Acoustics & Vibration

20
DELTA Danish Electronics, Light & Acoustics is an independent organisation, affiliated to the Danish Academy of Technical Sciences (ATV). DELTA Acoustics & Vibration Building 356 Akademivej DK-2800 Lyngby Denmark Tel. (+45) 45 93 12 11 Fax (+45) 45 93 19 90 www.delta.dk The report must not be reproduced, except in full, without the written approval of DELTA. 80rap-uk-a AV 2004/99 Page 1 of 20 DEL TA Acoustics & Vibration Comprehensive Model for Sound Propagation – Including Atmospheric Refraction Client: Nordic Noise Group 30th December 1999 REPORT

Transcript of Acoustics & Vibration

Page 1: Acoustics & Vibration

DELTA Danish Electronics, Light & Acoustics is an independent organisation, affiliated to the Danish Academy of Technical Sciences (ATV).

DELTAAcoustics & Vibration

Building 356AkademivejDK-2800 LyngbyDenmark

Tel. (+45) 45 93 12 11Fax (+45) 45 93 19 90www.delta.dk

The report must notbe reproduced,except in full,without the writtenapproval of DELTA.

80rap-uk-a

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DELTA Acoustics & Vibration

Comprehensive Model for Sound Propagation – IncludingAtmospheric Refraction

Client: Nordic Noise Group

30th December 1999

REPORT

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DELTA Acoustics & Vibration, 1999-12-30

Birger Plovsing Jørgen Kragh

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TitleComprehensive Model for Sound Propagation – Including Atmospheric Refraction

Journal no. Project no. Our ref.AV 2004/99 P 8219 BP/JK/bt

ClientNordic Noise Groupc/o Environmental and Food Agency of IcelandP.O. Box 8080IS-128 ReykjavikIceland

Client ref.Tór Tomasson

SummarySee page 4.

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Contents

Summary.......................................................................................................................... 4

1. Introduction...................................................................................................... 5

2. Air Absorption.................................................................................................. 6

3. Terrain and Screen Effects.............................................................................. 63.1 Flat Terrain......................................................................................................... 73.2 Terrain with a Single Screen ............................................................................ 113.3 Two Screens..................................................................................................... 123.4 Fresnel Zone..................................................................................................... 133.5 Transition between Propagation Models.......................................................... 143.6 Atmospheric Turbulence.................................................................................. 163.7 Finite Screens................................................................................................... 17

4. Scattering Zones ............................................................................................. 17

5. Reflections....................................................................................................... 17

6. Conclusions ..................................................................................................... 19

7. References ....................................................................................................... 20

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SummaryIn the present report it has been shown that the Nordic comprehensive model for straightline propagation elaborated previously can be modified to include the effect of a moder-ately refracting atmosphere by replacing the straight rays by curved rays. The curvedrays are predicted according to the heuristic model principle.

In the case of a strongly refracting atmosphere the effects of multiple ground reflections(downward refraction) and shadow zones (upward refraction) may turn up. In thesecases the refraction problem can no longer be solved by simple geometrical modifica-tion of rays but calls for a real extension of the models. Such an extension has been pro-posed in the present report.

In the heuristic model the actual sound speed profiles has to be approximated by a linearsound speed profile. The task of elaborating a procedure for approximating a non-linearsound speed profile by an equivalent linear profile has been described in a parallel re-port.

The method for including effects of refraction has not been validated by comparisonwith measurements but comparison with results predicted by the Parabolic Equationmethod for flat terrain has been carried out satisfactorily. However, further validation bymeasurements or accurate prediction methods is strongly needed and might lead tomodel adjustments.

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1. IntroductionIn 1998 a comprehensive model for propagation of sound in an atmosphere without sig-nificant refraction has been elaborated and is described in [1]. In this model, which isbased on geometrical ray theory combined with theory of diffraction, the sound rays areassumed to follow straight lines.

