June 2011

NASA/CR-2011-217158 

The Fast Scattering Code (FSC): Validation 

Studies and Program Guidelines 

Ana F. Tinetti and Mark H. Dunn

Mark H. Dunn, Consultants, Yorktown, Virginia 
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June 2011 

NASA/CR-2011-217158 

The Fast Scattering Code (FSC): Validation 

Studies and Program Guidelines 

Ana F. Tinetti and Mark H. Dunn

Mark H. Dunn, Consultants, Yorktown, Virginia
Available from: 

NASA Center for AeroSpace Information 

7115 Standard Drive 

Hanover, MD 21076-1320

443-757-5802 
TABLE OF CONTENTS Page 

Introduction ...................................................................................... 1

Background ...................................................................................... 1

The Fast Scattering Code (FSC) ........................................................... 1

 Numerical Instabilities – Symmetry .................................................. 3

Analytical Solution for the Scattering of Sound by a Sphere ........................... 4

Configuration Description ............................................................... 6

 Spheroids ................................................................................. 6

 Source Surface Construction ...................................................... 7

 Source surface size ........................................................... 7

 Source surface shape ........................................................... 7

 Flat Plate ................................................................................. 8

 Source Surface Construction ...................................................... 8

Results and Discussion ........................................................................ 8

Sphere ........................................................................................... 8

 Acoustically Hard Surface ........................................................... 9

 Acoustically Soft Surface ........................................................... 17

Oblate Spheroids ............................................................................. 18

Flat Plate ...................................................................................... 24

 Flat Plate with Square Edges ........................................................... 24

 Flat Plate with Rounded Edges ...................................................... 27

 Flat Plate with Sharp Edges ........................................................... 29

 Effect of Edge Geometry on Simulated Noise Scattering ....................... 31

Summary .......................................................................................... 32

Sphere .......................................................................................... 32

Oblate Spheroids ............................................................................. 32

Flat Plate ...................................................................................... 33

Guidelines for Successful Application of the FSC to General Scattering Problems ..... 33

Solution Quality ............................................................................. 33

 Scattering Surface Definition ........................................................... 33

 Scattering Surface Discretization ...................................................... 33

 Source Surface Discretization ........................................................... 33

 Source Surface Placement............................................................... 34

 Source surface size ............................................................... 34

 Source surface shape ............................................................... 34

 Use of Symmetry ........................................................................ 34

Solution Accuracy/Convergence ........................................................... 34

 Sound through Solid Surfaces ...................................................... 35
 Convergence Metrics .................................................................... 36

 Acoustic Power through a Sphere ....................................................... 37

Grid Effects on Solution Accuracy ...................................................... 38

Application to Partial Configurations ...................................................... 38

Problem Setup – A Roadmap ............................................................... 39

References ...................................................................................... 40
1 

1. INTRODUCTION 

The Fast Scattering Code (FSC)1,2 is a frequency domain noise prediction program developed at 

the NASA Langley Research Center (LaRC) to simulate the acoustic field produced by the 

interaction of known, time harmonic incident sound with bodies of arbitrary shape and surface 

impedance immersed in a potential flow. The code uses the equivalent source method (ESM) to 

solve an exterior 3-D Helmholtz boundary value problem (BVP) by expanding the scattered 

acoustic pressure field into a series of point sources (Ns) distributed on a fictitious surface placed 

inside the actual scatterer. The FSC discretizes the scattering surface into Nc collocation points to 

produce a dense, over-determined system of linear equations of size Nc x Ns; the unknown source 

strengths are adjusted so that a prescribed surface boundary condition is satisfied using least 

squares methods. The FSC has been validated using analytical3,4 and experimental data5,6, and 

applied successfully to various problems of interest to the aeroacoustics community7-9. 

The ESM is the most computationally efficient of the boundary methods currently used in 

acoustics (finite difference, finite element, boundary element), as it produces a smaller linear 

system whose solution requires a fraction of the computational resources; it is easy to implement, 

robust and stable and is devoid of singularities. The accuracy of the solution depends, however, 

on various factors: 1) the distribution and number of collocation points and equivalent sources on 

their respective surfaces, 2) the relative size of the source surface, and 3) configuration 

symmetry. 

It is the purpose of this work to provide additional code validation studies and determine the 

range of code parameters that produce accurate results with minimal computational costs. 

Systematic noise prediction studies are presented in which monopole generated incident sound is 

scattered by simple geometric shapes – spheres (acoustically hard and soft surfaces), oblate 

spheroids, flat disk, and flat plates with various edge topologies. Comparisons between FSC 

simulations and analytical results and experimental data are presented. 

2. BACKGROUND 

2.1. The Fast Scattering Code (FSC)

The FSC has been designed as an aeroacoustic analysis tool for assessing global effects on noise 

radiation and scattering caused by changes in configuration (geometry, component placement) 

and operating conditions (background flow, excitation frequency). It can be used as a component 

of an aircraft system noise prediction package, such as ANOPP II, or as standalone software. 

