equi val entnet work by usi ng Bergeron' s method w th ci r -cui t const ant s whi ch correspond to the soundf i el d. Then the t i me and space charact eri s-t i cs of the whol e systemare obtai ned bysol vi ng the node equat i ons. Cal cul at i ons i nt hi s method consi st of si mpl e addi t i on andsubt ract i on whi ch are sui t abl e f or recentsupercomput ers. A hi gh- speed computati on i svery ef f ecti ve f or a t hree- di mensi onal t reat -

ment of a sound f i el d i n a compl ex shape.Thi s method can easi l y be f ormul ated si nce i tuses onl y two ki nds of parameters ( equi val entvol t age and current ) so that compl ex condi -t i ons of sound medi a (e. g. , wal l s, f l ower ,cei l i ng) and boundari es can easi l y be f ormu-l ated.

Reference [ 1 3 ] descri bes the appl i cat i onof Bergeron' s method to a three- di mensi onalsound f i el d i n rel at i on t o a uni f i ed anal ysi sof a sound f i el d and an el ect roacoust i c trans-ducer . Thi s paper descri bes f eat ures and j us-t i f i cat i on of the three- di mensi onal equi va-

l ent ci rcui t appl i ed to t he sound f i el d i n aroom

Sect i on 2 descri bes a three- di mensi onalequi val ent - ci rcui t net work whi ch represent s at hree- di mensi onal sound f i el d. Sect i on 3 de-scr i bes a practi cal appl i cat i on of the methodto the sound f i el d i n a room Secti on 3. 1examnes f undament al characteri st i cs of amodel of a cubi c room Secti on 3. 2 descr i best i me- response wavef orms whi ch can expl ai n theacoust i c nature of t he cubi c room and sec-t i on 3 . 3 gi ves a quant i t at i ve represent at i onof reverberat i on t i me (whi ch i s an i mportantdesi gn parameter) and i t s j ust i f i cat i on.

2. Represent ati on of Three- D mensi onalSound Fi el d Usi ng an Equi val ent

Ci rcui t

2.1 Treatment of wave equat i ons usi ngcubi c- l at t i ce net work

To represent a sound f i el d usi ng at hree- di mensi onal l at t i ce net work, l et usconsi der an equi val ent ci rcui t as shown in

74



Fi g. l(a). I n t hi s net wor k, t wo adj acentgri ds ar e connect ed by a one- di mensi onalt ransm ss i on l i ne hav i ng a char ac te r i s t i ci mpedance and pr opagat i on t i me whi ch cor r e-spond t o a pl ane- sound wave f i el d i n ai r ( nol oss and no di st or t i on) . A r esi st ance whi chcor r esponds t o t he medi um condi t i on i s di vi d-ed i nto t wo par t s, and each part i s connect edto the t wo si des of t he l i ne. A concent r at edconduct ance i s connect ed i n t he cent r al node;

L , R, C and G are i nduct ance, r esi st ance,capaci t ance and conduct ance per uni t l engt h,r espec t i vel y, Ad i s the l engt h of each l i ne.

The f undament al equat i on of a one- di men-s i onal l oss l ess l i ne i s r epresent ed by

Equi val ent c i r cui t

( l a)

Sound f i el d

wher e V and I

wave, r espect i vel y, and u i s an ar bi t r ar y co-or di nat e whi ch i s ei t her x, y or 2 .

ar e a vol t age- wave and cur r ent -U

Curr ent I

I nduct ance L

Capaci t ance C

Char a ct er i s t i c i mpedance ( one- di -mensi onal l i ne) 2

Resi st ance R

Conduct ance G

By r epr esenti ng t he vol t age- drop al ongR ( l oss i n the medi um r epr esented by t wo con-cent r ated r esi st ances at t he two si des of t hel i ne) wi t h a cur r ent at t he cent er node( whi ch i s an average of t he t wo val ues), Eqs.(l a) and ( l b ) can be t r ansf ormed i nto di f f er-ence equati ons f or t he x-di r ect i on:

Par t i c l e vel oc i t y u

Densi t y of medi um p

Reci pr ocal numberof bul k modul us 1 / 3 x

Char a ct er i s t i ci mpedance ( pl anewave) 3 P x

Acoust i c res i s -t ance r

Acoust i c conduc-

t ance $713

1-{ V(J:-Ad, 9, I , t ) - V ( X , , Z, ) }Ad

Si m l ar equat i ons hol d i n t he y and z di r ec-t i ons , respect i vel y . By repr esent i ng t he

cur r ent at t he cent r al poi nt wi t h t he val ueof cur r ent at t he cent r al node, and by r ear -r angi ng ( i ncl udi ng addi t i on and subt r act i on)t hem accor di ng t o t he condi t i ons of cur r entcont i nui t y , t he fo l l owi ng equat i on i s ob-t ai ned :

1-{ V(x -Ad , y, Z, )+ V(x+Ad , , Z , t )AdZ

+ V ( X , -A d , Z, )+ V (X , +Ad, 2, >

+ V ( S , , z-Ad, t ) + V ( X ,, z+Ad, t )

! Vol t age V Sound pr essur e

*0

T

&Ad

Fi g. 2. One- di mensi onal Berger on' s ex-pr essi on.

