Acoustic boundary layer influence on scale model simulation of sound propagation: Theory and numerical examples

Journal of Sound and Vibration (1986) 105(2), 321-337

ACOUSTIC MODEL

BOUNDARY

SIMULATION AND

LAYER

INFLUENCE

OF SOUND

ON

PROPAGATION:

NUMERICAL

SCALE THEORY

EXAMPLES

M. ALMGREN Department of Building Acoustics, Chalmers University of Technology, S-412 96 Giiteborg, Sweden (Received 30 January 1985, and in revised form 3 May 1985)

Scale model simulation of sound propagation above a solid surface will give a systematic and calculable error in the predicted sound field, because the acoustic boundary layer

above the surface has an apparent admittance which is not invariant under scaling. The typical error is approximately 5 dB depending on the geometrical configuration, scale and frequency. The effect of the acoustic boundary layer admittance is negligible for sound propagation above an acoustically soft surface (e.g., grassland). One may moreover note, with reference to scale model simulation of concert hall acoustics, that the absorption coefficient of a solid surface increases with frequency.

1. INTRODUCTION

Scale model simulation has been used for many decades in room acoustics and environmental acoustics because it is a convenient tool to use when testing theoretical sound propagation models and when designing a concert hall or predicting noise immission. When scaling the dimensions and wavelength in a ratio 1: s, the frequency must be scaled in the ratio s: 1. The effects of geometrical divergence, scattering by small objects, and diffraction of the sound rays from the edges of large obstacles are then invariant under scaling. However, the effects of sound absorption in the air and reflections at absorptive surfaces are not (see, e.g., the report by Almgren [l]). Drying the air or using another gas than air for the atmosphere are two ways of overcoming the air absorption problem (the papers by Rapin and Favre [2] and by Ishii and Tachibana [3]). Compensation for the large sound absorption at high frequencies is possible also with a time and frequency dependent amplification (see reference [l] and the paper by Theile and Kiirer [4]). In a project at the Department of Building Acoustics at Chalmers University of Technology, the technique for scale model simulation of outdoor sound propagation has been investigated [l]. An understanding of the principles of acoustical scale modelling and a detailed knowledge of the theories of sound propagation can facilitate the practical use of the technique, since sources of errors can be found, quantified and corrected for. Therefore, in this study, a very simple scale model experiment was simulated and investigated theoretically: namely, an experiment in which the purpose is to predict the sound pressure level from a point source above a plane, rigid and solid surface. It is assumed that the source is an ideal point source, that the sound pressure is measured at a point, and that the surface is ideally plane and rigid. Then viscosity and heat conduction in the air will influence the result. The development of the calculation of the losses due to viscosity and heat conduction dates back to the nineteenth century. Helmholtz [5] studied theoretically the influence 321 0022-460X/86/050321 + 17 $03.00/O

0 1986 Academic Press Inc. (London) Limited

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M. ALMGREN

of air friction on sound motion, and Kundt [6] demonstrated experimentally that the sound speed in tubes is lower the thinner the tube and the lower the frequency. However, the measured sound speed was still lower than predicted by the Helmholtz theory. Kundt suggested that this discrepancy could be explained by the heat exchange between the air and the tube walls. Heat conduction was then included in the theory by Kirchhoff [7]. The dependence of the sound speed variation with frequency and tube radius was confirmed but lack of data for the coefficient of thermal conductivity inhibited a quantitative comparison of actual values. Plane wave reflection at a plane and solid surface, with account taken of the viscous and thermal conduction effects at the boundary, was theoretically studied by Konstantinov [8] and subsequently by Cremer [9]. Konstantinov’s analysis was in terms of acoustic, viscous, and thermal waves which are coupled at the boundary to satisfy the boundary conditions. The viscous and thermal waves die out exponentially from the boundary. The acoustic boundary layer was defined by Cremer as a layer at the surface above which the amplitudes of the viscous and thermal waves are less than l/e times their respective value at the boundary. Both Konstantinov and Cremer calculated an apparent specific acoustic admittance, reflection factor, and absorption coefficient for the acoustic boundary layer with plane waves incident from 0 to 90 degrees. In a limited frequency range, Cremer and Miiller [lo, 111 also determined the sound absorption coefficient averaged over all angles of incidence. The absorption coefficient is of interest when studying, e.g., the absorption of empty reverberation test rooms and sound reflectors in a concert hall [ll]. The theory for the acoustic boundary layer has also been presented tutorially by Morse and Ingard [12] and by Pierce [13, see pp. 508-534-J The effects of the acoustic boundary layer on the propagation characteristics in ducts have been experimentally verified, as discussed in reference [ 131. To the author’s knowledge, no measurements have been made to see how well the sound field close to a Bat solid surface can be calculated. The reason for this might be that the effects are rather small in the audio frequency range. scale model measurements above a painted concrete Nicolas et al. [14] performed surface and found that this hard surface could not be considered as an infinite impedance surface leading to reflection with no phase shift. They did not explain the phenomenon physically, but fitted the calculated sound pressure level relative to free field to the measured data using the effective flow resistivity as the parameter in Chessel’s ground model [ 151. The finite impedance phenomenon in the experiment by Nicolas et al. was explained by Embleton et al. [16] as being caused by heat conduction and viscous effects in a thin boundary layer adjacent to the surface. They referred to Konstantinov’s expression for normal incidence impedance and found the equivalent effective flow resistivity in Chessel’s ground model. The object of this paper is to show that a systematic error is made when scale model simulation of sound propagation above a solid surface is performed, and that this error can be predicted. Therefore it is possible to compensate for the error.