Inputs to the code are the scattering surfaces and their impedance, excitation frequency, local 

density, speed of sound and Mach number, and an incident sound field (generated by the code or 

provided by the user). The total acoustic field (incident plus scattered) is calculated at a userspecified collection of observers (see Figure 1). Details on the solution methodology and 

program usage are found in references 1 and 2. 
2 

Figure 1.- FSC input/output. 

Discretization of the scattering surfaces produces a rectangular system of linear equations of size 

Nc x Ns (FSC matrix); the procedure, which follows Nyquist frequency limitations, assumes that 

Nc is proportional to Sk2 (where S is the surface area of the scatterer, and k the excitation wave 

number of the incident sound), thus ensuring that the mesh spacing is small enough to capture 

the incident pressure fluctuations. The proportionality constant depends on a user-selected 

number of points per wavelength (Nw) and on flow speed. Through numerical experimentation, it 

has been determined that 1) Nc and Ns should be uniformly distributed on their respective 

surfaces, and 2) Ns ~ 1/3 Nc is adequate for the majority of applications. Computer memory 

consumption for large-scale commercial transports limits the original version of the FSC (v2.0) 

to problems with moderate excitation frequencies. Simulations for high frequency/large 

scattering surface area combinations yield immense linear systems that can only be solved with 

supercomputers and the development of numerical algorithms that drastically reduce 

computational resource utilization. 

In FSC v3.0 the dense, linear least squares solution approach of v2.0 was replaced with an 

iterative conjugate gradient method (CGM) featuring fast multipole methods (FMM) and fast 

matrix-vector multiplication algorithms to accelerate matrix solution and field calculations. 

Benchmarking against results obtained previously with FSC v2.0 indicated that a 95% reduction 

in computer memory was attained10, effectively increasing the maximum frequency range 

achievable for a given configuration using a single processor by a factor of 3. In addition, up to a 

four-fold reduction in computer time was demonstrated. 

Even with the substantial improvements in computer resource utilization achieved with FSC 

v3.0, simulations of the acoustic field generated by the interaction of very high frequency noise 

sources and large-scale configurations would require prohibitive amounts of computer memory 

and time. To obtain solutions within the limits of available computer resources, multi-processor 

(MP) versions of the FSC have been developed. Both versions feature the use of a CGM iterative 
3 

linear algebra solver with direct matrix-vector multiplication; v3.1 uses OpenMP constructs for 

use in shared memory platforms, and v3.2 includes MPI function calls for use in distributed 

memory platforms. Every element of the FSC matrix is calculated as needed during each 

iteration; thus, computer memory requirements are minimal at the expense of longer execution 

times. Because matrix ill-conditioning may occur during solution of highly symmetrical 

configurations (see next section), a new option that takes advantage of double symmetry in 

exterior BVPs has been added to versions 3.1/3.2 (isym = 2). The simulations presented in this 

report made use of this option. 

2.1.1. Numerical Instabilities – Symmetry

When applying the ESM to highly symmetrical configurations, advantage should be taken of all 

possible levels of symmetry to 1) reduce potential problems associated with matrix illconditioning caused by the presence of multiple identical elements in the matrix11, and 2) to 

make the solution process more efficient. For the purposes of this discussion, a configuration or 

system consists of all the scattering surfaces of interest plus the source of incident sound. 

Consider the sphere/monopole system depicted in Figure 2a – the sphere, of radius a equal to 5.0 

meters and centered at the origin, has been discretized such that one hemisphere is the mirror 

image of the other, with respect to both the x-z and y-z planes: the top half is identical to the 

bottom half (symmetry about x = 0), and the right half is identical to the left half (symmetry 

about y = 0). If the acoustic source is placed either on the x or z axis, then the sphere/source 

system is symmetric (case 1); if the source is moved to any other location, the system is no 

longer symmetric (case 2). The total (incident plus scattered) acoustic field resulting from the 

solution of a system consisting of the entire sphere and a source with f = 1000 Hz located 1.0a

away from the surface, is depicted in Figure 2b on a 7.5 m radius ring of observers surrounding 

the sphere: the field is given by the black line for case 1 (source 1 located on the z-axis, as shown 

in Fig. 2a), and by the red line for case 2 (source 2 located at a point 45o from the axes in the allpositive octant). Note from the figure that, as expected, the field for case 1 is symmetric with a 

well-defined shadow zone behind the sphere; for case 2, the acoustic field displays marked 

asymmetries and incorrect minimum/maximum levels. Most likely, different solutions would be 

obtained for varying source placements. 

For higher source excitation frequencies, at least one level of symmetry must be included in the 

discretization of the surface if a proper convergence behavior is to be established during solution 

of highly symmetric systems. This is necessary because the magnitude of the boundary error 

increases with the number of equivalent sources being used11. For case 1 with a hard-wall sphere 

surface, only one level of symmetry (about y = 0) was sufficient; employing an additional level 

of symmetry (about x = 0) did not accelerate convergence, but it substantially reduced computer 

resource utilization by requiring discretization of only one quarter of the sphere. Two levels of 

symmetry were required for a proper solution of case 1 with an acoustically soft sphere surface – 

it is surmised that the ill-conditioning of the problem increases with the increased complexity of 

the surface boundary condition when non-zero admittances are used. 
4 

 

Figure 2. Effects of a non-symmetric grid/source system, f = 1000 Hz, M∞ = 0.0. Observer field 

is a 7.5 m radius ring around the sphere. 