- 6 V(x ,Y, z, t ) }

=(Lz+R) -3GV(x ,y, z, t )

Ad )

( Y ) ( Ad 1( 2 ( Ad )

( Ad )I

+(Lc&+Rc-&{a 1 ( X - ~ , 2,

+ v x+-, y,z, t + v x, --,z, t

+ v x, +-, 2, +v x, , 2 - 7

+v I, , + T

75



Thi s equat i on deduces t o a wave- mot i on equa-t i on when Ad i s reduced to bei ng i nf i ni t el ysmal l . Then

+3RGV(x,y, z, t ) ( 4 )

Tabl e 1 shows t he cor r espondence of ci r cui tconstants t o parameter s i n t he sound f i el d;1/(3x) and g/3 i n the tabl e are der i ved byequal l y di vi di ng l/x and g of t he medi um i nt ot hr ee par t s on each l i ne of t he cubi c gr i d,and by f orm ng t he l i ne- const ant s t ogetherwi t h p and r . Ther ef or e, by r epl aci ng somet er ms of Eq. ( 4 ) wi t h par amet er s of t he soundf i el d and t he medi um const ant , Eq. ( 4 ) be-comes t he f ol l owi ng wave equat i on of soundpressure:

Si m l ar l y, the equat i on f or t he par t i c l e ve-l oci t y can be deduced. Thus, a wave- mot i onf i el d can be r epr esented by t aki ng i ncrement sof t he equi val ent ci r cui t ( shown i n Fi g. 1)

adequatel y smal l . The node- equat i on i n Ber-ger on' s met hod at each t i me i ncr ement can bededuced by usi ng t he equat i on of cur r ent con-t i nui t y at each node, when vol t age, cur r ent ,char act eri st i c i mpedance of a l i ne and pr opa-gati on t i me are def i ned (as i n Fi g. 2). Thechar act eri st i c i mpedance of each l i ne i s

z=m=*=Jzo

wher e Zo i s t he charact eri st i c i mpedance of a

pl ane wave. Si m l arl y, the wave- pr opagati ont i me At i s gi ven by

A t = A d m = A d m = t o / J ( 6b)

wher e At i s t he pr opagati on t i me of a pl anewave.

0

2. 2 Sound pr essur e di st r i but i on i nt hr ee- di mensi onal f r ee sound f i el d

When a sound sour ce i s s et on an arbi -t r ary si ngl e node, t hi s can be r egarded as ani deal nondi r ect i onal poi nt sound sour ce. Thei nt ens i t y of t he sound i n a f r ee f i el d i s re-ver sel y pr oport i onal t o the square of t he di s-t ance f r omt he sound sour ce, and uni di r ect i on-al . Theref ore a sound pr essur e i s r ever sel ypr oport i onal to t he di st ance. Sound pressureP at a di st ance separat ed f r omt he soundsour ce by a shor t di st ance Ad i s gi ven by

Pi=Po/ d ( 7 )

Fi g. 3 . Thr ee- di mensi onal f r ee sound-f i el d model .

-40.11 2 4 8 16 32

Distance ( X /Ad)

F i g. 4 . Compar i son of st eady- st at esound- pr essur e di st r i but i on i n t hr ee-di mensi onal f r ee- sound f i el d model be-

t ween comput ed ( ci r cl es) an? deri vedtheore t i cal l y.

wher e Po i s t he sound pressure at t he sour ce

on an arbi t r ary node. Sound pressur e P at a

di st ance nAd i n a di scret e gri d i s gi ven byn

P,=Po/(nAd)=Pdn ( 8 )

wher e n i s an arbi t r ar y i nt eger ; i . e. , asound pr essur e at an ar bi t r ar y poi nt on t hegr i d i s det erm ned by t he r ef erence sound

pr essur e at t he sour ce P and n. Theref or e,a sound pressur e Pr at an arbi t r ary node (i,j , k ) i s gi ven by

r=J( i- O ) Z + ( j j o y + ( k-a)z

wher e i, and k ar e t he number s of a di s-crete poi nt i n t he x, y and z di r ect i ons, and

j o and k ar e t hose of t he sound sour ce.io 0

76



Tabl e 2. Dat a of cubi c r oom ( whi ch i s on t he bot t om of t he model , i . e. ,

1 I

Fi g. 5. I nst ant aneous val ues of si nu-

soi dal waves pr opagat i ng i n t hr ee-di -mensi onal f r ee sound- f i el d model .

To conf i r m t hi s , t he di s t r i but i on of ast eady-st at e sound pr essur e i n a f r ee f i el dcaused by a poi nt sound sour ce shown i n Fi g.3 was i nvest i gat ed. Consi der i ng a symmet r yt o t he sound sour ce, t he act ual anal yt i calmodel was a cube havi ng a vol ume of one-ei ght h t he t ot al vol ume. Consi der that t he

sound sour ce S i s i n the cent er of t he cen-t r al pl ane ABCD, and t hat a cube has a vol -ume of one- ei ght h t he t otal vol ume wi t h t hebot t om pl ane of STCU. Then pl anes STCU, STOV

and USVW ar e symmet r y boundar i es, and t he re-mai ni ng pl anes have f r ee boundar i es. The co-ordi nat es used ar e al so s hown i n Fi g. 3,

wher e the or i gi n i s on t he sound sour ce.Mat ched i mpedances ar e used f or f r ee bounda-r i es of t he model ( ot her boundar y condi t i onsar e expl ai ned l at er i n sect i on 3 ) . Themat ched i mpedance f or E q . (6a) does not al -ways sati sf y t he mat chi ng condi t i ons wi t h awave wi t h an arb i t rary i nc i dent a t t he f i ni t eboundar y, si nce t he matchi ng i mpedance i s f ora pl ane wave wi t h a nor mal i nci dent angl e.Ther ef or e, a r egi on surr ounded by 35Ad i sused f or the anal ysi s. To r educe t he r ef l ec-t i on of waves f r omt he remai ni ng nodes, con-duct ances whi ch i ncr ease exponent i al - l yup tot he adm t t ance ( r eci pr ocal number of f i el di mpedance) of a f r ee boundar y ar e connect ed,so t hat t hese act as a s ound- wave absorber i nt he model [ 131.

