Dispersion and absorption of structure-borne sound in acoustically thick plates

Applied Acoustics 41 (1994) 97-111

Dispersion and Absorption of Structure-Borne Sound in Acoustically Thick Plates J. H. R i n d e l The Acoustics Laboratory, Technical University of Denmark, Building 352, DK-2800 Lyngby, Denmark (Received 14 October 1992; accepted 22 December 1992)

A BS TRA C T Building constructions made from concrete, light-weight concrete and brickwork are often so thick that the propagation of structural sound is shifted from bending waves to shear waves within the frequency range of building acoustics. This means that the dispersion function for the phase speed will be a complicated function of the frequency, which among other things influences the modal density. For such so called acoustically thick plates a theoretical basis is presented which has been confirmed through measurements of phase speed as a function of the frequency for a number of wall and slab elements. The loss factor has also been measured, using the time reverse decay method in order to avoid measuring errors from the third octave band filters. Measured acoustic data for clinker concrete and porous concrete are given.

1 INTRODUCTION Measuring methods for structure-borne sound and calculation models for sound transmission usually assume that vibrations travel like bending waves, i.e. the phase speed is assumed to be proportional to the square root of the frequency. However, this is not correct for many structures in buildings. Typical partition walls and slabs are relatively thick compared to the wavelength of the structure-borne sound within the frequency range of interest; they are acoustically thick plates. Such plates exhibit a more complicated dispersion function than thin plates. 97 Applied Acoustics 0003-682X/94/$06.00 © 1994 Elsevier Science Publishers Ltd, England.

Printed in Great Britain

J. H. Rindel

98

The main purpose of the present work is to establish and to verify as simple a basis as possible for structure-borne sound in thick plates. The experimental work has been concentrated on clinker concrete and porous concrete, partly because walls and slabs made from these materials were expected to behave like thick plates, and partly because--at least in Denmark--there is a great need for a better understanding of how these materials behave acoustically. This is particularly true in connection with flanking transmission. Throughout this paper the term velocity refers to the vibration of particles or surfaces, whereas the term speed is used for the propagation of waves. 2 PHASE SPEED OF WAVES IN A PLATE 2.1 Types of waves

Two types of waves are essential for the propagation of structure-borne sound in plates: bending waves (phase speed cB) and shear waves (phase speed cs). These phase speeds are given by: 1 cB = q ~ m

(1)

Cs -- ~

(2)

where to is the angular frequency, B is the bending stiffness and m is the mass per unit area. G is the shear modulus and h is the thickness of the plate. Note that the shear wave has a frequency-independent speed, whereas the speed of bending waves increases with frequency. For typical building materials the shear wave speed is in the range from 750 m s-1 (porous concrete, 660 kg m 3) to 2000 m s-~ (dense concrete, 2300 kg m-a), which corresponds to about two to six times the speed of sound in air. The actual wave in a thick plate will in general be a combination of bending and shear waves (a so-called Mindlin wave). Kurtze & Watters 2 have set up an equation for the effective phase speed cBeff, assuming the shear stiffness and the bending stiffness to be connected in parallel: 24

Cs C Beff

+ CBCBeff 42 _ c 4 ~ = 0

(3)

The solution of this yields Cgefr=Z

2 +2

1+

4ccs 4 [~l

(4)

The result is illustrated as a function of frequency in Fig. 1. It is seen

Dispersion and absorption of sound in thick plates

: es

1000

~

c

............

~

........

-y>,,

10

•

: /. ._. . ' /

•

100

99

....................... ire

fs

100

1000

10000

Frequency. Hz

Fig. 1. Effective phase speed as a function of frequency (dispersion curve) for transverse waves in a thick plate, calculated from eqn (4). The phase speeds for bending waves, cB, and for shear waves, Cs, are shown as asymptotic curves. This theoretical example corresponds to a plate with speed of shear wave equal to three times speed of sound in air, fc = 80 Hz and fs = 720 Hz.

that cB is an asymptote at low frequencies, as in cs at high frequencies. The actual wave, which could be called the effective bending wave, is some mixture of the two pure types. A somewhat simpler approximation of eqn (4), which is accurate within +1%, is: 1 1-1/3

2.2. Critical frequency

The critical frequency f~ is by definition the frequency where c~ff equals c, the speed of sound in air. The critical frequency can be related to pure bending waves only if cs >> c; however, for the building materials considered in the present paper the critical frequency is low enough for this assumption to be reasonable, so: d

_ (6)

fc=~ ,o h where Kc is introduced as a convenient material constant, the coincidence number, and h is the thickness of the plate• In the following it will be useful to express the phase speed of bending waves given by eqn (1) in terms of the critical frequency: cs = (7)

