Efficient modeling and experimental validation of acoustic resonances in three-dimensional rectangular open cavities

Applied Acoustics 74 (2013) 949–957

Contents lists available at SciVerse ScienceDirect

Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust

Efficient modeling and experimental validation of acoustic resonances in three-dimensional rectangular open cavities Santiago Ortiz a, Cyprien Le Plenier b, Pedro Cobo a,⇑ a b

Centro de Acústica Aplicada y Evaluación No Destructiva (CAEND), CSIC, Serrano 144, 28006 Madrid, Spain Ecole Nationale Supérieure d’Ingénieurs du Mans (ENSIM), Rue Aristote, 72085 Le Mans Cedex 09, France

a r t i c l e

i n f o

Article history: Received 14 November 2011 Received in revised form 20 September 2012 Accepted 21 January 2013

Keywords: Open cavity resonances Image Source Model

a b s t r a c t The goal of this paper is to investigate the acoustic resonances of a three-dimensional open cavity by a fast and efficient method in the time domain. This method models the time response in any point as the convolution of the source waveform with the impulse response of the cavity, which, in turn, is obtained as a sequence of attenuated and delayed impulses coming, the first from the real, and the subsequent from the mirror imaged sources (Image Source Model). This method, which main advantages with respect to others that work in the frequency domain are that the results for all frequencies can be calculated at once, the time domain data is recovered directly, and the computational cost does not increase with frequency, can easily provide the frequency response at each point of the cavity by Fourier transform. From these frequency responses, the relevant resonances of the cavity can simply be obtained. Illustration and experimental validation of the proposed method is presented by its application to a rectangular open cavity. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Closed and open cavities are both of engineering interest and fundamental importance in Acoustics. Sound in closed cavities is the core of Room Acoustics [1,2]. The interaction of a flow with an open cavity is of crucial importance in musical wind instruments, like acoustical guitars or organ pipes. But it is likely in the context of aeroacoustic noise control where the coupling between acoustic resonances and self-excited flow oscillations becomes especially relevant [3–11]. A complex feedback mechanism between the pressure oscillations in the shear layer above the cavity and the standing waves inside the open cavity leads to resonant tones as high as 170 dB [12]. Self-sustained unstable vortical flow triggers resonance modes of the cavity, and these acoustic modes also generate more vorticity [13]. Since, generally, the net transfer is in favor of the acoustic mode, it is more often told about cavity noise as the undesirable consequence of the flow over an open cavity [13]. However, other detrimental results of this flow-acoustic coupling are structural fatigue and drag increase. Aerodynamical noise is omnipresent in the context of air and ground transportation, penalizing the acoustic comfort inside airplanes and vehicles, and increasing the environmental noise in the vicinity of airports. Wheel wells and Auxiliary Power Unit com-

⇑ Corresponding author. Tel.: +34 915618806; fax: +34 914117651. E-mail address: [email protected] (P. Cobo). 0003-682X/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.apacoust.2013.01.007

partments are sources of high level cavity tones during landing and takeoff airplane operations. The gap between two wagons and the shallow cavity accommodating the pantograph produce cavity noise in high speed trains. Open sunroofs and open windows generate aeroacoustic noise in cars. Whilst the fundamental mechanisms underlying the coupling between the flow instability and the underneath open cavity have been elucidated by many investigators [3–13], the suppression of this feedback loop in a robust and reliable way still remains a challenge [12]. Promising active control techniques for the flow-induced oscillations [12,14–16] and noise [17] have been proposed that complements well known passive control techniques. A critical issue for an active noise control system is the location of the secondary sources and error sensors [18]. Nodal lines of acoustic modes might be avoided. Sources situated on a nodal line of an acoustic mode will be unable to excite such a mode. Sensors laying on the nodal line of a mode will not detect it. Therefore, an active control system with either a secondary source or an error sensor located on a nodal line will be inherently incapable to cancel the corresponding acoustic mode. As a consequence, for the accurate design of an active noise control system for cavity tones, the acoustic resonances of the open cavity should be known in advance. Some recent studies have demonstrated that the frequencies of the acoustic cavity tones generated by flow are practically independent of the velocity flow at low Mach numbers [19,20]. Thus, for noise control purposes, it is desirable to know in advance the acoustic resonances of the open cavity.