F i gur e 4 shows t he di st r i but i on ofst eady- st at e sound pr essur es on t he x- axi s

STCU pl ane), t he hori zont al axi s i s a normal -i zed val ue of t he di st ance f r omt he sour ce( us i ng Ad), and t he ver t i cal ax i s i s t he nor -mal i zed sound pr essur e ( usi ng t he val ue at apoi nt separat ed f r om t he sound sour ce by A d ) .The ci r c l es i n F i g. 4 show cal cul at ed val ues,and t he sol i d l i ne shows theor et i cal val ues.The t heor et i cal val ues are a l ogar i t hm ct r ansf ormat i on of a r eci procal number of adi st ance whi ch gi ves a t heor et i cal sound pressur e l evel i n a f ree- sound f i el d. Thi s f i g-ur e shows t hat t he cal cul ated sound pressureval ues al most f ol l ow t he theoreti cal att enua-t i on. Fi gur e 5 shows di st r i but i on of soundpr essur e of an i nst ant aneous val ue when t hesound f i el d r eached a st eady st at e, showi ngt he propagat i on of t he wavef r ont t owar d i nf i -ni t e di s t ance wi t h a concent r i c -c i r c l e pat -t ern . These r esul t s j us t i f y appl i cat i ons oft he pr oposed met hod t o f undament al pr opaga-t i on charact er i st i cs of sound waves i n at hr ee- di mensi onal f i el d.

3. Thr ee- Di mensi onal Anal ysi s of SoundFi el d i n R o o m

3 .1 Model t o anal yze

To j ust i f y t he pr oposed met hod, t he st a-t i onar y- wave charact eri st i c of sound wave i na cubi c r oom due to mul t i pl e r ef l ect i ons andi t s r ever ber at i on- t i me charact er i s t i c weresi mul ated s i nce t hese ar e f undament al acous-t i c char ac te r i s t i c s of the room F i gur e 7shows Ber ger on’ s expressi on of t he cent r alpl ane (ABCD) i n t he cubi c r oom shown i n Fi g.6. The cubi c r oom has di mensi ons of 50Ad x

50Ad x 50Ad ( wher e Ad i s an i nt er val of t hespace di f f erence) and has no openi ng. Thesound sour ce i s a poi nt , and i s s i t uat ed i nt he cent er of a wal l (ROEH pl ane). The ori -gi n of coor di nat es coi nci des wi t h t he soundsour ce. The act ual di mensi ons of t he modelar e det er m ned by t he wavel engt h of sound anddi scret e- t i me A t . I f t he f undament al f r e-quency f i s 100 Hz, t he di scr et e- number n perwavel engt h i s 20, and t he sound speed c i s340 m s , t hen Ad i s det erm ned by E q . (6b).Dat a of t he r oom ar e obt ai ned as shown i n Ta-bl e 2. I t i s poss i hl e to l i m t t he ac tualanal ysi s regi on t o a cuboi d of one- f our t h t her oomhavi ng a bott om pl ane of STCD, by con-si der i ng a symmetr y condi t i on t o t he soundsource. Ther ef or e, t he anal ys i s regi on i st he whol e s pace sur r ounded by t hese symmet r i -cal boundari es and boundar i es of t he cei l i ng.

To r epr esent an act ual sound f i el d i n a

room by t he model , i t s wa l l s , cei l i ng andf l oor must have spec i f i c acoust i c condi t i ons( sound- r efl ect i on and absorpti on) . To naket he model s i mpl e, l et us assume t hat a l l t hewal l s, cei l i ng and f l oor have a uni f orm bound-ar y condi t i ons, so t hat a t er m nat i on l oad-

-

7 7



- 0Ad-Fi g. 6. Model of a cubi c room f or si m

ul at i ng acoust i c behavi or .

3 . 2 Di s t r i but i on charact er i s t i cs ofsound pressur e i n a r oom

Unl i ke a f r ee sound f i el d such as i nsec t i on 2. 2, t he di st r i but i on of a sound-pressur e i n t he model shown i n Fi g. 6 hascl ear peaks and di ps due t o st at i onary wavesi n a cl osed cubi c r oom dependi ng on t he wave-l ength of an i nput wave. To c onf i rm t hespace charact eri st i c of t he sound pr essur edi s t r i but i on, an anal ys i s of t he model ( de-scr i bed i n the pr evi ous secti on) wi t h a poi ntsound- sour ce of 100- Hz si nusoi dal wave ( wave-l engt h 11. 5 Ad) was car r i ed out . A l l t heboundar i es of t he r oom wer e assumed t o havean aver age sound- absorpt i on coef f i ci ent of0. 2 uni f orm y. Pl ane STCD was used f or t heobser vat i on. Fi gur e 8(a) shows t he di st r i bu-t i on of envel opes of i nst ant aneous maxi mumsound- pr essur e at a st eady st ate. Fi gur e 8

(b) shows t he same r esul t s but usi ng cont ourl i nes of t he sound pr essur e. These f i gur esexpl ai n t he posi t i ons of peaks and di ps.

