Wind noise measured at the ground surface

Wind noise measured at the ground surface

Jiao YuCenter for Industrial and Medical Ultrasound, Applied Physics Lab, University of Washington, 1013 NorthEast 40th Street, Seattle, Washington 98105-6698

Richard Raspet, Jeremy Webster,a) and JohnPaul AbbottDepartment of Physics and Astronomy and National Center for Physical Acoustics, University of Mississippi,University, Mississippi 38677

(Received 21 May 2010; revised 16 November 2010; accepted 22 November 2010)

Measurements of the wind noise measured at the ground surface outdoors are analyzed using the

mirror flow model of anisotropic turbulence by Kraichnan [J. Acoust. Soc. Am. 28(3), 378–390

(1956)]. Predictions of the resulting behavior of the turbulence spectrum with height are developed,

as well as predictions of the turbulence-shear interaction pressure at the surface for different wind

velocity profiles and microphone mounting geometries are developed. The theoretical results of the

behavior of the velocity spectra with height are compared to measurements to demonstrate the

applicability of the mirror flow model to outdoor turbulence. The use of a logarithmic wind velocity

profile for analysis is tested using meteorological models for wind velocity profiles under different

stability conditions. Next, calculations of the turbulence-shear interaction pressure are compared to

flush microphone measurements at the surface and microphone measurements with a foam covering

flush with the surface. The measurements underneath the thin layers of foam agree closely with the

predictions, indicating that the turbulence-shear interaction pressure is the dominant source of wind

noise at the surface. The flush microphones measurements are intermittently larger than the predic-

tions which may indicate other contributions not accounted for by the turbulence-shear interaction

pressure. VC 2011 Acoustical Society of America. [DOI: 10.1121/1.3531809]

PACS number(s): 43.28.Ra [JWP] Pages: 622–632

I. INTRODUCTION

The research presented in this paper is an extension of

earlier research into the calculation of wind noise contribu-

tions from the measured atmospheric turbulence spectra and

wind velocity. Raspet et al.1 developed theories relating the

wind noise measured by screened and unscreened micro-

phones to meteorological measurements at the height of the

microphone. Reference 1 also established fitting and analysis

procedures for the turbulence spectra. The turbulence models

were then used by Yu et al.2 in a preliminary analysis of the

wind noise spectrum measured in a microphone mounted

flush with the ground surface. The analysis adapted a mirror

flow model of anisotropic turbulence and mathematical anal-

ysis of Kraichnan3 for turbulent boundary layer flow to the

problem of wind noise pressure fluctuations measured under

atmospheric turbulence.

In Ref. 2, predictions of the surface pressure fluctuations

were prepared with three models of the wind velocity pro-

files: single exponential, double exponential, and logarith-

mic. The logarithmic profile was developed since it is often

used in meteorology to describe the wind velocity profile

under neutral conditions.4 The wind velocity profile was

only measured in the first few centimeters above the surface.

All measurements were made with the microphone dia-

phragm flush mounted in a smooth plate. For some measure-

ment sets, the measured and predicted pressure fluctuation

spectra agreed closely, and for others the measured levels

were considerably higher than the predictions. It was

observed that a thin porous layer over the microphone

reduced the measured pressure spectral levels to approxi-

mately the predicted levels.

The research reported, herein, extends and improves the

investigation of Ref. 2 in the following ways:

(1) Predictions of the behavior of the vertical and longitudi-

nal turbulence spectra with height resulting from the mir-

ror flow model developed by Kraichnan3 are derived and

compared to the measured turbulence spectra to demon-

strate that the mirror flow model is realistic for outdoor

measurements (Secs. II A and IV A).

(2) A new method of integration is developed which elimi-

nates convergence problems encountered using the

theory of Ref. 2 and is used to derive new expressions

for the pressure fluctuation spectrum under logarithmic

velocity profiles. This method also allows for the trunca-

tion of wind velocity profiles in order to model regions

of low wind speed under the roughness length and in

foams (Secs. II B 1, II B 3, and IV B).

(3) The pressure fluctuation spectra resulting from a multi-ex-

ponential model of the wind velocity profile are derived.

The multi-exponential model can be fit to arbitrary meas-

ured and predicted profiles and is used to investigate

whether the assumption of a logarithmic wind velocity

profile is sufficient for data analysis of our data sets (Secs.

II B 2 and III).

(4) Predictions and measurements for the microphones moun-

ted under the thin layers of metal foam are investigated in

a)Author to whom correspondence should be addressed. Electronic mail:

[email protected]

622 J. Acoust. Soc. Am. 129 (2), February 2011 0001-4966/2011/129(2)/622/11/$30.00 VC 2011 Acoustical Society of America

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.39.62.90 On: Tue, 02 Sep 2014 10:45:23

addition to the predictions and measurements for flush

microphones (Secs. II B 4 and IV B).

(5) The wind velocity profile is measured up to 2.0 m height

(Sec. IV B).

(6) The effect of different layer thicknesses on the pressure

spectra measured under thin layers of foam is investi-

gated (Sec. IV C).

This research presents a much more complete investiga-

tion of the wind noise levels measured at the ground than the

preliminary research presented in Ref. 2.

