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American Institute of Aeronautics and Astronautics

1

Wave Confinement: Long-Distance Acoustics Propagation

through Realistic Environments

Subhashini Chitta

1

, John Steinhoff

2

and Andrew Wilson

3

Wave CPC Inc, Tullahoma TN 37388

and

Frank Caradonna

4

AFDD, Moffett Field, CA 94035

A new Eulerian method, "Wave Confinement", is presented for propagating short

acoustic signals over long distances through realistic environments, without numerically

induced spreading. Effects due to atmospheric and ground variations are automatically

captured, enabling more accurate acoustic predictions on modest computational grids.

Terrain effects are captured using an immersed boundary technique that avoids any need to

fit meshes to complex topography, which is especially important in real (3-D) cases.

I. Introduction

Several methods have been developed in the past to solve the linear wave equation. These methods tend to fail or

become complicated in realistic environement where the propagation is dominated by atmospheric and ground

effects. For example, Greens function methods, which are well known, require a closed form propagator function

and can only be applied for simple homogeneous media. Over long distances, such Green’s functions can be poor

approximations, as the propagation environment is often dominated by reflecting or refracting inhomogeneities,

which is a major challenge in long distance propagation. High frequency approximations such as Ray Tracing are

widely used, but fail at moderate or low frequencies, where diffraction effects become important. Another group of

methods, including Parabolic methods, involve complicated algorithms to accommodate wide angle propagation,

and are not feasible for realistic environments with multidirectional propagation. Eulerian methods can treat these

effects, but suffer from numerical dissipation and cannot be used for propagation over many wavelengths. Several

advanced techniques have been developed to reduce this limiting dissipation error, such as higher order

discretization. However, this does not eliminate the problem and cannot be used at boundaries.

A new endeavor to eliminate this error is termed “Wave Confinement” (WC), which uses Nonlinear Solitary

Waves (NSW’s) to propagate the solution. These NSW’s obey the wave equation and are numerically non-

dissipating and non-dispersing. Their application to the long-distance propagation of acoustics from a compact

source is the subject of this paper. (We assume the acoustic field is specified.)

There are many areas where the method described can have important applications. These include seismic

exploration, radar tracking or aircraft communication in complex environements, cell phone communications, and

many others. This paper describes one specific application: computation of rotorcraft acoustics at large distances.

This is vital for determining the detectability on the battlefield and is the the subject of a current SBIR project.

However, the method, and its implementation in a computer code, can be used with little substantive change for

many of the problems described above.

Our approach to solve the long distance wave propagation problem is to separately treat the near field acoustic

prediction and far field propagation components. For the far field, we use the WC method, described below and in

Ref.1. For the near field, we use any of the widely available near field analyses (for example, OVERFLOW, GT-

hybrid, HELIOS, FUN3D, etc., for rotorcraft aerodynamic noise prediction). With such codes, the near field is

1

Research Scientist

2

Chief Research Scientist, B. H. Goethert Professor (ret)

3

Research Scientist

4

Emiretus Scientist

American Institute of Aeronautics and Astronautics

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limited by computer memory and computational times to a region whose overall size is of the order of several wave

lengths. There are a number of methods that could be used to propagate the acoustic waves from the near field

(where the medium properties are close to uniform) to the far field. One possible approach would be to simply

interpolate the (filtered) inner solution onto the outer coarse grid, and propagate it indefinitely using WC. However,

with this approach, the computed acoustic waves (which can be much wider than the physical wave due to grid

restrictions) only preserve the speed and pressure integral through each cross section of the physical wave form,

while the remaining information is lost. (With any other discrete Eulerian approach, even this is not possible.) In this

way, Wave Confinement can give a general propagation distribution of a few acoustic wave properties, but no fine

details of the waveform.

However, a promising new alternative has been devised to couple the inner CFD region and outer acoustic

region. This involves a surface of interpolation (usually a sphere) between the two regions. We call this an