Tine= 350 ms

Fi g. 7. Ber ger on’ s expr essi on f or cubi cr oom shown i n Fi g. 6.

r es i st ance Rb i s used i n t he equi val ent c i r -

cui t whi ch gi ves a ref l ect i on condi t i on.Thi s val ue i s gi ven by

Rb= Z ( 1 1 )

wher e rn i s t he rat i o of a st at i onar y wave

whi ch gi ves t he sound pressur e r ef l ect i on co-ef f i c i ent , and Z i s t he char act er i st i c i mpe-dance of t he sound f i el d. A s a part i cul arcase, when rn = 1, t hi s equat i on gi ves amatchi ng condi t i on whi ch r epr esents a f r eeboundar y. Fi gur e 7 show a t hree- di mensi onalequi val ent c i r cui t of t he pol e ABCD ( shown

i n Fi g. 6 ) , usi ng t he f oredescri bed condi -t i ons and a vol t age source Ei ( equi val ent t o thepoi nt soundsour ce) wi t h i t s i mpedance R and

l i ne i mpedance Z.i

0

9

R

-0

a ?\ ”xv

0

40 . 0

(x/Ad)

Average absorbi ng coeff i cent : Z = O . 2

Fr equency : = l O O Hz

Dstri buti on of maxi mumsound pressure

Contour of maximum sound pressure( a( b )

Fi g. 8. Sound- pr essur e di st r i but i on atst eady- st at e f or s i nusoi dal waves i n

r ect angul ar r oom

78



SPL DI S T RI B UT I ON Ti me=350ns9

1

lic = 325-__-heoretical- m p u Led

Average absorbing coefficient : 6 = 0 . 2

Frequency : = l O O HzRoom cons tant : Rc=325(ni')

Fi g. 9 . Compar i son of soundpr essur e di s-t r i but i on at st eady st ate on x- axi s bet weenconput ed and t heor et i cal r esul t s. Theor et i -cal val ues ar e cal cul ated under assumpti on

of di f f used sound f i el d.

a4

Fi g. 10. Bound- measuri ng poi nt s f or r ever-ber at i on t i me i n r ect angul ar- r oommodel .

(d ) P(26.14.1) ATT:-23.4 ( D B )?-

-I0

f :-ez -

?

o!oo ' m.00' 4 0 . 0 0 ' clO.00 w.m' 1b.m n0.00 i4o.w 1~0.00 1ao.00 m.00

, , I I I I

T l H E r I I S ) (Y.10' 1 -Aver age absorbing coeffi cient : [r ~ 0 . 2

Center f requency : c = 100Hz 1/3 oct.( a) : I ncident waves( b )- d ) : Response waves

Fi g. 11. Ti me r esponses of sound press ures f or t one- bur st waves.

79



Time= 40 DS

Tine= 60 DS

Time= 80 DS

(x/Ad)

Tine= 250 BS

(x/Ad)

Tine= 500 DS

P l I0.00 Ib.00 i0.w ' n .w a 4i .W rbP60

(x/Ad)

Average absorbing coefficient a=0.2

Center frequency : fc=lOOHz

F i g . 1 2 . D i s t r i b u t i o n o f i n s t a n t a n e o u s soun d p r e s s u r e on STCD

p l a n e d u e t o t o n e - b u r s t waves.

C on sp ic uo us p e ak s on t h e c e n t r a l l i n e (x-a x i s ) , w a l l (DCGH p l a n e ) a nd c o r n e r s of t h e

room a re o b s e r v e d . F i g u r e 9 shows t h e var i a -

t i on o f in s tan tan eou s maximum sound p re ss u r e

on t h e x - a x i s ( i . e . , d i s t a n c e d ec a y) u s i n g a

r e l a t i v e so un d p r e s s u r e l e v e l . The s o l i d

l i n e i n t h e f i g u r e s hows t h e c a l c u l a t e d va l -

u e s , a nd t h e d o t t e d l i n e s ho ws t h e d i s t a n c e -

decay cu rv e when a d i f f u s e d s o u n d - f i e l d i s

assumed with a room constant of R = 325 .m2.

T h i s f i g u r e show s c l e a r l y p o s i t i o n s o f p e ak s

a n d d i p s ; n o t e a deep d ip near t h e center

( s l i g h t l y c l o s e d t o t h e s ou nd s o u r c e ) .

a t t e n u a t i o n a t t h e d e e p es t d i p i s about 40 dB,

w h i l e t h a t a t a peak i s o n l y 20 dB, indepen-d e n t l y of t h e d i s t a n c e f r om t h e s ou nd s o u r c e .

However, it i s c e r t a i n t h a t t h e g e ne r al p a t -

t e r n of t h e a t t e n u a t i o n i s v e ry d i f f e r e n t

f r o m t h a t i n a f r e e s o u n d - f i el d , a nd t h a t t h e

p a t t e r n i s w i t h i n a r e a s o n a b l e r a n g e w i t h t h e

in f lue nce o f t he room cons t an t wh ich i s d e t e r -

mined by t he mean sound-a bso rp t ion co e f f i -

c i e n t . I t i s c o nf i r me d t h a t t h e p ro p o se d

method can w e l l s i m u l a t e a fundamenta l sound-

f i e l d c h a r ac t e r i s t i c of a cub ic room; e .g . ,

t h e d i s t r i b u t i o n o f s ou nd p r e s s u r e i n a c u b i c

c

The

room surrounded by boundaries ( w a l l s , f l o o ra n d c e i l i n g ) w i t h a un i fo rm sound-abso rp t ion

c o e f f i c i e n t , and t h a t t h e d i s t r i b u t i o n of

s o u n d p r e s s u r e i s n o t u n i f or m .

