numerical and experimental studies on the effects of stage risers
TRANSCRIPT
Applied Acoustics 70 (2009) 588–594
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Applied Acoustics
journal homepage: www.elsevier .com/locate /apacoust
Numerical and experimental studies on the effects of stage risers
Yosuke Yasuda a,*, Ayumi Ushiyama b, Shinichi Sakamoto c, Tetsuya Sakuma a
a Graduate School of Frontier Sciences, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa-shi, Chiba 277-8563, Japanb Central Research Laboratory, Daiwa House Industry Co., Ltd., 6-6-2 Sakyo, Nara 631-0801, Japanc Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan
a r t i c l e i n f o a b s t r a c t
Article history:Received 14 June 2008Received in revised form 30 June 2008Accepted 7 July 2008Available online 12 August 2008
PACS:43.55.KA43.55.Mc
Keywords:Stage riserBoundary element methodMode expansionVibrationDiffraction
0003-682X/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.apacoust.2008.07.001
* Corresponding author. Tel./fax: +81 4 7136 4820/E-mail address: [email protected] (Y. Yasud
The acoustic effects of stage risers, especially on the sound of lower string instruments, are numericallyand experimentally analyzed. To discuss the effects of the vibration of riser’s boards due to the mechan-ical force from an instrument, a structural–acoustical coupling approach is applied based on the modeexpansion and the boundary element technique. Measurement results of the mechanical force from realinstruments are used in the numerical study. The vibration of the top board of a riser affects the soundfield only around the natural frequencies of the board and the cavity of the riser. In contrast, the acousticdiffraction due to the riser affects the sound field in a wide frequency range. The riser’s sideboard thatfaces to receiving points increases the sound pressure levels because it reflects waves diffracted at theriser’s edge to the front. To verify the numerical results, the effects of acoustic diffraction due to risersare especially investigated in detail through a scale model experiment.
� 2008 Elsevier Ltd. All rights reserved.
1. Introduction
In many concert halls, stage risers are often used for the wind sec-tion, for the back of the string section in the orchestra, and for soloistsin concertos for lower string instruments such as cellos and doublebasses. Stage risers have visual effects to make it easy for the conduc-tor and the orchestra players to see each other, and to give the audi-ence a better view of the orchestra. Regarding the acoustic effects, itis said that risers can lessen the seat-dip effect (low frequency atten-uation in concert halls), and that risers enhance the sound of lowerstring instruments. The latter phenomenon is often called ‘resonantsupport’, on which we focus in the present paper.
Several researchers have tried to physically confirm this effectof risers through experiments and measurements. Fuchs and Cre-mer [1,2] measured with a cello the change of acoustic radiationcaused by a riser in an anechoic chamber, and mentioned thatthe change was small at most frequencies, except around 64 Hzand 150 Hz where 6 dB and 2 dB decreases of acoustic radiationwere observed, respectively. Beranek et al. [3] measured soundpressure level at the audience area of a hall, where a cello and adouble bass were played on a rigid stage and a riser on it. It waspointed out that the riser increased the sound pressure level on
ll rights reserved.
4821.a).
the whole, while the effects varied widely even between adjacentsemitones. Askenfelt [4] carried out more detailed experiments si-milar to those by Beranek et al., and confirmed the effects of risersto increase sound pressure level of the lower string instruments.Although such measurements and experiments have been carriedout, the mechanism of the effects has not been sufficiently clari-fied. It is difficult to clarify the mechanism only through experi-ments or measurements, because the effects of risers arecomplicated phenomena caused by various factors such as acousticreflection and diffraction by risers, and acoustic radiation due tothe vibration of the risers’ boards. Moreover, the vibration of theboards is affected by mechanical forces from endpins of the instru-ments and acoustic loading.