A simple approach to include the influence of weather is to model the effect of refrac-tion by curved sound rays. For simple sound speed profiles it may be assumed that thesound speed varies linearly with the height above the ground in which case the soundrays will travel along circular arcs. Such a simplification is the basis of the "heuristic"model proposed by L'Espérance [2]. Via the ray curvature the heuristic model in a sim-plified way combines the principles of linear ray acoustics with the effect of theweather.

The heuristic model by L'Espérance [2] has been investigated during 1998 as describedin [3]. However, in the model only propagation over a flat terrain has been considered.Therefore, the possibilities of using the heuristic principle in case of screens has beenstudied in [4] leading to a proposed model.

The aim of the Nordic project is to develop prediction models with sufficient accuracyfor "good-natured" weather. Good-natured weather is weather where the sound speed asa function of the altitude is either decreasing or increasing monotonically without sig-nificant jumps in the sound speed gradient. Most often good-natured weather is repre-sented by an approximately logarithmic sound speed profile. The heuristic model con-cept is not expected to be applicable in case of irregularly shaped sound speed profiles.The range of application of the heuristic approach is more thoroughly discussed in [5].

Considerations concerning the possibility of using the heuristic principle in the compre-hensive model was initially discussed in 1998 [6]. The present report contains the finalprinciples elaborated during 1999 for including weather effects in the comprehensivemodel.

The very crucial point in the heuristic model concept is the procedure for approximatinga non-linear sound speed profile by an equivalent linear profile. It has been decided tokeep the description of the elaborated procedure in a separate report [5] as the procedurestill is object of discussions and possible future improvement. Comparison between pre-dictions made by the method outlined below and predictions by the Parabolic Equationmethod can also be found in [5].

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2. Air AbsorptionIn [1] the effect of air absorption is determined for a homogeneous atmosphere based onair temperature and relative humidity. For an inhomogeneous refractive atmosphere thetwo parameters may vary along the propagation path. In the method described in [7]which is used in [1] a method is presented for a layered atmosphere but this method hasbeen found too advanced, partly because predictions shall be made for good-naturedweather only, partly due to the great uncertainty of weather parameters.

As a simple approach it is proposed to use the average values of weather parametersalong the rays calculated according to the heuristic model extended to include screens.Spatial information on relative humidity will seldom be available and a value obtainedin a single or a few points will therefore often be used as representative of the entirepropagation path. However, the average temperature along the ray path can easily beobtained from parameters which are already available in the heuristic model for otherpurposes. This concerns the travel time τ and the length of the ray R between source andreceiver (or screen tops). The average sound speedc along the path can be calculatedby Equation (1) and hence the average temperaturet in °C corresponding to this soundspeed can be determined by Equation (2).

τRc = (1)

15.27305.20

ct2

−���

����

�= (2)

3. Terrain and Screen EffectsThe comprehensive model for terrain and screen effects assuming straight line propaga-tion described in [1] can easily be modified to include curved rays according to the heu-ristic model principle as long as the number of rays in the base models remains un-changed. This is fulfilled when the weather conditions are not causing multiple groundreflections (strong downwind) or shadow zones (strong upwind). In these cases the re-fraction problem can no longer be solved by simple geometrical modification of rays butcalls for a real extension of the models.

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In the comprehensive model a prediction for a complex terrain is carried by one of threepropagation models named “flat terrain”, “valley-shaped terrain” or “hill-shaped terrain”or by a combination of the three models.

The three propagation models are according to [1] founded on three base models:

1) Flat terrain

2) One screen with a flat reflecting surface before and after the screen

3) Two screens with a flat reflecting surface before, after and between the screens

Therefore, prediction for any complex terrain is always the result of predictions by thethree base models combined in a suitable manner by the Fresnel-zone interpolation prin-ciple and the model transition principles described in [1]. This implies that, if the prob-lem of using curved rays is solved for each of these basic models, and the Fresnel-zoneinterpolation principle and model transition principles are modified to deal with curvedrays, the problem has been solved for the entire comprehensive terrain and screen effectmodel.

3.1 Flat TerrainAs in the non-refraction case the model for flat terrain and moderate refraction containstwo rays (a direct and a reflected ray) as shown in Fig. 1 and 2.