II. THEORY

The geometry of the measurement and calculation is illus-

trated in Fig. 1. The microphones are mounted vertically so

the diaphragms are flush in the rigid surface and are either

open to the flow or under a thin layer of the open pore metal

foam. The rigid surface or the foam surface is flush with the

ground surface and the ground surface is level for a long dis-

tance. In meteorological terminology, the measurement has

good fetch. The vertical wind fluctuations approach zero as

the rigid surface is approached, so that the turbulence becomes

increasingly anisotropic as the surface is approached.

Section II A develops predictions of the behavior of the

vertical and longitudinal one-dimensional (1-D) turbulence

spectra with height above the surface for comparison with

measurements. Section II B derives predictions for the pres-

sure fluctuation spectra from two realistic types of wind ve-

locity profiles and the measured wind velocity spectra above

the surface.

A. The vertical and longitudinal spectral model forturbulence height dependence

The mirror flow model by Kraichnan simulates the verti-

cal anisotropy of the boundary layer turbulence by assuming

that the anisotropic velocity field eVað~x; tÞ for the boundary

layer flow can be expressed in terms of the velocity field

Vað~x; tÞ of homogeneous isotropic turbulence occupying the

upper and lower half space as

eV1ð~x; tÞ ¼ 2�1=2 ½V1ð~x; tÞ þ V1ð~x �; tÞ� ;eV2ð~x; tÞ ¼ 2�1=2 ½V2ð~x; tÞ � V2ð~x �; tÞ� ;eV3ð~x; tÞ ¼ 2�1=2 ½V3ð~x; tÞ þ V3ð~x �; tÞ� ; (1)

where the auxiliary variable~x � satisfies

x�1 ¼ x1; x�2 ¼ �x2; x�3 ¼ x3 :

Correlation functions of the longitudinal or vertical turbu-

lent velocity for the flow above a boundary surface are formed

and expressed in terms of the correlation functions for homo-

geneous isotropic flows. By applying temporal and spatial

Fourier transforms with respect to x01 � x1; x03 � x3; and t0 � t

and by applying the symmetry of R11 and R22, we obtain

eR11ðx02; x2;~j;xÞ ¼ <11ðx02 � x2;~j;xÞþ <11ðx02 þ x2;~j;xÞ ;eR22ðx02; x2;~j;xÞ ¼ <22ðx02 � x2;~j;xÞ� <22ðx02 þ x2;~j;xÞ ; (2)

where ~j is the wave vector in the plane parallel to the bound-

ary, ~j ¼ k1e1 þ k3e3 ; and <abðx2;~j;xÞ is the real part of the

Fourier transform Rabðx2;~j;xÞ:

<abðx2;~j;xÞ ¼1ffiffiffiffiffiffi2pp

ð1�1

cosðk2x2ÞR abð~k;xÞdk2: (3)

Here R11ð~j;xÞ and R22ð~j;xÞ are the Fourier transforms of the

longitudinal and vertical correlation functions for the homoge-

neous and isotropic turbulence. Batchelor5 has derived relations

between the correlation functions of the isotropic homogeneous

turbulence and their energy spectrum tensor Uabð~k Þ and the

three dimensional velocity spectrum function E(k):ð1�1

Rabð~k;xÞdx ¼ 4p2Uabð~kÞ ; (4)

where

Uabð~kÞ ¼EðkÞ½k2 � kakb�

4pk4:

In Refs. 1 and 2, it was demonstrated that the measured lon-

gitudinal velocity spectrum above the surface outdoors can

be fit to a revised von Karman form

F111ðk1Þ ¼

C

½1þ ðk1kÞ2�5=6; (5)

where k1 is the wave number in the direction of flow, and the

C and k are the fit parameters. The form for E(k) correspond-

ing to Eq. (5) is given by1,6

EðkÞ ¼ 55C

18

ðkkÞ4

½1þ ðkkÞ2�17=6: (6)

We integrate the correlation functions eR11ðx02; x2; ~k;xÞand eR22ðx02; x2; ~k;xÞ over x and k3 and use Eqs. (3), (4), and

(6) to simplify. The final results for the longitudinal and ver-

tical turbulence spectra measured at height x2 areFIG. 1. (Color online) Coordinate system for the setup with the foam and

the mean longitudinal velocity profile.

J. Acoust. Soc. Am., Vol. 129, No. 2, February 2011 Yu et al.: Wind noise measured at the ground surface 623


eU11ðx2; k1Þ ¼110

9pCk4

ð10

ð10

ðk22 þ k2

3Þ cos2ðk2x2Þ½1þ ðkkÞ2�17=6

dk2dk3;

eU22ðx2; k1Þ ¼110

9pCk4

ð10

ð10

j2 sin2ðk2x2Þ½1þ ðkkÞ2�17=6

dk2dk3: (7)

When x2 is large, Eq. (7) recovers the spectra for homo-

geneous isotropic turbulence. As x2 approaches zero, the hor-

izontal spectrum approaches twice the free space value and

the vertical spectrum goes to zero. The spectra calculated for

the mirror flow model will be compared to measurements in

Sec. IV A.