As for the effects on the sound of lower string instruments,some studies on the stage floor have been done. Sakagami et al.[5–9] theoretically analyzed sound fields including infinite elasticplates of various configurations. They studied on the effects ofmechanical point force from lower string instruments [6,8,9], ofair-back cavity [5,7–9], and of periodic ribs without mechanicalforces [7]. Nakanishi et al. [9] gave consideration of the effects ofrisers based on the theoretical analysis of infinite plates [8]. Thesestudies, however, are fundamentally for the stage floors, thus it isnot possible to directly apply the findings to clarify the effects ofrisers, which have finite vibrating surfaces and edges diffractingthe sound waves. Recently, Nakanishi et al. [10,11] have
Y. Yasuda et al. / Applied Acoustics 70 (2009) 588–594 589
theoretically analyzed the effects of the riser’s constructions on theacoustic gain. In these studies, however, the riser was placed in arigid baffle as all of the surfaces were flat, to investigate only theeffect of vibration from the riser’s top board. Thus, the effects ofthe riser’s shape were not considered.
In the present paper, we numerically and experimentally inves-tigate the acoustic effects of stage risers on the sound of the lowerstring instruments. In Section 2, we numerically analyze the effectsof risers of various configurations. Firstly, we describe a numericalmethod for structural–acoustical coupling problems, based on themode expansion and the boundary element technique. Secondly,we show measurement results with cellos and a double bass todetermine the relationship between the acoustic radiation of theinstrument and its mechanical force to the riser. These resultsare used for numerical study. Thirdly, we discuss the effects of con-figurations and board vibration of risers through numerical study.In Section 3, we especially investigate the effects of the diffractionby the riser in detail through a scale model experiment, to verifythe numerical results.
2. Numerical study
2.1. Numerical method
To solve structural–acoustical coupling problems, the boundaryelement technique and the mode expansion method are applied tothe sound field and the vibration field of elastic boards, respec-tively. The time dependence of exp(�jxt) is used throughout. Asillustrated in Fig. 1, a riser of 1.2 � 0.9 � H m3 size made of thinboards is placed on an infinite rigid floor with a point source at s(0, 0, 0.7 + H) located right above the center of the riser, and a pointforce at f (0, 0, H), the center of the riser’s top board. The pointsource and the point force are regarded as a simplified lower stringinstrument. The top board of the riser is assumed to be elastic withuniform thickness, and the sideboards to be rigid.
Consider a sound field that satisfies the three-dimensionalHelmholtz equation, with the boundary C and a point source at apoint s. In this field, the sound pressure p(rp) at a point p on theboundary is described by the Kirchhoff–Helmholtz integral equa-tion in the hypersingular form as
12
opðrpÞonp
¼ �jxqQoGðrp; rsÞ
onp
þZ
CpðrqÞ
o2Gðrp; rqÞonponq
� opðrqÞonq
oGðrp; rqÞonp
!dSq; ð1Þ
where o/onp denotes the normal derivative at a point p, Q is the vol-ume velocity of the point source, and G is the Green’s function. In a
point source s (0, 0, 0.7 + H)
infinite rigid floorz = 0
H
x
y
0.7
a = 1.2
b = 0.9
z
riser
point forcef (0, 0, H)
Fig. 1. Geometry of a point source, a point force, a riser and an infinite rigid floor.
semi-infinite field where an infinite rigid floor exists, the Green’sfunction is given by
Gðrp; rqÞ ¼expðjkjrp � rqjÞ
4pjrp � rqjþ expðjkjr̂p � rqjÞ
4pjr̂p � rqj; ð2Þ
where r̂p denotes the mirror image vector of rp with respect to theinfinite rigid floor. By applying Eq. (1) to the riser made of thinboards, the following expression can be obtained [12]:
opðrpÞonp
¼ �jxqQoGðrp; rsÞ
onpþZ
C0~pðrqÞ
o2Gðrp; rqÞonponq
dSq; ð3Þ
where ~p is the sound pressure difference between the two sides ofthe board, and C0 is one of the surface of the boards. Boundary con-ditions are written as
opðrqÞonq
¼ jxq0vpðrqÞ ¼ x2q0wpðrqÞ; for elastic boards;0; for rigid boards;
(ð4Þ
where vp and wp are the normal velocity and the normal displace-ment of the elastic board, respectively, and q0 is the air density.Substituting Eq. (4)to Eq. (3) and discretizing the boundary lead to
S � ~p ¼ x2q0
wp
0
� �þ jxq0Qd; ð5Þ
where
Sij ¼Z Z
ej
o2Gðri; rqÞonionq
dSq; ð6Þ
di ¼oGðrs; riÞ
oni; ð7Þ
~p is the sound pressure difference vector between the two sides ofthe board, and wp is the displacement vector of the elastic board.