SRp1

p2

ΨG

QFigure 1Ray model for flat terrain and downward refraction.

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SR

p1

p2

ΨG

QFigure 2Ray model for flat terrain and upward refraction.

Modifications according to the heuristic principle concern the travel time τ and distanceR from the source to the receiver (τ1 and R1 for the direct ray and τ2 and R2 for the re-flected ray) and the ground reflection angle ψG as described in [2]. The free-spaceGreen's function R-1ejkR where k is the wave number, is modified by replacing kR by2πfτ where f is the frequency. The spherical reflection coefficient Q is calculated basedon the modified values of R2 and ψG.

In case of strong downward refraction additional rays will occur in the model for flatterrain. A method for including the effect of multiple rays in excess of the two rays al-ready included in the modified ray model has been developed in [8]. The contributionfrom the multiple reflection model is added incoherently to the contribution of the modi-fied comprehensive model.

In the case of strong upward refraction no ray reaches the receiver in the model for flatterrain, resulting in an acoustical shadow zone as shown in Fig. 3.

S Shadowzone

Figure 3Acoustical shadow zone.

In [4] it has been proposed that the difficult problem of a meteorologically generatedshadow zone in the case of propagation over flat terrain is considered analogous to theproblem of predicting sound levels in the shadow zone behind a diffracting wedge in

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case of a non-refracting atmosphere. The task has been to elaborate a method based onthis idea which fits into the structure of the comprehensive model and which does notintroduce a discontinuity at the edge of the shadow zone. It has also been necessary inthe elaboration of a method to consider the applicability in the presence of screens.

At the edge of the shadow zone the direct and reflected ray become identical and grazethe ground. The grazing angle of the reflected ray becomes 0 in this case. Therefore thesound pressure at the receiver can be expressed by Equation (3) as p1 = p2. Q is deter-mined using the curved path length R of the direct ray and ψG = 0. The horisontal dis-tance dSZ from the source to the reflection point when the receiver is at the edge of theshadow zone can be calculated by Equation (4) given that the total horisontal propaga-tion distance is substantially less than the ray radius of curvature.

( )Q1pp 1 += (3)

dhh

hd

RS

SSZ +

= (4)

To avoid discontinuities it has been decided to divide the ground effect in the shadowzone into a reflection effect contribution ∆LG and a shadow zone shielding effect contri-bution ∆LSZ as shown in Equation (5).

SZG L∆L∆L∆ += (5)

∆LG is calculated based on Equation (3). p1 is calculated using the path length R for theray from the source to the receiver disregarding the ground as shown in Fig. 4. Q is cal-culated on the basis of R and ψG = 0. Although R will increase somewhat the more thereceiver moves into the shadow zone, the effect on p1 and Q and therefore on ∆LG willbe almost negligible.

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S R

hSZ

dSZ

Figure 4Ray used to calculate the reflection effect contribution in the shadow zone.

The shielding effect ∆LSZ is determined using a modification of the Hadden/Pierce so-lution applied in the in the Nordic propagation model for wedge-shaped screens [9]. TheHadden/Pierce solution is a four rays model where the finite impedance of the wedge istaken into account by applying a spherical reflection coefficient to the image rays. Thefirst term in the Hadden/Pierce solution (n = 1 in Equation 3.16 of [9]) is interpreted asthe direct diffraction ray and the other three terms (n = 2, 3, and 4) are interpreted asreflected diffraction rays. As the ground reflection has already been included in the term1+Q in the calculation of ∆LG the contributions of the reflected rays should be omittedin the calculation of ∆LSZ. Furthermore the equation used to calculate ∆LSZ should benormalized to produce a value of 0 for at 180° wedge.

When the receiver is in the shadow zone, the vertical distance hSZ between the groundsurface and the ray from the source S to the receiver R at the horisontal distance dSZ

from the source as shown in Fig. 4 is used to define an equivalent wedge. The top of thewedge is placed hSZ above the line from the S to R at the distance dSZ from the sourcemeasured along the line, and the wedge legs are passing through S and R as shown inFig. 5.