950

S. Ortiz et al. / Applied Acoustics 74 (2013) 949–957

Several researches have reported numerical methods to provide the eigenvalues (resonant frequencies) and eigenfunctions (normal modes) of open cavities without flow. Tam [21] obtained the frequencies of the acoustic modes of a 2D rectangular cavity by numerical integration of the motion equations after Laplace and Fourier transforms. More recently, Koch [22] and Hein et al. [23] proposed to find the acoustic resonances of 2D and 3D rectangular open cavities by solving the Helmholtz wave equation by finiteelement methods. This method provides the resonant frequencies and the normal modes of the open cavity and needs to implement absorbing conditions (PML) at the boundaries of the numerical domain to avoid unphysical reflections at such computational boundaries. Therefore, this method results to be high-demanding from a computational point of view. The main objective of this paper is to propose a faster, lessdemanding and efficient method to reveal the acoustic resonances and modes of an open cavity. As usually done in Room Acoustics, the method provides the time response of any source-microphone pair through the convolution between the cavity impulse response with the loudspeaker time waveform [24]. The impulse response of the cavity is obtained by the application of the Image Source Method [25,26]. As compared to the method proposed by Koch and Hein et al., the ISM method does not require PML boundary conditions, and computational cost does not increase with frequency. The layout of the paper is as follows. In Section 2, the main features of the ISM are reviewed. Section 3 describes the 3D rectangular open cavity built for numerical/experimental comparison, as well as the measurement method. Section 4 provides experimental validation of numerical predictions. Concluding remarks are given in Section 5.

fectly rigid except for one which absorbs the incident sounds completely, Fig. 1b. Compared to a closed room, the image space consists now of only two levels, as the opening of the cavity generates no images. With the above assumption, the original Allen and Berkley equation can be easily modified to obtain the pressure time response for a point source S = (x, y, z) and receiver R = (x0 , y0 , z0 ) inside an open cavity. The original equation for the case of six reflecting surfaces with angle independent wall absorption is [25]:

2. Image Source Modeling

where the unnecessary terms mentioned above have been removed, rp = (xx0 + 2qx0 ,yy0 + 2jy0 ,zz0 + 2kz0 ) remains the same as in Eq. (1), and rr is modified to

The time response between a source and a receiver inside the open cavity is calculated by using the simple and efficient Image Source Model (ISM in the following). Cavity resonances can be easily predicted by applying Fourier transform to convert this time signal in a frequency response function. The source-receiver time response is obtained as the convolution of the source waveform with the cavity impulse response [24]. The classical algorithm by Allen and Berkley [25] provides the impulse response of a room of any dimension and reflection characteristics. This algorithm has been adapted to, and optimized for, an open cavity, including also the time domain directivity function of the source. Fig. 1 shows how the real and mirror-imaged sources are spatially arranged in a closed room and an open cavity. The solid box represents the real cavity. Although the actual image space is three-dimensional, a two-dimensional view is illustrated for the sake of clarity. The impulse response of an open cavity is modeled in the same way as that of a closed room, Fig. 1a, with all walls per-

pðt; S; RÞ ¼

1 X 1 X 1 X 1 X 1 X 1 X

jnqj

bx1

jnj

jljj

jlj

jmkj

bx2 by1 by2 bz1

jmj

bz2

q¼0 j¼0 k¼0 n¼1l¼1m¼1



djt  ðjr p þ r r j=cÞj ; 4pjr p þ r r j ð1Þ

0

where the b s are the reflection coefficients at the six boundaries, rp = (xx0 + 2qx0 , yy0 + 2jy0 , zz0 + 2kz0 ) represents the vectors created in the image space and rr = 2(nW, lL, mD), being (n, l, m) the integer vector triplet for higher order reflections and (W, L, D) the room dimensions. jmj In an open cavity the reflection coefficient bz2 no longer exists, P and the sum over m orders of reflection in the z-direction, 1 n¼1 , does not extend infinitely any longer, so that it reduces to two levels, Fig. 1b. Therefore, Eq. (1) can be simplified to

pðt; S; RÞ ¼

1 X 1 X 1 X 1 X 1 X

jljj jlj jkj bxjnqj bxjnj 1 2 by1 by1 bz1

q¼0 j¼0 k¼0 n¼1l¼1



djt  ðjr p þ r r j=cÞj ; 4pjr p þ r r j

rr ¼ ð2nW; 2lL; 2kDÞ;