B. Turbulence-shear interaction pressure spectra

1. Basic formulation

In this section, we apply and improve the method devel-

oped by Kraichnan to predict the turbulence-shear interaction

pressure at the ground. In Kraichnan3 and other studies7–12 of

pressure fluctuations in boundary layer flows, it is assumed

that the turbulent-shear interaction pressures are larger than

the turbulence-turbulence interaction pressures since the gra-

dient of the mean wind speed with height is larger than any

other gradient of fluctuating wind velocity components. The

source equation for the pressure fluctuations due to the inter-

action of the turbulence with a shear is then given by

r2pð~x; tÞ ¼ � 2qsðx2Þ@ eV2=@x1; (8)

where p is the pressure fluctuation, s(x2)¼ dU1/dx2 is the

vertical gradient of the mean longitudinal velocity, eV2 is the

vertical turbulent velocity component, and q¼ 1.2 kg/m3 is

the density of air.

For the mirror flow model given in Sec. II A, and under

the boundary conditions that the normal pressure gradient is

zero at the rigid boundary and the pressure is bounded at in-

finity, Kraichnan3 derives a formula for the pressure fluctua-

tion spectrum at the surface:

pð0;~j;xÞj j2 ¼ 4ð2pÞ�3=2q2k21j�2

ð10

ð10

e�jðx2þx02Þ

� sðx02Þsðx2Þ½<22ðx02 � x2;~j;xÞ� <22ðx02 þ x2;~j;xÞ�dx2dx02: (9)

In Kraichnan3 and Yu et al.,2 a coordinate transform

converting the double integration on x2 and x02 to integration

on s¼ x2þ x02 and t¼ x02� x2 is applied for a specified s(x2)

form. Here a different method of performing the integration

is used which retains the original coordinates x2 and x02 in

the integration. This method allows us to directly truncate

the integrals in regions where the mean velocity is modeled

as zero. Applying Eq. (3) and the trigonometric function

relation cos a� cos b¼�2 sin [(aþb)/2] sin[(a� b)/2], Eq.

(9) becomes

pð0;~j;xÞj j2 ¼ 2q2k21

p2j2

ð1�1

R22ð~k;xÞdk2

�ð1

0

e�jx02 sðx02Þ sinðk2x02Þdx02

�ð1

0

e�jx2 sðx2Þ sinðk2x2Þdx2: (10)

In Kraichnan,3 the corresponding equation is integrated

over x, k1, and k3 to give the total mean square pressure con-

tribution at the surface. In order to develop an expression for

the pressure spectrum measured along the direction of flow

due to the turbulence-shear interaction, Eq. (10) is integrated

over x and k3 only. Applying the relations given for homo-

geneous isotropic turbulence in Eq. (4) and the three dimen-

sional velocity spectrum function form in Eq. (6), we obtain

the form of the predicted turbulence-shear interaction pres-

sure spectra at the surface measured by a flush microphone

in a rigid plate under different assumed velocity profiles

using the turbulence spectral form Eq. (5) as

pð0; k1Þj j2 ¼ 440Ck4q2k21

9p

ð10

ð10

dk2dk3

½1þ ðkkÞ2�17=6

�ð1

0

e�jx02 sðx02Þ sinðk2x02Þdx02

�ð1

0

e�jx2 sðx2Þ sinðk2x2Þdx2: (11)

In Yu et al.,2 the prediction of the spectrum of the turbu-

lence-shear interaction pressure contribution at the boundary

surface for a logarithmic profile used the s and t representa-

tion described above following Eq. (9). Comparisons of the

logarithmic profile prediction extended to the ground surface

with the method described above showed that the integrals

using the s and t representation did not converge uniformly

as the surface was approached.

2. Application to a multi-exponential profile

Kraichnan used a single exponential to model the mean

wind velocity profile. A single exponential function cannot

accurately model the wind profile outdoors. In this section, a

multiple exponential form which can be used to fit a wide

variety of wind velocity profiles is developed. For atmos-

pheric surface layer flows, the mean velocity profile is usu-

ally modeled as approaching zero at a roughness length x0

and is zero below this height. The multiple exponential

mean velocity profile has the form

U1ðx2Þ ¼U 1�

Xn�1

i¼0

Aie�biðx2�x0Þ � 1�

Xn�1

i¼0

Ai

!e�bnðx2�x0Þ

!; x2 � x0;

0 0 � x2 < x0: (12)

8><>:624 J. Acoust. Soc. Am., Vol. 129, No. 2, February 2011 Yu et al.: Wind noise measured at the ground surface


This form guarantees the mean longitudinal velocity is

zero at and below the roughness length and is U at large

heights. The velocity gradient is then

sðx2Þ ¼Xn

j¼0

sje�bjðx2�x0Þ; x2 � x0;

0 0 � x2 < x0 ; (13)

8><>:where

sj ¼UbjAj; j ¼ 0; 1; :::; n� 1;

Ubn 1�Xn�1

i¼0

Ai

!; j ¼ n:

8><>: (14)

Eq. (11) becomes

pð0;k1Þj j2 ¼ 440Ck4q2k21

9p

Xi¼n;j¼n

i¼0;j¼0

sisjeðbiþbjÞx0

�ð1

0

ð10

dk2dk3

½1þ ðkkÞ2�17=6

ð1x0

e�ðjþbiÞx02

� sinðk2x02Þdx02

ð1x0

e�ðjþbjÞx2 sinðk2x2Þdx2: (15)

In Sec. III, the multiple exponential model is used to

investigate the effect of the variation in wind velocity profile

with atmospheric stability conditions on the predicted pres-

sure fluctuation spectrum.