As for the vibration field of the elastic boards, the equation ofmotion for the board is
ðDr4 �x2qphÞwpðrÞ ¼ ~ppðrÞ þ FpðrÞ; ð8Þ
where Fp is the external force, ~pp is the sound pressure difference be-tween the two sides of the elastic board, D = Eh3(1 � jg)/12(1 � m2)is the flexural rigidity of the board, with E the Young’s modulus, mthe Poisson’s ratio, h the thickness, g the loss factor, and qp the den-sity of the board. Assuming a rectangle elastic board of a � b m2 sizeto be simply supported, wp, ~pp and Fp are expressed by the modeexpansion with the modal function sin(mpx/a)sin(npy/b) as
wpðrÞ ¼X1m¼1
X1n¼1
Wmn sinmpx
a
� �sin
npyb
� �; ð9Þ
~ppðrÞ þ FpðrÞ ¼X1m¼1
X1n¼1
ðePmn þ FmnÞ sinmpx
a
� �sin
npyb
� �; ð10Þ
where Wmn, ePmn and Fmn are the expansion coefficients for wp, ~pp
and Fp, respectively. Substituting Eqs. (9) and (10) into Eq. (8) andconsidering the orthogonality of the modal functions yield the fol-lowing expression:
Dmp
a
� �2þ np
b
� �2� �2
�x2qph
" #Wmn ¼ ePmn þ Fmn: ð11Þ
If the external force Fp(r) is only a point force F at f (xf,yf), multiply-ing Eq. (10) by the function sin(mpx/a)sin(npy/b) and integrating itover the surface of the elastic board Cp lead toZ Z
Cp
~ppðrÞ sinmpx
a
� �sin
npyb
� �dxdyþ F sin
mpxf
a
� �sin
npyf
b
� �¼ ab
4ðePmn þ FmnÞ: ð12Þ
r
cello / double bassFFT Analyzer(ONO SOKKI DS2100)
590 Y. Yasuda et al. / Applied Acoustics 70 (2009) 588–594
After elimination of Wmn, ePmn and Fmn by substitution of Eqs. (11)and (12) into Eq. (9), discretization of Eq. (9) corresponding to thatof Eq. (5) finally leads to
wp ¼ B � A � T � ~pp þ B � A � f; ð13Þ
where
Tij ¼Z Z
ej
sinmipx
a
� �sin
nipyb
� �dxdy; ð14Þ
Aij ¼4dij
ab D mipa
� 2 þ nipb
� 2� �2
�x2qph � ; ð15Þ
Bij ¼ sinmjpxi
a
� �sin
njpyi
b
� �; ð16Þ
fi ¼ F sinmipxf
a
� �sin
nipyf
b
� �; ð17Þ
and dij is Kronecker’s delta. Substitution of Eq. (13) into Eq. (5)yields
S�x2q0
B � A � T 00 0
�� �� ~p ¼ jxq0Qdþx2q0
B � A � f0
� �:
ð18Þ
Solving Eq. (18) gives the sound pressure difference between thetwo sides of each boundary element. Substitution of the values intothe following integral equation gives sound pressures at a point r inthe sound field
pðrrÞ ¼ �jxq0QGðrr; rsÞ þZ
C0~pðrqÞ
oGðrr; rqÞonq
dS: ð19Þ
Force Transducer(B&K Type 8200)
Charge Amplifier(B&K Type 2635)
DAT Recorder(SONY PC208A)
Sound Level Meter(ONO SOKKI LA-1350)
unit: [m]
h
bridge
glass wool
concrete
carpetconcrete
0.1
Fig. 2. Geometry and block diagram for measurement. r = 2.1 and h = 0.6 for cellos,r = 1.78 and h = 0.75 for a double bass.