S R

RhSZ

dSZ

RSZ

Figure 5Wedge used to calculate the shielding effect contribution in the shadow zone.

The diffracted sound pressure pSZ for the wedge in Fig. 5 can be predicted by Equation(6) where DSZ is the diffraction coefficient defined in [9] but including only the firstterm in the Hadden/Pierce solution (n = 1). If ∆LSZ is defined as the diffracted soundpressure level determined on the basis of Equation (6) relative to the level for a 180°

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wedge, ∆LSZ can be calculated by Equation (7). R is the direct distance from the sourceto the receiver and RSZ is the distance over the top of the wedge. The constant 2 ensuresthat ∆LSZ is 0 for a 180° wedge where DSZ becomes 0.5.

SZ

jkR

SZSZ ReDp

SZ

= (6)

���

����

�=

SZSZSZ R

RD2log20L∆ (7)

3.2 Terrain with a Single ScreenFor moderate downward and upward refraction the model for a single screen consists of4 rays as shown in Fig. 6 and 7 in the same way as in the model for non-refracting at-mosphere.

p ,p1 3

SR

p ,p1 2

p ,p3 4p ,p2 4

Q1 Q2

ΨG,1 ΨG,2

Figure 6Ray model for a single screen and downward refraction.

p ,p1 3

SRp ,p1 2

p ,p3 4p ,p2 4

Q1 Q2

ΨG,1 ΨG,2

Figure 7Ray model for a single screen and upward refraction.

The screen model for a refracting atmosphere proposed in [4] which is based on the heu-ristic model approach, concerns modification of the travel time τ and distance R be-

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tween source, screen top and receiver and of the ground reflection angles before andafter the screen in the same way as for flat terrain. In addition to that modified diffrac-tion angles are determined on the basis of curved rays for use in the Hadden/Pierce dif-fraction solution [9].

In the case of strong downward refraction additional rays will occur due to multiple re-flections before or after the screen. A method for including the effect of multiple rays inexcess of the 4 rays already included in the ray model has been developed in [8]. Thecontribution from the multiple reflection model is added incoherently to the contributionof the modified comprehensive model.

In the case of strong upward refraction the ray from the source to the top of the screen orfrom the top of the screen to the receiver may be blocked by the ground, resulting inacoustical shadow zones. The problem of shadow zones in the case of a screen has beensolved analogously to flat terrain. If a shadow zone occurs before, after, or on both sidesof the screen, the combined ground and screen effect is determined by Equation (8) to(10), respectively. The sound pressures p1 to p3 correspond to the rays defined in Fig. 7and p0 is the free-field sound pressure. Q1 and Q2 are the spherical reflection coefficientsbefore and after the screen. ∆LSZ,1 is the excess shielding effect on the source side of thescreen and is calculated using the same procedure as for flat terrain but with the receiverplaced on the top of the screen. ∆LSZ,2 is in the same way the excess shielding effect onthe receiver side of the screen but with the source placed on the top of the screen.

( ) 1,SZ10

231 L∆Q1p

Qpplog20L∆ ++���

����

� += (8)

( ) 2,SZ20

121 L∆Q1p

Qpplog20L∆ ++���

����

� += (9)

( ) ( ) 2,SZ1,SZ210

1 L∆L∆Q1Q1pplog20L∆ ++++= (10)

3.3 Two ScreensFor moderate downward and upward refraction the model for double screens includes 8rays as in the model for a non-refracting atmosphere.

The heuristic modifications in the double screen model concern the travel time τ anddistance R between source, screen tops and receiver, the ground reflection angles be-

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fore, after and between the screen and the diffraction angles in the same way as for thesingle screen case.

In case of strong downward refraction the effect of additional rays in excess of the 8rays already included in the ray model will follow the principle outlined in [8]. In thesame way as for the single screen case the contribution from the multiple reflections isadded incoherently to the contribution of the modified comprehensive model.