“acoustic” or “source” surface with a radius of a few wavelengths, which surrounds the acoustic source. (If this

distance is “large enough,” the waves will propagate closely along local normals at the source surface.) As a result,

the wave at any point in space (outside the acoustic sphere) derives from a small region of a specific point on the

sphere. This is the basis for the present far-field propagation method that does not lose waveform information even

under refraction and reflection. A schematic of this new idea is shown in Figure 1. This uses Wave Confinement to

propagate the coordinates of each point on the acoustic surface (where the waveform is completely known) instead

of the entire complex waveform. The concept of using initial source coordinates as conserved variables was

published previously2, as “dynamic surface extension”. In later papers, the method was renamed, “closest point

method”3. With this new approach, details of physical waveforms are not numerically propagated: only locations of

the origin of a known waveform. This small amount information is accurately preserved during the propagation

using Wave Confinement. A point in the far field can then be associated with a specific point on this initial sphere,

which can then be used to compute the distant acoustic field. Our unique approach involves computationally

emitting a short computational wave (CW) from a set of points on the acoustic sphere and propagating them through

a grid (Cartesian, though others are possible) using the acoustic wave equation modified to propagate conserved

solitary waves. The components of this CW (or the information “carried”) include the x, y, and z coordinates of the

emission point on the acoustic sphere, and total amplitude.

Figure 1: Schematic of the problem

An Eulerian propagation of these quantities (with no computational losses) can then be used to determine the

point of origin and geometric attenuation at every point in the computational space – which can be used to determine

the acoustic field. This computation requires propagation of a CW indefinitely with no loss, while remaining

compact, i.e., spanning only a few grid cells. Conventional CFD methods cannot do this because they always have

some dissipation. There is no other way known to the authors of performing this operation with conventional finite

difference schemes over long distances.

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II. Propagation Model

The basic enabling concept of this work is to compute propagating CW’s that follow wave front solutions of the

wave equation using the method of Wave Confinement (WC). The idea is to “capture” thin features over a few grid

cells on a computational grid by using model discretized equations. This is similar to shock capturing (as opposed to

“shock fitting” that uses markers to define the geometry), and goes back to the work of Lax and others3. The thin

features are treated as nonlinear solitary waves that can propagate on the computational lattice indefinitely with no

numerical dissipation, and can act as “wave packets” propagating sub-grid-scale details.

This model is a discrete Eulerian method in which the wave equation is modified by the addition of a term that

results in propagating solitary waves without changing the basic conservation properties of the equation. These

conservation properties include the line integral normal to the wave front of the quantity to be transmitted, and the

centroid of that solution. This can then be used to determine the time of arrival and propagation direction. The

essential solitary wave property of the CW’s (in addition to these integral properties) is that they do not have the

dispersive and diffusive truncation error inherent in conventional Eulerian methods: A confined solitary wave may

undergo distortions due to local truncation effects, but due to the shape preservation property, it always returns to its

(approximate) asymptotic shape with no error accumulation – as a result the pulse never decays. The other unique

property of these solitary waves is that they have the same structure on the grid, independent of the actual grid

interval size. Because of this property the grid can be made as coarse as necessary, consistent with overall accuracy

(for example, to resolve atmospheric variations).

A. Wave Confinement

The wave equation for a travelling wave of speed, c, is (with no background wind)

222

2c

t

(1)

If the above equation is numerically solved using a conventional, discrete finite difference scheme, it will develop

discretization errors. Conventional Eulerian schemes4 suffer some dissipation no matter what the order is. Many

higher order schemes have been proposed to reduce the numerical diffusion/dispersion, but they only reduce the

error if the number of grid points across the pulse, N is relatively large. As the present work involves treating thin

pulses—approximately 2-3 grid cells in size—the method cannot be made more accurate by increasing order. To

keep the pulses confined on the discrete domain, numerical dissipation/dispersion must be counter-acted by adding a

new term to the advection equation that will not interfere with the essential properties of the pulse. This term is

called the “Confinement” term and hence the method is named “Wave Confinement”.

The discretized form can only conserve a limited number of physical quantities of the pulse corresponding to the

limited number of grid nodes across the pulse. These include total amplitude (normal to the wave) and the centroid

speed. The modified form of equation (1), with the confinement term, when discretized, is:

1 1 2 2

, , ,

2

n n n

i j i j i j E

(2)

where,

2

n

EF

,

ct

h

,

t

is the time step, h is the grid cell size and n labels the time step. The idea is

that the second order form of F acts as a permanent “pulse shaping” term and the quantities of interest are conserved.

A simple form of F can be taken as

F

(3)

where

,

are constants and

is the harmonic mean given as a sum over neighboring nodes:

1

1 1 1 1

,,

1 1 1

,, 27

n

i j k

n

i j k

(4)

To demonstrate long distance propagation with no increasing numerical errors, a diverging (2-D) circular wave of

speed

0.23

is simulated on a

2

(128)

cell grid with periodic boundary conditions. Confinement values used

American Institute of Aeronautics and Astronautics

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where

0.2

,

0.3

. In Figure 2, the circular wave for different time steps is presented. It can be seen that the

wave persists on the grid with no plottable error.

