Sound Diffraction Modeling of Rotorcraft Noise …...Sound Diffraction Modeling of Rotorcraft Noise Around Terrain James H. Stephenson Ben W. Sim US Army Aviation Development Directorate

Sound Diffraction Modeling of Rotorcraft Noise Around Terr ain

James H. Stephenson Ben W. SimUS Army Aviation Development Directorate

Hampton, VA Moffett Field, CA

Subhashini Chitta John SteinhoffWave CPC, IncTullahoma, TN

ABSTRACTA new computational technique, Wave Confinement (WC), is extended here to account for sound diffraction aroundarbitrary terrain. While diffraction around elementary scattering objects, such as a knife edge, single slit, disc, sphere,etc. has been studied for several decades, realistic environments still pose significant problems. This new techniqueis first validated against Sommerfeld’s classical problem of diffraction due to a knife edge. This is followed bycomparisons with diffraction over three-dimensional smooth obstacles, such as a disc and Gaussian hill. Finally,comparisons with flight test acoustics data measured behinda hill are also shown. Comparison between experimentand Wave Confinement prediction demonstrates that a Poissonspot occurred behind the isolated hill, resulting insignificantly increased sound intensity near the center of the shadowed region.

INTRODUCTION

Helicopters are widely used in many applications such ascommercial and private transportation, medical emergency,tourism, evacuation and rescue, etc. While many advance-ments have been made throughout the years, one area thatrequires further attention is propagation of aerodynamicallygenerated noise over long distances (thousands of wave-lengths). The main sources of rotorcraft noise are due tothe main rotor and tail rotor; both of which produce lowerfrequency sounds that are especially capable of propagat-ing over significant distances causing community annoyancecomplaints across a wide area (Ref.1).

As a result of helicopter acoustic emissions, several re-strictions (Refs.2, 3) have been imposed to limit the flightoperations to specific times and locations, posing an imme-diate requirement for noise mitigation measures. A criticalstep toward community noise reduction is to develop a com-putationally fast noise propagation tool that can account foratmospheric and ground effects, including diffraction. Sucha tool will be of great importance in assessing the acousticimpact on populated areas and for finding flight trajectorieswith optimal noise performance. The focus of this paper is onone important propagation feature, diffraction, which is notcurrently well modeled for the problem of interest.

Diffraction is a naturally occurring phenomenon that al-lows waves (including acoustic, electromagnetic, seismic, wa-ter waves, etc.) to propagate around objects. Some examplesof diffraction include the ability to hear people around cor-ners, optical effects that result in “silver lining” or iridescenceof opaque objects, and water wave propagation through break-waters. Similarly, when a helicopter flies near a hill or other

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large obstacle such as a building, significant noise levels areobserved due to diffraction of sound waves around the ob-stacle. This is depicted schematically in Figure1, where thewaves from source, S, continue to propagate into the shadowregion despite the obstruction.

s

Region of Diffraction

Edge Source

Shadow Region Boundary

Fig. 1. Schematic depiction of diffraction due to a Gaus-sian obstruction.

There are several existing aeroacoustic methods that arewidely used to solve the above problem but have a broad rangeof physical and numerical limitations, which restrict their ap-plicability (Refs.4–6). Some of these methods are based on aninhomogeneous wave equation derived by Lighthill (Ref.4),where the computational domain is split into a nonlinearsource region where a turbulence model is used to evaluatenoise sources, and an acoustic region where integral methodssuch as Kirchhoff (Ref.6) or Ffowcs-Williams-Hawkings For-mulation 1A (F1A) (Ref.7) are used for propagation. Thesepropagation methods can be used for long distances but areonly feasible for uniform media with no scattering topograph-ical features such as buildings or hills.

A closed form equation is usually required in the conven-tional use of integral methods, to account for wave propaga-tion from each point on an acoustic source surface in an as-sumed medium to a specified observation point. Thus, foreach observation point, the equation has to be integrated overthe entire source surface. Reflective and refractive effects re-quire a separate integration over the source surface, whichre-sults in a nested integral for each observation point. Account-

1

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ing for each of these effects quickly becomes cumbersome tosolve. In such cases, it is more appropriate to use discretiza-tion methods to automatically account for the effects due tothe environmental factors, including atmospheric conditions(wind, temperature and humidity gradients), terrain (topog-raphy, ground impedance) without changing the equations.The price for this generality is that the equations can only besolved over a very limited region since they are restricted tofiner grid sizes to reduce numerical dissipation and dispersionerrors. Fine grid sizes quickly exceed the memory require-ment beyond the capacity of most computers when distancesor frequencies increase.

A reasonable alternative to this problem is to use high fre-quency approximations, such as eikonal or ray tracing. Thesemethods are numerically fast but do not account for sounddiffraction effects in an environment with non-flat ground(Ref. 8). To overcome the flat-ground limitation, parabolicmethods (Ref.9) or Geometric Theory of Diffraction (GTD)(Ref. 10) have been coupled with conventional ray tracingtechniques. However, it is well known that the parabolicmethods tend to become computationally complex and expen-sive in three dimensions. The latter model, GTD, is compu-tationally cheaper but uses numerical fitting on geometric (2-D) slices through a three dimensional terrain from source toreceiver. This method further restricts principle features ofthe terrain to either: flat, concave, convex, thin screens, orwedges (Ref.11). This is not physically appropriate in a gen-eral sense, and has implications on the accuracy of the finalsolution.