3.3 Si m u l a t i o n s o f r e v e r b e r a t i o n

F i n a l l y , s i m u l a ti o n s o f r e v e r b e r a t i o n ,

which i s a n i m po r t an t p h y s i ca l f a c t o r t o e v a l-

u a t e t he s o u n d f i e l d i n a room, a r e examined.

The sound f i e l d a s shown i n F i g . 6 with tone -

b u r s t wa ves ( i n p u t ) w a s u s e d f o r t h e m od el .

The tone-burst waves a r e gene r a ted by app ly -

in g a Hamming window ove r e ver y s i x waves of

s i n u s o i d a l w a ve s s o t h a t t h i s fo rm s a 113-

o c t a v e b a nd w i t h a c e n t r a l f r e q ue n c y o f 100

H z 1141. It was a s sumed t h a t a l l t h e bounda-

r i e s o f t h e room ( w a l l s , f l o o r an d c e i l i n g )

h a ve a un i f o r m s o u nd - ab s or b i ng c o e f f i c i e n t a t

e a c h t e s t c a s e . F i v e cases having sound-ab-

s o r b i n g c o e f f i c i e n t s of 0 . 10 , 0 .1 5, 0.20 an d

0.30 w e r e t e s t e d . To r e p r e s e n t e ac h v a l u e ,

l o a d r e s i s t a n c e a t t h e m od el b o u n d a r i e s w a s

chosen .

80



Tine= 2ooo 6

la

t 2-

M -2%

=r)

-0 d-

0 -v)

w *t 2-9

0 , -o

0

9 -

0

9nh

-0+XI?v

0

0

0 . 0 0 10.00 2 0 . 0 0 3 0 . 0 0 4 0 .0 0 50.00

( x / A d )

Average absorbi ng coeff i cient : i i =O . 2

Cent er f r equency : .L = 100 I l z

Fi g. 13. Di st r i but i on of soundpr es-sur e on STCD pl ane due to t one- burst -

waves. LOG OF P O U E R : P ( 2 6 . 1 4 . 1 )

F i gur e 10 shows t he observat i on posi -t i ons (9 poi nt s) on t he pl ane ABCD of t hemodel . Fi gur e 11 shows t i me- r esponses ofsound pr essure at sever al poi nt s wi t h anaverage-absorbi ng coeff i ci ent of 0. 2. The

scal e of wavef orms i n Fi g. 11 i s nor mal i zedusi ng t he i nst ant aneous maxi mum sound pr es-sure. The di f f er ence of r el at i ve soundpr es-sur e bet ween t he maxi mum sound pr essur e ateach poi nt and t he maxi mum i nst ant aneoussound pr essur e i s s hown besi de each wavef or mt aki ng 0 dB at t he sound sour ce. Fi gur e 11

(a) shows t he i nci dent waves, and Fi gs. l l (b),(c) and (d) show r esponse waves. Each wave-f orm shows cl ear l y t hat t hi s i s t he r esul t oft he di r ect wave f r omt he sound sour ce andmany r ef l ected waves f r om t he wal l s, f l oorand cei l i ng. Not abl y, Fi gs. l l (a) and (c),whi ch ar e t he r esponse wavef or ms at a peak

and di p of a s t andi ng wave on t he cent r all i ne of t he room r espec t i vel y , show c l ear l yt he di f f erence i n concent r at i on of energy.Fi gur es l l (c) and (d) show f l att er echos.Fi gur e 12 shows di st r i but i ons of i nst ant ane-ous s ound pr essur e on t he cent r al pl ane(STCD) of t he r oom due t o t he t one- bur stwaves. These i l l ust r ate wel l t he decay ofwaves wi t h pr opagat i on and r efl ect i ons. I tt akes about 500 ms to compl et e decay of t her efl ect ed waves. Fi gur e 13 shows cont ourl i nes of t he sound press ure ( maxi mumval ues)on t he STCD pl ane af t er 2 s. Thi s shows amode of st andi ng waves whi ch i s a c har act er -i s t i c of a rectangul ar r oom al t hough t her ear e some di f f er ences i n patt erns compar edwi t h t he same ki nd of i l l us t r at i on [ Fi g. 8

(a)] f or t he st ati onar y si nusoi dal waves.

LOG OF R T E N R G Y : P ( 2 6 . 1 4 . 1 )

:

- 8% &

- 0

--i

0.00 40 .00 80 .00 120.00 160.00 ,

0

0

! I I I I I I I I

0.00 40 .00 80 .00 120.00 160.00 ,

10.00

lO.00

10.00

T I M E ( ? I S ) ( X 1 0 ' 1

( a )

( b )

( c )

Squared sound pressure.Logari t hmtransformati on (reset tim : 5ms)

Reverberati on decay curves (resol uti on: 2. 5ms)

Fi g. 14. Exampl e of r ever ber at i on de-cay cur ve obt ai ned by i mpul se i nt egr a-

t i on met hod.The r everberat i on t i me i s usual l y meas-

ur ed by r adi at i ng a whi t e noi se havi ng a band-wi dth of 1- oct ave or 113- oct ave i n a soundf i el d, and by st oppi ng i t when t he sound-f i el d reached a st eady st ate. An average has some var i at i on due t o t he f l uctuat i on ofval ue of r esul t s obt ai ned by mul t i pl e measur e- a sound sour ce. Schr oeder [ 15] pr oposed ament s ( order of t ens) ar e empl oyed usual l y met hod ( known as " t he i nt egr at ed i mpul sef or a r everber at i on t i me, si nce each r esul t method" [16]) i n whi ch an ensembl e- aver age of