2. THEORY When describing outdoor sound propagation, the sound pressure relative to free field is a relevant quantity. The field from a distributed source can be calculated as an integral of the contributions from a distribution of point sources over the source volume (see reference [ 131, pp. 163-165). Therefore it is of interest to study the field from a point source.

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BOUNDARY

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IN

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323

MODELS

There are two ways to calculate the sound field at a receiver point from a point source above an infinite plane surface, henceforth denoted the plane wave reflection model and the spherical wave reflection model. The definition of geometrical parameters is shown in Figure 1.

Figure 1. Definition of the geometry for the problem. distance. height; R,,,, horizontal source-to-receiver

2.1.

THE

PLANE

WAVE

REFLECTION

*, Source;

/I, receiver;

h,, source

height;

h, receiver

MODEL

plane wave reflection model is a simple ray acoustics model (see Figure 2) in which it is assumed that the sound pressure at the receiver point can be calculated as the sum of the instantaneous sound pressures from the two rays. Here the complex notation for signals is used with the time factor exp (-iwt) commonly used in sound propagation studies. The

Figure 2. 8 is the angle of incidence, reflected ray.

R, is the length of the direct ray, and R, is the length of the ground

The two rays have to be summed with correct phase. The phase difference is due to the path length difference, the phase shift at the reflection, and possibly other mechanisms depending on which parameters relating to sound propagation that are taken into account. A good presentation of ray acoustics has been given by Pierce [13, see pp. 371-423, 495-SOlJ. The amplitude of each ray at the receiver point can be calculated by taking into account the divergence, the magnitude of the reflection coefficient for the regected ray, and other mechanisms. The effect of refraction in an inhomogeneous atmosphere is not taken into account here. The effect of dissipative mechanisms in the air on amplitude and phase is shown below. In the plane wave reflection model it is assumed that the wavefronts are locally planar and that the reflection coefficient for incident plane waves Rp can be used. The reflection coefficient for plane waves can be calculated from the normalized surface admittance, p, and the angle of incidence, 0, as %=(cos O-/?)/(cos 0+/3).

(1)

p is defined as P=

PC/Z,

(2)

324

M. ALMGREN

where Z is the surface impedance and pc is the characteristic impedance of the air. The surface impedance is here defined as Z = -p/(v

- n).

(3)

n is the outward normal vector of the surface, p and v are the complex sound pressure and the complex particle velocity of the acoustic field at the surface. The sound pressure relative to free field, with the dissipative mechanisms in the air neglected is then easily found: pJp,-- = 1 + R,(R,/

R,) eikcRzeR1),

(4)

where k is the wave number w/c. The effect of sound absorption in air can be included by introducing a complex wavenumber (see reference [13], p. 558) into equation (4) 1 -i

(f/.L,i)'

flf,i

NZo(QA)max,i 2

2

l+ (f/_L,i)'

l+(f/f,i)z

1’

(5)

where the notation conforms with reference [17]. (A list of symbols is included in Appendix 2.) The imaginary part of the wavenumber is the absorption coefficient (Y for sound absorption in air. In reference [l] it was shown that (Y could be calculated from the formulas given in the American standard ANSI S1.26 [ 171 with an excellent agreement between the calculated and the measured sound pressure level decay with distance in the octave bands 2-125 kHz. The formulas for the calculation of (Yare compiled in reference [l] together with a Fortran subroutine. The relaxation processes in air cause a slight dispersion of the sound waves. A small correction, E, to the phase velocity in air can be calculated from

(6) where E can be deduced from equation (5). An examination of the frequency dependence of the phase velocity correction shows that E is asymptotically zero at low frequencies, increases with frequency, and approaches a constant value at high frequencies. Numeri= 251 Hz, &, = 20 900 Hz at a temperature of 20°C the cally, ( .sc,Jmax= 0.14 m/s and fr,,, relative humidity of 30% RH and a static atmospheric pressure of 1.013 * 10’Pa. An examination of equations (l)- (4) shows that the admittance or the impedance and the angle of incidence must remain invariant under scaling, if the model experiment is to predict the sound pressure relative to free field correctly. Further, equations (5) and (6) indicate that the effect of sound absorption in the air will not be correctly scaled if the same medium is used in the two cases. 2.2. THE SPHERICAL WAVE REFLECTION MODEL Thomasson [ 181 has derived an exact integral solution of the scalar wavefield produced by a point source in the presence of an infinite impedance boundary in a homogeneous loss-free atmosphere. His solution (equations (32)-(41) in [18]) is rewritten here to give the sound pressure relative to the free field sound pressure: PR/Pff = 1+ Q(R,IRJ

eik(RZ-R’)+ PsurdPfiv

(7)

