PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Sound propagation in a forest based on 3D multiple scattering
theories
Vladimir E. OSTASHEV1; D. Keith WILSON2; Michael B. MUHLESTEIN3;
Keith ATTENBOROUGH4
1-3

U.S. Army Engineer Research and Development Center, Hanover, NH, USA
4Open University, Milton Keynes, UK

ABSTRACTThis paper overviews recently developed approaches for forest acoustics based on a three-dimensional (3D)
multiple scattering theory. The mean sound field is calculated with the effective wavenumber which
represents the scattering by trunks, branches, and the canopy. The radiative transfer equation is formulated
appropriately to forest acoustics; this result enables calculations of the mean sound intensities transmitted andbackscattered from a stand of trees. The correspondence between sound propagation in a turbulent
atmosphere and a forest is outlined. This correspondence, and the existing theory of the interference of thedirect and ground reflected waves in a turbulent atmosphere, enable analysis of a similar effect in a forest,
where these waves are scattered by forest elements. Numerical examples illustrating application of the 3D
multiple scattering theory for calculating the mean sound pressure and mean sound intensity are presented.
The 3D multiple scattering theory can also be used to study pulse propagation in a forest.
Keywords: Forest acoustics, Multiple scattering, Radiative transfer theory

1. INTRODUCTION
Forest acoustics is important in several applications such as noise reduction by a stand of trees andlocalization of sound sources, e.g., reference (1). Due to multiple scattering by trunks and branches,
attenuation in the canopy, micrometeorology, and interaction with the impedance ground, soundpropagation in a forest is a very complicated phenomenon. Despite significant efforts, there were nosatisfactory prediction methods based on first principles.
Recently, a 3D multiple scattering theory was applied to forest acoustics and the results obtained
were reported in references (2-5). The 3D multiple scattering theory assumes that different scatterers
in a forest have random locations. Then, the propagating sound field becomes a random field. This
stochastic approach is often used in wave propagation in complex media. Using this approach, closed
form equations for the mean sound pressure and mean sound intensity are derived. The 3D multiple
scattering theory is a rigorous approach and enables predictions in realistic environments. The theory
is related closely to sound propagation in a turbulent atmosphere. The results known in the latter field(6) can be used to advance forest acoustics. The main goal of this paper is to overview the 3D multiple
scattering theory as applied to forest acoustics. Note that the theory might not be applicable and needs
to be modified for a forest planted in a regular pattern.
The paper is organized as follows. In Sec. II, scattering properties of the canopy and trunk layers in
a forest are considered. Section III provides an approach for calculating the mean sound field. In Sec.
IV, the radiative transfer equation (RTE) is introduced, which enables calculations of the mean soundintensity. Section V considers the RTE in the high-frequency approximation. The results presented aresummarized in Sec. VI.

1

Vladimir.Ostashev@colorado.edu
D.Keith.Wilson@erdc.dren.mil
3Michael.B.Muhlestein@erdc.dren.mi
4
Keith.Attenborough@open.ac.uk
2574

2. SCATTERING CROSS SECTIONS IN A FOREST
In a 3D multiple scattering theory, a key quantity is the scattering amplitude ݂൫࢔Ԣǡ ࢔൯ of a scatterer,
which is defined as follows. Consider a plane wave propagating in the direction of the unit vector ࢔,
which is incident on the scatterer. The scattered wave can be expressed as a product of the Green ’s
function in free space, the amplitude of the incident wave, and the scatte ring amplitude ݂൫࢔Ԣǡ ࢔൯,where the unit vector ࢔Ԣ is in the direction of the scattered wave. Thus, ݂൫࢔Ԣǡ ࢔൯ characterizes theamplitude of the wave scattered in the direction of the vector ࢔Ԣ.
Equations for the statistical moments of the sound field in a forest contain the differential scatteringcross section (DSCS), ߪୢ ሺ࢔Ԣǡ ࢔ሻ, and the total cross section (TCS), ߪሺ࢔ሻ. These cross sections may be
expressed in terms of the scattering amplitude asߪୢ ሺ࢔Ԣǡ ࢔ሻ ൌ ߥȁ݂ሺ࢔Ԣǡ ࢔ሻȁଶ ǡߪሺ࢔ሻ ൌͶߨߥ݂ሺ࢔ǡ ࢔ሻ.
݇(1)Here, ߥis the number of scatterers per unit volume and ݇ is the sound wavenumber in free space. It is
often useful to express the TCS as a sum of the scattering cross section (SCS), ߪ௦ ሺ࢔ሻ, and absorption
cross section (ACS), ߪ௔ ሺ࢔ሻ:(2)ߪሺ࢔ሻ ൌ ߪ௦ ሺ࢔ሻ+ߪ௔ ሺ࢔ሻ.
The SCS characterizes the loss of energy of a sound wave propagating in the direction of the unit
vector ࢔ due to sound scattering in all directions. It is obtained by integrating the DSCS over the unit
vector ࢔ᇱ :(3)