Frequency [Hz]
π
−π
Phas
e of
R (
f ) [
rad]
Am
plitu
de o
f R
(f )
[N
/(Pa
• m
)]
0
1
2
3
4
0
40 100 500
cello 1
double basscello 2
a
b
Fig. 3. Difference in R(f) measured with two cellos and a double bass: (a) amplitudeand (b) phase. The results for fundamental components are shown.
2.2. Measurement of mechanical force from lower string instruments
In order to appropriately study the effect of a riser on the soundof a lower string instrument, it is necessary to examine the rela-tionship between the acoustic radiation of the instrument andthe mechanical point force from the endpin of the instrument tothe riser. Nakanishi et al. [9] measured a mechanical point forcefrom a cello to a wooden riser, and calculated this relation frommeasured acceleration and driving point impedance of the board’svibration. In this measurement, however, a complicated methodwas adopted to separate the direct sound from measured oneincluding reflection and radiation from the riser; the radiatedsound from the riser was calculated backward with the measuredacceleration and the driving point impedance. Moreover, in thismeasurement the results depend on the condition of the riser be-cause the mechanical point force should be changed by the drivingpoint impedance of the riser. (This problem was pointed out in [9].)To cope with these problems, we conducted a measurement with-out a riser. An endpin of an instrument was placed on a mass ofconcrete and the mechanical point force was measured directlyusing a force transducer. Since the amplitude of the driving pointimpedance of the concrete mass is considered to be larger thanthat of wooden risers, the point force is not underestimated.
Another point of the measurement in [9] is that this was oneexample with only open strings of a cello and a riser, thus the dif-ferences among different instruments of the same kind (e.g.,among cellos), among different kinds of instrument (between cel-los and double basses), among strings, and among fundamentaland harmonic tones have not been clarified. In our measurement,cellos and a double bass were used to clarify these differences onthe relationship between the radiation and the mechanical pointforce of the instrument.
2.2.1. Measurement setupFig. 2 shows the geometry and the block diagram for measure-
ment. A rectangular mass of concrete, 153 � 152 � 71 mm, wasfixed on a concrete floor paved with thin carpet in an anechoicchamber, and a point force from an instrument was applied on aforce transducer mounted on the concrete mass. The other surfaceof the floor was paved with glass wool not to reflect the sound. Twocellos and a double bass were played by two amateur players (acellist and a bassist). Measurement was done after practice forplayers. When long tones of each note of the chromatic scale wereplayed, sound pressure at a receiving point and the point forcewere measured simultaneously. Players were requested to playeach note as steady as possible. Frequency spectrum was obtainedby Fourier transform of each tone, and the sound pressure P(f,r)and the point force F(f) at fundamental and harmonic frequenciesf were given, where r was the distance between the bridge of theinstrument and the receiving point. Assuming the instrument asa point source, namely P(f,r) = A(f)exp(jkr)/4pr, the relation be-tween A(f) and F(f) are defined as
Rðf Þ ¼ Fðf ÞAðf Þ ¼
expðjkrÞ4pr
Fðf ÞPðf ; rÞ ; ð20Þ
A(f) = �jxqoQ corresponding to the above numerical method.