The persistence of computational difficulty of diffractionmodeling, even after decades of research, is a major inhibitorto assess accurate acoustic footprints of rotorcraft. An accu-rate method to solve diffraction can also help generate new op-erational guidelines for flight paths and maneuvers that min-imize noise levels in populated areas and improve land-useplanning. So, a new method that is both computationally fastand accurate is required for propagation of rotor noise overrealistic distances and terrain.

It has been well established that grid-based methods aremore general in implementation for acoustic propagation overlong ranges, but they are currently limited to short ranges(Ref. 12). An improvement that can eliminate this limitationof conventional discrete methods would be a rational approachto solve the problem of interest. A promising improvement isWave Confinement (WC). WC is a new finite difference for-mulation of a basic formalism that, to the authors’ knowledge,has not been used before in this context. Wave Confinementuses nonlinear solitary waves as basis functions to determinethe wave fronts, as treated by Whitham (Ref.13). Thus, withWC, the evolving acoustic field from a point source can be ac-curately represented as these propagating wave fronts, whichobey the wave equation.

Even though Wave Confinement uses finite differences,it produces stable, asymptotic solutions, unlike conventionaldiscretization methods that eventually decay the solutionevenwith higher order accuracy. This improvement is made pos-

sible by adding a nonlinear term to the wave equation, whichdoes not interfere with propagation dynamics, but controlsthewidth of the solution, while conserving the essential integralsof the problem. Wave Confinement has already been provenuseful in long range acoustic propagation including effectsdue to temperature and wind gradients with arbitrary topog-raphy (Ref.14). In this paper, implementation and validationof a new capability that automatically accounts for diffractionis discussed.

METHOD DESCRIPTION

The linearized acoustic wave equation,

∂ 2t φ = c2∇2φ (1)

whereφ is a scalar and c is the speed of sound, is solved usinga new grid-based method described below. This involves (a)Wave Confinement for propagation of acoustic wave fronts asasymptotic solutions, (b) Dynamic Surface Extension (DSE)to compute a mapping function between source and far fieldand (c) Scaling Law (for Diffraction) to apply a correctionfactor to adjust the amplitude of wave fronts to that of physicalwaves.

Wave Confinement

Conventional discretizations of Equation1 are linear andbased on polynomial expansions (with coefficients deter-mined by Taylor expansion, perhaps with numerical disper-sion minimization). The difficulty with the resulting dis-cretization errors is that they accumulate and continuallygrow. In these circumstances, higher order methods are of-ten necessary; however, they only delay, rather than eliminate,error accumulation.

By contrast, WC entails a discretization which contains dy-namic terms that relax the solution to an equilibrium in theframe of the propagating function. Therefore, error accumu-lation does not occur; higher order methods become unneces-sary since these solutions are stable and can propagate withoutspreading or dispersing. These solutions are called nonlinearsolitary waves, which are well known to arise from a balancebetween nonlinear and dispersive effects (Ref.13). The dy-namic terms added to the RHS of Equation1 to produce sta-ble, nondissipative waves are:

E = ∂t[

∇2(µφ − εΦ)]

(2)

whereΦ is a nonlinear harmonic mean defined as

Φni, j,k =

∑+1l=−1 ∑+1

m=−1∑+1o=−1

(

φ ni+l, j+m,k+o

)−1

27

−1

. (3)

Here,ε andµ represent the diffusion coefficient and numer-ical coefficient, respectively, which stay constant duringtheentire computation. Equation1 (with the dynamic terms de-fined in Equation2) in discretized form is then written as

φ n+1i, j,k = 2φ n

i, j,k −φ n−1i, j,k + v2(∇2φ )+aδ−

n [∇2(µφ − εΦ)] (4)

2

where,

∇2 (•) = δ 2i (•)+δ 2

j (•)+δ 2k (•)

δ 2i (•)n = (•)i+1−2 (•)i + (•)i−1

δ−n (•)n = (•)n − (•)n−1,

and v = c∆th , a = ∆t2

h2 , ∆t is the time step, andh is the gridcell size. Equation4 describes all the features of propagationsuch as reflection, refraction, atmospheric and ground absorp-tion, as well as diffraction. Taylor expansion of this equationcan be simplified to an eigenfunction equation with a fixedeigenvalue, whose solutions are nonlinear solitary waves,thedetails of which are provided in Ref.15.

This leads to a highly intuitive interpretation where the mo-tion of the physical wave can be represented by evolving wavefronts generated from WC. The main idea that makes this ap-proach feasible is that the position, arrival time (phase),andwavelength of a nonlinear solitary wave are essentially unaf-fected by discretization error; the profile neither disperses nordiffuses due to discretization effects. Instead, the wave re-mains concentrated over a small number of grid cells1, mak-ing it possible to consider discrete Eulerian methods as a prac-tical approach for tracking wave fronts over long distances.

This allows the Wave Confinement method to use coarsergrids than required by conventional resolution considerations,while accounting for the effects of varying atmospheric andtopographic features. Since WC is a grid-based method, itis fairly simple to accommodate varying ground and atmo-spheric properties. A unique advantage of WC is that it doesnot need body conforming grids as any topography can besimply immersed in a Cartesian grid. Further, Wave Confine-ment employs a simple zero contour or level set representationof the surface and can easily accommodate complex shapeswith little computational effort.

