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Wave Confinement: Long-Distance Acoustics Propagation through Realistic Environments


Abstract and Figures

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.
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American Institute of Aeronautics and Astronautics
Wave Confinement: Long-Distance Acoustics Propagation
through Realistic Environments
Subhashini Chitta
, John Steinhoff
and Andrew Wilson
Wave CPC Inc, Tullahoma TN 37388
Frank Caradonna
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
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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)
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
pulsesapproximately 2-3 grid cells in sizethe 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
, , ,
n n n
i j i j i j E
   
  
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
 
 
are constants and
is the harmonic mean given as a sum over neighboring nodes:
 
1 1 1 1
1 1 1
,, 27
i j k
i j k
  
  
  
  
  
  
To demonstrate long distance propagation with no increasing numerical errors, a diverging (2-D) circular wave of
is simulated on a
cell grid with periodic boundary conditions. Confinement values used
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. 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
i j k
for all grid nodes, where
,, 0
i j k
, 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:
 
  
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
, where
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
. 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
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
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
is delta-like function,
, and (i, j, k) are the coordinates of the each grid cell that lies within the
initial CW with amplitude,
. 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
   
where F is the confinement term (previously defined7) and l = 0, 1, 2, 3.
For a source location, (
), the attenuation due to geometric spreading,
, at each far field point, is
computed using the relation,
is the source strength.
When there are multiple wave passes at any point due to reflection, the above mentioned expectation values (
) and geometric attenuation factor,
, 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,
are computed as
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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,
,, ,,
cos i j k
i j k i j k
 
