Explanation of Acoustical Principles

Introduction

The purpose of this page is to present and explain the different acoustic principles used in Code_TYMPAN.
But also to show the consistency of the theoretical results and those of Code_TYMPAN using simple models.

Model

Model description

_images/Model_case_0.png
The reference model consists of a source and a receiver without any obstacle or object around but with a ground.

Name

Pos X

Pos Y

Pos Z

Units

Source

-50

0

2

m

Receiver

50

0

2

m

Symbol

Value

Units

Comment

Distance S/R

100

m

Distance ref

1

m

Source height

2

m

Receiver height

2

m

dp

100

m

distance S->R Projected on the ground

Calculation parameters for 3 reference modelizations

All acoustic calculations were carried out under homogeneous conditions.

Ground reflection case without screen

Model 1 : acoustic propagation with ground reflection and without screen

Symbol

Value

Units

Comment

Gs

0

ground factor 9613-2 source region

Gr

0

ground factor 9613-2 receiver region

Gm

0

ground factor 9613-2 intermediate region

σe

20000

kRayls, flow resistivity

e

1

m

ground thickness

dp1

50

m

ground distance source -> reflection point; Descartes’ law: tan β = hs/dp1 = hr/(dp-dp1)

cos β

0.999200959

cosine of the ground reflection angle

sin β

0.03996803

length

100.079968

m

reflected path length

Screen case

_images/Model_case_1.png

Model 2 : acoustic propagation with ground reflection and thin screen between the source and the receiver

Name

Value

Units

Comment

Single thin screen

Position

(-25, 10) à (-25,-10)

Screen/source distance

25

m

Screen height

8

m

e

0.3

m

screen thickness

9613-2 calculation parameters

Symbol

Value

Units

Comment

a*

0

m

component of the distance parallel to the edge of the screen between the source and the receiver (9613-2)

ll

10

m

perpendicular horizontal dimension SR left

lr

10

m

horizontal dimension perpendicular SR right

dSS,top

25.71

m

distance between the source and the first diffraction edge for the vertical path

dSR,top

75.24

m

distance between the second diffraction edge and the receiver for the vertical path

dSS,lat

26.93

m

distance between source and first diffraction edge for side paths

dSR,lat

75.66

m

distance between second diffraction ridge and receiver for side paths

Note

The component a corresponds to the component of the distance along the y axis. In our example, it is zero, but in general it is not.

Reflection case

_images/Model_case_2.png

Name

Value

Units

Comment

Position

(-52,10) à (-52,-10)

β_reflex

0.0

rad

angle of incidence

ρ_reflex

1.0

acoustic reflection coefficient

l_min

8.0

m

minimum reflective length dimension

dS,O

1.85

m

source distance to reflection point

dO,R

101.85

m

distance from reflection point to receiver

DIr

0

source directivity index -> receiver image

Acoustic

Source spectrum

Source directivity

Frequency

Lw(fc) dB re 1 pW

W(fc) W

Module

Phase

16 Hz

0

1E-12

1

0

20 Hz

0

1E-12

1

0

25 Hz

0

1E-12

1

0

31.5 Hz

0

1E-12

1

0

40 Hz

0

1E-12

1

0

50 Hz

80

0.0001

1

0

63 Hz

80

0.0001

1

0

80 Hz

80

0.0001

1

0

100 Hz

80

0.0001

1

0

125 Hz

80

0.0001

1

0

160 Hz

80

0.0001

1

0

200 Hz

80

0.0001

1

0

250 Hz

80

0.0001

1

0

315 Hz

80

0.0001

1

0

400 Hz

80

0.0001

1

0

500 Hz

80

0.0001

1

0

630 Hz

80

0.0001

1

0

800 Hz

80

0.0001

1

0

1 kHz

80

0.0001

1

0

1.25 kHz

80

0.0001

1

0

1.6 kHz

80

0.0001

1

0

2 kHz

80

0.0001

1

0

2.5 kHz

80

0.0001

1

0

3.15 kHz

80

0.0001

1

0

4 kHz

80

0.0001

1

0

5 kHz

80

0.0001

1

0

6.3 kHz

80

0.0001

1

0

8 kHz

80

0.0001

1

0

10 kHz

80

0.0001

1

0

12.5 kHz

0

1E-12

1

0

16 kHz

0

1E-12

Lw

93.80

dB

Atmospheric absorption

This part details the global and model-specific parameters for the calculation of atmospheric absorption