3. Application to a logarithmic profile

The wind velocity outdoors often varies approximately

logarithmically with height in the surface layer. Under the

roughness length, the average velocity is assumed to be zero.

The logarithmic profile also approximates the wind velocity

profile for unstable but mechanically dominated turbulence

near the ground surface.4 For a logarithmic profile, the mean

velocity has the form

U1ðx2Þ ¼a ln x2

x0

� �; x2 � x0;

0 0 � x2 < x0; (16)

(

which gives a mean velocity gradient of

sðx2Þ ¼ax2; x2 � x0;

0 0 � x2 < x0: (17)

�Eq. (11) becomes

pð0; k1Þj j2 ¼ 440a2q2k21Ck4

9p

ð10

ð10

dk2dk3

½1þ ðkkÞ2�17=6

�ð1

x0

e�jx2 sinðk2x2Þx2

dx2

��ð1

x0

e�jx02 sinðk2x02Þ

x02dx02

�: (18)

4. Predictions for microphones under porous foamlayers

The normal pressure derivative is assumed to be zero at

the rigid plane surface. Figure 1 displays the coordinate sys-

tem for the measurement and calculation with the foam. The

plane where the microphone is mounted is chosen as the

zero height (x2, x02¼ 0). The wind profile above the foam is

assumed to be identical to the measured profile above the

grass. For open celled foam, it is reasonable to assume that

there are vertical turbulent velocity fluctuations inside the

foam but that the horizontal velocity will be small and negli-

gible. The vertical component of the turbulent velocity goes

to zero on the rigid plane surface (a flat plate) where the

microphone is flush mounted (see Fig. 1). The wind velocity

profile is set to zero at the roughness height above the foam

surface. This modifies the definitions of the velocity profiles

and gradients by the foam thickness. We do not expect this

to realistically model the details of the wind velocity profile

near the surface. However, at moderately low wave numbers,

the dominant source region is found to be several centi-

meters to several meters above the surface.13

III. EFFECTS OF ATMOSPHERIC STABILITY

The wind velocity profile measurements used in this pa-

per are limited in height to about 2.0 m. Our measurement

setup is portable and the measurements are made at an active

airport, which limits the height of the measurement. Close to

the ground, the measured profiles typically follow a logarith-

mic behavior down to the roughness length with small devia-

tion, although higher measurements might display systematic

variations due to the changing stability conditions. In this

section, we investigate the sensitivity of the pressure fluctua-

tion spectrum to deviations of the wind velocity profile from

logarithmic due to atmospheric stability effects. We base this

study on measured values for the wind velocity profile and

turbulence spectrum from the ground to 2.0 m. The details of

the measurement are described in Sec. IV. We will fit the

measured profiles to standard forms of the wind velocity pro-

file under differing stability conditions consistent with the

measurement days in this study. The atmospheric modeling is

based on the material from Ref. 4.

The mean velocity profile for unstable air satisfies

U1ðx2Þ ¼ a lnx2

x0

� wm

x2

L

� �� ; (19)

where x0 is the roughness length, x2 is the height, and L is

the Monin-Obukhov length which is negative for unstable

air. The more negative x2/L is, the greater the contribution of

convective turbulence. For the commonly used Businger-

Dyer form in unstable air, the expression for wm is

wm ¼ ln1þ x2

2

� 1þ x

2

� 2" #

� 2 arctan xþ p2; (20)

where x¼ (1� 16x2/L)1/4. When L¼�1, the neutral condi-

tion is recovered.



The Monin-Obukhov length, L, depends on wind speed,

heat flux, and roughness. The combined effect of wind and

heat flux can be described by the Turner Classes, which can

be estimated by Table 6.4 of Panofsky and Dutton.4 Turner

Classes 2 and 3 are appropriate for most of our measure-

ments. The roughness length of the grass at the measurement

site is estimated to be z0¼ 0.01 m. Figure 6.6 of Panofsky

and Dutton provides an approximate range for L between

�9.0 and �40.0 m. We take L¼�10.0 m, L¼�40.0 m, and

L¼�1 m as examples in our analysis.

Figure 2(a) displays measured wind velocity profile data

and fits from the ground up to 2.0 m using Eqs. (19) and (20)

for L¼�10.0 m, L¼�40.0 m, and L¼�1 m. Table I lists

the values of a and x0 obtained from the fits displayed in Fig.

2(a). The similarity of the fits in Fig. 2(a) indicates that the

atmospheric stability condition has little effect on the wind

velocity profile in the first few meters above the ground.

Figure 2(b) displays the three profile fits extended up to

100.0 m height. As the profile is extended to greater heights,

the difference between the three cases becomes obvious. The

neutral condition has larger velocities and velocity gradients

than the unstable conditions above 5.0 m. The roughness

lengths are 0.020, 0.017, and 0.016 m for L¼�10.0 m,

L¼�40.0 m, and L¼�1 m, respectively.