ð2Þ

ð3Þ

where the index k replaces now the index m, which can only take two values: k = 0, being the second level of the image source space, and k = 1, representing the image sources in the first level, Fig. 1b. The terms in the indexes n and l continue extending infinitely as they run over x- and y-directions, where the cavity is closed. 2.1. Efficiency and accuracy of the ISM In order to implement the ISM model it is necessary to define an interruption criterion, otherwise the algorithm would loop infinitely; note the sums over the indexes n and l in Eq. (2). The criterion used for this purpose is the image source order ISN ðnx ; ny ; nz Þ, commonly used in Room Acoustics. The number of image sources, NIS, for a rectangular closed room can be calculated with the expression [1]

Fig. 1. Image source space for (a) a rectangular room and (b) an open cavity. The bold box is the real room or cavity.

951

S. Ortiz et al. / Applied Acoustics 74 (2013) 949–957

NIS ¼ 2nx  2ny  2nz ; ct int W

ð4Þ ct int L

; nz ctDint ,

where nx ¼ ; ny ¼ c is the speed of sound, tint is the interruption time and (W, L, D) are the cavity dimensions. Assuming that the image space in an open cavity is as discussed previously, Fig. 1b, Eq. (4) can also be simplified as

NIS ¼ 2nx  2ny  2:

ð5Þ

Fig. 2 shows the number of image sources (NIS) growth as function of time for a closed and open rectangular room cavity (Eqs. 4 and 5), with dimensions (W, L, D) = (52, 32, 38) cm. As it can be observed, the rate of growth is slower for an open cavity, making it less computationally demanding. The accuracy of the ISM is also determined by the NIS, most specifically by the interruption time tint. This time has been determined using the measured impulse responses at 10 different positions inside the cavity. We have plotted the energy curve [1] at these positions and selected the maximum time for an energy decay of 100 dB, Fig. 3. Assuming that any contribution after this point could be neglected, Table 1 shows the truncation time tint, the nx value, the ny value and the total number of image sources, NIS, calculated using Eq. (5) for a cavity with dimensions (W, L, D) = (52, 32, 38) cm.

Fig. 3. ISM interruption time determined from energy decay measurements.

Table 1 Interruption criteria parameters for the ISM. Rectangular open cavity with dimensions (W, L, D) = (52, 32, 38) cm. tint (s)

nx

ny

NIS

0.25

165

268

353760

2.2. Computational implementation A code that simulates the Acoustics of a rectangular open cavity has been created with MATLAB. The groundwork on which the program is based is the Fortran routine included in Allen and Berkley [25]. The original algorithm has been modified according to the simplification contained in Eq. (2). The main program consists of three nested ‘‘for’’ loops that extend over nx, ny, and nz (only 1 and 0). For each iteration (path segment image source-receiver), the corresponding delay ||rp + rr||/ c (where c is the speed ofpsound), ffiffiffiffiffiffiffiffiffiffiffiffi spreading factor 1/||rp + rr||, and  (where a  is the frequency-depenreflection coefficient b ¼ 1  a dent absorption coefficient), are computed. Also, the source directivity is taken into consideration in the ISM by multiplying the spreading factor by Wðhs Þ, where hs is the angle between two vectors: the image source-receiver vector and the direction of preference, Ds , which is a normal vector in the direction of the acoustic axis of the loudspeaker, Fig. 4. The output of the program is a sequence of delayed Dirac delta functions, which are arranged according to the travel time and attenuated by the wall absorption on each reflection. In order to effectively allow the exact representation of non-integer time de-

Fig. 4. Source directivity function. Ds is the direction of preference of the original source, D0s is the direction of preference of the image source, Rs is the sourcereceiver vector, and hs is the source-receiver angle.