In addition to tracking position and phase of these non-linear solitary waves, Wave Confinement also provides an at-tenuation factor for each wave front2 arriving at an observer.This allows WC to calculate acoustics amplitude associatedwith: (a) geometrical distance of travel, (b) terrain acousticsimpedance (for ground reflections), (c) atmospheric sound ab-sorption, and (d) sound diffraction. An important note hereis that (b), (c) and (d) are frequency-dependent effects. Atthis initial stage of the computational chain, Wave Confine-ment derived attenuation factors are strictly valid only forthe selected wavelength (see Footnote1). To accurately ac-count for rotorcraft acoustics that contain a broad range offre-quencies, attenuation factors derived from WC are “adjusted”with frequency-dependent scaling laws as explained in ensu-ing sections.

1The wavelength of the nonlinear solitary wave used incurrent studies typically covers five to seven grid cells.

2There may be multiple wave fronts due to reflec-tions/refractions that arrive at a single observer.

Dynamic Surface Extension

If the source is omnidirectional, computation of phase and at-tenuation factor are enough to construct the acoustic signatureat any far field point. However, rotorcraft noise has angu-lar dependence (i.e., nonuniform sound source), which can becaptured directly by propagating the waveform on the grid orby using Dynamic Surface Extension (DSE) (Refs.14, 16).The former is not feasible since grid-based methods dissipatethe waveform. The latter, which uses a mapping function thatmaps each point inR3 to the source surface to compute thewaveform, is not numerically dissipative/dispersive. This in-volves propagating a set of conserved variables (e.g., initialemission angles) from known points on the source sphere tofar field locations in the same way as the scalar,φ , shownin the previous section. These conserved variables propagateon the characteristics, or lines that are normal to the evolv-ing wave fronts (see Figure2), and therefore, stay invariant inthat direction. So, at any far field point,(x,y,z), a set of emis-sion angles (qr,qs) corresponding to each pass of the wave arecomputed.

qr (θ , ψ , t)

qs (θ , ψ , t)

(x, y, z)

Fig. 2. Dynamic surface extension mapping of sourcesphere to destination location.

The acoustic signal at any far field point will then be

p(x, t) = ∑i=1,n

Ri (x)∗ p(ψi (x) ,θi (x) , t + τi (x)) (5)

wherex is location in(x,y,z), R is the attenuation factor,iis the index of the arrival,ψ and θ are azimuth and eleva-tion angles respectively,τ is the emission time, andn is thenumber of arrivals. Dynamic Surface Extension was later re-named by others as “closest point method” (Ref.17); however,the concept of mapping the surface along propagation paths isidentical. With this new approach, details of physical wave-forms are not numerically propagated, only locations of theorigin where the waveforms are known. This approach hasalready been validated for long range propagation, includingrefraction and multiple reflections (Ref.14). Validation of thismethod for diffraction will be demonstrated in this paper.

DIFFRACTION

According to Huygens’ principle of wave theory (Ref.18),each point on a wave propagates as a point source. This is ex-pressed asA = A0

r e−iωt , whereA0 is the source strength,r isthe distance of propagation andω is the frequency. For exam-ple, in Figure3a, the wave front att2 = t1+ dt is constructed

3

D

t2 = t1 + dtWave front att1

(a)

Plane

Wav

e

Gaussian Hill

P

Q

(b)

Fig. 3. Description of diffraction using Huygens’ Principlefor a (a) 2D and (b) 3D case.

by applying the principle of composition to the wavelets gen-erated by each point on the wave front at an earlier time, t.These wavelets cancel each other at oblique angles of inci-dence in free space. However, when there is an obstruction,only part of the wave travels unimpeded. These unimpededwavelets interact with the obstacle, essentially acting asnewsources. The oblique wave portions no longer cancel eachother; instead, they form a secondary wave that propagatesinto the shadow region. This is defined as a diffracted wave.

The total field associated with a scattering object is thesum of the incident field, reflected field, and diffracted field.High frequency approximations can accurately predict inci-dent and reflected fields (if surface normals are specified withreasonable accuracy). However, diffraction is a frequencydependent problem that cannot be solved with a high fre-quency assumption. Sommerfeld, Keller, Kirchhoff and otherresearchers (Refs.10,19,20) have carefully studied this phe-nomenon and presented solutions for diffraction due to a knifeedge, single slit, double slit, disk, sphere, etc. Althoughthesemethods work quite well for simple scattering objects, theyare not feasible in realistic environments, where obstacles arenot well defined and no closed form solutions exist.

Another attempt to solve the diffraction problem was tocouple the high frequency approximation to Geometric The-ory of Diffraction (Ref.11). As previously discussed, how-ever, this does not account for realistic spreading. For exam-ple, Figure3b shows a plane wave propagating over a Gaus-

sian hill, each point on the wave front acts as a point sourceand the information is propagated inall directions. The sec-ondary waves in this case are spherical and not confined to asingle plane. Therefore, the amplitude at a point (P) is due tothe wavelets from all the planes. This also proves that solvingthe diffraction problem as 2D slices neglects spherical spread-ing.