2.2.2. Measurement resultsFig. 3 shows R(f) for fundamental components, measured with
two cellos and a double bass. Results for two cellos are not
Y. Yasuda et al. / Applied Acoustics 70 (2009) 588–594 591
different from each other in amplitude and in phase below 300 Hz.The difference between cellos and a double bass is great only atlow frequencies and those at which R(f) has peaks. It is also seenthat the phase varies with frequencies, whereas the amplitude isroughly constant, about less than 1, except at low frequencies.The amplitude of driving point impedance of real wooden risersis considered to be smaller than that of the concrete mass usedin the measurement, thus the amplitude of R(f) with a riser is alsoprobably smaller. This view is in accordance with the results in [9],in which the amplitude is about 0.1 except at frequencies whereR(f) (T(x) in [9]) has peaks. The amplitude of R(f) obtained here,which is less than about 1, is regarded as the upper limitation ofthat with risers. Fig. 4 shows R(f) measured with different fourstrings of a cello, C2, G2, D3, and A3. The difference both in ampli-tude and in phase is small, thus, there is no need to consider thedifference in vibration characteristics of the instruments causedby different strings. Fig. 5 shows R(f) of fundamental and harmoniccomponents of a cello. There is small difference in amplitude atmost frequencies except at frequencies where R(f) has peaks. Thedifference in phase is also small at all frequencies less than about300 Hz. Thus, it is not quite necessary to consider the differencebetween fundamental and harmonic components.
2.3. Numerical setup
Sound fields including risers of various configurations wereanalyzed by the numerical method previously mentioned, in con-
a
b
Fig. 4. Difference in R(f) measured with different strings of a cello: (a) amplitudeand (b) phase. The results for fundamental components are shown.
a
b
Fig. 5. Difference in R(f) between fundamental and harmonic components of acello: (a) amplitude and (b) phase.
sideration of the above measurement results for R(f). Four typesof model were defined as illustrated in Fig. 6. In type 0 no riserwas placed on an infinite rigid floor. In type 1 only the top boardof a riser was placed above the floor. In type 2 and type 3 thetop board and sideboards were placed on the floor, and thesetwo types were different in height. The sideboards were rigid intypes 2 and 3. The amplitude of R(f) was considered to be jR(f)j 6 1,and four phases of R(f), h = 0, p/2, p, and 3p/2, were examined.Other parameters for the study were the stiffness of the top boardof risers and the point force. In the implementation, quadratureconstant elements of 5 cm width, which was less than 1/10 ofthe shortest wavelength for the calculation, were used. Receivingpoints were fixed at (x, 0, 1.2), where x = 3, 5 and 10. Characteris-tics of the top elastic board were as follows: E = 1010 (N/m2),qp = 600 (kg/m3), h = 0.03 (m), g = 0, and m = 0.2. In all of the fol-lowing figures in this section, sound pressure levels are shownwithout direct sound and normalized by those of type 0.
2.4. Results and discussion
2.4.1. Effects of board’s vibrationFig. 7 shows normalized sound pressure levels at the receiving
point (10, 0, 1.2) for three cases of type 2: with a rigid top board,with an elastic top board without a point force, and with the elastictop board with a point force. It is seen that the effects of top board’sstiffness appear around the natural frequencies of the riser’s topboard and the cavity. The point force enhances not only the ampli-tude of peaks but also dips around the natural frequencies, thus,the point force from the lower string instrument cannot makeacoustic gains in a wide range of frequency. The effects appearmainly only around the natural frequencies of the board. Fig. 8shows the effects of R(f) for type 2 with an elastic top board and
Type 0 Type 3 (H = 0.2)Type 2 (H = 0.1)Type 1 (H = 0.1)
point forceinfinite rigid floor
elastic boardrigid board
Fig. 6. Illustration of four types of model.
-10
0
10
100Frequency [Hz]
Nor
mal
ized
SPL
[dB
]
50063
rigid elastic (with apoint force)elastic
(without apoint force)
x = 10
natural freq. of the boardnatural freq. of the cavity
Fig. 7. Effects of a point force and top board’s rigidity of a riser on sound pressurelevels without direct sound at a receiving point (10, 0, 1.2) in type 2. jR(f)j = 0.2 andh = 0. The sound pressure levels are normalized by those of type 0.
-10
0
10
100Frequency [Hz]
Nor
mal
ized
SPL
[dB
]
500
x = 10
63
|R(f)| = 0.2θ = 0
|R(f)| = 1θ = 0
|R(f)| = 0.2θ = 3π / 2
natural freq. of the boardnatural freq. of the cavity
Fig. 8. Effects of the amplitude and phase of R(f) on sound pressure levels withoutdirect sound in type 2 with an elastic top board and a point force, at a receivingpoint (10, 0, 1.2). The sound pressure levels are normalized by those of type 0.