In case of strong upward refraction shadow zones may occur before, after, or betweenthe screens. The combined ground and screen effect is determined by equations follow-ing the same principle given by Equation (8) to (10). Instead of the three possible com-binations for a single screen expressed by Equation (8) to (10), there will be seven pos-sible combinations in case of a double screen. The excess shielding effect before thefirst screen and after the second screen is calculated corresponding to a single screenand the excess shielding effect between the screens is calculated by replacing source andreceiver by the screen tops.

3.4 Fresnel ZoneAs mentioned earlier, prediction by the propagation model for a valley- and hill-shapedterrain is the result of predictions by the three base models combined according to theFresnel-zone interpolation principle. This implies that the Fresnel-zone interpolationprinciple has to be modified to deal with curved rays. It is assumed that the Fresnel-zoneellipsoid has to be bent so that its axis of rotation follows the curved ray. As shown inFig. 8 the size of the Fresnel-zone is calculated using the algorithms for straight linepropagation but for a modified image source position S’’ and a modified receiver posi-tion R’. The angle between the line S’’R’ and the ground surface is equal to the groundreflection angle ψG of the curved ray. The distance from the reflection point to S’’ isequal to the distance to the source S measured along the curved ray, while the distancefrom the reflection point to R’ in the same way is equal to the curved distance to the re-ceiver R.

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S’

S

S’’

R

R’

R2

R2

R1

R1

R1

ΨG

Figure 8Determination of the Fresnel-zone in case of curved rays.

3.5 Transition between Propagation ModelsThe transition between the models for flat terrain and non-flat terrain depends in thenon-refraction case on the change in ∆R2 corresponding to the average deviation of theterrain ∆h from the equivalent flat terrain (as described in [1]). Same principle is used inthe refraction case but ∆R2 is determined on the basis of curved rays.

The transition between screened and unscreened terrain as well as between the singleand double screen cases are in the non-refraction case based on the path length differ-ence ∆l and on the height of the screen above the ground compared to the wavelengthand to effective width of the sound field at the screen as described in [1]. The pathlength difference is defined as the difference in length of the path from source to the re-ceiver via the top of the screen and the length of the direct path. If the top of the screenis below the line-of-sight the path length difference is represented by a negative value.

In the refraction case, the transition terms which are based on the height of the screenare unchanged whereas the term based on the path length difference is modified to takethe curved rays into account. In the light of the very approximative nature of the transi-tion equations, the calculation of the path length difference has been simplified as de-scribed in the following.

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The geometrical parameters used when predicting the path length difference ∆l are de-fined in Figure 9 (for downward refraction). The top and bottom point of the screen aredenoted T and G, respectively. The intersection between the screen (or vertical exten-sion of the screen) and the direct ray path (curved due to refraction) is denoted Pwhereas the intersection with the line-of-sight is denoted L.

SR

T

P

L

G

Figure 9Definition of geometrical parameters in the calculation of path length difference.

In the case of downward refraction (P above L) ∆l is calculated by Equation (11) if T isabove L and by Equation (12) if T is below L. In the case of upward refraction (P belowL) ∆l is calculated by Equation (13) if T is above L and by Equation (14) if T is belowL.

PRSPTRSTl∆:GLGT −−+=≥ (11)

TRSTPRSPSR2l∆:GLGT −−−−=< (12)

TRSTPRSPSR2l∆:GLGT ++++−=≥ (13)

PRSPTRSTl∆:GLGT ++−−=< (14)

If the relative sound speed gradient a’ [2] is positive (downward refraction) and hS isless than or equal to hR, PG is determined by Equation (15). Ψ’G is the angle betweenthe ray and the horizontal direction at the source or receiver which ever is the lowest [2]and dSCR is the horizontal distance from the source to the screen. If hS is greater than hR

the same equation is used except dSCR is replaced by d-dSCR and hS is replaced by hR. d isthe horizontal distance from source to receiver.

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0'ahh RS >∧≤ :

S

2

SCRG

2

G

h'a

1d'a

'ψtan'ψcos'a

1PG +−��

���

� −−���

����

�= (15)

If a’ is negative (upward refraction) and hS is greater than or equal hR, PG is deter-mined by Equation (16). If hS is less than hR the same equation is used except dSCR isreplaced by d-dSCR and hS is replaced by hR. d is the horizontal distance from the sourceto the receiver.