ߪ௦ ሺ࢔ሻ ൌ න ߪௗ ሺ࢔ᇱ ǡ ࢔ሻ ݀ߗሺ࢔ᇱ ሻǤସగHere, ݀ߗሺ࢔ᇱሻ is the solid angle in the direction of the vector ࢔ᇱ . The ACS has a similar meaning but ispertinent to sound absorption rather than scattering. The ACS accounts for sound absorption in a forest
such as visco-thermal dissipation in foliage.
There are two approaches for determining the DSCS and TCS in a forest: (i) The cross sections canbe measured experimentally as explained in references (3,4). (ii) Trunks can be modeled as finite
vertical cylinders, branches as finite slanted cylinders, and the canopy layer as diffuse scatterers.
Using these scatterers, different realistic forests can be built.
As an example of the second approach, we provide modeling of the trunk and canopy layers, which
will be used later in the paper. Trunks are modelled as solid finite vertical cylinders with the scattering
amplitude ݂൫࢔Ԣǡ ࢔൯ from reference (7). The DSCS is then (3):ஶߪௗ ሺ࢔Ԣǡ ࢔ሻ ൌ ߥ ฬ݄ܾ݇ ሾ݄݇ሺ  ߠԢ െ  ߠሻȀʹሿ ෍ଶ‫ܤ‬௠ ሺ݉ሺ߮ᇱ െ ߮ሻሻฬ Ǥ

(4)

௠ୀ଴

Here, ݄ is the length of a cylinder, ܾ is its radius,  ሺ‫ݔ‬ሻ is the sinc function, ߠ and ߮ are the polar
and azimuthal angles of the unit vector ࢔, and ߠԢ and ߮Ԣ are those of the unit vector ࢔Ԣ. The
functions ‫ܤ‬௠ are given byሺଵሻ‫ܤ‬௠ ൌߝ௠ ᇱ‫ܪ‬௠ ሺܾ݇  ߠሻᇱ ሺܾ݇‫ܬ‬௠ ሺܾ݇  ߠሻ ൥ ߠ ‫ܬ‬௠ ሺܾ݇  ߠԢሻ െ  ߠԢ ‫ܬ‬௠ ߠ ᇱ ሻ ሺଵሻᇲ൩ǡʹ‫ ܪ‬ሺܾ݇  ߠሻ

(5)