, where
,,i j k
ri,j,k =√〈xi,j,k2+yi,j,k2+zi,j,k2 is the radius at a point, (
), has good agreement with the analytical solution. As explained, (
) 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
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
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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
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)
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
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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2Steinhoff, J., Fan, M., and Wang, L., “A new Eulerian method for the computation of propagating short acoustic
and electromagnetic pulses,” Journal of Computational Physics, Vol. 157, No. 2, 2000, pp. 683706.
3Osher, S., and Fedkiw, R., “Level Set Methods and Dynamic Implicit Surfaces,” Applied Mathematical Sciences,
Vol. 153, 2003.
4Anderson, D.A., Tannehill, J.C., and Pletcher, R.H., “Computational Fluid Mechanics and Heat Transfer”,
Hemisphere Publishing Corp., Washington D.C., USA, 1984..
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6Levy, M. F., “Parabolic equation modelling of propagation over irregular terrain,” Electronics Letters, Vol. 26, No.
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... [23][24][25][26][27][28] For Self Amplified Spontaneous Emission (SASE) and collective non-linear instability in Free-Electro-Lasers (FEL) the cylindrical shell region ( fig. 3) may comprise of the undulator magnets in a wiggler of an X-ray laser. [29][30][31][32] The topic of wave confinement is of topical interest, [33][34] here we have discussed the generation of maximally focused waves, but similar considerations apply for imaging elements as well. From the interference point of view both Arago and Fresnel lenses are similar, but in Arago the emphasis on the extremal regions. ...
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Optics is limited in the 'ray-approximation'-inclusion of wave properties result in additional phenomena and applications; interferometers and diffraction gratings are two manifestations of such non-geometric, physical optics. Incidentally, the most precise measurement ever, at one part per 10^21 in the (2017) Nobel winning discovery of gravitational waves was achieved with an interferometer. Amendments to the properties of the medium promise negative refractive index meta-materials, perfect imaging, light cloaking, and other ultra-natural marvels. Attention to photon phase, correlations, statistics and wavelength independent phase shifts result in singular optics, quantum optics and anholonomy. Here we present another possibility, namely 'Arago-optics' to maximize the efficacy of a device by strategically deploying the key qualities along its perimeter. For instance, in conventional sources, waves are generated with maximum intensity at the core; whereas in an Arago-source, intensity is minimal or zero at the center, but highest on villus stretches at the margins. We reason that for a given size and energy output, this radiation profile, produces the highest concentration of energy at the focus, with the maximal confinement of the wave packet. Likewise, the utmost detector resolution is attained when sensitivity is highest on the perimeter and less at the center. This concept holds beyond ultra-focus and Gaussian beams, but generally applies to beams of 'waves' that show constructive and destructive interference. The idea is particularly well suited for a fresh integration of geometry and topology with electronics and materials into real-time wave engineering.
The linear wave equation represents the basis of many linear electromagnetic and acoustic propagation problems. Features that a computational model must have, to capture large scale realistic effects (for over the horizon or “OTH” radar communication, for example), include propagation of short waves with scattering and partial absorption by complex topography. For these reasons, it is not feasible to use Green’s Function or any simple integral method, which neglects these intermediate effects and requires a known propagation function between source and observer. In this paper, we describe a new method for propagating such short waves over long distances, including intersecting scattered waves. The new method appears to be much simpler than conventional high frequency schemes: Lagrangian “particle” based approaches, such as “ray tracing” become very complex in 3-D, especially for waves that may be expanding, or even intersecting. The other high frequency scheme in common use, the Eikonal, also has difficulty with intersecting waves.
A new method is described to compute short acoustic or electromagnetic pulses that propagate according to geometrical optics. The pulses are treated as zero thickness sheets that can propagate over long distances through inhomogeneous media with multiple reflections. The method has many of the advantages of Lagrangian ray tracing, but is completely Eulerian, typically using a uniform Cartesian grid. Accordingly, it can treat arbitrary configurations of pulses that can reflect from surfaces and pass through each other without requiring special computational marker arrays for each pulse. Also, information describing the pulses, which are treated as continuous surfaces, can be available throughout the computational grid, rather than only at isolated individual markers. The method uses a new type of representation, which we call “Dynamic Surface Extension.” The basic idea is to propagate or “broadcast” defining fields from each pulse surface through a computational grid. These fields carry information about a nearby pulse surface that is used at each node to compute the location of the pulse surfaces and other attributes, such as amplitude. Thus the emphasis is on the dynamics of these propagating defining fields, which obey only local Eulerian equations at each node. The Dynamic Surface Extension representation can be thought of as dual to level set representation: The defining fields involve single valued variables which are constant at each time along lines that are normal to the evolving surface, whereas level set techniques involve a function which has constant values on the evolving surface and neighboring surfaces. In this way the new method overcomes the inability of level set or Eikonal methods to treat intersecting pulses that obey a wave equation and can pass through each other, while still using only single-valued variables. Propagating thin pulse surfaces in 1-D, 2-D, and 3-D that can reflect from boundaries and pass through each other are computed using the new method. The method was first presented as a new, general representation of surfaces, filaments, and particles by J. Steinhoff and M. Fan (1998, Eulerian computation of evolving surfaces, curves and discontinuous fields, UTSI preprint).
This book is intended to serve as a text for introductory courses in computational fluid mechanics and heat transfer for advanced undergraduates and/or first-year graduate students. The first part of the book presents basic concepts and provides an introduction to the fundamentals of finite-difference methods, while the second part is devoted to applications involving the equations of fluid mechanics and heat transfer. A description is given of the application of finite-difference methods to selected model equations, taking into account the wave equation, heat equation, Laplace's equation, Burgers' equation (inviscid), and Burgers' equation (viscous). Numerical methods for inviscid flow equations are considered along with governing equations of fluid mechanics and heat transfer, numerical methods for boundary-layer type equations, numerical methods for the 'parabolized' Navier-Stokes equations, numerical methods for the Navier-Stokes equations, and aspects of grid generation.
The finite difference implementation of the parabolic equation method provides a numerical solution to the problem of diffraction of radiowaves by irregular terrain in the presence of atmospheric refraction effects. The method has been validated by comparisons with theory and measured data.
A method to model tropospheric radiowave propagation over land in the presence of range-dependent refractivity is presented. The terrain parabolic equation model (TPEM), is based on the split-step Fourier algorithm to solve the parabolic wave equation, which has been shown to be numerically efficient. Comparisons between TPEM, other terrain models (SEKE, GTD, FDPEM), and experimental data show predominantly excellent agreement. TPEM is also compared to results from an experiment in the Arizona desert in which range-dependent refractive conditions were measured. Although horizontal polarization is used in the implementation of the model, vertical polarization is also discussed