Weather parameters

Symbol

Value

Physical values

R

8.31441

Ideal gas constant

Absolute_zero

273.15

Temperature 0°C in K

M

0.029

Molar mass of air

y

1.41

Ratio of the specific heats respectively at constant pressure and volume

Reference atmospheric pressure

101325

Pa

Reference sound pressure

293.15

K

Reference sound pressure

0.00002

Pa

Reference sound power

1E-12

W

Specific model

Symbol

Value

Units

Temperature

20

°C

Pressure

101325

Pa

Hygrometry

70

% humidity in the air

Atmospheric attenuation calculation

Atmospheric attenuation comes from molecular relaxation phenomena on the path of the wave.
It is proportional to the distance and depends on the conditions of temperature, pressure and relative humidity of the air.

Atmospheric attenuation in Iso 9613-1 Standard

The attenuation by atmospheric absorption is given by:
_images/Attenuation_1.png
With f the frequency, i the initial point, t the point considered and s the sound propagation distance in m.
α is the pure sound attenuation coefficient in decibels per meter.

Atmospheric attenuation in default Solver

Calculations up to atmospheric absorption strictly identical to what is in the standard
Calculation of attenuation according to the formula below applied by frequency third octav.
_images/Attenuation_2.png

Atmospheric attenuation in Code_TYMPAN

The calculations are identical to the note of the default solver. Note that in the Code_Code_TYMPAN implementation: * Modulus and phase calculation for each Aatmi third octav band. * The calculation being made in spectrum of p²(fc), the attenuation is squared.

Implementation

  • void AtmosphericConditions::compute_absorption_spectrum(): Calculates the atmospheric absorption spectrum α for the weather conditions of the model.

  • AtmosphericConditions::compute_length_absorption(double length) const: Returns the atmospheric attenuation spectrum in explicit state (physical quantity), corresponds to the Aatmi(f) of the DefaultSolver.

  • void TYChemin::calcAttenuation(const TYTabEtape& tabEtapes, const AtmosphericConditions& atmos): Calculates the atmospheric attenuation from the power spectrum of the source taking into account the directivity and as a function of the path length.

_images/Attenuation_3.png

  • OSpectre TYTrajet::getPEnergetique(const AtmosphericConditions& atmos): Calculates the squared sound pressure spectrum, p²(fc). In energy, p²(fc) = W(fc)*divGeom*Aatmi(fc)²

Atmospheric calculation

The objective is to present the calculation of the various parameters of atmospheric attenuation based on what is implemented in Code_Code_TYMPAN.
The calculation proposed below relates to a specific use case described in the “Model” and “Weather” parts.

Sound celerity calculation, c

Symbol

Value

Units

Comment

tk

293.15

K

Temperature in Kelvin

c

344.25

m/s

Sound speed
In Code_TYMPAN :
double AtmosphericConditions::compute_c() const

Impedance calculation

Symbol

Value

Units

Comment

ρ

1.21

kg.m-3

Air density

z

415.02

kg.m-2s-1

acoustic specific characteristic impedance. Usually denoted z0 = ρ*c

Note

In Code_TYMPAN ρ, the density of air, is calculated from the molar mass of air M and the weather conditions according to the following ideal gas formula: ρ = PM / RT

In Code_TYMPAN :
double AtmosphericConditions::compute_z()

Atmospheric absorption

Mole fraction of water vapor

Symbol

Value

Units

Comment

T01

273.16

K

Isothermal temperature at the triple point

Relative pressure

1

Pa/Pa

Relative pressure

C

-1.63712658

Exponent C, see Appendix B2 ISO 9613-1, dimensionless quantity

Psat

2336.630453

Pa

Saturating vapor pressure

h molar

1.614252472

%

Mole fraction of water vapor
In Code_TYMPAN :
double AtmosphericConditions::compute_hm() const