lays for each image source, the interpolation process recommended by Lehmann and Johansson [27] has been incorporated. Multiple impulse responses from a source S = (x, y, z) to a grid of receivers ri = (xi, yi, zi) evenly distributed inside the cavity are computed. The convolution of the source (a loudspeaker, for instance) waveform with the corresponding impulse response provides the time domain pressure p(t, ri) at each point of the simulation grid ri = (xi, yi, zi). The Frequency Response Functions (FRFs) between the source and receivers are computed as FRFðf ; ri Þ ¼ 20 log jIfpðt; ri Þgj. The resonant frequencies of the cavity are identified as the peaks of the FRF’s. The block diagram of the computer implementation of the method is illustrated in Fig. 5. A comparison of the implemented image source algorithm with experimental results is presented in Section 3, and the parameters used for this simulation are listed in Table 2. Note that b is the same for all five reflecting surfaces and a is the average of the octave band absorption coefficients measured, Table 3.

3. Experimental setup

Fig. 2. Image source order (NIS) growth as function of time for a closed and open rectangular cavity with dimensions (W, L, D) = (52, 32, 38) cm.

A three-dimensional rectangular open cavity was built to compare numerical predictions of time responses at points inside the cavity with measurements. This Section describes the cavity, as well as the measurement method to characterize the acoustic field inside it.

952

S. Ortiz et al. / Applied Acoustics 74 (2013) 949–957

Fig. 5. Block diagram of the implemented ISM for open cavities.

Table 2 Parameters used for the ISM simulation. Sampling frequency (kHz)

Open cavity dimensions (W, L, D) cm

Source position (x, y, z) cm

Receiver array cm

b

a

20

(52, 32, 38)

(26, 0, 35)

Dx = Dy = Dz = 31

0.94

0.11

speaker-microphone pair has a non-flat frequency response. Therefore, when it is used to characterize the acoustic field in the open cavity, peaks of this non-flat frequency response could be misinterpreted as resonances of the cavity. To avoid that, the loudspeakermicrophone frequency response will be first equalized by inverse filtering. 3.2. Equalization by inverse filtering of the loudspeaker The loudspeaker-microphone system can be modeled as a linear filter. Let h(t) be the impulse response of the loudspeaker-micro-

Table 3 One-octave band absorption coefficients of the cavity walls. Frequency (Hz) Absorption coefficient

125 0.09

250 0.09

500 0.10

1k 0.10

2k 0.13

4k 0.14

3.1. The rectangular open cavity A 3D open cavity was made with medium density fibreboard (MDF) panels of thickness 3 cm. The open cavity, Fig. 6, has dimensions (W, L, D) = (53, 32, 38) cm, thus resulting in L/D = 0.84 and W/ D = 1.4. A 4 inch Sonavox Honeycomb loudspeaker is located at the center line of the front wall, with coordinates (x, y, z) = (26.5, 0, 35) cm. Eight ½ inch FONESTAR 2210 electrect microphones were used to measure the loudspeaker-microphone time responses in the cavity. Fig. 7 shows the impulse response and the frequency response of the loudspeaker-microphone pair measured in the anechoic room of the Centre for Applied Acoustics and Non Destructive Testing (CAEND). As it can be seen, the loud-

Fig. 6. The 3D rectangular open cavity.

953

S. Ortiz et al. / Applied Acoustics 74 (2013) 949–957

Fig. 7. Impulse response (a), and frequency response (b), of the loudspeaker–microphone pair.

phone system, and H(f) its Fourier transform. When the loudspeaker is driven by a conventional voltage, xc(t), the microphone receives a conventional waveform, yc(t), such as

Y c ðf Þ ¼ Hðf ÞX c ðf Þ

ð6Þ

where Xc(f) and Yc(f) are the Fourier transforms of the input and output signals, respectively. Instead of driving the loudspeaker with a conventional voltage and wait for its response, let us prescribe the shape of the radiated spectrum received by the microphone, Ys(f), and calculate the voltage which should drive the tweeter, Xs(f). According to the linear system theory, this should be

X s ðf Þ ¼

Y s ðf Þ : Hðf Þ

ð7Þ

To avoid instabilities at the notches of H(f), a positive constant, C2, must be added to the denominator (regularization). Thus

X s ðf Þ ¼ Y s ðf Þ

H  ðf Þ jHðf Þj2 þ C 2

ð8Þ

:

The electrical signal which should drive the loudspeaker is

( 1

xs ðtÞ ¼ I

Y s ðf Þ

H  ðf Þ

) ð9Þ

jHðf Þj2 þ C 2

where I1 stands for inverse Fourier transform. When the loudspeaker is driven by xs(t) the system responds with

hðtÞ xs ðtÞ ¼ ys ðtÞ:

jY s ðf Þj ¼

1 f0 ¼ f2 þf 2

is the central frequency

B ¼ ðf2  f1 Þ is the bandwidth

:

ð11bÞ

For symmetric pulses around its center, the phase spectrum WX sðf Þ ¼ 0 (zero-phase pulses).The parameter g in Eq. (11a) determines the trade-off between the main lobe and the lateral oscillations of the pulse [28]. As g ? 0, the main lobe of the pulse narrows and the amplitude of the secondary oscillations increase. In the limit the time pulse becomes a sinc function. In the range g > 1, the pulse presents wider main lobe but the secondary oscillations are smaller instead. Inverse filtering can be combined with the MLS method [29,30] for measuring short impulse responses. In this case the MLS signal is pre-emphasized by the inverse filter of Eq. (10), with the zerophase cosine-magnitude spectrum of Eq. (11) [31,32]. Therefore, equalization by inverse filtering depends on four parameters: (f1, f2) the lower and upper frequencies of the band, g, the main-side lobes trade-off parameter, and the regularization constant C2. This equalization process was applied to the loudspeaker-microphone system in the anechoic room. Fig. 8 shows the time response and frequency response of the loudspeakermicrophone system after equalization by inverse filtering with parameters (f1, f2, g, C2) = (109 Hz, 9082 Hz, 0.1%, 0.5% of the spectral maximum). It can be seen that inverse filtering was successful to equalize the frequency response of the loudspeaker-microphone system. 3.3. Directivity pattern of the loudspeaker

ð10Þ

Cosine-magnitude pulses have a magnitude spectrum given by

(

with

  A cosg pðf Bf0 Þ

f 1 6 f 6 f2

0

f < f1 ; f > f2

ð11aÞ

The directivity function of the loudspeaker is needed for the Image Source Model. To measure this function the loudspeaker was placed on an outline ET2/ST2 turntable. A time response was measured for a fixed turn angle of the turntable each 5°, between 90° and 90°. Therefore, 37 time responses were measured between

954

S. Ortiz et al. / Applied Acoustics 74 (2013) 949–957

Fig. 8. Impulse response (a), and frequency response (b) of the loudspeaker-microphone system after equalization by inverse filtering.

90° and 90°. The turn angle was controlled visually by means of a digital camera. Time responses were measured in the horizontal (h) and vertical (/) planes of the loudspeaker. Let us define the directivity pattern of the loudspeaker as the peak value of the time response y(h, /), so that

Wðh; /Þ ¼

max ðabsðyðh; /ÞÞ max ðabsðyð0; 0ÞÞ

ð12Þ

The measured directivity function has resulted to be symmetrical with respect to the acoustical axis of the loudspeaker, W ¼ WðhÞ, where h is the angle between the acoustical axis and the microphone position (see Section 2). Fig. 9 shows the 3D time directivity function of the Sonavox loudspeaker to be incorporated to the ISM described in Section 2.

Fig. 9. 3D time directivity function of the Sonavox loudspeaker.

3.4. Absorption coefficient of the inner walls Absorption coefficient of the cavity walls is needed for calculating the theoretical time responses at each position by the ISM. Two cylindrical samples of diameters 29 and 100 mm were cut to be measured in a B&K 4206 impedance tube with a valid frequency range from 50 to 6400 Hz. Table 3 summarizes the resulting oneoctave band absorption coefficients of the cavity walls. 3.5. Measurement procedure Time responses between the loudspeaker and microphones located at the points of a grid with Dx = Dy = Dz = D = 31 cm, Fig. 10, were measured. Thus, each plane of the grid contains 160

Fig. 10. Measurement grid inside the cavity. Red points represent microphone positions and the grey one stands for the loudspeaker position.