௠

ሺଵሻ

where, ߝ௠ is the Neumann factor, ‫ܬ‬௠ is the Bessel function, ‫ܪ‬௠ is the Hankel function of the firstkind, and primes above these functions denote derivates with respect to the argument. Although theseformulas appear to be involved, they are relatively easy to implement numerically. The SCS, ߪ௦ ሺ࢔ሻ,can be obtained with Eq. (3). Since trunks are assumed as solid cylinders, ߪ௔ ൌ Ͳ and ߪሺ࢔ሻ ൌ ߪ௦ ሺ࢔ሻǤIn the considered example, branches can be modeled as solid finite slanted cylinders. The DSCS is
still given by Eqs. (4) and (5) with some transformation of the angles. Since the canopy layer usuallyconsists of many small scatterers such as leaves, needles, and twigs, it can be modelled by diffuse
scatterers. For such scatterers, the DSCS ߪୢ ሺ࢔Ԣǡ ࢔ሻ does not depend on the vectors ࢔Ԣ and ࢔ .
Visco-thermal attenuation and induced vibration in the canopy layer may be modelled by a finite valueof the ACS, ߪ௔ .
In a forest, the scattering amplitude and cross sections usually depend on the spatial coordinatesࡾ ൌ ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݖ‬ሻ. In what follows, this dependence is accounted for by including ࡾ as an argument of thecorresponding quantities, e.g., ݂ሺࡾǢ ࢔Ԣǡ ࢔ሻ, ߪୢ ሺࡾǢ ࢔Ԣǡ ࢔ሻ, and ߪሺࡾǢ ࢔ሻǤ

575

3. MEAN SOUND PRESSURE
In a 3D multiple scattering theory, the sound pressure in a forest can be written as(6)‫݌‬ሺࡾሻ ൌ ‫݌ۦ‬ሺࡾሻۧ ൅ ‫݌‬෤ሺࡾሻǤ
Here, the brackets ‫ ۄ ۃ‬indicate ensemble averaging, ‫݌ۦ‬ሺࡾሻۧis the mean sound pressure (also termed
the coherent sound field), and ‫݌‬෤ሺࡾሻis the sound-pressure fluctuation. The mean sound pressure can becalculated using the well-known parabolic equation (PE) in a non-random medium,௖మడబ‫ ۧ݌ۦ‬ൌ Ͳ,ቂʹ݅݇ డ௫ ൅ οୄ ൅ ݇ ଶ ቀͳ ൅ ሺ௖ ା௖ሻ
మ ቁቃ
బ

(7)

if the sound wavenumber ݇ is replaced with the effective wavenumber ݇ୣ୤୤ which accounts for soundscattering in a forest. In this equation, the ‫ݔ‬-axis is in the direction of sound propagation, ȟୄ ൌ
డమ
డ௭ మ

డమ

డ௬ మ

൅

, ܿሺࡾሻ is the sound speed, and ܿ଴ is its reference value. The effective wavenumber is given byʹߨߥ(8)݂ሺࡾǢ ࢔ǡ ࢔ሻǤ݇The effective wavenumber is also used in a 2D multiple scattering theory to predict the mean soundpressure, e.g., references (1,8,9). The effective wavenumbers for 2D and 3D cases coincide forhorizontal sound propagation and differ for slanted propagation, see reference (2).
Effective numerical algorithms for solving Eq. (8), which account for atmospheric stratificationand sound interaction with the impedance ground, have been developed in atmospheric acoustics , e.g.,reference (6). These algorithms can be readily applied to sound propagation in a forest. As an example,
Fig. 1 depicts the complex amplitude of the mean sound pressure, ‫ ۄܣۃ‬ൌ ሺെ݅݇‫ݔ‬ሻ‫ۄ݌ۃ‬, due to a point
unit strength source. In the figure, dashed horizontal lines indicate the trunk and canopy layers, whichare 12 m and 25 m high. For this example, the sound frequency is 1 kHz, the source height is 1 m, thenumber of trees per unit area is ܰ ൌ ߥ݄ ൌ ͲǤͲͷ1/ଶ , and the tree radius is 0.09 m. These parameters
and the scattering amplitudefor solid finite vertical cylinders (7) enable us to calculate ݇ୣ୤୤ in thetrunk layer. In the canopy layer, we assume that ሺ݇ୣ୤୤ െ ݇ሻ is that in the trunk layer multiplied by afactor 1.5. The relaxation model (6,10) with parameters pertinent for a fermentation/humus layer in aforest (11) is used to assess sound interaction with the ground, see reference (3) for more details. In
Fig.1, four subplots depict gradually increasing complexity of sound propagation. Figure 1(a)corresponds to a source located above an impedance ground in a homogeneous atmosphere. In Fig.
1(b), a 12 m high trunk layer is added to a homogeneous atmosphere. As a result, the complex
amplitude in this layer and above is significantly attenuated starting at the range of about 200 m. In Fig.
1(c), a 13 m high canopy layer is added to the previous geometry. This results in significant soundattenuation with height. Finally, in Fig. 1(d), the sound speed has a vertical gradient 0.12 1/s which
results in downward refraction.
The mean sound pressure attenuates approximately exponentially with propagation range and isapplicable for a relatively small range. Beyond this range, the 3D multiple scattering theory provides
approaches for calculating the mean sound intensity as explained in sections 4 and 5.
݇ୣ୤୤ ሺࡾǡ ࢔ሻ ൌ ݇ ൅