Figure 2: Circular wave propagation

B. Treatment of Boundaries

The reflection approximation described below is consistent with the first order treatment of the wave front by

WC. WC has an important advantage over conventional methods for these cases: Unlike existing finite difference

methods, which must be higher order to prevent numerical error, WC does not suffer from numerical diffusion and

hence, complex configurations can be treated using very efficient, simple lower order discretizations, with accuracy

consistent with the specification of the physical boundary. Also unlike, for example, ray tracing, the reflected wave

is directly computed without having to compute a surface normal at each point of reflection.

This new approach employs a simple contour or “level set” representation of the surface and can easily

accommodate complex topography with little computational effort, unlike complicated mapping5 or adaptive grid

methods6. For example, reflection from a (3-D) boundary defined by

( , , ) 0f x y z

, only requires setting

,, 0

n

i j k

for all grid nodes, where

,, 0

i j k

f

, at each time step, (n).

A thin (about 2-3 grid cells) boundary region is maintained using this “immersed” boundary approach because

the confinement term contracts the pulse in the normal direction. Also, “staircase” artifacts do not appear because

WC has a smoothing effect in the tangential direction. This can be seen in Figure 3, where an initially smooth,

straight wave at an angle is reflected from a boundary not aligned with the grid, and after reflection, becomes

smooth and straight again after propagation over only a few grid cells.

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Figure 3: Plane wave reflection from immersed surface

An important point is that the (locally irregular) boundary automatically results in smooth reflected waves with

WC. This would require interpolation along the boundary if ray tracing were used. Scattering from a more

complicated ground (for example, a Gaussian half-hill) is shown in Figure 4 for a source located at (0, 300) (in 2-D).

Figure 4: Reflection from a Gaussian half hill. Blue: direct wave, red: reflected wave

At low frequencies, the high frequency ray tracing approximation fails and the waves undergo diffraction.

However, WC still gives meaningful results. As shown in Figure 5, diffracted and creeping waves propagate into the

shadow region of an obstruction, which is not possible to capture using Ray Tracing. The wave fronts shown in the

below figure are the computational wave fronts with a (computed) diffraction pattern corresponding to the “central

frequency” of the computed wave (~ 2 - 3 grid cells wide), independent of the frequency of the physical waves. A

correction to account for the frequency is applied to the amplitude to compute physical diffracted waves (Work in

progress). More results will be shown in the presentation.

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Figure 5: Wave front at a sequence of consecutive time steps

C. Propagation through background wind

One of the important factors that affect acoustic wave propagation in the atmosphere is the planetary boundary

layer. In the presence of non-uniform background wind, the rays bend upwards or downwards depending on the

wind gradient and propagation direction. To accommodate this feature, an additional step is added to the

computation. The wave is advected according to the difference equation:

tU

(5)

This will take into account the advection due to the background wind, U. In this paper, the atmosphere is

assumed to be stratified (i.e., wind and other atmospheric variables only vary with altitude). A simple example for

exhibiting the influence of the wind gradient is shown below in Figure 6. It can be seen that due to the presence of

wind, each part of the wave advects with a different speed and distorts the shape.

Figure 6: Left: Boundary layer background wind profile. Right: CW with and without background wind.

D. Propagation of structural details

First, the acoustic variables (computed using an existing CFD method or experiment) from the near field

computation are saved at each point on the outer acoustic surface, r = KR. This function is

x ,t

s

p

, where

s

x

defines the points on the emission sphere. These could be defined with only one independent variable in 2-D and

two in 3-D, for example, angles in spherical coordinates. Instead, we use Cartesian points (i, j, k), which involve one

additional coordinate to avoid discontinuities.

All the information defining the initial state of the acoustic field is contained in

x ,t

s

p

. The important point

is that, due to the short wavelength (compared to total distance), acoustic information at each point in the far field at

each time step can be defined, or “read”, for a corresponding emission time and acoustic surface point. The acoustic

waveform at each

xs

propagates unchanged (assuming no dispersion or dissipation) throughout the far field along

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each “ray”, or normal to the wave surface. Thus, at any point in the far field at any time, we have a mapping or line

linking it to a point on the acoustic surface. The acoustic pressure stored at that point on the acoustic sphere as a

function of

x ,t

s

is then directly related to the pressure at the far field point, only translated by the travel time and

scaled by a simple geometric attenuation factor.