The three profiles with zero mean velocity under the

roughness length are then fit with the multiple exponential

profile from the roughness length up to 100.0 m height with

[bj]j¼0¼ 10j�3, n¼ 6. Setting the power of the exponential

function in Eq. (12) to powers of ten and using U and the A’s

as the fitting parameters is an effective method of fitting a

given measured profile to the multiple exponential form. A

few terms obtain a good fit to a variety of profiles. Table II

lists the fitting parameters obtained for L¼�10.0 m,

L¼�40.0 m, and L¼�1 m, respectively.

Figure 3 shows the pressure spectra predicted from the

three different stability conditions using the multiple expo-

nential profile fits. All the predictions use the C and k values

derived from measurement: C¼ 7.638 and k¼ 7.419.

The two unstable condition pressure spectra are a little

bit lower than the neutral atmospheric condition pressure

spectra at very low wave number. This is consistent with

Fig. 2(b), because the pressure fluctuation source is propor-

tional to the velocity gradient. The stability conditions for

the expected range of Monin-Obukhov lengths do not have a

large effect even at moderately low wave number. The maxi-

mum spectral level difference between the unstable condi-

tion and neutral condition is about 5 dB. At high wave

number, the slight difference between the three predictions

is due to the variation of the roughness length values, x0,

obtained from the fits. The difference between the unstable

condition and neutral condition on the turbulence-shear

interaction pressure spectrum is minor.

We have not taken measurements under stable condi-

tions—all the measurements have been taken during the day-

time, under windy conditions. The range modeled here

(L¼�10.0 m to L¼�1 m) should encompass all our data.

At the lower limit of measured wave number in this paper

(0.1 to 0.2 m�1), the difference between neutral conditions

and unstable conditions is on the order of 2 to 3 dB. Hence,

FIG. 2. (a) The measured profile and the fits for L¼�10.0 m, L¼�40.0 m,

and L¼�1 m. (b) The three profiles extended up to 100.0 m height. The

dotted, dashed, and solid lines are the fits for L¼�10.0 m, L¼�40.0 m,

and L¼�1 m, respectively. The black squares are the measured profile in

the first 2.0 m.

TABLE I. Values of a and x0 obtained from the fits to the wind velocity

profile models, Eqs. (19) and (20).

L (m) a x0 (m)

�10.0 1.30 0.020

�40.0 1.18 0.017

�1 1.13 0.016

TABLE II. Fit parameters of the multiple exponential form, Eq. (12), to the

three extended profiles from the roughness length up to 100.0 m height.

L (m) U (m/s) A0 A1 A2 A3 A4 A5

�10.0 11.7 0.360 0.032 0.142 0.220 0.203 0.057

�40.0 14.8 0.476 0.013 0.143 0.179 0.152 0.040

�1 14.4 0.277 0.176 0.178 0.179 0.150 0.045

626 J. Acoust. Soc. Am., Vol. 129, No. 2, February 2011 Yu et al.: Wind noise measured at the ground surface


the logarithmic fit of our measured profile near the ground

can be used to provide a reasonable prediction of the pres-

sure fluctuation spectrum over most of the wave number

range.

IV. MEASUREMENT AND ANALYSIS

Three types of measurements have been performed. The

first series of measurements was taken to investigate whether

the mirror flow model accurately simulates the behavior of

the vertical spectrum of the turbulence as the ground surface

is approached. We also measure and present the behavior of

the longitudinal turbulence spectra as a function of height

above the ground. The second series of measurements was

taken to investigate the pressure fluctuation levels measured

by the flush mounted microphones at the ground surface

simultaneously with the pressure levels measured under 2.54

cm of the porous metal foam. The third series of measure-

ments was taken to investigate the effect of foam thickness

on the measured pressure fluctuation spectral level. All data

presented were taken at Clegg Field in Oxford, MS, from

February to July in 2009. Clegg Field is a large flat open

area with mowed grass about 4–6 cm deep. Panofsky and

Dutton4 indicate such a surface has a roughness length corre-

sponding to approximately 0.01 m.

A. Investigation of the turbulence spectrum model

In order to investigate whether the mirror flow model is

a valid model for the behavior of the vertical and longitudinal

turbulence spectra with height, two sets of turbulence spectra

data were taken. In the first, a reference longitudinal wind ve-

locity spectrum was taken using a Gill Instruments R3A-100

Ultrasonic Research Anemometer mounted at 2.0 m above

the ground surface. The spectrum measured at 2.0 m is

approximately homogeneous and isotropic in the wave num-

ber range of interest. The Gill Anemometer (Gill Instruments

Ltd., Hampshire, England) has an internal sampling rate of

100 Hz. Since Eq. (5) requires a measurement of the wind

velocity spectrum in the direction of flow, the anemometer is

aligned so that the u1 direction is the mean direction of the

incident wind. If the wind shifts during the course of a mea-

surement, we perform a coordinate transform on the data to

assure that the mean of the u3 component of the velocity is

zero. Then the spectrum of the u1 component is fit to Eq. (5)

in order to get the C and k values used in calculating the lon-

gitudinal and vertical turbulence spectra for flow at lower

heights. The fitting method is described in Ref. 1.