To demonstrate this property, wave propagation over aGaussian hill is computed using both 2D and 3D approxima-tions, shown in Figure4. For 2D, a cylindrical source, equiv-alent to a line source in 3D, is propagated separately in eachy-slice, using a 2D wave equation. There is no interactionbetween the slices and the spreading is confined to a specificslice. It is as if the information from only one point (Q) isreaching the far field point (P) in Figure3b.

Y

1000

0

-1000 0 1000 2000

x (ft)

y (

ft)

z

x

(a) 1000

0

-1000 0 1000 2000

x (ft)

y (

ft)

(b)

Fig. 4. Diffraction computation (a) solved as 2D slices vs(b) 3D. Red identifies high sound intensity, while blue islow sound intensity.

For the 3D case, a line source is propagated using the threedimensional wave equation. The amplitude at a point, P, takesinto account spherical spreading from all points Q, which isin agreement with Huygens’ Principle. As can be seen in Fig-ure4, the amplitude computed with the 2D slice assumption issignificantly different from the 3D solution. Note that the am-plitude is much stronger behind the object at the center of theshadow region, due to constructive interference of the wavesemanating from the edge surface. This is called a Poissonspot, which is not observed in the 2D computation, and hasimplications to the flight test data.

Knife Edge Diffraction

The proposed method is first validated against the classicdiffraction problem of a plane wave propagating over a per-fectly reflecting semi-infinite plane as shown in Figure5. Theexact solution of this problem was originally solved by Som-merfeld (Ref.19) in the frequency domain. A number of otherresearchers subsequently developed solutions using a variety

4

sIncident ray Illuminated Region

Shadow RegionDiffracted Ray

d2

h

d1Edge

Fig. 5. Schematic of a wave diffracting due to a knife edge.

of methods such as Green’s functions, Fourier/Laplace trans-forms, etc.

Here we will use Wave Confinement to solve this problem,with the following computational setup. The knife edge isconsidered to be along the y-axis (vertical axis), extending toy= 1000 ft, and is positioned atx= 1000 ft along the horizon-tal axis. The computational plane wave is initialized atx = 0ft, which is then propagated from left to right using Equa-tion 4 as a codimension one structure,φ =A0(sechα(~x−~x0)).Here,~x0 is the centroid (position) andα defines the width (orthe central wavelength) of the computational wave.α is afunction of the confinement parameters,ε and µ . The timeevolution of these propagating computational waves is shownin Figure6, which demonstrates that unlike conventional highfrequency approximations that solve ODEs along the ray, thelinear wave equation does not discontinuously decrease am-plitudes at the shadow boundary.

1�� �� Wa��������

x (ft)

y (

ft)

Edge

ν = -1

ν = -2

ν = -3

0

2500

2000

2000

1500

1000

1000

500

3000

Fig. 6. Time evolution of codimension one surfaces propa-gating (to the right) over a knife edge using Wave Confine-ment.

The diffraction loss (Lc) is computed using the relations,

Lci, j,k = 20 log

(

Ai, j,k

A0

)

(6)

andAi, j,k =

∫

φi, j,k dt. (7)

As mentioned previously, Lc correspond to the wavelength ofthe computational wave, which is scaled and compared to the

approximated analytical diffraction loss (La) defined by Ref.21 as,

La =

20log(0.225/ |v|), v <=−2.4

20log

[

0.4−

√

(

0.1184− (0.38−0.1v)2)

]

,

−2.4< v <=−1

20log(0.5e0.95v), −1< v <= 0

20log(0.5+0.62v), 0< v <= 1

0, otherwise.

(8)

Here,ν is a nondimensional variable defined as

ν = h

√

2λs

(

1d1

+1d2

)

, (9)

whered1 is the normal distance between source (S) and theedge, andd2 is the normal distance between the observationpoint and the edge, as depicted in Figure5.

Since the incident wave is assumed to be planar (i.e.,

d1 → ∞), ν reduces toν = h√

2λsd2

. h is defined as,h= y−y0

wherey is the height of the observation point andy0 is theheight of the edge (1000 ft in Figure6). So, below the edge,h< 0=⇒ v< 0. For example, in Figure6, contour lines corre-sponding tov =−1,−2,−3 are shown, all of which are belowthe edge of the semi-infinite plane.

Both Lc andLa are independent of wavelength when theyare plotted as a function ofv, which means Figure7 is true forall wavelengths. It can be seen in Figure7 that the computeddiffraction loss (Lc) is in good agreement with the analyticallydetermined diffraction loss (La), validating the ability of WaveConfinement to capture diffraction for the knife edge case.

dB

Loss

Analytical (A)

Computer (Lc)

0

0

-5

-10

-15

-20

-4 -3 -2 -1 1 2

Analytical (La)

Computed (Lc)

ν

Fig. 7. Diffraction loss plotted as a function ofν , plotted atx = 2000 ft from Figure 6.

5

Scaling Law

While WC gives an accurate solution, it is not feasible to re-peat Equation4 for each frequency separately. Thus, a scalinglaw is required. This scaling law is used to correct the ampli-tude of the computational wave,Ac, for any physical wave-length, λp. This is generalized to an arbitrary shape of theobstacle with reasonable accuracy, and is defined as,

Ap = Ac

√

λp

λc(10)

where Ap is the amplitude of physical wave andλc is thewavelength of the computational wave.