4. MEAN SOUND INTENSITY
The mean sound intensity is given by ‫ ܫ‬ൌ ‫݌ۦ‬ሺࡾሻ‫ כ݌‬ሺࡾሻۧ. Substituting with Eq. (6), the mean
intensity can be written as(9)‫ܫ‬ሺࡾሻ ൌ ‫݌ۦ‬ሺࡾሻۧ‫ כ݌ۦ‬ሺࡾሻۧ ൅ ‫݌ۦ‬෤ሺࡾሻ‫݌‬෤‫ כ‬ሺࡾሻۧǤThe first term on the right-hand side ‫ܫ‬௖ ሺࡾሻ ൌ ‫݌ۦ‬ሺࡾሻۧ‫ כ݌ۦ‬ሺࡾሻۧ is termed the coherent sound intensity
and the second term ‫ܫ‬ௗ ሺࡾሻ ൌ ‫݌ۦ‬෤ሺࡾሻ‫݌‬෤‫ כ‬ሺࡾሻۧ the diffuse sound intensity.
In the radiative transfer theory, the mean intensity can be expressed in terms of the radiance,‫ܫ‬ሺࡾሻ ൌ ‫׬‬ସగ ‫ܬ‬ሺࡾǡ ࢔ሻ݀ߗሺ࢔ሻ.

(10)

The radiance ‫ܬ‬ሺࡾǡ ࢔ሻ is defined as the average energy flux within a unit solid angle ߗ in the directionof the unit vector n. It follows from Eq. (10) that the mean intensity is obtained by integrating theradiance over all directions of the vector n. The radiance can be found by solving the RTE:

576

డ

ቀ࢔ ή డࡾቁ ‫ܬ‬ሺࡾǡ ࢔ሻ ൅ ߪሺࡾǢ ࢔ሻ‫ܬ‬ሺࡾǡ ࢔ሻ ൌ ‫׬‬ସగ ߪୢ ሺࡾǢ ࢔Ԣǡ ࢔ሻ‫ܬ‬ሺࡾǡ ࢔Ԣሻ ݀ߗሺ࢔Ԣሻ.

(11)

Figure 1 – Complex amplitude of the mean sound pressure versus range and height. The geometries of soundpropagation are explained in the text.
Solutions of the RTE are well developed in many fields of physics and can be readily applied toforest acoustics. The RTE accounts for diffraction. Similar to the mean sound intensity, the radiancecan be expressed as a sum ‫ܬ‬ሺࡾǡ ࢔ሻ ൌ ‫ܬ‬௖ ሺࡾǡ ࢔ሻ ൅ ‫ܬ‬ௗ ሺࡾǡ ࢔ሻ, where ‫ܬ‬௖ ሺࡾǡ ࢔ሻ is the coherent radiance and‫ܬ‬ௗ ሺࡾǡ ࢔ሻ is the diffuse radiance. Reference (3) provides RTEs for the coherent and diffuse radiances.
Consider an example of using the RTE in forest acoustics. Let a plane sound wave be normally577