ACOUSTIC

BOUNDARY

LAYER

IN

SCALE

325

MODELS

where

-27Tk~HC’[kRhor(l -P2)“21 e-w(hs+hn) if Re (l/R,)

Psurface _

{

‘yo} >

1

Im {p}

and

> 0

eikRt

(84

Pff

0

otherwise

Hg’ is a Hankel function. For large arguments, the asymptotic expansion of Hc’ can be used which simplifies the numerical evaluation of equation (8a) and the interpretation of the surface wave term, leading to

_ I

_p*)l/*)ll*(l/R~~~)

(grkp*/i(l

Psurface _ Pff

(l/R,)

eik(l-P2)1/2R,o, e-iW(hs+h,)

eikRl

(8b)

ifRe{y,}~1andIm{p}>O

b

otherwise I=

Q = l -2kR,PI,

I0

W=(cos

a‘(eekR2’/ W”‘) dt,

0-tP)2+2it(l+/3

cos 0)--t’. (9-11)

The square root of W is defined such that Re { W”2} ~o=-~cos~+(1-~2)“2sinf?,

~0 >O

ifRe{yo}>1andIm{/3}<0,t~t, otherwise

9

(12)

t,=-Im{~}[Re{~}+cos~]/[l+cos(~)Re{/3}]. (13914)

The square root of 1 -p2 is defined such that Re((1 -p’)“‘}>O.

(15)

The solution can be interpreted only with difficulty in a ray acoustics approach. The first term in equation (7), however, represents the ray that goes directly from the source to the receiver. If Q in equation (7) is identified as a reflection coefficient for spherical waves, then the second term appears roughly as a specularly reflected ray. The ray approach is not correct here since in the ray the phase of the complex sound pressure would depend on R, only as exp (kR,), while Q has a complicated variation with distance along the ray. Likewise, the surface wave term can be interpreted as a ray only as a gross approximation, because of the cylindrical wave factor

(16) and the exponential

decrease of the amplitude with height e klmlPl(%+%),

1m

{pj


(17)

In the case of an ideally hard surface it is practically impossible to excite the surface wave. The imaginary part of the admittance is less than zero, but the condition in equation (8) that the real part of y. shall be greater than one leads to the condition (hs+hR)lRhor
(18)

where a is the small number defined by a = Re {p} = Im {jl} for the acoustic boundary layer admittance in equation (19). If, for instance, the horizontal distance is 1000 meters and the frequency is 1000 Hz, then the sum of the source and receiver heights must be

326

M. ALMGREN

than 4 x lo-’ meters, or, in a scale model experiment, if Rhor is 10 meters and f is 100 kHz, then hs + hR must be less than 1 x lo-’ meters. Of course it is practically impossible to realize a point source placed so close to a surface. There are several simplified solutions to the problem of determining the sound field from a point source above an impedance surface (see, e.g., the papers by Thomasson [19], Chien and Soroka [20,21], and Kawai et al. [22]). In this study Thomasson’s exact solution has been used because generality is maintained whereas in a simplified solution this is frequently decreased by some restrictions. Moreover, the computation time is not a limiting aspect in this context. From equations (7)-( 15) it is deduced that it suffices to keep the surface admittance and the angle of incidence invariant under scaling in order to obtain a true prediction of the sound pressure relative to free field. However, one must compensate for the effect of sound absorption in air on the wavenumber. less

2.3.

THE

ADMITTANCE

OF

A

PLANE

SOLID

SURFACE

WITH

AN

ACOUSTIC

BOUNDARY

LAYER

Pierce [ 13, see pp. 508-5341 has given a systematic derivation of the acoustic boundary layer theory. The main result of his compilation of the theory, which is of importance for this study, is that a plane, solid surface presents an apparent non-zero admittance to an incoming plane wave. The formula for the acoustic boundary layer admittance, equation ((19a) below), will here be used in the calculation of the sound field from a point source above a flat rigid surface, from equations (7)-( 15) above. Then it will be possible to deduce the influence of the acoustic boundary layer on the outcome of a scale model experiment. From reference [13, see pp. 508-5341 or reference [12] and in the current notation and with the time factor exp (-iot) the normalized surface admittance is given by P=(l/JZ)(l-i)(wj.~/pc*)“~[sin~B+(y-l)/(Pr)”2].

(I9a)

Pr is the Prandtl number, /..u+/K, and y is the ratio of the specific heats. For air in the temperature range 0” to 40°C ( y - 1)/a is 0.48 (calculated from data given in reference [23]). The viscosity of air is [13, see pp. 508-5341 CL= 1.846 x 10-5( T/300)3’2(300+ 110.4)/( T+ 110.4), where T is the absolute temperature becomes

(20)

in degrees Kelvin. For air at 2O”C, equation (19a)

p = 2.01 x 10m5(1 -i)fi[sin2

8 +0*48].

(I9b)

The derivation of equation (19a) can be found in reference [ 13, pp. 519-5291 or reference [12, pp. 281-2941. In short, the linearized fluid dynamic equations, with viscosity and heat conduction taken into account, can be solved by separating the wave field into three fields which are uncoupled except at the boundaries, interfaces and sources. One expects a disturbance in an extended space to be made up primarily of the ordinary acoustic wave field except near such perturbations. At a rigid boundary the superposition of the wavefields satisfies the boundary conditions: i.e., the normal and the tangential fluid velocities do all vanish at the boundary. The viscous and the thermal wave die out exponentially with distance from the boundary. The normal velocity component (into the surface) of the acoustic wavefield divided by the pressure gives the surface admittance.