Atmospheric absorption spectrum

Symbol

Value

Units

Comment

frO

53173.95674

Hz

Oxygen relaxation frequency

frn

460.9906921

Hz

Nitrogen relaxation frequency

Frequency

Order

Exact frequency

f²

α (dB.m-1) atmospheric absorption coefficient

k (m-1) acoustic wave number

16 Hz

0

15.8

251.2

5.75329E-06

0.289273495

20 Hz

1

20.0

398.1

9.11237E-06

0.364173754

25 Hz

2

25.1

631.0

1.44272E-05

0.458467594

31.5 Hz

3

31.6

1000.0

2.2828E-05

0.577176504

40 Hz

4

39.8

1584.9

3.60861E-05

0.726622168

50 Hz

5

50.1

2511.9

5.69587E-05

0.914763112

63 Hz

6

63.1

3981.1

8.96923E-05

1.151618528

80 Hz

7

79.4

6309.6

0.000140718

1.449801829

100 Hz

8

100.0

10000.0

0.000219517

1.825192365

125 Hz

9

125.9

15848.9

0.000339472

2.29778105

160 Hz

10

158.5

25118.9

0.000518179

2.892734954

200 Hz

11

199.5

39810.7

0.000776135

3.641737544

250 Hz

12

251.1

63095.7

0.001132366

4.584675937

315 Hz

13

316.2

100000.0

0.001596482

5.771765042

400 Hz

14

398.1

158489.3

0.002160413

7.266221682

500 Hz

15

501.2

251188.6

0.002797920

9.147631123

630 Hz

16

631.0

398107.2

0.003478850

11.51618528

800 Hz

17

794.3

630957.3

0.004194039

14.49801829

1 kHz

18

1000.0

1000000.0

0.004977810

18.25192365

1.25 kHz

19

1258.9

1584893.2

0.005921435

22.9778105

1.6 kHz

20

1584.9

2511886.4

0.007184447

28.92734954

2 kHz

21

1995.3

3981071.7

0.009016418

36.41737544

2.5 kHz

22

2511.9

6309573.4

0.011800116

45.84675937

3.15 kHz

23

3162.3

10000000.0

0.016125980

57.71765042

4 kHz

24

3981.1

15848931.9

0.022911167

72.66221682

5 kHz

25

5011.9

25118864.3

0.033582957

91.47631123

6.3 kHz

26

6309.6

39810717.1

0.050353926

115.1618528

8 kHz

27

7943.3

63095734.4

0.076620551

144.9801829

10 kHz

28

10000.0

100000000.0

0.117507392

182.5192365

12.5 kHz

29

12589.3

158489319.2

0.180531919

229.778105

16 kHz

30

15848.9

251188643.2

0.276238144

289.2734954
In Code_TYMPAN :
void AtmosphericConditions::compute_absorption_spectrum(), calcule le spectre d’absorption atmosphérique α.
void AtmosphericConditions::compute_k(), calcule le spectre du nombre d’onde k.

Atmospheric attenuation

Aatmos

Frequency

16 Hz

0.999934

28.927350

20 Hz

0.999895

36.417375

25 Hz

0.999834

45.846759

31.5 Hz

0.999737

57.717650

40 Hz

0.999585

72.662217

50 Hz

0.999344

91.476311

63 Hz

0.998968

115.161853

80 Hz

0.998381

144.980183

100 Hz

0.997476

182.519237

125 Hz

0.996099

229.778105

160 Hz

0.994052

289.273495

200 Hz

0.991104

364.173754

250 Hz

0.987048

458.467594

315 Hz

0.981788

577.176504

400 Hz

0.975434

726.622168

500 Hz

0.968301

914.763112

630 Hz

0.960740

1151.618528

800 Hz

0.952862

1449.801829

1 kHz

0.944302

1825.192365

1.25 kHz

0.934099

2297.781050

1.6 kHz

0.920614

2892.734954

2 kHz

0.901401

3641.737544

2.5 kHz

0.872970

4584.675937

3.15 kHz

0.830558

5771.765042

4 kHz

0.768147

7266.221682

5 kHz

0.679337

9147.631123

6.3 kHz

0.560055

11516.185278

8 kHz

0.413902

14498.018294

10 kHz

0.258501

18251.923651

12.5 kHz

0.125124

22977.810498

16 kHz

0.041573

28927.349543
In Code_TYMPAN :
OSpectre AtmosphericConditions::compute_length_absorption(double length) const, calcule le module Aatmos.
void TYChemin::calcAttenuation(const TYTabEtape& tabEtapes, const AtmosphericConditions& atmos), calcule la phase Aatmos