S. Ortiz et al. / Applied Acoustics 74 (2013) 949–957

955

Fig. 13. Experimental and numerical FRF at the microphone located at (3.1, 14.4, 16.5) cm [on the nodal line of modes (1, 0, 0) and (2, 1, 0)]. Fig. 11. Setup for the measurements of loudspeaker-microphone time responses in the open cavity.

and maximum frequency sampling of 250 kHz, driven by MATLAB. In this case, MUIRSA was configured to acquire the time response between the loudspeaker and eight microphones, sampled at 20 kHz, using a MLS signal of order M = 16. Therefore, 10 sets of

Fig. 12. Experimental and numerical FRF at the microphone located at (3.1, 2.0, 16.5) cm.

Table 4 Modal frequencies of the open cavity picked up from the peaks of the FRF at point (3.1, 2.0, 16.5) cm. Peak at FRF

Mode (l, m, n)

fexp (Hz)

fthe (Hz)

E (%)

1 2 3 4 5 6 7 8 9 10 11

(0, 0, 0) (1, 0, 0) (0, 1, 0) (2, 0, 0) (2, 1, 0) (0, 2, 0) (2, 2, 0) (4, 1, 0) (4, 2, 0)

157 374.5 575 680.9 863.4 1103 1290 1430 1642 1768 2273

151.4 Not seen 549 659.8 854.5 1090 1270 1431 1646 1763 2255

3.6 4.5 3.1 1.0 1.2 1.6 0.1 0.2 0.3 0.8

points, 16 along the x-axis times 10 along the y-axis. Since there are 11 planes along the z-axis, the whole grid contains 1760 measurement points. However, because of the cavity symmetry, only half of these points need to be measured. Notice that if four points are required to resolve a wavelength, only modes with frequency less than c/(4D) = 2760 Hz will be reliably measured. A self-developed Virtual Instrument called MUIRSA was used to measure the time responses between the loudspeaker and the microphones in the cavity. MUIRSA consists of a DAQ NI PCIMIO-16E card with eight input channels and two output channels

Fig. 14. Experimental (a) and numerical (b) mode (0, 1, 0).

956

S. Ortiz et al. / Applied Acoustics 74 (2013) 949–957

Fig. 15. Experimental (a) and numerical (b) mode (2, 0, 0). Fig. 16. Experimental (a) and numerical (b) mode (2, 2, 0).

measurements will be necessary to characterize a plane, and 110 to complete the study of the acoustic field inside the cavity. Fig. 11 shows the setup for the measurement inside the anechoic chamber. Two baffles were added at the front and rear open edges of the cavity to avoid interference peaks between the frontward and backward radiation of the loudspeaker. The above explained procedure provides the time domain pressure p(t, ri) radiated by the loudspeaker at each point ri = (xi, yi, zi) of the measurement grid. From these, the Frequency Response Functions (FRFs) between the loudspeaker and the microphones can be easily obtained by

FRFðf ; ri Þ ¼ 20 log jIfpðt; ri Þgj;

ð13Þ

where I stands for Fourier transform. The plots of these FRF for a given frequency, fm, at a plane will image the acoustic field at this frequency in such a plane. If fm is a modal frequency, this plot will outline closely the corresponding acoustic mode. 4. Results A comparison between numerical and experimental results inside the cavity will be accomplished in this section. Bear in mind that the numerical model runs for the moment inside the cavity. To image the acoustic field above the open wall of the cavity, the

diffraction at the open edges of the cavity might have to be taken into account. As stated above, the modal frequencies of the cavity are picked up from the peaks of the FRF in a microphone inside the cavity. Let us choose the microphone at coordinates (3.1, 2.0, 16.5) cm. Fig. 12 shows superimposed the theoretical and experimental FRF at this position. Notice the vertical offset of the numerical FRF with respect to the experimental one, due to the non-calibration of the measurement system because our main interest in this work is to calculate the resonant frequencies and normal modes of the cavity. As it can be seen, there is a good correspondence between the experimental and the numerical FRFs, except for some peaks which are observed in the experimental FRF but they are not in the theoretical one (374 Hz, 2355 Hz, etc.). These correspond to frequencies of axial odd modes (m, 0, 0), with m = 1,3,... While the theoretical point source is exactly on a nodal line of these modes, the real one has a size (4 in.) enough to slightly excite them. Table 4 summarizes the frequencies of the first eleven modes at this point taken out from the peaks of the theoretical and experimental FRFs. The relative error calculated as Eð%Þ ¼ jðfexp  fthe Þ=fexp j100 is also included in the last column of Table 4. Note that the relative error decreases as the frequency increases, which is expected since the ISM model is more adequate for mid and high frequencies. The first mode corresponds to the