ACOUSTIC

BOUNDARY

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SCALE

MODELS

327

The assumptions made in the derivation lead to the following conditions of validity for equations (19): f<< 10”Hz; ( KpCp):{:

<<

w

( PcpK)airK ( PcpK)solid,

I’*( ~CP)solid(Volume/surface afea),,,i+

(21a, b)

The conditions (21b) are well fulfilled by plane sheets of steel, glass, and wood fibre. The apparent admittance of a plane rigid surface can be used when studying the sound field quantities outside the acoustic boundary layer. Within the boundary layer, the tangential particle velocity caused by the viscous waves and the fluctuating temperature caused by the thermal waves must be taken into account. The boundary layer thicknesses are I,,,, = (2p/l0p)“> = 2.2 x lo-3/fl (22a) for the viscous wave and 1therm=

(2K/OPCp)“2

=

Ivis,J(

Pr)“’

=

2.6 X lo-‘/fl

(22b)

for the thermal wave: i.e., the boundary layer thickness is approximately 3 x 10-‘/G. The parameters in equations (22) were evaluated for air at 20°C. The boundary layer thickness obviously does not scale linearly with length (or with the inverse of the frequency). In equation (19), one sees that the admittance increases with the square-root of the frequency. Hence, the admittance of the acoustic boundary layer will not be correctly scaled, since in a correct scaling the admittance at the scaled frequencies must have the same value as at the full scale frequencies. For an acoustically soft surface, on the other hand, the usual admittance will dominate over the acoustic boundary layer admittance, even in the frequency range associated with scale model experiments (see Figure 9). The admittances of the actual surface and of the boundary layer can simply be added (see reference [12]). The reflection coefficient for incident plane waves is found by using equation (1). The magnitude of the reflection coefficient is minimum, 0.414, and the absorption coefficient is maximum, 0.828 for 8,i,=(77/2)-4*3

X

10m5G

(23)

in air, which is found after setting the derivative with respect to the angle of incidence equal to zero. The phase angle of the reflection coefficient is 7r/2 when the magnitude is minimum. At grazing incidence, i.e., when 0 = 7r/2, I$, equals -1 or equivalently the phase of R equals n: At normal incidence the phase has a small value. 3. DISCUSSION As stated in the introduction, the object of this paper is to show that a systematic error is made when a scale model simulation of outdoor sound propagation above a solid surface is performed, and that this error can be predicted. Therefore it is possible to compensate for the error. The systematic error is the difference between the calculated sound pressure level relative to free field at full scale and the corresponding calculated level in a simulated scale model experiment. Some examples are shown in Figures 3 and 4. Figures 3(a)-(d) show the calculated sound pressure level relative to free field from a point source above a flat and solid surface at full scale 1: 1 and at model scale 1: 100 for some geometrical configurations. The full scale dimensions in these examples are 200 m horizontal distance and 0.1 to 3 m equal source and receiver heights.

O-

10 -

-40

-30

100

-

(c)

I 300

/ 500

500

I

,

Full scale frequency

I

300

I

I

200

I

Full scale frequency

I 200

I 1000

1000

I

I

and model stole

700

1

I

and model stole

/ 700

frequency

1

1

(Hz)

I

,

10000

I I 7000 10000

(Hz)

I

50007000

1100

5000

I100

I 3000

I 2000

Full stole

I

,

I 3000

frequency

I 2000

-40. 100

20

-401 100

Cd)

(b)

I

, 500

1

300

I

I

frequency

500

,

/

scale frequency

t 300

Full scale

200

I

/

Full

I 200

1

Full scale

1

1000

I

/

and model scale

700

I

I

1

1 3000

I

I

/

3000

frequency

2000

/

Full scale

/

frequency

I 2000

ond model scale

I / 7001000

I I

I

I

(Hz)

/ 100 (Hz)

50007000

/lOO

1

I

I I 5000700010000

I

10000

Figure 3. (a) Calculated sound pressure level relative to free field from a point source above a plane solid surface with the acoustic boundary layer at full scale and at model scale 1: 100. The difference between the two curves is the systematic error inherent in scale modelling. The temperature was taken as 15°C in the full scale case and 20°C in the model scale case. The horizontal source-to-receiver distance is 200 m in full scale and 2 m in model scale. The source and receiver heights were equal, 3 m in full scale and 0.03 m in model scale. (b) As (a), but with 1.5 m full scale and 0.015 m model scale source and receiver heights. (c) As (a), but with 0.5 m full scale and 0.005 m model scale source and receiver heights. (d) As (a), but with 0.1 m full scale and 0.001 m model scale source and receiver heights.

_F

-10 ,o (D ., 5 -2o?!

8 c

0 E P .‘D

zo-

-401 100

20

ACOUSTIC

(ep) rwi

aw

01 awoial

d7

BOUNDARY

LAYER

IN

SCALE

MODELS

329

330 The

M.

sound

pressure

level was calculated

ALMGREN

as

L prelff= 20 lois IPRIP,,l.