incident on the forest edge, with a receiver located at the other edge of the forest. The forest is modeledwith four layers: ground, trunks, canopy, and open air. This geometry is pertinent to sound attenuation
by a stand of trees. Parameters typical for a temperate conifer forest are considered: the height of the
canopy layer is 25 m, the height of the trunk layer is 12 m, the number of trees per unit area is 0.05
1/ଶ , and the tree radius is 0.09 m. Furthermore, the sound frequency is 1 kHz and the height of thereceiver is 2 m. Using these parameters, the TCS and DSCS in the trunk layer can be calculated with
Eqs. (2)-(5). Currently, scattering and absorption properties of canopy layers are mostly unknown. For
concreteness, we assume that ߪୟ ൌ ߪୱ and the TCS ߪ ൌ ߪୟ ൅ ߪ௦ is equal to ߪሺ࢔ሻ in the trunk layer forthe vector ࢔ in the direction of the incident wave. Sound interaction with the ground is modelledsimilar to that in Fig. 1.

Figure 2 – (a) Normalized sound intensities transmitted through the forest. (b) Normalized sound intensity
backscattered from the forest.
For the considered example, the RTE, Eq. (11), is solved in a modified Born approximation. Figure2(a) depicts the mean, coherent, and diffuse sound intensities, normalized by the incident intensity,transmitted through the forest versus the forest length. The coherent intensity exponentially attenuateswith increasing forest length. The diffuse intensity is zero at the forest edge, reaches a maximum at74.5 m, and then decreases with the forest length. The mean sound intensity is a sum of the coherent
and diffuse intensities. Figure 2(b) depicts the mean sound intensity backscattered from the forest. The
backscattered intensity monotonically increases with increasing forest length and reaches a plateau at
about 140 m.
The RTE is valid for both low and high frequencies. Sound propagation in the atmosphere is usuallyconsidered in the high-frequency approximation. In this approximation, the RTE simplifies as
explained in the next section.

5. HIGH-FREQUENCY APPROXIMATION
The high-frequency approximation corresponds to narrow-angle sound propagation in a
predominant direction, which we align with the ‫ݔ‬-axis. In this case, the RTE can be reduced to thefollowing PE, see reference (5):
ቈ

߲
݅ ߲ଶ
െ
൅ ߪୟ ሺ‫ݔ‬ǡ ࢘ሻ ൅ ܳሺ‫ݔ‬Ǣ ࢘ǡ ࢘ௗ ሻ቉ ‫ܤ‬ሺ‫ݔ‬Ǣ ࢘ǡ ࢘ௗ ሻ ൌ ͲǤ߲‫߲߲࢘࢘ ݇ ݔ‬ௗ(12)Here, ‫ܤ‬ሺ‫ݔ‬Ǣ ࢘ǡ ࢘ௗ ሻ ൌ ‫݌ۦ‬ሺ‫ݔ‬ǡ ࢘ଵ ሻ‫כ݌‬ሺ‫ݔ‬ǡ ࢘ଶ ሻۧ is the spatial correlation function of the sound field, ሺ‫ݔ‬ǡ ࢘ଵ ሻand ሺ‫ݔ‬ǡ ࢘ଶ ሻ are two points of observation located at the same range ‫ݔ‬, ࢘ௗ ൌ ࢘ଵ െ ࢘ଶ is the vector
between the observation points in the plane perpendicular to the ‫ݔ‬-axis, ࢘ ൌ ሺ࢘ଵ ൅ ࢘ଶ ሻȀʹ is thegeometrical center of these two points, and ߪୟ corresponds to the ACS in the direction of the ‫ݔ‬-axis.
The mean sound intensity is obtained by setting ࢘ௗ ൌ Ͳ in the correlation function: ሺ‫ݔ‬ǡ ࢘ሻ ൌ‫ܤ‬ሺ‫ݔ‬Ǣ ࢘ǡ ࢘ௗ ൌ Ͳሻ. In Eq. (12), the function ܳሺ‫ݔ‬Ǣ ࢘ǡ ࢘ௗ ሻ is given by578ܳሺ‫ݔ‬Ǣ ࢘ǡ ࢘ௗ ሻ ൌͳන ߪୢ ሺ‫ݔ‬ǡ ࢘Ǣ ࣄȀ݇ሻ൫ͳ െ ௜ࣄή࢘೏ ൯݀ ଶ ߢǡ
݇ଶ