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J. H. Rindel

2.3 Shear wave cross-over frequency

The shift from bending waves to shear waves can be characterised by a cross-over frequency f~, which is defined as the frequency where cB equals % i.e. at the intersection of the two asymptotes shown in Fig. 1.

q ,Vm This cross-over frequency may be used to rewrite the phase speed of the effective bending wave, eqn (4), as follows:

CBeff =c ,~,f~/-- ~l + 1~ ~

1+

(~)2

(9)

A consideration of whether a plate is acoustically thick or not should be related to the shear wave cross-over frequency. Traditionally the guideline has been that the plate is thick if the thickness h exceeds AB/6, where the bending wavelength is AB = cB/f. Thus the traditional limiting frequency for a thick plate is:

l [C,~2

k = ~[. t ~ ) : f c

(C c.)2

(10)

The use of this and eqn (8) will be discussed later in relation to the experimental results.

3 M O D A L DENSITY IN R E C T A N G U L A R PLATES The relatively complicated frequency dependence of CBen- means that the natural frequencies of the normal modes cannot be calculated directly as for a thin panel. The condition for a normal mode in a simply supported rectangular plate with dimensions lx and ly is: ~ t2ft 2

(mt2+(n12

where m and n are integers. The solutions to eqn (11) yield the natural the lowest natural frequency for a simply supported plate is f~. At low frequencies it will usually be allowable to assume bending waves, so the lowest natural frequencies can be calculated from:

frequencies fmn and

C2 [ [ m ~ 2

[n~21

fmn = --4fc tt --lv) -t-fly--)I

(12)

It is interesting to observe from this formula that the first natural frequencies are widely spread. The interval of two octaves between f~

Dispersion and absorption of sound in thick plates

101

and f22 will contain only two natural frequencies, namely f~2 and f21- If third octave bands are considered there will be bands without any modes below f22 = 4fll. For f > 4f~ (at least) the modal density may be approximated by a statistical approach and the average number of modes below a given frequency f is found to be: 3 f2 N ( f ) = 7rS ~T-(13) C Beff

where S is the area of the simply supported rectangular plate. It follows from this that the modal density is:

dN_ 2rrS 2f-f- (1 df CB~

f

dCBeff]

(14)

c~ff df )

It is seen that the slope of the phase speed versus frequency must be taken into account. The factor within the brackets can vary between 0-5 for pure bending waves and 1 for shear waves. For the acoustically thick plates considered here there will be two asymptotic approximations to eqn (14): dN ~ (rrS(fJ c2) 0 c < f J 2 )

df

12rrS(fl~ ) i f > L / Z )

(15)

The modal density is constant at low frequencies but increases with frequency above a cross-over frequency, which is found to be f/2. The observations in this section should be of special interest to those who want to use SEA for calculations of building structures. Statistical methods should only be used at frequencies higher than four times the fundamental natural frequency of the plate, and above a certain frequency the modal density is no longer constant.

4 THE G R O U P SPEED A N D ITS APPLICATIONS

4.1 Group speed in a plate The energy in a wave propagates with a speed which is called the group speed. For a frequency band with the arithmetic average angular frequency toa the group speed is (see Morse & Ingard 4 p. 478):

dto [d ~]-1 __CBeff(tOa)[ 1 Cg

~

tdto cBeerJ,o,

tOa [dCBcfr] ]-l Cacfr(tOa) ~, dto/,~d

(16)

where c~fr is the phase speed. For shear waves, i.e. above the cross-over frequency f~, the phase speed is a constant equal to cs, and so is the

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102

group speed. For bending waves, however, the group speed will be twice the phase speed. This means that the cross-over frequency for the group speed is only£/4, and approximately: f2¢B cg -'- ~

=

2c f'[f/f,.

cs

0c < J~/4) (f > A/4)

(17)

The cross-over frequency j~ is given by eqn (8). 4.2 Calculation of boundary losses In a plate with steady state average rms vibrational velocity v and energy travelling equally in all directions, the incident power on the perimeter U is (see Cremer et al. s p. 426): Pinc ---

1 Um(v2)Cg

(18)

m is the surface density and Cg the group speed. Let ot be the ratio of absorbed energy at the boundary, then the loss factor related to boundary losses is: where

Pabs _

otPin c

rlbnO -- wE

_

wmS(v2)

¢ggOt

2wefS

(19)

where E is the steady state energy of vibration in the plate. In this expression the group speed Cg may be taken from the approximation of eqn (17). 4.3 Transmission loss between two plates Let plate 1 be the excited plate and plate 2 the receiving plate. Under the assumption that the energy travels equally in all directions the incident power on the connection edge with the length l is: 1 lml(v2)Cgl

einc = ~-

(20)