(24)

of the surface (i.e., here the where pR/p,-- is given by equations (7)-( 15). The admittance admittance of the acoustic boundary layer) was determined by using equations (19) and (20) both in the full scale and model scale cases. The sound pressure level versus frequency in Figure 3(a) shows the characteristic series of interference minima and maxima. The troughs in the model scale curve, when compared with the full scale curve, are shallower and wider and they are shifted towards lower frequencies due to the larger admittance at high frequencies. When the source and receiver heights are lowered (see Figures 3(b)-(d)), the troughs are shifted towards higher frequencies and they become even wider. The systematic error is particularly large at the interference troughs and increases with frequency below the first troughs. Note that the deviation in the low frequency part of the diagrams (the part which is associated with most environmental noise problems) does not increase when the source and receiver come closer to the ground, which is expected from the simple plane wave reflection model in equation (4). The magnitude of the frequency response function IpR/pffl, was averaged in one-thirdoctave bands with centre frequencies 500-8000 Hz full scale and 5-800 kHz model scale, under the assumption of a constant (white) pressure magnitude in each band (see Figures 4(a)-(d)). The error is not at all negligible for scale model simulation of environmental noise. With, e.g., 200 m horizontal distance and 3 m equal source and receiver heights, the error is 2-6 dB in the bands 500-1250 Hz (full scale). The error decreases with decreasing source and receiver heights and is l-3.5 dB in the same bands for a 0.5 m height. Similar calculations were carried out for the horizontal distances 100 and 400 m and source and receiver heights 1.5 m (full scale dimensions). The error was then 1 to 4 dB in the 100 m case and 2 to 7.5 dB in the 400 m case in the one-third octave bands 500-1250 Hz (full scale). In the scale model experiments considered in this study [ 1,241, the upper limiting frequency is about 200 kHz, but the results at higher frequencies may be of value in other

Frequency (kHz) Figure 5. Calculated sound pressure level, averaged over one-third octave frequency bands, relative to free and with acoustic boundary field from a point source above a plane solid surface with zero admittance ( -), layer admittance (+). R,,, = 2 m, h, = h, = 0.03 m.

ACOUSTIC

BOUNDARY

LAYER

IN

SCALE

331

MODELS

ultrasonic applications. The influence of the acoustic boundary layer admittance in a scale model experiment is illustrated in Figure 5. There the calculated sound pressure level relative to free field is shown with and without the acoustic boundary layer for one of the geometrical configurations in Figure 3. In the audio frequency range, the boundary layer effect is negligible for many cases. In Figure 6 the full scale case corresponding to the model scale configuration in Figure 5 is shown, and in Figure 7 the source and receiver are closer to the ground. For some source and receiver combinations and source frequency spectra, the acoustic boundary layer admittance may be an explanation for unexpected measurement results.

5 t

-201 400 500

700

IOCKJ

2000

7000

3000 4000

10000

Frequency(Hz)

Figure 6. Calculated sound pressure level, averaged over one-third octave frequency bands, relative to free and with acoustic boundary field from a point source above a plane solid surface with zero admittance (e), layer admittance (+). Rho, = 200 m, h, = h, = 3 m.

10

I

I

I

I

I

I

I

,

Frequency(Hz)

Figure 7. Calculated sound pressure level, averaged over one-third octave frequency bands, relative to free and with acoustic boundary field from a point source above a plane solid surface with zero admittance ( -s--), layer admittance (-Q-). The curve designated + is calculated by using the plane wave reflection model in section 2.1. R,,, =200m, hs=h,=O.lm.

M.

332

ALMGREN

The simple plane wave reflection model from section 2.1 is not sufficient (which is a well known fact), when calculating the sound pressure level relative to free field from a point source above a surface with non-zero admittance. This is also illustrated in Figure 7. In the calculations presented above the sound speed and the wavenumber were determined at 15°C in the full scale cases and at 20°C in the model scale cases, because in a real scale model experiment the laboratory temperature will not be equal to the outdoor temperature. This difference was tested and it was found to give a negligible effect (less than 0.5 dB in the one-third octave bands 500-8000 Hz for 3 m source and receiver heights and 200 m horizontal distance). The sound absorption processes in air will also give a systematic error in the calculated sound pressure level relative to free field from a point source above an impedance surface. The error in the predicted level can roughly be estimated from

AL=2010g[(1+e-‘m’k”R~PR~‘)/2].

(25)

Say that the error is negligible if -AL is less than 0.5 dB. This is then the case if cx( R2 - R,) is less than 0.12, where (Y= Im {k} is the sound absorption coefficient in the air. This is “generously” satisfied by the configurations in this study. The error will be the largest for the configuration with 2 m horizontal distance, 0.03 m source and receiver heights and the frequency 1 MHz. Then a( R2 - R,) is 0.017 for typical atmospheric conditions. Likewise the small correction to the phase velocity in air, see equation (6), has a negligible influence on the phase of the sound pressure. (wE/c~)( R, - R,) is only 7 milliradians for the same case as above. Figure 8 shows the normalized admittance of a solid surface with an acoustic boundary layer, calculated according to equation (19) for grazing incidence in the audio frequency range. The flow resistivity in the Delany and Bazley empirical model for the impedance of fibrous absorbent materials [25] was fitted to give the same real part of the admittance at 1 kHz as the acoustic boundary layer admittance. The resulting flow resistivity was 1.8 x lo8 kg/s m3 and the calculated normalized admittance is shown also in Figure 8.