The second set of turbulence spectrum measurements

are of the spectra at different heights using six Applied Tech-

nologies Anemometers mounted at 10, 50, 90, 110, 150, and

190 cm above the ground. The internal sampling rate of the

anemometers is 10 Hz. For the longitudinal or vertical turbu-

lence spectrum measurements, the measurement axes of all

the six anemometers were placed along the wind direction to

measure the longitudinal turbulence velocity or along the

direction perpendicular to the ground to measure the vertical

turbulence velocity.

All velocity data were acquired simultaneously for 1800 s

(30 min). The data from the Gill anemometer and the six 1-D

anemometers were collected at sampling rates of 100 and

10 Hz, respectively. All of the sensors were connected to a

National Instruments data acquisition card controlled by a pro-

gram written in LABVIEW. After acquisition, all data analysis

was done with MATLAB, except for the graphical fittings done

using ORIGIN.

The wind velocity power spectra were generated following

the method of Ref. 1. For the investigation of the height de-

pendence of the vertical and longitudinal turbulence spectra,

each run was broken into non-overlapping blocks of 256 sam-

ples as a compromise between good averaging and good resolu-

tion. Each block was detrended and windowed before its power

spectral density (PSD) was calculated. The average of the block

PSDs was calculated and converted from frequency to wave

number space using Taylor’s frozen turbulence hypothesis and

the convection velocity UC in the direction of flow,

Fvðk1Þ ¼UC

2pF0vðf Þ; (21)

where k1 ¼ 2pfUC

, Fv is the PSD of the turbulent velocity in

wave number space, and F0v is the PSD in frequency space.

A value of 0.7 times the mean stream wise velocity

measured by the Gill Anemometer at 2.0 m is used for the

convection velocity, UC. This choice is supported by the

cross spectral and cross correlation data in many studies14–19

and in our correlation measurements outdoors which deter-

mined a value of approximately 0.72 for the ratio of the con-

vection velocity to the mean wind speed measured away

from the surface, independent of the frequency. Figure 4 dis-

plays the comparisons of measured and predicted longitudi-

nal (a) and vertical (b) turbulence power spectra at different

heights. The predictions for different heights are generated

from Eq. (7). Table III lists the measurement information,

U1, UC and fit parameters C and k for Fig. 4.

From Fig. 4(a), it can be seen that the measured and pre-

dicted longitudinal turbulence spectra are all very similar

except at the 10 cm height. Measurements at all the other

heights are close to the prediction in both magnitudes and

FIG. 3. The predicted pressure spectra for the three stability conditions: The

dotted, dashed, and solid lines are the fits for L¼�10.0 m, L¼�40.0 m,

and L¼�1 m, respectively.



spectral slopes. The longitudinal turbulence spectrum is not

very sensitive to the height, and the measurements and the

predictions are close. Figure 4(b) displays the reduction of

the measured and predicted vertical turbulence spectra as the

ground is approached. The predicted spectral plots all level

off at low wave numbers as do the measured spectra. There

is a deviation at high wave number for 90 cm height mea-

surement. We have not observed this behavior in other data

sets. Overall the similarity of the measured and predicted

vertical turbulence spectra is satisfying.

In summary, there is a reasonable agreement between the

predictions using the mirror flow model and measurements of

the height dependences of the longitudinal and vertical turbu-

lence spectra. These measurements provide support for the

use of the models developed here to predict the wind noise

measured at and beneath the surface under turbulent wind

fields outdoors.

B. Investigation of the spectral models for turbulence-shear interaction pressure at and beneath the surface

This experiment measures the turbulence power spec-

trum, the wind velocity profile, and the pressure fluctuations

at the ground surface. The pressure spectra are then pre-

dicted from the measured turbulence spectrum and the wind

velocity profile fit to the logarithmic profile, and compared

to the measured pressure fluctuation spectra. The turbulence

data was collected with a Gill Anemometer mounted at 1.0

m height. The wind velocity profile measurements were

again taken using six 1-D Anemometers mounted at heights

of 10, 50, 90, 110, 150, and 190 cm with the measurement

axes aligned with the wind.

Pressure measurements were taken with Bruel & Kjær

(B&K, Nærum, Denmark) type 4193 1/2 in. microphones

powered by a Nexus brand conditioning amplifier. The fre-

quency response for the microphone drops off below 0.07

Hz, but the actual low frequency cutoff of the pressure data is

0.1 Hz set by the high pass filter in the Nexus. Figure 5 dis-

plays the microphone arrangement for simultaneous flush and

foam covered microphone measurements. One microphone

was mounted flush in a flat acrylic plate placed flush with the

ground surface. The other microphone was mounted in an

acrylic plate beneath a 2.54 cm thick sheet of aluminum

foam mounted flush in the same surface as the flush micro-

phone. All foam used in the measurements here are 40 pores

per inch (ppi) aluminum foam with a porosity of about 94%.

The piece used in this section is 2.54 cm thick and is approxi-

mately 35 cm by 40 cm. The microphone mounted flush with

the surface of the plate is mounted upwind from the foam.