(13)

where ߪୢ ሺ‫ݔ‬ǡ ࢘Ǣ ࣄȀ݇ሻ has the following meaning. In the narrow-angle approximation, the DSCSߪୢ ሺ‫ݔ‬ǡ ࢘Ǣ ࢔Ԣǡ ࢔ሻ depends only on the difference between the vectors ࢔Ԣ and ࢔: ߪୢ ൌ ߪୢ ሺ‫ݔ‬ǡ ࢘Ǣ ࢔ᇱ െ ࢔ሻ. In
this formula, ࢔ᇱ െ ࢔ ൎ ࢔ᇱୄ െ ࢔ୄ, where ࢔ୄ is the component of the vector ࢔ perpendicular to the‫ݔ‬-axis. Then, ߪୢ ൎ ߪୢ ሺ‫ݔ‬ǡ ࢘Ǣ ࢔ᇱୄ െ ࢔ୄ ሻ ൌ ߪୢ ሺ‫ݔ‬ǡ ࢘Ǣ ࣄȀ݇ሻ, where ࣄ ൌ ݇ሺ࢔ᇱୄ െ ࢔ୄ ሻ. Thus, ߪୢ ሺ‫ݔ‬ǡ ࢘Ǣ ࣄȀ݇ሻ isthe DSCS in the narrow-angle approximation.
Equation (12), which is termed the second-moment PE, also describes the correlation function of
the sound field in a turbulent atmosphere. In this case, the DSCS, ߪୢ ሺ‫ݔ‬ǡ ࢘Ǣ ࣄȀ݇ሻ, is expressed in termsof the spectra of atmospheric turbulence and ߪୟ ൌ Ͳ, see reference (5) for details. Effective numerical
techniques for solution of the second-moment PE have been developed in atmospheric acoustics(6,12,13) and can be used to study sound propagation in a forest. This approach can be generalized toaccount for sound refraction in a forest.
Sound propagation in a turbulent atmosphere has been investigated relatively well and issummarized in reference (6). The results from this reference and the correspondence between soundpropagation in a turbulent atmosphere and forest can be used to advance forest acoustics. As an
example, consider the effect of trees on the interference of the direct and ground -reflected waves. For
typical ground impedances of forest floors and certain frequencies and ranges, this interference could
result in significant reduction of the sound pressure level. In practice, however, this reduction does not
fully occur since scattering by trees diminishes the coherence between the direct and ground -reflected
waves. A similar phenomenon exists (and is well studied) in atmospheric acoustics, where turbulence
reduces coherence between these waves. Using the similarity between sound propagation in aturbulent atmosphere and forest, we rigorously account for the effect of trees on the interference
between the direct and ground-reflected waves, see reference (5). For the mean-squared soundpressure, we have‫ ۧ כ݌݌ۦ‬ൌ

ଵ
ோభమ

൅

ଵ
ோమమ

൅

ଶȁ࣬ȁ஼ౙ౥౞ோభ ோమሺ݇ሺܴଶ െ ܴଵ ሻ ൅ ߚሻ.

(14)

Here, ܴଵ and ܴଶ are the path lengths of the direct and ground-reflected waves, ࣬ ൌ ȁ࣬ȁሺ݅ߚሻ isthe spherical-wave reflection coefficient, and ‫ܥ‬ୡ୭୦ is the coherence factor which describes the loss ofcoherence between the direct and ground-reflected waves due to scattering in a forest. The coherence
factor is expressed in terms of DSCS,‫ܥ‬ୡ୭୦ ൌ  ቈെ