-0 006 100

I 200

I 300

I 500

I 1 7001000

I 2000

1 3000

1 5000

\ \ 10000

Frequency (Hz)

Figure 8. Calculated normalized admittance for a plane solid surface. -, The acoustic boundary layer normalized admittance, equation (19), grazing incidence 0 = 90”, and temperature 20°C; - - -, the Delany-Bazley empirical model [29] with the flow resistivity 1.8 x IO* kg/s m3.

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IN

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333

Figure S shows that the flow resistivity model will not give the correct frequency dependence of the admittance. Further, the acoustic boundary layer admittance is dependent on the angle of incidence. Thus, it is better to use the calculated acoustic boundary layer admittance as a lower limit to the ground admittance rather than to use an upper limit to the empirical effective flow resistivity, as is done in reference [16]. The admittance of an acoustically softsurface will dominate over the acoustic boundary layer admittance, even at high frequencies. A typical grassland admittance is shown in Figure 9, together with the acoustic boundary layer admittance. The grassland admittance was calculated by using the empirical model of reference [25] and the effective flow resistivity 2 x lo7 kg/s m3, which is scaled in the ratio 100: 1 from the typical value given by Embleton et al. [16]. Obviously, the surface admittance due to the acoustic boundary layer is negligible in this case.

o-5 02 01 005 T5 002 O-01 0005

Frequency

(Hz)

Figure 9. The magnitude of the calculated normalized admittance for a plane surface. -, The acoustic boundary layer normalized admittance, equation (19), grazing incidence 6 = 90”, and temperature 20°C; ---, the Delany-Bazley empirical model [29] with the flow resistivity = 2 x 10’ kg/s m3, which is a typical grassland value [16] when scaled 1OO:l.

4. CONCLUSIONS

Scale model simulation of sound propagation above a rigid solid surface will give a systematic error in the predicted sound field. The deviation depends on the acoustic boundary layer, the admittance of which is not invariant under scaling. Typical errors are l-7 dB depending on the geometrical configuration, scale and frequency. The effect of the acoustic boundary layer is to broaden the troughs in the relative pressure versus frequency curve and to shift them towards lower frequencies. In the audio frequency range the effect is negligible for most environmental noise problems of interest, but if unexpected results emerge in problems concerning sound propagation above a hard surface, the explanation may perhaps be found in the acoustic boundary layer admittance. In the ultrasonic frequency range it is more obvious that the acoustic boundary layer influences the sound field, since the associated admittance increases with the square root of the frequency. The reason for the phenomenon is that the ordinary acoustic field wriables cannot satisfy the boundary conditions (zero normal and tangential velocity and zero temperature

334

M.

ALMGREN

fluctuations) at the rigid, flat and solid surface. Two other wavefields, the amplitudes of which decrease exponentially from the boundary, are generated and are coupled to the acoustic wavefield at the boundary. The phase shift between the incident and the reflected acoustic wave is then associated with an energy loss which leads to an apparent admittance. A more physical explanation is perhaps that at the boundary of a solid surface the tangential velocity must equal zero. Layers of air will then slide against one another and due to the internal friction or viscosity of the air, a part of the energy associated with transversal velocity fluctuations is transformed into heat: i.e., disorganized motion of the molecules. Further, the heat transfer between a small volume element of air in the compression or rarefaction phase and the solid is much larger than the heat transfer between neighbouring small volume elements of air in the free field, since the solid is a much better heat conductor. Thus, when the volume element is compressed, the temperature increases relative to the solid which leads to a heat transfer into the solid and then the internal energy is reduced which means that the air density, which would otherwise be adiabatically related to the sound pressure, is reduced somewhat. In the rarefaction phase, the temperature is decreased relative to the solid and heat is conducted from the solid to the volume element of air and the density is decreased. In total, energy is lost due to the irreversible processes of internal friction and heat conduction. This study combines the results of the theory of the reflection of plane waves in a flat solid boundary with the exact solution of the sound field from a point source above a plane impedance boundary. From the exact solution and from the simple ray acoustics solution as well it is concluded that the sound field above a surface with non-zero admittance is correctly scaled if the admittance and the angle of incidence can be kept invariant under scaling. However, the acoustic boundary layer admittance, which is easily calculated from equations (19a) or (19b), is not invariant under scaling since its magnitude increases with the square root of the frequency. The apparent admittance of a solid ground surface is usually negligible in comparison

with the admittance of an acoustically soft surface, e.g., a grassland surface, even in the frequency range associated with scale model experiments. It is common, in the field of outdoor sound propagation, to treat the ground surface as locally reacting, but for a solid ground surface the acoustic boundary layer admittance is dependent on the angle of incidence, and hence the surface is not locally reacting. The effect of the sound absorption in air, as well as the correction of the phase velocity (due to the vibrational relaxation of the molecules in the air), have been neglected in this study when predicting the sound pressure level relative to free field, because the path difference between the direct and the ground-reflected ray was small compared to the inverse of the absorption coefficient (for atmospheric absorption) with the geometrical configurations used. However, such an effect can easily be taken into account for other configurations. One may moreover conclude, with regard to the scale model simulation of concert hall acoustics, that the acoustic boundary layer leads to absorption coefficients of solid surfaces such that the coefficients depend on the angle of incidence and increase with frequency.