81



0

DD

9

I I I I I I I ~

' O 1 . 0 O ' 40.00 80.00 120.00 160.00

T I M E (MS) ( X I O ' 1

0I

0.00 40.00 80.00 120.00 160.00T I I I I I I 1 I I

T I M E ( f l S ) ( X I O 1 1

10.00

10.00

Fig. 15. Examples of cu rv atu re reve r-

bera t ion-decay curves observed .

r e v e r b e r a t i o n c u r v e i s obtainel from impulse

res pon ses . The ensemble-average of squ are of

sound pressure i s given by

<S2()>=N/%( t )dt (12)

where S ( t ) i s t h e so u n d p re s su re a t a measur-

i n g p o i n t , h ( t ) i s t h e i m p u l se r e sp o n se b e -

tween the sound sourc e and th e measuring

p o i n t i n c l u d i n g a microphone, loudspeaker and

f i l t e r s , and N i s t h e power of whi t e no i s e

per un i t bandwidth. By modifying Eq. ( 1 2 ) ,

and by normal izing i t w i th t h e t o t a l e ne rg y,

th e ensemble-average of s quar e of sound pres -

s u r e i s given by

T h e re f o r e, t h e r e v e r b e r a t i o n t i m e can be rep-

r e s e n t e d q u a n t i t a t i v e l y by a p p ly i n g a l oga-

r i t h m i c c om p re ss io n t o t h e r e v e r b e r a t i o n -

decay curve R (t)which i s give n by Eq. (1 3) ,

and by approximat ing i t by a s t r a i g h t l i n e

w i t h t h e l e a s t - sq u a re s m e th od . S i n c e t h e r e -

verbera t ion-d ecay curve ob ta ined f rom a tone-

burst response becomes an ensemble-average of

reverbera t ion -decay curve ob ta ined f rom th e

w h it e n o i s e i n t h e same bandwidth assuming

t h a t t h e sound f i e l d i s l i n e a r . T h e r e f o r e ,

t h e r e v e r b e r a t i o n t i m e can be de t ermined a s a

band-l im i ted one which i s obtained by calcu-l a t i n g t h e p re v i o u s l y o b t a i n e d t i m e- resp o nse

waveform. A r e v e r b e r a t i o n t i m e u s u a l l y i s

defin ed by approx imat ing a decay curve of -5

t o -35 dB wi th a s t r a i g h t l i n e . However, i n

t h i s e x pe r im e n t, a decay curve of -10 t o -60

dB was used f or t h i s purpose , s i nc e an ade-

qu ate ly wide dynamic range i s obta ined . F ig-

u re s 1 4 (a ) t o ( c ) show t h e r e su l t s of c a l c u -

l a t i o n of t h e response waveform [shown i n F ig .

l l ( d ) ] a t p o s i t io n s P (26 , 14, 1 i n F ig . 1 0 ) .

F igure 14(a) shows the squared sound pressure ,

Fig. 14(b) shows i t s l o g a r i t h m t r a n s fo rm a t i o n ,

and Fig. 14(c ) shows th e reverb erat ion-d ecay

curve . A r i s e t i m e ofl O A t

(5.0 ms) was usedfo r t h e l o g a r i th m t r a n s fo rm a t i o n , a nd a n i n -

t e g r a t i o n i s 5At (2.5 m). The response wave-

form shown i n F i g . l l ( d ) i n d i c a t e s a n e c h o,

and t h e d i s t r i b u t i o n of i t s energy can be con-

f i rm ed i n F i g . 1 4 (a ) . T he l o g a r i t h m t r a n s fo r -

m a ti o n shown i n F i g . 1 4 (b ) i n d i c a t e s t h e

energ y de cays down to -70 dB, con firm ing an

adequate dynamic range . The rev erh era t ion -

decay curv e shown i n Fig. 14(c ) i s a lmost

s t r a i g h t e x ce p t f o r a n i n f l e c t i o n p o i n t n e a r

2000 ms. This i s d ue t o a d i r e c t e f f e c t of

t h e r e s o l u t i o n o f t h e i n t e g r a t i o n wh ic h i s

t r un c at e d w i th t h e f i n i t e t i m e . I n t h i s e x-

ample, an adequa te decay curv e can be ob-

t a i n e d w it h t h i s o r d e r of i n t e g r a t i o n r e s o l u -

t i o n . Figur e 15 shows examples of reve rber a-

t ion-decay curv es which have obvious bends:

a bend around 40 ms i n F ig . 15 (a ) , and a bend

i n the whole shape in F ig . 15( b) . The wave

t h e o ry [ 1 7 , 181 su g g e s t s : a ssum in g t h e a x i a l

waves (one-dimensional wave), tangent ial

waves (two-dimensional wave) , and ob li qu e

( three-d imens iona l wave) i n a re c t a ng ula r

room, i t s reverberat ion-decay waveform i s a f -f e c t e d b y t h e se w av es i n d i f f e r e n t ways s o

t h a t t h e c u r ve i s n o t s t r a i g h t . T he e xa mp le s

shown i n F i g . 1 5 c o n f irm t h i s t h e o ry .