Equation10 can be checked using an analytical solutionfrom the knife edge diffraction case previously described.Weassume a computational (or reference) wavelength of 200 ft,and calculateAp (for all wavelengths) andA200 using theequationA• = 10La•/20, whereLa is defined in Equation8.Figure8 shows the ratios,λp/200 andAp/A200, for differentvalues ofh below the edge, as defined in Figure5. It canbe seen from Figure8 that Equation10 becomes more accu-rate further into the shadow region. So, a predetermined tableof ratios computed using the equations above can be used infuture simulations, with an assumption that the size of the ob-stacle is greater than the computational wavelength.

0�

0�

0

0�

50 00 50 �00

H����� � �50 ��H����� � ��5 ��H����� � �00 ��H����� � ��5 ��H����� � �50 ��

F�������� H��

√

λp200

Ap/A

200

Fig. 8. Scaling factor plotted as a function of frequency.Scaling factor is calculated at various heights beneath aknife edge, for a given 200 ft computational wave.

Propagation over realistic terrain

In realistic environments, there are no flat grounds or knifeedges. The scattering surfaces are not aligned to the compu-tational grid, and an accurate immersed boundary conditionis required. As mentioned earlier, Wave Confinement uses alevel set approximation to specify a boundary. This means

f ! "

f # "

f $ "

G%&'()

A*+,-. /23,46

Computational

789:;<

G%&'()

G%&'()

Fig. 9. Stencil used for computation.

that the boundary is represented by a function,f , which onany grid point is<, >, or= 0. Since Cartesian grids are usedhere, the physical boundary does not necessarily fall on gridpoints, as shown in Figure9. The practical implementationof this scheme is that the computational waves reflect fromthe computational ground. This can result in a small error inthe phase computation, which is approximately constant ev-erywhere and can be added during the post-processing. Forsimplicity, it is assumed here that the computational groundis the actual ground. Further, it is assumed that all reflectionsare specular in nature.

The derivatives at the ground are computed using the sten-cil shown in Figure9. At the ground, one or more grid nodesin the stencil are below the boundary, in which case, they areinterpolated using the ones above. This does not involve com-plicated logic since functionf automatically defines whethera grid point is above or below ground. Also, Wave Confine-ment uses low order discretizations. So, there is never morethan one grid point in each direction that needs to be inter-polated, which makes WC computationally cheaper than ex-isting grid-based methods. Also, despite the initial staircaseeffect, the reflected wave fronts quickly become smooth dueto tangential dissipation (Ref.14).

To demonstrate the immersed boundary condition, diffrac-tion of a normally incident plane wave due to a circular diskis shown. This is a canonical diffraction problem, the closedform solution of which is not as straightforward as that ofsemi-infinite half plane (Ref.22). The diffraction pattern de-pends on the ratio of perimeter to wavelength,2π

λ a or ka,wherea is the radius of the disk andk is the wavenumber ofthe plane wave. Forka >> 1, the diffraction pattern is calcu-lated using the computational setup shown in Figure10. Thedisk is aligned to they− z plane defined using the boundaryfunction, f = y2+ z2− a2 = 0, with radius (a) of 200 ft. Theplane wave is represented by the scalar,φ , and has a thicknessof 50 ft. As the wave propagates over the disk, each pointon the disk acts as an edge source, which constructively inter-feres to form a bright spot (Poisson spot), shown in Figure11.

The quantity,(

∫

φdtφ0

)2, which is a representation of Inten-

sity ratio, II0

is plotted on ay − z plane behind the disk in

6

D=>?

a

Fig. 10. Computational setup for scattering over disk.

y@B

zCE

2

I

J

KI

KL

KL KI J I L

Fig. 11. Plot of intensity on they− z plane behind a disk.

Figure 11, to show the Poisson spot. The intensity ratio isalso plotted as a function of z/a, at y = 0 forv = x λ

π a2 (or

N = 1v = 41.8) in Figure12. This is compared with an ana-

lytical solution from Ref.23. Note that this method does notneed to be used to compute the interference pattern in the il-luminated region (z/a > 1), since the incident wave has muchhigher amplitude. The solution for the proposed diffractionmodel is shown to have a qualitative agreement with the ana-lytical solution for locationsz/a< 1, with some discrepanciesat z/a < 0.4.

Ground topography can be more complex than elementaryobjects like disks and spheres. However, it is still possibleto immerse highly complex geometry in a Cartesian grid ina similar manner as described. Further, realistic ground canlead to more than one reflection. An example of multiple re-flections from a single point is shown in Figure13. The rayreflected from one part of the ground (flat ground) can reflectagain from another location (Gaussian hill). In such cases,there is a phase difference between both waves which can sig-nificantly affect the total signal unless the phase difference ismuch smaller than the wavelength.