Figure 6 displays the measurement arrangement. All velocity

and pressure data were acquired simultaneously for 900 s in

all runs. The pressure data were collected at a sampling rate

of 200 Hz. The wind speed data from each 1-D Anemometer

are averaged to give the mean wind speed at each height. The

measured wind velocity profile is fit to the logarithmic form

to determine the values of a and x0 [Eq. (16)].

FIG. 4. Comparisons of measured and predicted longitudinal (a) and verti-

cal (b) turbulence power spectra at different heights. The solid lines are the

predictions and the symbols represent the measured data for heights of 10

cm (þ), 50 cm (�), 90 cm (*), 110 cm (dot), 150 cm (square), and 190 cm

(circle). To make viewing the data easier, each data set and prediction were

multiplied by increasing factors of 4 starting with the 50 cm set, i.e., the 50

cm set is multiplied by 4, the 90 cm set by 16 etc.

TABLE III. Measurement information, U1, UC, and fit parameters C and kfor Fig. 4.

Figures Turbulence spectrum U1 (m/s) UC (m/s) C k

4(a) eU11ðk1Þ 2.10 1.47 9.04 20.97

4(b) eU22ðk1Þ 2.47 1.73 12.37 26.25

FIG. 5. (Color online) The flush mounted microphone at the surface and the

microphone mounted beneath the surface before and after the 2.54 cm thick

foam covering is placed.



Figure 7 displays results of the predicted and measured

pressure spectra for measurements on two different days.

Table IV lists the measurement information, U1, UC, and fit

parameters C, k, a, and x0 for preparing the predictions for

Fig. 7. All the measured profile fittings give roughness

length values close to that expected for mowed grass (0.01

m). Yu13 presents similar plots for four preliminary measure-

ments taken over 2 days and nine measurement sets taken

over 3 days. The results from these measurements are well

represented by the two sets presented herein.

Figures 7(a) and 7(b) represent the two different kinds

of results taken during our measurement program. The most

distinct difference between Figs. 7(a) and 7(b) is that the

measured levels for the flush and foam covered microphones

are very similar over a wide wave number range in Fig. 7(a),

while the flush microphone levels are roughly 15–20 dB

higher in Fig. 7(b). For all measurements presented in the

dissertation, 2 days (including 1 day for the preliminary

measurements) have results similar to Fig. 7(a); the other 3

days (including 1 day of the preliminary measurements) are

similar to Fig. 7(b). There is no apparent difference among

the 5 days in setting up the measurement device, and no

apparent weather condition difference can be categorized

between the two different kinds of results.

The measured spectral slope in the inertial range for

flush mounted microphones on low level days and for all

foam covered microphone measurements are well modeled

by the predictions with a value of about �5/3. The high level

flush microphone results are not as well behaved, but

roughly follow a �5/3 law in the inertial range. The predic-

ted spectra for the flush and under foam measurements are

very similar in the inertial region and source region. Both

increase proportionally to k21 in the source region and then

curve down in inertial range after reaching a maximum. At

high wave numbers, the foam covered microphones have

higher reductions than the flush microphones. The slight dif-

ference between the two predictions in each figure in the

middle wave number range is due to the small change of

coordinate system with respect to the wind velocity profile.

The large differences at high wave number are due to the

fact that the effective mean source region and the micro-

phone are separated by the additional thickness of the foam

for the microphone underneath the foam layer. The effective

source layer is closer to the surface at high wave number so

FIG. 7. Results of the predicted and measured pressure spectra for simulta-

neous measurements with a flush mounted microphone at the surface and a

microphone beneath the surface with a 2.54 cm thick foam covering on dif-

ferent days. The black solid and dashed lines are the measured and predicted

spectra for a flush microphone, respectively. The gray solid and dashed lines

are the measured and predicted spectra for a microphone beneath the foam

covering.

TABLE IV. Measurement information, U1, UC and fit parameters C, k, a,

and x0 for Fig. 7.

Figures Mic. U1 (m/s) UC (m/s) C k a x0 (m)

7(a) Flush, 2.54 cm foam 6.15 4.31 7.33 10.34 1.11 0.008

7(b) Flush, 2.54 cm foam 4.57 3.20 3.33 7.39 0.82 0.007

FIG. 6. (Color online) Measurement setup. The wind in the picture blows

from right to left.



the small additional separation has a larger effect on the high

wave number region of the spectrum.

For all our data, the foam covered microphone measure-

ments are well predicted both in magnitude and in spectral

slope for low and moderate wave number range that meas-

urements cover. An examination of all our data found that

the measured levels with the foam covered microphone

change consistently from figure to figure with the change of

the wind speed, while the flush microphone measurement

does not follow the wind speed variation systematically. We

can find no correlations between the meteorological condi-

tions and the presence or absence of the large error in the

flush microphone measurements. For example, the wind

speed for Fig. 7(b) is lower than for Fig. 7(a), but the flush

measurement in Fig. 7(b) has higher levels. The variation of

the difference between the flush and under foam measured

levels on different days cannot be due to the differences in

the atmospheric stability because the atmospheric stability

should influence them in the same way. We will examine

this contribution further in the conclusions.