S. Ortiz et al. / Applied Acoustics 74 (2013) 949–957

fundamental mode (rocking mode) of the open cavity [21]. The next eight modes have been identified as either axial or longitudinal modes along the (x, y) plane. At the frequencies of the last two peaks, the modal density is so high that it is difficult to assign them to separated modes. Fig. 13 illustrates the inherent inability of an active noise control system to cancel a mode with the error sensor on a nodal line of such a mode. Since the microphone at coordinates (3.1, 14.4, 16.5) cm is on a nodal line of modes (0, 1, 0) and (2, 1, 0), it does not pick up their frequencies, as clearly seen in Fig. 13. Finally, let us illustrate some of the revealed normal modes. As explained above, this is done by plotting the acoustic field at the frequencies of some of the peaks. Figs. 14 and 15 show the 3D experimental and numerical representation of the axial modes (0, 1, 0) and (2, 0, 0), respectively. A significant agreement is found between numerical and experimental modes, except for the top plane (the one closer to the open wall of the cavity). The explanation of this slight disagreement is that diffraction in the open edges of the cavity is included in the experimental data but excluded in the numerical ones. This fact is also observed in the longitudinal mode (2, 2, 0) shown in Fig. 16. 5. Conclusions A fast and efficient model, based upon the Image Source Model used in Room Acoustics, is proposed to characterize the acoustic resonances of open cavities. The model works in the time domain and computes the impulse response of the open cavity by the superposition of delayed impulses, coming from the real and the image sources. Measuring the source waveform, the time response at any point inside the cavity is then calculated through its convolution with the cavity impulse response. Once the time response is known, the frequency response at any point can be simply determined by Fourier transform. Then, the resonance frequencies are estimated as the relevant peaks of these frequency responses. Plotting the acoustic field at the resonant frequencies reveals the corresponding modes of the cavity. The proposed model has been experimentally validated on a three-dimensional rectangular cavity. Since the resonance frequencies are estimated from the relevant peaks of the frequency response functions, the loudspeaker-microphone system is first equalized by inverse filtering. A time directivity function has been defined and measured to be incorporated in the Image Source Model. The reflection coefficient of the walls has been also measured by the transfer function method. With all these amendment, a good agreement has been obtained between numerical predictions and experimental results. Experimental and numerical resonances differs less than 4% in the frequency resolution. The modes are also well outlined by the numerical model and the measurements. They only differ slightly at the above (open) plane of the cavity. This is due to the diffraction on the edges of the open cavity which is not still included in the model. Acknowledgement This research was supported by the Spanish Ministry of Science and Innovation through Projects Nos. TRA2008-05654-C03-03 and TRA2011-26261-C03-01.