0'ahh RS <∧≤ :

S

2

SCRG

2

G

h'a

1d'a

'ψtan'ψcos'a

1PG +−��

���

� −−���

����

�−= (16)

Equations (15) and (16) are taken from [10].

3.6 Atmospheric TurbulenceA method for taking into account the energy scattered from atmospheric turbulence intothe shadow zone of a screen has been proposed in [4] and adopted in the comprehensivepropagation model for an atmosphere without refraction [1]. In the method the contri-bution of scattered energy is added incoherently to the contribution from the screenmodel. The prediction is based on geometrical parameters determined for the top of thescreen in relation to the line-of-sight.

It has been found that no modification should in principle be applied to the model of [4]to account for propagation in a refractive atmosphere. However, the model in [4] is pre-scribed to be used only when the top of the screen is above or at the line-of-sight be-cause the contribution has been assumed to be insignificant when the screen is below theline-of-sight. In case of upward refraction significant shadow zones may occur behind ascreen lower than the line-of-sight or even in the case of flat terrain. It will therefore benecessary to take into account the contribution from turbulent scattering also in thesecases.

It is proposed that when the top of the screen is below the line-of-sight, the contributionis predicted as if the screen top was at the line-of-sight (hOB = 0 in [4]).

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3.7 Finite ScreensFinite screens are in the comprehensive model taken into account by applying the solu-tion for an infinite screen but adding the contribution from sound diffracted around thevertical edges of the screen. This is done practically by adding an extra propagation pathfrom the source via the vertical edge of the screen to the receiver. The diffracted soundpressure for each side of the screen is calculated by multiplying the sound pressure cal-culated using the propagation conditions along the path by the diffraction coefficient ofthe vertical edge. The sound pressures diffracted around the vertical edges are addedincoherently to the sound pressure diffracted over the top of the screen.

As the path around the end of the screen is a breakline, the relative sound speed gradientused in the heuristic modifications may vary along the path. This is solved by calculat-ing a weighted average of relative sound speed gradient along the path by the samemethod as introduced for a reflected ray path (described in Section 5).

4. Scattering ZonesIn the model for predicting the effect of scattering zones one of the basic parameterswill be the distance through the scattering zone measured along the source-receiver raypath. In the case of a refracting atmosphere this distance has to be measured along thecurved path in stead. Otherwise the model will be identical to the model for a non-refracting atmosphere.

5. ReflectionsWhen the propagation path is a path generated by the reflection from an obstacle thedirection of propagation before and after the reflection will change relative to the direc-tion of the wind. The implication is that the relative sound speed gradient (denoted “a”in [2]) and consequently the radius of curvature of the circular rays may change at thereflection. If more than one reflection is involved the curvature of the rays may changeeach time the ray is reflected. In case of strong sound speed gradients the propagationmay involve complicated cases of simple downward and upward reflection combinedwith multiple ray propagation and shadow zone effects.

Initially it was expected that the effect of refraction could be taken into account for areflected path by changing the ray curvature piecewise depending of the direction ofpropagation [6]. However, it has been found that the geometrical complexity increases

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very fast and the conclusion is that this approach in many cases hardly would producereliable results.

Considering the complexity of the problem combined with a complete lack of experi-mental knowledge a simple straight-forward approach has been proposed. The value ofthe relative sound speed gradient "a" and c(0) is determined for each segment of the re-flected ray path and the average value of "a" and c(0), weighted by the horizontal lengthof the segment di, is determined by Equations (17) and (18). Then, the propagation ef-fect for the entire reflected path is calculated assuming that the effect of refraction isrepresented by the average value of "a" and c(0). The variables of Equation (17) are de-fined in Fig. 19 in the case of a single reflecting surface. If the reflecting surface is closeto the source or receiver or the reflected ray path is close to the direct path this assump-tion appear to be quite reasonable, and in most cases where a reflection is contributingsignificantly compared to the direct sound these requirements are fulfilled. However, theapproximation in cases with downward refraction half the way and upward refractionthe rest of the way by no refraction will be very rough. Fortunately, such cases are oflittle practical importance unless the direct path is affected by excessive attenuationwhile the reflected path is not.