592 Y. Yasuda et al. / Applied Acoustics 70 (2009) 588–594
a point force. Both the amplitude jR(f)j and the phase h affect thesound pressure levels around the natural frequencies of the topboard, and the frequency range where the point force affects in-crease with jR(f)j. The effect of the point force in real situations is
-10
0
10
-10
0
10
100Frequency [Hz]
Nor
mal
ized
SPL
[dB
]
5
x = 3
x = 10
63
Fig. 9. Effects of riser’s sideboards and height on sound pressure levels without direct southose of type 0.
Fig. 10. Effects of riser’s sideboards and height on sound pressure level distribution withois on the plane of y = 0.
probably smaller than the case of jR(f)j = 1, because jR(f)j is consid-ered to be less than 1 based on the above discussion on the mea-surement results and the loss factor g > 0 for real risers. Theresults are similar at all receiving points in all types.
2.4.2. Effects of riser’s configurationsFig. 9 shows normalized sound pressure levels for three types of
riser having a rigid top board. In the case without sideboards (type1), small dips by interference of diffraction waves are seen,whereas in the cases with sideboards (types 2 and 3), there is awide range of acoustic gains approximately from 120 to 300 Hzat all receiving points, and the gains increase with the height ofthe riser. This effect of the sideboards is a contrast to that of thevibration of the boards, which is seen only around the natural fre-quencies. The phenomenon can be interpreted as follows: riserswithout sideboards do not reflect the energy of diffracted wavesto receiving points because the edge-diffracted waves turn aroundthe edges and go beneath of the risers, while risers with sideboardsefficiently reflect and propagate the energy to the front. Fig. 10shows sound pressure level distributions at 200 Hz for all typeswith rigid top boards. It is seen that the types considerably differin distributions, and that the existence of the sideboards and theincrease in height produce acoustic gain in sound pressure levelsin the horizontal direction. The results are similar at the frequen-cies where the gain is observed in Fig. 9. From these results, itcan be stated that one of the causes of the so-called ‘resonant
00
Type 1 Type 2
Type 3
x = 5
100Frequency [Hz]
50063
nd. The top boards are rigid in all types. The sound pressure levels are normalized by
ut direct sound at 200 Hz. The top boards are rigid in types 1, 2 and 3. Receiving area
Y. Yasuda et al. / Applied Acoustics 70 (2009) 588–594 593
support’ is efficient propagation of energy of diffracted waves tothe front by the sideboards of risers.
3. Experimental study
In the previous section it has been clarified that diffractedwaves by risers affect sound fields in a wide range of frequency.To verify this phenomenon, we investigated the effects of waves
Table 1Configurations of riser models
Parameter Type Configuration
Type X Without a riserType A a = 1.2, b = 0.9, H = 0.2 [m], with all sideboards
Sideboard Type SX With sideboards only in x-directionType SY With sideboards only in y-directionType SN Without sideboards
Width b (m) Type WL b = 1.8 (2 times)Type WL2 b = 3.6 (4 times)
Height H (m) Type HS H = 0.1 (0.5 times)Type HL H = 0.4 (2 times)
Depth a (m) Type DL a = 1.8 (1.5 times)Type DL2 a = 2.4 (2 times)
The coordinates of x, y and z are the same as shown in Fig. 1. x is the direction from ariser to receiving points.
Fig. 11. Comparison between numerical and experimental results of total sound pressu
-10
0
10
-10
0
10
125 100Frequency [Hz]
Nor
mal
ized
SPL
[dB
]
Type A Type SX
Type SY Type SN
Type A Type HS
Type HL
a
c
x = 5
x = 5
250 500
Fig. 12. Effects of riser’s configurations on total sound pressure levels: (a) the effects of tsound pressure levels are normalized by those of type X.
diffracted by various configurations of risers in detail through ascale model experiment.