We first initialize multicomponent CW’s,

0 1 2 3

( , , , )

, at r = KR, where

0

is delta-like function,

10

i

,

20

j

,

30

k

, and (i, j, k) are the coordinates of the each grid cell that lies within the

initial CW with amplitude,

0

. These are propagated into the far field using the wave equation (described for no

background wind, for simplicity),

2 2 2 2

t l l t

cF

(6)

where F is the confinement term (previously defined7) and l = 0, 1, 2, 3.

For a source location, (

0

i

,

0

j

,

0

k

), the attenuation due to geometric spreading,

f

A

, at each far field point, is

computed using the relation,

0

t

fsource

dt

AA

(7)

where

source

A

is the source strength.

When there are multiple wave passes at any point due to reflection, the above mentioned expectation values (

i

,

j

,

k

,

) and geometric attenuation factor,

f

A

, for each pass are computed and saved separately. A simple

logic that will identify each pass is used so each arrival can be assigned an index, which can later be accessed to

compute waveforms corresponding to each arrival. Each of these computed waveforms can be sorted into a set of

“bins” by time of arrival relative to the time at source.

E. Validations

A first validation is shown in Figure 7, where the computed and analytical attenuation factor (1/r) are plotted. It

can be seen that WC can accurately compute the attenuation factor. (Note: A homogenous medium is used for

demonstration. More realistic effects are shown in later sections.)

Figure 7: Geometric attenuation (1/r)

As the CW’s pass over each grid node, expectation values,

i

,

j

,

k

are computed as

American Institute of Aeronautics and Astronautics

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0

f

t

t

dt

fdt

(8)

These remain unchanged along each ray (even when there is refraction and reflection). These will allow us to

determine the point of origin of the ray on the acoustic sphere. Figure 8 shows that the computed elevation angle,

0

,,

1

,, ,,

cos i j k

i j k i j k

kk

r

, where

,,i j k

r

ri,j,k =√〈x〉i,j,k2+〈y〉i,j,k2+〈z〉i,j,k2 is the radius at a point, (

i

,

j

,

k

), has good agreement with the analytical solution. As explained, (

i

,

j

,

k

) are the coordinates of the

point of origin (on the source sphere) corresponding to any far field point. Also, it can be seen that the constant

elevation angle (

) contours are equivalent to ray paths.

Figure 8: Constant elevation angle (

) contours plotted for xz plane. WC (dashed)

and Analytical solution (solid).

In addition, arrival times (

), which represent wave fronts or phase contours are computed and validated using

an analytical formulation as shown in Figure 9. At any far field point, the physical waveform is simply,

* x ,t

fs

Ap

.

Figure 9: Constant arrival time contours on xz plane. WC (dashed).

Analytical solution (solid).

The above method has been implemented in a realistic medium with variable speed of sound. For a linearly

increasing or decreasing speed of sound, the rays bend downwards or upwards respectively as expected. This effect

is accurately captured using source coordinate propagation and is demonstrated below. Equation (1) can be used for

propagation with no extra logic to accommodate refraction effects. For the linearly increasing speed of sound profile

shown in Figure 10, the source coordinates are plotted and compared to Ray Tracing. (A rapidly varying speed of

American Institute of Aeronautics and Astronautics

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sound profile is used only to demonstrate bending with low computational cost: for a realistic profile, these bending

effects only become significant at very long distances.)

It can be seen in Figure 10 that the bending computed using our source coordinate computation matches well with

Ray Tracing. The same computation is repeated for a negative speed of sound gradient and the source coordinates

are plotted against ray tracing in Figure 11. This shows that the new method can accurately capture the bending due

to varying speed of sound. One of the advantages of this new Eulerian method is that varying atmospheric properties

can be simply specified on the grid, unlike Ray Tracing, where the parameters need to be interpolated at every ray

location.

Figure 10: Bending due to positive speed of sound gradient. WC (dashed), Ray Tracing (Solid)

Figure 11: Bending due to negative speed of sound gradient WC (dashed), Ray Tracing (Solid)

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III. Discussion of Results

Far field computations for a near field solution (as an example, rotorcraft thickness noise, which is confined to

the rotor plane) saved on the source sphere are shown below. The problem setup is shown in Figure 12.