Figure14adepicts an omnidirectional acoustic wave prop-agating over an isolated hill. This figure shows a secondarywave front due to multiple reflections, as described in Figure13. Since Wave Confinement is a grid-based method, these

MNOPNQROSTUORVXMYM

0)

N = 41.8

2

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

00 0.5 1 1.5 2

z/a

(a)

Z[\

]^_`^bc_d

2

e

00 ghi e ehi 2

j k l

j k lg

(b)

Fig. 12. Intensity ratio from Figure 11, for y = 0 line. In-tensity ratio (a) computed using Wave Confinement, com-pared with an (b) analytical solution. (b) is digitally tracedfrom Ref. 23.

s

Fig. 13. Multiple reflections depicted using a Gaussian hill.

wave fronts need to be separated by at least 5 grid cells tocapture the phase difference between them (see Figure14b).If the grid cell size is increased for the same computation,there are not enough grid points to separate these waves, andthey merge as if there is no phase difference, shown in Figure14c.

The merger of incident and reflected wave is an accept-able approximation for waves on the same order as the com-putational wave, but is invalid for the problem of interest,where the physical waves are much shorter and phase dif-ferences become significant. In other terms, if the ampli-tudes of these two waves are|A1| and |A2|, respectively, theintensity is computed as(|A1|+ |A2|)

2. However, a physi-cally accurate calculation would yield|A1+A2|

2, which is

7

Y

m

x

(a) 200

nop

npp

50

0

qop

qnpp

nopp nrpp nspp ntpp

y

uvwy

{ |}~�

(b)

200

���

���

50

0

���

����

���� ���� ���� ����

y

����

� ����

(c)

Fig. 14.(a) Time snapshot of computational wave on fine grid. Close-up view of computational waves on a(b) fine and(c) coarse grid. Darker colors (red) indicate higher sound intensity.

|A1|2 + |A2|

2 + 2|A1| |A2|cosδ , whereδ is the phase differ-ence. The current implementation of Wave Confinement re-sults in this limitation, but it will be overcome in the future bysolving the reflected wave in a separate array.

FLIGHT TEST COMPARISON

Helicopter acoustic data from an AS350 SD1 vehicle were ac-quired in Sweetwater, NV in 2014. The flight test data includeacoustic measurements from behind an isolated hill. Figure15 shows the microphone locations with microphones behindthe isolated hill identified as microphones 27 thru 29. Thefull test description can be found in Reference24, while rele-vant parameters are provided here. The AS350 SD1 is a 3000pound civilian aircraft with a main rotor blade passage fre-quency of 20 Hz, with a tail rotor blade passage frequencyof 70 Hz. Acoustic data acquisition for each level flight case

started when the vehicle was approximately 4000 ft before themain microphone array (microphones 1−21), and terminatedapproximately 4000 ft after the main array.

Pressure time-series data from microphones 27 thru 29 arehigh-pass filtered using a 5th order Butterworth filter with acutoff frequency below 10 Hz. This high-pass filtering is re-quired to remove wind noise from the measurement data inorder to better identify the faint acoustic signals arriving fromthe vehicle.

Source Hemisphere

Source hemispheres from steady level flight conditions canbe created using the main microphone array. The hemisphereused for this paper has a radius of 100 ft, and comes froma 105 knot level flight condition with very low backgroundwind speeds (less than 1 knot at flight altitude). Pressure

8

Flight Path

Control TrailerWeather Balloon

LIDAR

Isolated Hill

1

500 ft

2111

24

27

2829

Fig. 15. Equipment locations with notional vehicle flightpath and local geography shown.

data are corrected for pressure doubling at the ground, andtransformed from time of reception to time of emission, de-Dopplerized, and corrected for spherical spreading (Ref.25).

Pressure time-series data from microphones 1-21 (sub-sampled to 12 kHz) are stored in half-second increments, with50% overlap, at the ‘average’ elevation and azimuth on thevehicle, as experienced during each half second time incre-ment. Figure16 is a Lambert projection of the overall soundpressure level of the hemisphere for this run. Each dot repre-sents a half second of stored pressure time-series data at everyquarter second (50% overlap) throughout the duration of thesteady level flight.

The AS350 SD1 main rotor rotates clockwise when viewedfrom above, so the Lambert projection begins with an azimuthof 0◦ at the tail, 90◦ azimuth is on the left side of the vehicle,while 270◦ is on the right side of the vehicle. Elevation beginsin the plane of the rotor (0◦) at the edge of the Lambert projec-tion and decreases radially such that directly beneath the rotor(−90◦) is represented by the point in the center of the Lambertprojection. The pressure time-series hemisphere data are usedto propagate signals to microphones 27 thru 29, from a knownand measured vehicle location.

-60 o -30 o

-90 o

0 o

270o

0o 0 o

90o

180

o

86

89

92

95

98

101

104

107

OA

SP

L[d

B]

Fig. 16. Lambert projection of the overall sound pressurelevel [dB], with markers identifying each segment of half-second pressure time-series data.

Far Field Computation

The computational-setup of the flight test for a fixed aircraftposition is described below:

First, the size of computational domain (in physical units)centered at microphone 11 is defined as shown in Figure17.Ground elevation data of≈ 150 ft resolution for this domainare imported from Google earth. Terrain data were rotated toalign with the flight direction and then linearly interpolated ona Cartesian grid (x,y) with nodes located every 20 ft and al-titude (z) calculated through linear interpolation from nearestneighbors. Interpolated altitude was grounded using the mea-sured GPS location of microphone 11 and verified against allmicrophone locations. The ground interpolation scheme wasaccurate within−2.2 to+ 0.9 feet for the main microphonearray and within 8 feet for the diffracted microphones.