d

Table 3 shows comparison of th e rev er-

b e r a t i o n t i m e s o b t a i n e d by t h e s i m u l a t i o n an d

t h o se o b t a i n e d b y E y r i n g ' s fo rm u la [1 9 ] . The

f i g u r e s i n t h i s t a b l e a r e o b ta in e d f rom an

a r i t h m e t i c a v e r a g in g of a l l t h e r e s u l t s ob-

t a i n e d a t ni ne po in t s shown i n F ig . 10 . Eyr-

i n g ' s fo rm u l a c a n n o t b e c o m p a re d d i r e c t l y

w i t h t h i s s i m u l a t i o n i n w hic h t h e l o a d resis-

t ance represen t ing an average sound-absorb ing

c o e f f i c i e n t i s u s e d f o r a l l t h e model bounda-

r i e s . The a i m of th e comparison i s t o exam-

i n e t h e r e l a t i o n s h i p b et we en t h e r e v e r b e r a t i o n

82



t i me deduced by Eyr i ng' s f or mul a and t hat ofsi mul at i on usi ng sound- absor bi ng i .n t he modeli n t he pr oposed met hod. The compar i son showst hat t he r everberat i on t i mes obt ai ned by t heproposed met hod ar e r easonabl e, and t hat t hemet hod wi l l be usef ul f or quant i t at i ve est i -mati on of t hei r val ues.

Abs. coef. Eyri ng' s

a

4 . Concl usi ons

Comput ed

(sec)

( 1) The cor r espondence of a t hree- di men-si onal sound f i el d to a t hr ee- di mensi onalequi val ent - c i r cui t consi st i ng of one- di men-si onal t r ansm ssi on l i nes and nodes i s de-scr i bed so t hat t he base of t he t hr ee- di men-si onal sound- f i el d s i mul at i on usi ng Ber ge-son' s met hod i s expl ai ned. The sound- decayl aw of r ever se- pr opor t i on of t he di st ancef r om a sound sour ce i n a f r ee sound f i el d i sshown by usi ng t he pr oposed met hod. Thi sconf i r ms t hat t he pr oposed met hod can si mu-l ate t he f undament al pr opert y of a f r ee sound-f i el d.

( 2 ) Based on t hi s r esul t , a f undament almodel of a thr ee- di mensi onal sound f i el d i n acubi c r oomwas const r ucted. The wal l s, f l oorand cei l i ng of t he r oomwer e repr esent ed bypur e r esi st ance f or t he f ormul at i on. T i mer esponses t o a si nusoi dal wave i nput and at one- bur st i nput were exam ned. The r esul t sshow successf ul l y t he char acter i s t i cs ofsound f i el d i n a r ectangul ar r oom and j ust i -f i es t he pr oposed method i ncl udi ng i t s boun-dar y condi t i ons.

( 3 ) To est i mate a r everberat i on t i mequant i t at i vel y, the response wavef or m to a

t one- bur st wave was cal cul ated usi ng t he i nt e-gr at ed i mpul se method. The resul t s wer e compared wi t h s i m l ar r esul t s obt ai ned by usi ngt he convent i onal met hod. The compar i sonshows t hat t he pr oposed met hod can be appl i edto a quant i t at i ve est i mat i on o f acoust i cpr oper t i es such as a rever ber at i on t i me, i . e. ,

1.

2 .

3 .

4 .

Tabl e 3. Compar i son of r everberat i on t i meobt ai ned by s i mul at i on and Eyri ng' s f ormul a

0.10 1 3.74 1 3.18

0.25 1.37

0.30 1 1.10 1 0.99

t he usef ul ness of t he proposed method f or t hesound f i el d i n a room i s conf i r med.

The pr oposed method i s ver y usef ul t oexam ne the pr opert i es of t he sound f i el d i n

a room whi ch was di f f i cul t t o anal yze by ana-l yt i cal met hod based on wave t heor y. The suc-cessi ve cal cul at i on used i n thi s met hod wi l li ncrease i t s ef f i c i ency wi t h t he recent de-vel opment of super comput er s.

Fur t her devel opment of t he met hod i s un-der consi der at i on i ncl udi ng t he compari son oft hi s met hod wi t h experi ment al r esul t s by t ak-i ng account of t he f r equency-s cat t er i ng char-act eri st i cs of boundar y condi t i ons and medi umcondi t i ons. Hence, t he method can be appl i edmor e preci sel y t o a wi der r ange of r oom acous-t i c pr obl ems.

Acknowl edgement . The aut hor s wi sh t oexpr ess t hei r t hanks t o Assi st ant Prof .Yut aka Fuj i shi ma ( El ectr i cal Depar t ment ,Tomakomai Techni cal Col l ege) f or di scussi onson t hi s work.

REFERENCES

Ter uj i Yamamot o. Exam nat i on of sound 5 .

character i st i cs of r ooms usi ng acoust i cmodel s. NHK Techni cal Resear ch, 25, 6,

A. Kr okst ad, S. Str omand S. Sor sdal .Cal cul at i ng t he acoust i cal r oom r esponseby t he use of a ray t r aci ng t echni que.J . Sound Vi b. , 8, 1, pp. 1 1 8 - 1 2 5 ( 1 9 6 8 ) . 6.

Yuko Ogawa and Tokashi Ni shi . Acoust i c-al desi gn of a r oom usi ng CAD. Resear chDat a, Ar chi t ectural Acoust i cs Comm t t eeof Acoust . SOC. , J apan, AA 8 2 - 3 3 ( 1 9 8 2 ) .

Possi bi l i t y of use of comput er f or acous-t i cal desi gn of a room and pr obl ems, AA

pp. 2 9 9 - 3 2 0 ( 1 9 7 3 ) .

Tsugumasa Takam ya and Fukushi Kawakam . 7 .

8 2 - 3 5 ( 1 9 8 2 ) .

Kazuhi r o I i da, Ydi chi Ando and J yun' i chiMaekawa. Char act er i st i cs of sound pr opa-gat i on over seat s , and i t s appl i cat i ont o s i mul at i on of t he sound f i el d i n ar oom Resear ch Dat a, Ar chi t ect uralAcoust i cs Comm t t ee of Acoust . SOC. ,J apan, AA68-04 ( 1 9 8 6 ) .