In the receiving plate the transmitted power must equal the absorbed power under steady state conditions, and the total loss factor of the receiving plate is: 2'2 _ P2,abs _ Ptrans rh f T 2 wE2 wm2S2(v2) (21) The transmission factor is then given by:

Y-

Ptrans_ 27r2 m2(v2) $2 2"2 Pi,c mj(v~ lCglT2

(22)

Dispersion and absorption of sound in thick plates

103

From this formula the transmission factor may be determined experimentally by measurement of the difference between the velocity levels in the two plates and the reverberation time in the receiving plate. The point that has usually been overlooked is the more or less frequencydependent group speed in the excited plate, Cg~,which may be taken from the approximation of eqn (17). In most building constructions the influence of the shear waves must be taken into account since the crossover frequency for the group speed is as low as fJ4. For the constructions actually investigated in the present work this cross-over frequency is between 200 and 900 Hz.

5 M E A S U R I N G T E C H N I Q U E FOR PHASE SPEED IN PLATES When a plate of infinite extent is excited by a harmonic point force at r = 0, the transverse velocity v of the plate is according to Morse and Ingard 4 p. 221: v(r, t) = Vo[Jo(kr ) + iNo(kr ) - Jo(ikr) - iNo(ikr)] exp ( - ioot) ~- v o

[ e x p [ i ( k r - 7r/4)1- exp(-kr)] exp(-itot)

(kr>> 1) (23)

where v0 is the velocity of the excitation point. The last expression is not valid in the near field.

5.1 Low frequencies At low frequencies the measuring technique is based on a measurement of the phase difference between one accelerometer at the excitation point and another accelerometer outside the near field in the plate at a distance r. In order to separate the direct wave from the disturbing reflections a hammer was used to generate an impulse excitation. The accelerometer at the excitation point was covered with a specially made brass cup, on the top of which the hammer beat was made. From eqn (23) the phase angle between the two signals is: ~ v , v0) = arctan kr-

sin (kr - 7r/4) cos (kr - 7r/4) - exp ( - k r )

Ir/4

(24)

(kr >> 1)

At low frequencies it was sometimes impossible to satisfy the requirement k r >> 1. Since ~ v , v0) ~ 0 for k r ~ O, the phase correction was taken as k r ¢r/4 for k r < 1, which was found to be an acceptably simple

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J. H. Rindel

approximation. Thus the final expression to determine kr from the measured phase angle is: kr ~- ~q~(v, v0)/(l - 7r/4) (q~(v, v0) + 7r/4

(q~ < 1 - ~'/4) (q~ > 1 - 7r/4)

(25)

The error due to this simplified correction is assumed to be negligible compared to the measuring uncertainty. The phase speed as a function of frequency is calculated from: ~o 2~rfr CBeff -- ~: -- (kr~ (26) where (kr) is taken from eqn (25). At higher frequencies the measured phase angle will exhibit one or more turns of 2~r, and after a few turns the method can no longer be used.

5.2 High frequencies At high frequencies a different and much more straightforward method was used. If the phase angle is measured between the velocities at two positions, both of which are at a sufficient distance from the excitation point, i.e. kr~ > 7r and kr 2 > 7r, then, from eqn (24): ~ v l , v2) = k(r2 - rl) and the phase speed is determined from: ~o 2~']' CBen- k - ~ v l , v2) (r2 - rj)

(27)

(28)

Again this m e t h o d m a y be used for one or more turns of the phase.

5.3 Measuring instruments The measurements were made with a two-channel real-time analyzer of type B&K 2133. The sampling time interval was 15 /xs, and the bandwidth was 1/12 octave. The distance between the pair o f accelerometers was 1-2 m in the low frequency range (with one accelerometer at the excitation point), and typically 0.5 m for measurements in the high frequency range.

6 M E A S U R I N G T E C H N I Q U E F O R LOSS F A C T O R The loss factor was measured in 1/3 octave bands using decay measurements in the range 100 to 5000 Hz. In order to avoid errors from the

Dispersion and absorption of sound in thick plates

105

reverberation of the filters the time reversal method 6 was used. The impulse response was recorded and time reversed before the 1/3 octave band filtering. Then the signal was squared and integrated to obtain a decay curve for each frequency band. The reverberation time was determined from a regression line between - 5 dB and - 3 5 dB for most measurements. The real-time analyzer type B&K 2133 was used for these measurements. The results from nine different positions on each plate were averaged, and from the average reverberation time T the loss factor was calculated as:

2-2 ~7

(29)

fT

w h e r e f i s the centre frequency of the 1/3 octave band. The limit for reliable results of reverberation time measurements using a filter bandwidth B has been shown to be 6 B T > 16 for forward analysis and B T > 4 for time reversed analysis. For 1/3 octave filters having a relative bandwidth of 0.23 this leads to the following upper limits for reliable measurements of loss factor: 0.032 0.127

2 . 2 B / 1 6 f ~~7 <

2 . 2 B / 4 f ~-

(forward) (time reversed)

(30)

The need for using the time reversal method was demonstrated in several of the results, especially at low frequencies.