At high wave number, our predictions model the effects

of larger reductions with the foam, but the predictions

decrease faster than the measurements. A better prediction at

high wave number would require a better understanding of

the profile close to the plate surface. The flat plate has a

shorter roughness length than the grass which surrounds it,

so there is a change in the surface layer profile when the

wind blows from the grass to the plate and this may influence

the results at high wave number.

C. Measurements under different foam thicknesses

The measurements from Sec. IV B were repeated with

different foam thicknesses as well as with an air gap between

the microphone and the foam. Four continuous runs were

taken on one day with the same microphone used in each

run. In the first three runs, the microphone was mounted

directly beneath a 1.27, 2.54, and 5.08 cm thick piece of alu-

minum foam, respectively. In the fourth run, the 1.27 cm

thick piece of aluminum foam was used with a 3.81 cm air

gap to the microphone face. All foams were 40 ppi alumi-

num foam with a porosity of approximately 94%.

Figure 8 displays the measured and predicted pressure

spectra for the four cases. Table V lists the measurement pa-

rameters used in preparing Fig. 8. Since the air gap, like the

foam, is assumed to have no horizontal average velocity, the

same calculation is used for the case with 1.27 cm thick foam

and a 3.81 cm thick air gap as with the 5.08 cm thick foam.

Figure 8 shows that all the foam covered microphone

measurements can be well modeled with our predictions in

the measured low and moderate wave number range. At high

wave number, the under foam predictions drop off faster

than the measurements. In Figs. 8(c) and 8(d), at the very

high wave number, the measured plots level off instead of

keep rolling down. The 5.08 cm thick foam (or foam plus air

gap) reduces the wind noise level sensed by the microphone

at that wave number range to lower than the background

noise, so the measured level follows the background noise

curve rather than decaying as wind noise. Excluding the

FIG. 8. Measured and predicted pressure spectra for a microphone directly beneath a 1.27 cm (a), 2.54 cm (b), 5.08 cm (c) thick foam covering and beneath a

1.27 cm thick foam with a 3.81 cm thick air gap (d) in four continuous runs. The solid line is the measurement and the dashed line is the prediction.



background noise effect, it is also found that the larger the

distance between the ground surface and the microphone

mounting plane, the more accurate the predictions are at

high wave number. This is reasonable, because the further

the microphone is located below the ground surface, the less

sensitive it will be to near surface details of the mean veloc-

ity profile. The good prediction in Fig. 8(d) is an indication

that our model assumption that there is no mean horizontal

velocity in the air gap beneath the foam is reasonable. The

foam does not have to extend all the way to the microphone

surface to provide additional wind noise reduction.

V. CONCLUSIONS

This paper provides predictions of the turbulence-shear

interaction pressure contributions to wind noise for a flush

microphone at the surface and a flush microphone in a sur-

face beneath a foam covering. Logarithmic and a multiple

exponential wind velocity profiles are used to develop pres-

sure spectrum predictions from the measured turbulence

spectrum. Investigations with the multiple exponential pro-

file model suggest that use of the logarithmic profile for

moderately unstable conditions is justified.

The height dependences of the longitudinal and vertical

turbulence spectra using the mirror flow model by Kraichnan

were investigated theoretically and were found to be in rea-

sonable agreement with the measurements. The high degree

of consistency provides a good support to the reliability and

effectiveness of our theory developed by using Kraichnan’s

assumptions and approach to model wind noise outdoors.

Our theory provides reliable predictions for all the runs

at low and middle wave number range of foam covered

microphone measurements. At high wave number, our model

overestimates the wind noise reduction and underestimates

the wind noise level. The refinement of the high wave num-

ber prediction would require more attention to evaluation of

the velocity field close to the surface.

The flush microphone measurements show large varia-

tions of the spectral levels relative to the under foam meas-

urements on different days and are not well predicted. It is

possible that the high level spectra are the results of a differ-

ent turbulence-shear interaction from that studied by

Kraichnan and this paper. All the indications suggest that the

largely varied levels for the flush microphone must originate

from the thin boundary layer next to the microphone. The

fact that the high level pressure has the same slope as the cal-

culated pressure fluctuations implies that the source is an av-

erage gradient interacting with a turbulence component. We

speculate that there are other contributions to the high level

pressure, which may be generated by a large average velocity

gradient interacting with longitudinal or transverse velocity

fluctuations near the surface.

The pressure fluctuations generated by a thin layer inco-

herent source would decay rapidly with distance. Some evi-

dence of rapid decay may be observed in Fig. 8. The

measurements all have a slightly higher level than predicted

for the 1.27 cm foam but the agreement is better for the 5.08

cm separation between the surface and the microphone.

Detailed measurements of the average and fluctuating veloc-

ity near the surface would be necessary for a quantitative

investigation of this contribution.

The research in this paper improves our understanding

of the basic characteristics and parameters of the flow that

influence the wind noise spectrum generated at and beneath

the surface. The success of the model developed for the

foam covered microphone lays a good theoretical basis for

investigating wind noise generated under porous layers and

under trees and bushes.

ACKNOWLEDGMENTS

This research has been supported by the U.S. Army

TACOM-ARDEC at Picatinny Arsenal, NJ.

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Figures Mic. U1 (m/s) UC (m/s) C k a x0 (m)

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Wind noise measured at the ground surface