Figure 12: Test case setup

An important capability of this new approach is to perform rapid computations for varying atmospheric and

ground conditions and produce results for an entire surface or volume. Particularly important is the ability to predict

noise at every point on the ground. One of the primary outputs is therefore a map of the sound pressure level (SPL)

on the ground surface beneath a source.

A. Varying atmospheric conditions

As mentioned earlier, the Wave Confinement method outlined in this paper can accommodate effects due to

variations in speed of sound and background wind. Initially, these profiles are assumed to be stratified (i.e.,

dependent only on altitude), but more complex environments can easily be accommodated (e.g., latest weather

information over the domain of interest).

In Figure 13, sound pressure levels (SPL’s) for 3 different speed of sound profiles are shown, which provides the

user a good understanding of the effect of refraction on far field acoustics. When rays bend down due to positive

gradient, the SPL’s are stronger since the thickness noise, which is dominant in the rotor plane, reaches the ground

quickly, and when rays bend up due to negative gradient, most information travels away from the ground and so the

SPL’s are not as strong and eventually create a shadow zone. (The profiles for varying speed of sound are not

realistic and only used for demonstration purposes.)

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Figure 13: Sound pressure levels for varying speed of sound (ft/s)

Same is repeated for propagation through background wind and is shown in Figure 14. When there is nonzero

wind, waves bend up in the upwind and form shadow zones. In downwind, waves undergo multiple “hops”, which

will let sound propagate farther. Figure 14 shows the SPL’s for propagation without and with wind. It can be

observed that when there is wind, the SPL’s become weak and form a shadow region behind the helicopter.

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Figure 14: Effects due to background wind (ft/s)

B. Ground Reflection

At a given receiver location, the total observed signal is the superposition of the direct wave from the source and

any reflected waves off of the ground and any significant nearby terrain. Near the surface of a relatively flat ground,

this simply results in the signal being twice as strong as it would otherwise be. Away from any surfaces (a flying

observer, for instance), the phases are different and so are the SPL’s. Figure 15 shows the sound pressure levels on a

horizontal plane 300 feet above the ground.

Figure 15: Sound pressure level contours for ground reflection due to direct (left) and reflected (right)

waves.

IV. Conclusion

There are many difficulties in predicting long range acoustics – this paper deals only with the methodological

problems with the goal of developing a method that is both general and practical. The work is motivated by the past

experience of how the field of aerodynamic prediction was revolutionized by the development of Eulerian methods,

replacing Lagrangian ones. However, a direct application of those particular Eulerian methods to far-field acoustics

is impractical because of prohibitive grid requirements. The use of such large grids derives from an implicit

assumption that the acoustic field is too complex for simple characterization and must therefore be solved in

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generality with a grid that resolves all physical waves; hence, the large cost. Beyond a certain radius, however, the

acoustic field simplifies, becoming amenable to characterization and simpler analyses. In these regions, the acoustic

field of a rotor is always grouped into well-defined packets whose spatial extent is far smaller than the propagation

distance. Using this information, we have devised a new type of Eulerian solution method.

We assume that any particular wave packet emanates from a specific location on a spherical source surface

(determined experimentally or computationally) and that the waveform at any point of the initial sphere determines

that of a wave packet emanating from that point. This point of origin is then a unique identifier of a wave packet.

Our approach then is to propagate this identity information rather than the physical waveform that it represents. This

identity information (in the form of coordinates on the initial sphere) is contained in a delta function that is

propagated from the source sphere.

The above process requires the ability to propagate a thin CW (a numerical equivalent of a delta function).

Classically, such propagation is impractical or perhaps impossible because it is not a physically resolvable quantity

that is propagated. However, we employ Wave Confinement, which propagates the wave as a solitary wave – this

has the unique property of propagating indefinitely with no numerical losses on coarse grids.

This is the basis for a new long-range acoustic prediction method under development. The far-field computation,

the primary object of this paper, appears to work as intended. We envision a computational capability with speeds

approaching real time while using computational resources that are compatible with a broad range of field

environments.

Acknowledgments

This work is being funded by the U.S. Army under a Small Business Innovation Research (SBIR) program

(contract number: # W911W6-12-C-0036). The authors would like to thank Dr. Mark Potsdam for this support.

Helpful conversations with Dr. Ben W. Sim and Dr. Fredrick Schmitz of AFDD are also acknowledged.

Development of WC method was partially funded by AFOSR under Dr. Arje Nachman.

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