ACKNOWLEDGMENT The author is grateful to Professor Tor Kihlman and Dr BjSm Petersson for valuable support and criticism. Dr Elizabeth Lindqvist is also thanked for constructive criticism of the manuscript. The study has been financially supported by the National Swedish Environment Protection Board.

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LAYER

IN SCALE MODELS

335

REFERENCES 1. M. ALMGREN 1983 Chalmers University of Technology, Department of Building Acoustics, Report F83-04. Scale model simulation of outdoor sound propagation. Measurement technique considerations. 2, J.-M. RAPIN and B. FAVRE 1977 Proceedings of Inter-Noise 77, A52-A59. Community noise evaluation by model studies. 3. K. ISHIIand H. TACHIBANA 1974 Proceedings of the 8th International Congress on Acoustics 610. Acoustic scale model experiment using medium of nitrogen gas. 4. G. THEILE and R. K~RER 1975 DAGA ‘75, 491-494. Einfacher Echtxeit-Kompensator zum Ausgleich der Dissipationsverluste bei Impulsmessungen in der Modell-Akustik. 5. H. HELMHOLTZ 1863 Verhandlungen des natur-historisch-medizinischen Vereins zu Heidelberg Reprinted in 3, 16-20. Ueber den Einfluss der Reibung in der Luft auf die Schallbewegung. Wissenschaftliche Abhandlungen 1, 383-387, Barth Leipzig, 1882. 6. KUNDT 19 Dee 1867 Preussische Akademie der Wissenschaften, Monatsbericht. Uber die Schallgeschwindigkeit der Luft in Rohren. 7. G. KIRCHHOFF 1868 Poggendorffs Annalen der Physik und Chemie 134, 177-193. Ueber den Einfluss der Warmeleitung in einem Gase auf die Schallbewegung. 1939 Journal of Technical Physics 9, 226-234. On the absorption of 8. B. P. KONSTANTINOV acoustical waves due to reflection at a hard boundary (in Russian). 9. L. CREMER 1946 Archivfir Elektronik und Ubertragungstechnik 2, 136-139. Uber die akustische Grenzschicht vor starren Wanden. 10. L. CREMER and H. A. MILLER 1976 in Die wissenschaftlichen Grundlagen der Raumakustik 2. Berlin: Springer, second edition. Die unvermeidliche absorption an starrer Wand. 11. L. CREMER and H. A. MILLER 1978 Die wissenschaftlichen Grundlagen der Raumakustik 1. Berlin: Springer, second edition. Unvermeidliche Absorptionsgrade. 12. P. M. MORSE and U. INGARD 1968 Theoretical Acoustics. New York: McGraw-Hill. See 270-294. 13. A. D. PIERCE 1981 Acoustics. New York: McGraw-Hill. 14. J. NICOLAS, T. F. W. EMBLETON and J.E. PIERCY 1983 Journalofthe Acoustical Society of versus theoretical prediction of barrier insertion America 73,44-54. Precise model measurements loss in presence of the ground. 15. C. I. CHESSEL 1977 Journal of the Acoustical Society of America 62, 825-834. Propagation of noise along a finite impedance boundary. 16. T. F. W. EMBLETON, J.E. PIERCY and G. A. DAIGLE 1983 Journal oftheAcoustica1 Society of America 74, 1239- 1244. Effective flow resistivity of ground surfaces determined by acoustical measurements. of absorption of 17. American National Standard ANSI S1.26 1978 Method for the calculation sound by the atmosphere. Acoustical Society of America. 18. S-I. THOMASSON 1976 Journal of the Acoustical Society of America 59, 780-785. Reflection of waves from a point source by an impedance boundary. 19. S-I. THOMASSON 1980 Acustica 45, 122-125. A powerful asymptotic solution for sound propagation above an impedance boundary. 20. C. F. CHIEN and W. W. SOROKA 1975 Journal of Sound and Vibration 43, 9-20. Sound propagation along an impedance plane. 21. C. F. CHIEN and W. W. SOROKA 1980 Journal ofSound and Vibration 69, 340-343. A note on the calculation of sound propagation along an impedance surface. 22. T. KAWAI, T. HIDAKA and T. NAKAJIMA 1982 Journal of Sound and Vibration 83, 125-138. Sound propagation above an impedance boundary. 23. Landolt-Bornstein, 6. A&age Zahlenwerte und Funktionen. IV. Band, Technik 4. Teil. Warmetechnik. Berlin: Springer, Bandteil a 1967. Bandteil b 1972. 24. M. ALMGREN 1984 Proceedings of the 4th FASE Congress 407-410, Sandejord, Norway. Scale model simulation of outdoor sound propagation at long distances. 25. M. E. DELANY and E. N. BAZLEY 1970 Applied Acoustics 3, 105-116. Acoustical properties of fibrous absorbent materials. 26. S-l. THOMASSON 1977 Journal of the Acoustical Society of America 61, 659-674. Sound propagation above a layer with a large refraction index.