957

References [1] Kuttruff H. Room Acoustics. London: Spon Press; 2009. [2] Pierce AD. Acoustics. An Introduction to Its Physical Principles and Applications. Melville, USA: ASA; 1989. [3] Plumbee HE, Gibson JS, Lassiter LW. A theoretical and experimental investigation of the acoustic response of cavities in an aerodynamic flow, WADD-TR-61-75. Dayton, OH: Wright-Patterson AFB; 1962. [4] Rossiter JE. Wind-tunnel experiments on the flow over rectangular cavities at subsonic and transonic speeds. aeronautical research council reports and memoranda, N° 3438, London; 1964. [5] East LF. Aerodynamically induced resonance in rectangular cavities. J Sound Vib 1966;3:277–87. [6] Nelson PA, Halliwell NA, Doak PE. Fluid dynamic of a flow excited resonance Part I: experiment. J Sound Vib 1981;78:15–38. [7] Nelson PA, Halliwell NA, Doak PE. Fluid dynamic of a flow excited resonance, Part II: flow acoustic interaction. J Sound Vib 1983;91:375–402. [8] Elder SA, Farabee TM, DeMetz FC. Mechanisms of flow-excited cavity tones at low Mach number. J Acous Soc Am 1982;72:532–49. [9] Burroughs CB, Stinebring DR. Cavity flow tones in water. J Acous Soc Am 1994;95:1256–63. [10] Disimile PJ, Toy N. Acoustical properties of a long rectangular cavity of constant cross-section immersed in a thick boundary layer. Int J Mech Sci 2004;46:1827–44. [11] Bres GA, Colonius T. Three-dimensional instabilities in compressible flow over open cavities. J Fluid Mech 2008;599:309–39. [12] Rowley CW, Williams DR. Dynamics and control of high-Reynolds-number flow over open cavities. Ann Rev Fluid Mech 2006;38:251–76. [13] Gloerfelt X, Cavity noise, Chapter 0, VKI Lectures: Aerodynamic noise from wall-bounded flows. Von Karman Institute; 2009. [14] Micheau P, Chatellier L, Laumonier J, Gervais Y. Stability analysis of active control of self-sustained pressure fluctuations due to flow over a cavity. J Acous Soc Am 2006;119:1496–503. [15] Cattafesta LN, Song Q, Williams DR, Rowley CW, Alvia FS. Active control of flow-induced cavity oscillations. Prog Aerosp Sci 2007;44:479–502. [16] Illingworth SJ, Morgan AS, Rowley CW. Feedback control of flow resonances using balanced reduced-order models. J Sound Vib 2011;330:1567–81. [17] Yoo SP, Lee DY. Time-delayed phase-control for suppression of the flowinduced noise from an open cavity. Appl Acous 2008;69:215–24. [18] Nelson PA, Elliott SJ. Active Control of Sound. London: Academic Press; 1992. [19] Lauterbach A, Ehrenfried K, Loose S, Wagner C. Microphone array wind tunnel measurements of Reynolds number effects in high-speed train aeroacoustics. Int J Aeroacous 2012;11:411–46. [20] Rodriguez Verdugo F, Stephens DB, Bennet GJ. Partially covered cylindrical cavity noise: resonance identification. In: C. Meskell, G. Bennet, (Eds.), FlowInduced Vibration, Dublin, Ireland; 2012. [21] Tam CKW. The acoustic modes of a two-dimensional rectangular cavity. J Sound Vib 1976;49:353–64. [22] Koch W. Acoustic resonances in rectangular open cavities. AIAA J 2005;43:2342–9. [23] Hein S, Koch W, Schöberl J. In: Acoustic resonances in a 2D high lift configuration and 3D open cavity, 26th AIAA Aeroacoustics Conference, Monterey, USA, May 2005, paper 2005–2867. [24] Vörlander M. Auralization of spaces. Phys Today 2009:35–40. [25] Allen JB, Berkley DA. Image method for efficiently simulating small-room acoustics. J Acous Soc Am 1979;65:943–50. [26] Suh JS, Nelson PA. Measurement of transient response of rooms and comparison with geometrical acoustic models. J Acous Soc Am 1999;105:2304–17. [27] Lehman EA, Johansson AM. Prediction of energy in room impulse responses simulated with an image-source model. J Acous Soc Am 2008;124:269–77. [28] Cobo P. Application of shaping deconvolution to the generation of arbitrary acoustic pulses with conventional sonar transducers. J Sound Vib 1995;188:131–44. [29] Rife DD, Vanderkooy J. Transfer function measurement with maximum length sequences. J Audio Eng Soc 1989;37:419–43. [30] Garai M. Measurement of the sound-absorption coefficient in situ: The reflection method using periodic pseudorandom sequences of maximum length. Appl Acous 1993;39:119–39. [31] Cobo P, Cuesta M, Pfretzschener J, Fernandez A, Siguero M. Measuring the absorption coefficient of panels at oblique incidence by using inverse filtered MLS signals. Noise Control Eng J 2006;56:414–9. [32] Cobo P, Fernández A, Cuesta M. Measuring short impulses responses with inverse filtered maximum-length sequences. Appl Acous 2007;68:820–30.