7 EXPERIMENTAL RESULTS

7.1 Phase speed Measurements of the phase speed dispersion curve were performed on six walls and six slabs, the main data of which are given in Table 1. As shown in the examples given in Figs 2 4 the shape of the curve was in reasonable agreement with the theoretical curve in Fig. 1. The examples are selected in order to show different levels of the shear wave speed. The measurements were not easy to do and no reliable results could be obtained below 300 Hz. The low frequency part of the curves was compared with the expected slope for bending waves, eqn (7), and from this the critical frequency and the coincidence number K c were calculated. F r o m the more or less horizontal part in the high frequency range the shear wave speed was determined. Both values are given in Table 1. The table also shows the limiting frequency for thick plates fh calculated from eqn (10) and the

J. H. Rindel

106

TABLE 1 Measured Data for Walls and Slabs

Material

Density h (kg/m 3) (mm)

Walls: Clinker concrete

K,. (m/s)

c~ (m/s)

.[~. (Hz)

J), {Hz)

,{74 (Hz)

1030 1 690 I 740

100 100 240

35 27 33

1 100 1 100 1 100

351 270 137

1127 867 440

897 690 350

660 660 660

100 100 150

55 53 53

750 750 750

547 530 355

617 597 400

650 630 422

1700 1750

180 240

34 34

1400 1400

190 140

540 398

787 580

Sandwich clinker concrete

1125 1 000 940

120 180 260

44 44 49

950 950 950

365 244 189

567 379 294

696 465 360

Hollow concrete

1700

185

18

1000

96

974

203

Porous concrete

Slabs: Clinker concrete

2000 ~-J j-

1000 E "o Q. t~

500

200

'

500

.

.

.

.

~

'

'

1000

2000

5000

Frequency. Hz

Fig. 2. Measured phase speed as a function of frequency for a 100-mm clinker concrete wall (1690 kg/m3). ( . . . . . ) bending wave approximation; ( - - - ) shear wave approximation.

Dispersion and absorption of sound in thick plates

107

2000

1000 E

500

f_

200

.

. 500

.

.

. . 1000

.

. . 2000

5000

F r e q u e n c y . Hz

Fig. 3. Measured phase speed as a function of frequency for a 150-mm porous concrete wall (660 kg/m3). ( . . . . . ) bending wave approximation; ( - - - ) shear wave approximation.

2000

.

,

.

.

.

.

,

'...-"

'

1000 E o o o. t~

500 a_

200

, 500

. . . .

, 1000

, 2000

, 5000

F r e q u e n c y . Hz

Fig. 4. Measured phase speed as a function of frequency for a 240-mm clinker concrete slab (1750 kg/m3). ( . . . . . ) bending wave approximation; ( - - - ) shear wave approximation.

108

J. H. Rimh'/

cross-over frequency for group speed jJ4. Both frequencies are supposed to give an upper limit for the bending wave assumption. So, it is interesting to observe that, with the exception of the hollow concrete slab, the two frequencies are of the same order of size (within +2/3 octaves). 7.2 Loss factor Examples of measured loss factors are shown in Figs 5 and 6. The wall elements in Fig. 5 had the dimensions 2.54 m × 1-80 m and they were mounted in a test set-up as the lower wall in a T-junction to a 180-mm clinker concrete slab. In these examples the lowest natural frequency f ~ was calculated to be between 25 Hz and 50 Hz, and the decay measurements were made without any problems. One o f the walls was measured with an elastic connection to the slab, and in this case the loss factor was clearly higher than in the other examples with rigid connections. The loss factor generally increases towards the low frequencies, but in this case the loss factor has a local maximum around 200-250 Hz, which may be related to viscous losses in the elastic connection. It should be noted that these results could not have been obtained if a traditional forward analysis had been used; some of the results are very close to the upper limit of time reversed analysis given by eqn (30).

"\.

/

\

\./

\'\ \

.05

\, %. / - / .

",

\

o

02

~,

"x.

.01

.005

' 125

' 250

'

' 500 Frequency.

Fig. 5.