For the purposes of this paper,x is defined as the directionof the flight path, withy to the left of the vehicle. The originis located at the reference microphone location (microphone11). The terrain data are then immersed in a 20 ft incremented

-2

-2

-4

-4-6

-6 -84

2

2-400-300-200-100

100200300

0

0

0

x(×

103

ft)

y (×103 ft)

Vehicle Location

27

2829

z[ft

](r

e.69

93ft)

Fig. 17. Computational domain with origin at microphone11, (reference altitude 6993 ft mean sea level). Contourlines are every 50 ft of elevation.

9

Cartesian grid (x,y,z) using the boundary function,

f =

{

0, z− zelevation <= 0

1, otherwise.(11)

The locations of the microphones used for comparisonare shown in Table1. There is an offset between mea-sured locations and computed location because the physicalground does not align with the grid. Second, the sourcesurface, represented by isotropic spherical wave is initial-ized at a radius of 170 ft, around the aircraft positioned at(−3701,−48.5,−252.2) identified as ‘Vehicle Location’ inFigure17.

Table 1. Microphone locations as measured versus nearestcomputational grid locations.

Mic # Measured (x, y, z ft) Computed (x, y, z ft)10 (2.1, 26.9, 1.6) (0, 20, 0)13 (3.3, -60., -3.2) (0, -60, 0)17 (-0.3, -235.6, -12.7) (0, -240, -20)27 (-1480.2 -5345.6 -280.9) (-1480 -5340 -280)28 (-1908.7 -6048.7 -316.8) (-1900 -6040 -320)29 (-1732.6 -5780.6 -303.1) (-1740 -5780 -300)

With the vehicle location, source noise, and terrain de-fined, Equation4 can now be solved to compute phase, at-tenuation factor (which includes the effect due to diffraction),and emission angles. For the purposes of this diffraction in-vestigation, the terrain is assumed to be a hard surface andatmospheric attenuation has been turned off. However, WaveConfinement is capable of accounting for each of these ef-fects (Ref.14).

The pressure time histories at microphones 10, 13 and17, which are in line of sight, are extracted using the map-ping function and compared to the measured data in Figure18. Computed and measured data are in excellent agreement,which is to be expected since the source surface is constructedusing the recorded data from these microphones. Further, thedata were forward propagated using the same atmospheric andground conditions as that of backward propagation used toform the spheres. This validates that the computational setupand the algorithm are in agreement with the source data forthe simple straight ray case.

Pressure signals from the source hemisphere are then prop-agated to microphones 27-29, behind the isolated hill. Figure19 shows the computed attenuation factor (for approximately5 Hz3) near these microphones. It was anticipated that soundintensity at microphones 27 and 28 would be higher than atmicrophone 29, which is confirmed by the measured data asshown in Figure20. It was anticipated that further behindthe hill, the sound intensity would decrease. However, as canbe seen from Figure19, microphone 28 is in a Poisson spot

3Wave front is 5-7 grid cells wide, with grid cells spacedevery 20 feet. This results in an approximately 4 to 5.5 Hzwave.

-0.2

0.0

0.2

-0.2

0.0

0.2

Pre

ssur

e [P

a]

0.00 0.02 0.04 0.06 0.08 0.10time [s]

-0.4

-0.2

0.0

0.2

MeasuredComputed

Fig. 18. Comparison of computed versus measured pres-sure time-series data for microphone (top) 10, (middle) 13,and (bottom) 17. Every 30th time stamp of the computedsignal is shown for clarity.

Mic 27Mic 29

Mic 28

-7000-6000-5000y [ft]

-4000

-3000

-2000

-1000

0

x [ft

]

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

Fig. 19. Computed attenuation factor (x10−4) for an ap-proximately 5 Hz wave propagating around the isolatedhill, with microphones 27 thru 29 identified. Black linesare 50 foot elevation lines from Figure17.

formed by the hill. This resulted in a higher measured valuethan that seen by microphone 29, which was more “line ofsight”. The prediction of this Poisson spot not only helps val-idate the proposed propagation method, it elucidates the ap-parent anomaly seen in the measured data, and confirms that2D methods are not adequate for real life scenarios.

Now, the frequency spectrum at each of the isolated hillmicrophone locations is calculated using the closest availabledata point on the source hemisphere. Table2 shows the dif-ference in azimuth and elevation angles from available source

10

50 100 150 200 250 300 350Frequency [Hz]

3

9

15

21

27

33

39

45

Pre

ssur

e [d

B]

Microphone 27Microphone 28Microphone 29

Fig. 20. Measured spectral data of microphones 27, 28 and29.

data (steady level flight over microphones 1 thru 21) and re-quired emission direction.

Table 2. Computed emission angle compared with closestavailable measured source location.

Mic # Computed emission anglesClosest data points27 (248.1, -5.78) (249, -11)28 (254.4, -6.1) (255, -12)29 (251.1, -5.74) (252, -11)

Since microphone 27 is close to the shadow boundary, thescaling factor is≈ 1 for all frequencies. Figure21 shows thecomparison of the propagated data versus measured spectra.The computed data are close to measured values, with someover predictions at the first tail rotor harmonic (70 Hz), andathigher frequencies around 225 Hz.