ஶ
‫ ݔ‬ଵ
න
݀ߟ
න
݀ߢ௭ ߪୢ ൫ߢ௬ ǡ ߢ௭ ൯൫ͳ െ ݁ ௜ఎ఑೥ ௛ೞೝ ൯቉ǡ
݇ଶ ଴
ିஶ

(15)

where ‫ ݔ‬is the propagation range, ݄௦௥ ൌ ʹ݄௦ ݄௥ Ȁሺ݄௦ ൅ ݄௥ ሻ, and ݄௦ and ݄௥ are the source and receiverheights. For sound scattering by trunks and ݄௦௥ ൑ ݄, Eq. (15) reduces to‫݄ܾܰݔ‬௦௥(16)൰Ǥ݄
Equations (14-16) enable us to access the effect of trees on the interference of the direct andground-reflected waves. As an example, the solid line in Fig. 3 depicts the sound pressure level (SPL)
relative to that in free space, ͳͲሺ‫ܴۧ כ݌݌ۦ‬ଵଶ ሻ, versus sound frequency ݂. The propagation range is
100 m, the source and receiver heights are 1 m and 2 m, the height of the trunk layer is 12 m, thenumber of trees per unit area is 0.05 ିଶ, and the tree’s radius is 0.09 m. The ground impedance is thesame as that assumed in Figs. 1 and 2. The other two lines in Fig. 3 correspond to the cases when there
is no forest (in Eq. (14), ‫ܥ‬ୡ୭୦ ൌ ͳ) and when the direct and ground-reflected waves are added
incoherently (‫ܥ‬ୡ୭୦ ൌ Ͳ). It follows from Fig. 3 that without the forest, the interference results inmaxima and minima of the SPL as a function of frequency. With the forest, the interference minima are
reduced by several dB due to the coherence loss between the direct and ground-reflected waves,resulting in an apparent increase in the SPL. However, the ground effect interference pattern ispredicted not to be completely suppressed since, as shown by several measurements (1), there remains
some coherency between direct and ground-reflected paths in a forest. When these waves areincoherent, the maxima and minima are completely suppressed, and the SPL only slightly depends onthe frequency (Fig. 3).
‫ܥ‬ୡ୭୦ ൌ  ൬െ

579

Figure 1 – SPL relative to that in free space for the interference of the direct and ground-reflected waves in aforest.

6. CONCLUSIONSThis paper overviewed application of a 3D multiple scattering theory to forest acoustics (2-5). In
this theory, the differential scattering cross section (DSCS), total cross section (TCS), scattering crosssection (SCS), and absorbing cross section (ACS) are utilized to characterize different scatterers in aforest such as trunks, branches, and the canopy. Approaches were outlined to access these crosssections.
The mean sound pressure ‫݌ۃ‬ሺࡾሻ‫ ۄ‬may be calculated with a standard parabolic equation (PE) if thesound wavenumber in free space, ݇, is replaced with the 3D effective wavenumber, ݇ୣ୤୤ , which
accounts for sound scattering in a forest. The mean sound intensity ‫ܫ‬ሺࡾሻ was expressed in terms of theradiance ‫ܬ‬ሺࡾǡ ࢔ሻ , which satisfies the radiative transfer equation (RTE). In the high-frequency
approximation, the RTE reduces to the second-moment PE for the correlation function of the soundfield ‫ܤ‬ሺ‫ݔ‬Ǣ ࢘ǡ ࢘ௗ ሻ; the mean sound intensity is obtained by setting ࢘ௗ ൌ Ͳ in the correlation function.
The correspondence between sound propagation in a turbulent atmosphere and forest was outlined andused to predict the effect of trees on the interference of the direct and ground-reflected waves. The
methods for calculating the mean sound pressure and intensity were illustrated by numerical examples.
Propagation of continues sound signals was considered. A 3D multiple scattering theory can also be
used to predict pulse propagation in a forest as described in reference (4). 3D multiple scattering
theory appears to be a very suitable approach for forest acoustics.

ACKNOWLEDGEMENTS
This research was sponsored by the U.S. Army Engineer Research and Development Center,
Geospatial Research and Engineering business area. Permission to publish was granted by Director,
Cold Regions Research and Engineering Laboratory.

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