K. Seki guchi , S . Ki mur a and T. Sugi yama.Appr oxi mat i on of i mpul se r esponset hrough comput er s i mul at i on based on f i -ni t e sound r ay i nt egr at i on. J . Acoust .

T. Ter ai . On cal cul at i on of soundf i e l ds ar ound t hr ee- di mensi onal obj ect sby i ntegral equat i on met hod. J . SoundVi b. , 69, 1, pp. 7 1 - 1 0 0 ( 1 9 8 0 ) .

SOC. , J(E), 6, 2, pp. 1 0 3 - 1 1 5 ( 1 9 8 5 ) .

8 3



8.

9.

10.

11.

1 2 .

13 .

A. Craggs. The use of si mpl e three-di mensi onal acoust i c f i ni t e el ement s f ordeterm ni ng the natural modes and f re-quenci es of compl ex encl osures. J .

Sound Vi b. , 23, 3, pp. 331-339 (1972).

Shi n Kagam and I chi ro Fukai . Formul a-t i on of t hree- di mensi onal sound f i el dusi ng boundary el ement method. Trans.I . E. C. E. , J apan, J70-A, 1 , pp. 110-115

(1987).

Nori nobu Yoshi da, I chi ro Fukai andJ uni ch Fukuoka. Transi ent anal ysi s of

two-di mensi onal Maxwel l equati on usi ngBergeron' s method. Trans. I . E. C. E. ,J apan, J62-B, 6, pp. 511-518 (1979).

Masahi ro Sato, Nori nobu Yoshi da andI chi ro Fukai . Anal ysi s of t i me- responseof two-di mensi onal el ast i c body usi ngBergeron' s method. Trans. I . E. C. E. ,J apan, J70-A, 11 , pp. 1515-1523 (1987).

I sao Nakamura. D scussi on on el asti cwaves usi ng equi val ent ci r cui t . J our .of the Acoust i cal Soci ety of J apan, 36,

Aki ra Takagi , Nori nobu Yoshi da and

I chi ro Fukai . Uni f i ed anal ysi s of t i me-

4 , pp. 185-193 (1980).

14 .

15 .

16 .

1 7 .

18.

19 .

response of a sound f i el d i ncl udi ngsource system Trans. I . E. C. E. , J apan,

Tadashi Taki se and Ken Matsudai ra.Measur i ng apparatus i n roomacoust i csusi ng tone- burst f roml oudspeaker . Re-search Data, Archi t ectural Acoust i csCommt t ee of Acoust . SOC. , J apan, AA 78-

21 (1978) .

M R. Schroeder . New method of measur-i ng reverberati on t i me. 3. Acoust . SOC.

F. Fawakam, K. Ni i m and K. Yamaguchi .A t ool f or roomacoust i c measurement .J . Audi o Eng. SOC. , 2, 10 , pp. 739-748

(Oct. 1981).

P. M Morse and R. Bol t . Sound waves i nrooms.2 , p. 111 (1944).

Yoshi mut su Hi rata. Anal ysi s of soundf i el d i n a roomtaki ng account of low-degree di f f usi on. Techni cal Repor t ,I . E. C. E. , J apan, EA70-12 (1970).

J yun' i chi Maekawa. RoomAcoust i cs, p.34, Kyor i t su Publ . , Tokyo (1978).

J68-A, 7 , pp. 688-695 (1985).

Am., 37, p. 409-412 (1965).

Revi ews of Modern Physi cs, 16,

AUTHORS (f roml ef t t o r i ght )

Hi demaro Shi moda graduated i n 1972 f romEl ect r i cal Depart ment , Facul ty of Ecgi neer i ng,Hokkai do Uni versi t y. He was empl oyed by Shi mzu Corporat i on i n 1973, and engaged i n oceano-graphi c measurement s at Oceanographi c Depart ment , Techni cal Research Laborator i es, Shi mzu Cor-porat i on. Si nce 1984, he has been engaged i n research at Acoust i c Group of the same l aborato-r i es on acoust i c measurement s, and numeri cal anal ysi s of sound f i el ds.

Nori nobu Yoshi da graduated i n 1965 f romEl ectroni c Depart ment , Facul t y of Engi neer i ng,Hokkai do Uni versi t y, and obtai ned a Master' s degree f romt here i n 1968. He t hen j oi ned NEC thesame year , wher e he worked on CAD i n t he I ntegrated- Ci rcui t Depart ment . He has a Dr . of Eng.degree. He was appoi nted Assi st ant (El ect r i cal Depart ment ) on the Facul t y of Engi neer i ng, Hok-

kai do Uni versi t y i n 1969; Lecturer, i n 1983; and Associ at e Prof essor , i n 1984. He i s engagedi n research on numer i cal anal ysi s of el ect romagnet i c f i el ds.

I chi ro Fukai graduated i n 1953 f romEl ectr i cal Depar t ment , Facul ty of Engi neer i ng, Hok-kai do Uni versi t y, obt ai ned a Master ' s degree f romt here i n 1956, and has a Dr . of Eng. degree.Af ter j oi ni ng Techni cal Research I nst i t ut e, J apanese Defense Agency, and Techni cal - TeachersTrai ni ng Col l ege of Hokkai do Uni versi t y, he was appoi nted Prof essor of the Facul t y of Engi neer-i ng at Hokkai do Uni versi t y i n 1977.

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analysis of sound fields in rooms using

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