There is an error in the near field data used for computationsince no data are available at the required emission angles,so the closest measured data points are used. Further, sincethe source data were measured many seconds further into therun, it is possible that the tail rotor forces have changed forthis nominally steady flight, affecting the emitted noise signal.Future work will look into subsequent steady level flight runsto identify if this natural unsteadiness is affecting the results.

Contrary to microphone 27, microphones 28 and 29 are inthe shadow region of the hill, where the scaling factor is not1. For these cases, Equation10 is used. The frequency spec-trum for microphones 29 and 28 is plotted in Figures22 and23, respectively, against measured data. Both figures showthe computed data with and without scaling. The scaling lawimpacts frequencies above 5 Hz, with higher frequencies in-curring lower attenuation factors, resulting in lower sound in-tensity.

21

24

27

30

33

36

39

42

21

35030025020015010050

Pre

ssur

e[d

B]

Frequency [Hz]

MeasuredComputed

Fig. 21. Microphone 27 spectra, measured data comparedwith propagated (computed) data.

3

9

15

21

27

33

39

45

Pre

ssur

e [d

B]

50 100 150 200 250 300 350Frequency [Hz]

3

9

15

21

27

33

39

45

Pre

ssur

e [d

B]

MeasuredComputed

Fig. 22. Measured versus propagated (computed) data formicrophone 29. The upper plot includes the computeddata without scaling factor applied and the lower plot con-tains the computed data with the scaling factor applied.

Spectral data from the scaled computation are in goodagreement with measured data, although there is a slight over-prediction at the first tail rotor harmonic. This overpredictionat the tail rotor harmonic is attributed to the lack of adequate

11

3

9

15

21

27

33

39

45

Pre

ssur

e [d

B]

50 100 150 200 250 300 350Frequency [Hz]

3

9

15

21

27

33

39

45

Pre

ssur

e [d

B]

MeasuredComputed

Fig. 23. Measured versus propagated (computed) data forMicrophone 28. Top is computed data without scaling fac-tor applied, while bottom has the scaling factor applied.

source noise data for this condition. There are more deviationsfrom measured data for microphone 28 as shown in Figure23,which will be studied in the future.

Computational errors also exist at microphones 28 and 29,because of the low frequency assumption of the computationalwave. As explained before, if there are not enough grid pointsbetween two wave passes, the signals can merge due to dis-cretization effects, losing the ability to capture the details ofeach pass separately and compute interference. If multiplereflections are present in the signal, the implemented compu-tational setup is not currently able to identify them.

CONCLUSIONS

The wave equation (in PDE form) accounts for all the prop-agation effects such as refraction, multiple reflections, anddiffraction. When discretized and numerically evaluated,thepropagating waves incur numerical dissipation, which plays adetrimental role in propagation problems. Artificial dissipa-tion is especially problematic when the range is on the orderof thousands of wavelengths. For this reason, finite differ-ence methods are replaced by high frequency approximations,which solve an ODE along each ray rather than a collectionof rays (or a wave front). Although this is a reasonable ap-proximation for many wave propagation problems, ray tracingtends to fail when diffraction effects are dominant.

Wave Confinement provides a method for solving the waveequation for cases where ray tracing techniques fail. WaveConfinement is a discretization technique that solves the linearwave equation, where the solutions (nonlinear solitary waves)are asymptotically stable. This makes WC a viable choicefor long range propagation problems. The asymptotic solu-tions are representations of physical waves and can be used topropagate details of short waveforms through Dynamic Sur-face Extension. Since diffraction depends on the frequency,each frequency component of the waveform needs to be re-solved separately, which is computationally expensive. Toavoid that problem, a physics based scaling law is used totransform information from the computational wavelength toall the physical frequencies within the waveform.

Since the wave length of the nonlinear solitary wave is ap-proximately 5 Hz (for the flight test computations shown inthis paper), the scaling factor is 1 at 5 Hz and is< 1 for fre-quencies greater than 5 Hz. The scaling law plays an impor-tant role in capturing the frequency dependent diffractionphe-nomena, without which Wave Confinement would artificiallypropagate higher frequency wave fronts into the shadow re-gion. The scaling law that was developed in this paper cor-rectly shielded higher frequency sounds and resulted in qual-ity comparisons with classic diffraction problems as well asflight test acoustics data.

Comparison of Wave Confinement with analytical solu-tions shows that this new idea is capable of accounting fordiffraction effects with reasonable accuracy. A flight testcom-parison is also completed, with a low frequency assumptionthat assumes the effects of phase differences between multi-ple reflected waves are negligible. For example, two wavepasses with small phase difference are treated as one pass. Fu-ture work will show the capability to compute each pass sep-arately to avoid merging. The flight test comparison showedreasonable agreement with measured data. Further, the propa-gation method was able to help explain the seemingly anoma-lous readings from a microphone that was placed in a Poissonspot of an isolated hill.

ACKNOWLEDGMENTS

We acknowledge the U.S. Army SBIR program and AATDfor funding this work under contract #W911W6-12C-0036.We also acknowledge Dr. Frank Caradonna for his valuablesuggestions and insights, Dr. Arje Nachman for funding underAFOSR contracts and Dr. Eric Greenwood for his insight andassistance on this project.

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