' 1000

' 2000

'

' 4000

Hz

Measured loss factor as a function of frequency for three walls, which were the lower wall in a T-junction to a slab of 180-mm clinker concrete. ( . . . . . ) 100-mm clinker concrete wall (1690 kg/m 3) with rigid connection to the slab; (. . . . . . ) 100-mm porous concrete wall (660 kg/m 3) with elastic connection o f 4-mm P U R to the slab; ( - - - ) 150-mm porous concrete wall (660 kg/m 3) with rigid connection to the slab.

Dispersion and absorption of sound in thick plates

.2

,

•

,

,

109

,

"'-~ .05

t~""~l

o

\

",, \ o J

.02

.01

t

.005 125

250

i

i

t

t

500

1000

2000

t

i

4000

Frequency. Hz

Fig. 6. Measured loss factor as a function of frequency for three slabs without connecting walls. (©... G) 180-mm clinker concrete slab (1700 kg/m3); ( D - . - 12) 240-ram clinker concrete slab (1750 kg/m3), ( V - - - V ) 180-ram clinker concrete sandwich slab (1000 kg/m3).

Examples of measured loss factors for slabs are shown in Fig. 6. The slabs were long single elements used also for the phase speed measurements, so there were no connecting walls. The dimensions were 5.00 m × 1.20 m, which means that the lowest natural frequency fLl was calculated to be 90 Hz, 115 Hz and 155 Hz, respectively, in the three examples. Due to the large aspect ratio the distribution of the natural frequencies was not as uniform as for the walls, and the decay measurements were quite difficult in the frequency range 100 Hz to around 500 Hz. In several cases it was impossible to obtain any reliable decay curves in some of the 1/3 octave bands, which can be explained by the absence of modes in those frequency bands. This is the reason for the interruptions in some of the curves in Fig. 6. Probably these difficulties would not have been observed if traditional forward analysis has been used; in that case the response of the filters would mask the absence of reverberation in the structure. At high frequencies (above 1000 Hz) the measured loss factors for all walls and slabs were about the same relatively low value of 0-01. This may be interpreted mainly as internal losses.

8 CONCLUSIONS A plate may be called acoustically thick if structure-borne sound does not propagate as bending waves in the entire building acoustic frequency

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J. H. Rindel

range. Walls and slabs of clinker concrete and porous concrete have been shown to be typical examples of this. As measuring method for the phase speed as a function of frequency has been developed, and this has been used to verify a theoretical model for the dispersion function. This assumes a gradual shift from bending waves at low frequencies to shear waves at high frequencies. A shear-wave cross-over frequency J~ has been introduced in order to indicate approximately the shift in phase speed. One of the consequences of the theoretical model is that the shift in group speed occurs two octaves below the shift in phase speed, i.e. at3'~/4, which should be used as an upper limit for the assumption of bending waves. Another consequence is that the statistical modal density function will change around a frequency fJ2. Loss factors have been measured in 1/3 octave bands, and in several cases values as high as 0.12 were measured, which was made possible by use of the time reversed decay method. In general the modal density was very low in the low frequency range. In some cases measurements were impossible in certain frequency bands, which may be explained from a lack of natural modes in those bands. These difficulties are taken as a sign of the reliability of the measuring method.

ACKNOWLEDGEMENTS This work was supported by the Danish Technical Research Council (STVF), and it was part of a joint project with the Danish Technological Institute in Aarhus (DTI). The materials used for the measuring objects were kindly delivered by H + H Industri A/S, Fibo A/S and LemvighMtiller & Munck A/S. The numerous measurements were performed with great care by Per Andersen, MSc, who also assisted in the development of the measuring methods.

REFERENCES 1. Ver, I. L. & Holmer, C. I., Interaction of sound waves with solid structures. In Noise and Vibration control, ed. L. L. Beranek. McGraw-Hill New York, 1971, Ch. ll. 2. Kurtze, G. & Watters, B. G., New wall design for high transmission loss or high damping. J. Acoust. Soc. Am., 31 (1959) 739-48. 3. Rindel, J. H., Prediction of sound transmission through thick and stiff panels. Proc. Inst. Acoust., 10(8) (1988) 119-26.

Dispersion and absorption of sound in thick plates

111

4. Morse, P. M. & Ingard, K. U., Theoretical Acoustics. McGraw-Hill, New York, 1968. 5. Cremer, L., Heckl, M. & Ungar, U., Structure-Borne Sound, 2nd edn. Springer-Verlag, Berlin, 1988. 6. Jacobsen, F. & Rindel, J. H., Time reversed decay measurements. J. Sound Vibration, 117(1) (1987) 187-90.