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M. ALMGREN

APPENDIX

1: NUMERICAL

COMPUTATION

for a PDP-11/34 for the calculation of the were written in FORTRAN of the plane surface and the sound pressure relative to free field. Complex arithmetic with single precision was used, except for the path difference RZ- R,, where double precision was used, since it is obvious that R, and R2 are nearly equal when the source and receiver are close to the surface. The calculation of the sound pressure relative to free field from a point source above an impedance surface (equations (7) - ( 15)) was tested by comparing with Thomasson’s calculation [26], and the calculation by Kawai et al. [22]; see Table Al. The levels calculated by Thomasson and Kawai et al. deviate from one another with at most 0.14 dB. The levels obtained from the routine used by the author deviate from Thomasson’s maximally by 0.31 dB and by 0.19 dB from the results by Kawai et al. The deviation is probably due to the limited precision in the computations, but it was considered acceptable for the purpose. Programs

admittance

TABLE

Al

Sound pressure level relative to free field as calculated with Thomasson’s exact solution [ 181 by Thomasson [ 261, by Kawai et al. [22] and by the author; the effects on the wavenumber of the sound absorption processes in the air have been neglected; the source-to-receiver distance is 20 m and the source and receiver heights are both 0.1 m; the normalized admittances have been calculated with Thomasson’s ground model, equation (59) of reference [26], with the parameters a = 1.0, b = O”, c = 10 000 Hz and d = 800 Hz in this study, the wavenumber was calculated as 0/20+04JT, where the temperature 293 “K (20°C) was chosen L

Lr horn (dB) 50 63 79 100 126 159 200 252 317 400 504 635 800 1008 1270 1600 2016 2540 3200 4032 5080

7.90 8.60 9.53 10.68 11.98 13.00 12.38 5.71 - 16.50 -32.11 -42.02 -47.06 -42.68 -37.36 -33.20 -30.04 -28.08 -27.91 -25.82 -22.47 -21.06

L Kawa

(dB) 7.89 8.60 9.52 10.68 Il.98 13.00 12.38 5.71 -16.50 -32.07 -41.90 -47.11 -42.82 -37.42 -33.23 -30.06 -28.10 -27.93 -25.82 -22.47 -21.06

7.89 8.58 9.48 10.67 11.96 12.99 12.39 5.84 -16.38 -31.92 -41.71 -47.07 -42.90 -37.50 -33.30 -30.13 -28.16 -28.01 -25.91 -22.54 -21.14

ACOUSTIC

BOUNDARY

APPENDIX a

c, co 5

?_L Hh”( z) h,, ha I k L, &s/f Lxm> Lx n l+ P Pff PR

Psurfacc

Q

R ho, 4 4,

T

R,

Cl

V vphare

W

P Y ii E

0, %l,” K

P P w

LAYER

IN

SCALE

MODELS

337

2: SYMBOLS

real or imaginary part of the acoustic boundary layer admittance (-) speed of sound, speed of sound at zero frequency (m/s) specific heat at constant pressure = l-006 x lo3 J/kg “K for dry air at 20°C the specific heat at constant volume (J/kg “K) frequency, vibrational relaxation frequency of O2 or N, Hankel function of complex argument z (-) source and receiver heights (m) integral in Thomasson’s solution (-) wave number (m-‘) level, level relative to free field (dB) boundary layer thickness due to heat conduction and viscosity (m) unit vector, normal to the surface (-) the Prandtl number, =~c~/K -0.714 for air at 20°C sound pressure (Pa) sound pressure at a receiver point in free field (Pa) sound pressure at the receiver point (Pa) sound pressure due to the surface wave at the receiver point (Pa) spherical reflection factor (-) horizontal source-to-receiver distance (m) reflection coefficient for plane waves (-) lengths of the direct and ground reflected rays (m) absolute temperature (“K) time (s) or integration variable (-) parameter in Thomasson’s solution (-) particle velocity in the sound field (m/s) phase velocity of the sound wave (m/s) parameter in Thomasson’s solution( -) surface impedance (Pa s/m) coefficient for sound absorption in air (Nepers/m) classical plus rotational contributions to the absorption coefficient (Nepers/m) maximum absorption in a distance of one wavelength, caused by vibratiohal relaxation of substance i (Neper) normalized surface admittance PC/Z (-) = c,/ c, = 1.402 for dry air at 20°C parameter in Thomasson’s solution (-) sound pressure level difference (dB) correction of the real part of the wavenumber, due to the vibrational relaxation of molecules in the air (-) angle of incidence, the angle of incidence for which the magnitude of the reflection factor is minimum (radians) coefficient of thermal conductivity, =2.567x lo-* W/m “K at 20°C viscosity, =1*813 x 10’ kg/ms for air at 20°C density of the air (= 1.20 kg/m3 at 20°C) frequency (radian/s)