3.1. Experimental setup
The experiment was conducted using 1/4 scale models in aroom, which was enclosed by absorptive materials. The floor ofthe room was rigid. The arrangement of the experiment was thesame as that of the numerical study as shown in Fig. 1, exceptfor the height of the sound source, which was fixed 0.9 m fromthe floor independently of the riser’s height. Table 1 shows config-urations of riser models investigated in this experiment. In type X,no riser was placed on the rigid floor, which corresponded to type 0in Fig. 6. In type A, a 1.2 � 0.9 � 0.2 m3 riser with all sideboardswas placed, which was the same as type 2 in Fig. 6. In the othereight types, one of the parameters, i.e., sideboards, width, heightand depth, was different from type A. For all models, the top boardand the sideboards were made of a 2.5-mm-thick MDF board and10-mm-thick spruce boards, respectively, assumed to be rigid. Re-sponses were measured at each receiving point with a 1/4-in.omnidirectional microphone using the swept-sine method. By16-times synchronous averaging of these measured responses, aresponse with high S/N ratio was obtained. Sound energy for 1/3octave bands was obtained by filtering this response in which lat-ter reflected waves from the room were cut off.
re levels in type HL. The sound pressure levels are normalized by those of type X.
0 125 1000Frequency [Hz]
Type A Type WL
Type WL2
Type A Type DL
Type DL2
b
d x = 5
x = 10
250 500
he sideboards, (b) of the width, (c) of the height, and (d) of the depth of a riser. The
594 Y. Yasuda et al. / Applied Acoustics 70 (2009) 588–594
3.2. Results and discussion
In all of the following figures of experimental results, 1/3 octaveband sound pressure levels including direct sound are shown. Thesound pressure levels are normalized by those of type X. Receivingpoints are (x,0,1.2), which are the same as the numerical study.
Numerical and experimental results are compared in Fig. 11.Both results are quite similar. Results were also similar in the othertypes, independently of receiving points. Fig. 12a shows the effectsof the sideboards. The difference between types is not clear at lessthan 250 Hz, while at higher frequencies it is clearly seen thedifference between types A, SX and SY, SN; sound pressure levelsare larger in types A and SX that have sideboards in the directionof receiving points (x-direction). Regarding the effects of the width,clear differences were not seen at relatively near points to the riserx = 3 and 5, although at a far receiving point x = 10 sound pressurelevels become larger with the width of a riser below 125 Hz, asshown in Fig. 12b. These results support the previous discussionin the numerical study that diffracted waves propagated by thesideboard of the riser cause acoustic gains in low- and middle-frequency ranges. It can be also stated that this effect contributesthe total sound pressure levels including direct sound, as well asthose without direct sound. Fig. 12c and d shows the effects ofthe height and the depth, respectively. In both cases clear differ-ences are not seen except at relatively high frequencies, wherepositions of the edges in the direction of receiving points affectthe difference.
4. Conclusions
Sound fields including a stage riser and a lower string instru-ment have been numerically and experimentally analyzed in orderto clarify the acoustic effects of risers. The conclusions are summa-rized as follows:
(1) The vibration of the top board of a riser affects the soundfield only around the natural frequencies of the board andthe cavity of the riser.
(2) A mechanical point force enlarges peaks of sound pressurelevels at the natural frequencies of the board, but alsoenlarges dips at these frequencies. Therefore, mechanicalpoint forces from lower string instruments cannot make
acoustic gains in a wide range of frequency. The effect ofthe point force changes with the relation between the radi-ated sound and the mechanical point force of the lowerstring instrument.
(3) The configuration of a riser affects the sound field in a widerange of frequency, and the sideboard in the direction ofreceiving points increases the sound pressure levels becauseit reflects waves diffracted at the riser’s edge to the front. Atlow frequencies, sound pressure levels in the far fieldincrease with the width of the sideboard in the direction ofreceiving points.
Acknowledgement
The authors wish to thank Dr. Yokota (Kobayasi Institute ofPhysical Research) for their support in the experimental work.
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