=

== n

1ii

n

1iii

d

daa (17)

=

== n

1ii

n

1iii

d

d)0(c)0(c (18)

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*

R

S

S’

d1a1

a2

d2

R’

Figure 10Plan view illustrating different refrac-tion before and after reflection fromobstacle.

6. ConclusionsIn the present report it has been shown that the comprehensive model for straight linepropagation described in [1] can be modified to include the effect of a moderately re-fracting atmosphere by replacing the straight rays by curved rays. The curved rays arepredicted according to the heuristic model principle.

In a strongly refracting atmosphere the effects of multiple ground reflections (downwardrefraction) and shadow zones (upward refraction) may turn up. In these cases the refrac-tion problem can no longer be solved by simple geometrical modification of rays butcalls for a real extension of the models. Such extensions have been proposed.

In the heuristic model the actual sound speed profile has to be approximated by a linearsound speed profile. The task of elaborating a procedure for approximating a non-linearsound speed profile by an equivalent linear profile has been described in a parallel re-port.

The method for including effects of refraction has not yet been validated by comparisonwith measurements but comparison with results predicted by the Parabolic Equationmethod for flat terrain has been carried out satisfactorily. Further validation by meas-urements or accurate calculation is strongly needed and might lead to adjustments.

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The content of the report is a proposal which has not been discussed by the TechnicalCommittee. After such a discussion alterations may take place.

7. References[1] B. Plovsing, J. Kragh: ‘Prediction of Sound Propagation in an Atmosphere with-

out Significant Refraction, Outline of a Comprehensive Model’, DELTA Acous-tics & Vibration Report AV 1818/98, Lyngby 1998.

[2] A. L'Espérance, P. Herzog, G. A. Daigle, and J. R. Nicolas: ‘Heuristic model foroutdoor sound propagation based on an extension of the geometrical ray theory inthe case of a linear sound speed profile’, Appl. Acoust. 37, 111-139, 1992.

[3] S. Å Storeheier: ‘Nord2000: Sound propagation under simplified meteorologicalmodels: a heuristic approach’, SINTEF Report STF40 A98075, Trondheim 1998.

[4] M. Ögren: ‘Propagation of Sound - Screening and Ground Effect, Part 2: Re-fracting Atmosphere’, Swedish National Testing and Research Institute, SP RE-PORT 1998:40, ISBN: 91-7848-746-3, Borås 1998.

[5] B. Plovsing and J. Kragh: ‘Approximation of a Non-Linear Sound Speed Profileby an Equivalent Linear Profile’, DELTA Acoustics & Vibration Report AV2005/99, Lyngby 1999.

[6] B. Plovsing and J. Kragh: ‘Prediction of Sound Propagation in an Atmospherewith Significant Refraction, Considerations concerning the ComprehensiveModel’, DELTA Acoustics & Vibration Report AV 1819/98, Lyngby 1998.

[7] P. D. Joppa, L. C. Sutherland and A. J. Zuckerwar: ‘Representative frequency ap-proach to the effect of bandpass filters on evaluation of sound attenuation by theatmosphere’, Noise Control Eng. J. 44, 261-273, 1996.

[8] M. Ögren: ‘Multi reflected rays in a refracting atmosphere - Nord 2000 Progressreport’. SP Technical Note 1999:28, Borås 1999.

[9] M. Ögren: ‘Propagation of Sound - Screening and Ground Effect. Part I: Non-refracting Atmosphere’. Swedish National Testing and Research Institute, SP Re-port No. 1997:44, ISBN: 91-7848-703-X, Borås 1997.

[10] B. Plovsing and J. Kragh: ‘Wind Turbine Noise Propagation Model’, DELTAAcoustics & Vibration Report AV 1119/98, Lyngby 1998.