2.2. Analytical Solution for the Scattering of Sound by a Sphere 

The scattering of incident, time harmonic (

ti

e

 

time dependence), monopole generated sound by 

a sound absorbing sphere of constant admittance, A, is governed by the following boundary value 

problem (BVP): 

        Sxxxpkp



0

2 2



(1) 

  SxpA

n

p

ck

i

pB   



 







0

 00

(2) 

In (1-2), k , 

0

x



 is the spatial location of the incident monopole (

 ax0



), S denotes the 

sphere surface and S

+

 denotes the sphere exterior. It is assumed that the sphere is centered at the 

origin of the coordinate system and has radius a. 

To solve (1-2), the acoustic pressure is split into known incident and unknown scattered parts: 

inc ppp scat





  

(3) 

The incident pressure is the solution of the Helmholtz equation in unbounded space 

  0

2 2

inc inc xxpkp



     

(4) 

and is given by 

  0 0

0

0

4

1

xxR

R

e

xp

ikR

inc

 



 





. (5) 

Source 1

Source 2

(a) Sphere-source system (b) Total acoustic field at observer ring

 = 0.0o
5 

Combining (1-5), the scattered pressure is the solution of the BVP 

  

scat  scat  Sxpkp



0

2 2

(6) 

    

scat



inc

 SxpBpB



. (7) 

The spherical geometry of the BVP suggests an expansion of the solution as a series of spherical 

harmonics. Following the analysis of reference 12 (Section 4.2), the incident pressure is written 

     



 





 0











 inc xp xRE

 

(8) 

where 

  0

xikSE

 









 . (9) 

      









 jxR kr Y ,



(10) 

       



 





,

2

 hxS kr Y



. (11) 

In (9-10), 

  r ,, 

 are spherical coordinates associated with the Cartesian point 

x



, 



j

 are 

spherical Bessel functions of the first kind, 

  2

h

 are spherical Hankel functions of the second 

kind, and 

 Y

are spherical harmonics given by the formula 

      

      



 

 







i

Y cos eP

!4

!12

1,







(12) 

where 



P

are Legendre functions of degree 



and order 

 .

The scattered pressure is written as a superposition of singular harmonics 

     



 





 0











 scat xp xSC

 

(13) 

where the unknown coefficients 

 C

 are determined from the boundary condition (7) and the 

incident sound expansion (8). Let 

ik Ac   00

, (14) 

then 

   

       





 

  







E

h ka hk ka

j ka jk ka C

2 21

1









 



(15) 

Special function routines for calculating the formulas in (1-15) were taken from user supplied 

software of reference 13. 
6 

2.3. Configuration Description 

2.3.1 Spheroids

An important goal of the study was to assess the effects of excitation frequency and surface 

shape on shadow zone characteristics for similar geometries. To this end a sphere, two oblate 

spheroids of decreasing thickness, and a thin flat disk with a rounded edge, were placed with 

respect to the acoustic source so as to project the same geometric zone of silence. The 

configurations are depicted in Figure 3. The sound generator was a stationary monopole of unit 

strength located at (0, 0, 10) meters. The sphere was centered at the origin, and had a radius r

equal to 5.0 meters. The oblate spheroids (OS1 and OS2), also centered at the origin, were 

generated by adjusting the length of the semi-major (a) and semi-minor (b) axes of the sphere. 

Geometric characteristics for the scattering bodies are given in Table 1. 

Table 1. – Geometric data for oblate spheroids. 

Configuration Geometrical Parameters, m Source Surface Size

Sphere r = a = 5.0 0.9r

Oblate Spheroid 1 (OS1) a = 5.77, b = 1.147 0.80b

Oblate Spheroid 2 (OS2) a = 5.77, b = 0.38 0.40b

Flat Disk (rounded edge) r = 4.33, t/D = 0.035 0.033t

Figure 3.- Oblate spheroid geometries and their relative placement. 
7 

2.3.1.1 Source Surface Construction 

The geometry module of the FSC is used to 1) discretize the scattering surfaces into Nc

collocation points, and 2) generate/discretize the source surfaces into Ns equivalent sources. For 

simplicity, the module generates source surfaces by creating smaller versions of the scatterers 

utilizing a single user-supplied scaling parameter, srcalpha. The main disadvantages of using 

only one scaling parameter are that 1) portions of the source surface may not be interior to the 

scattering body, thus violating a requirement of the ESM for exterior BVPs; and 2) large 

localized differences in source surface size, relative to the scattering body, arise for elongated 

geometries. To alleviate these effects, the source surface is anchored to the scattering body at the 

leading and trailing edges for wing-like structures consisting of a collection of airfoils (wings, 

nacelles, pylons, blended wing bodies), and at the nose and tip for structures composed of two – 

top and bottom – arbitrarily shaped halves (fuselages, spheres, cylinders, etc.). The size and 

shape of the resulting source surface can substantially affect the quality of a solution. 

Source surface size – An adequate source surface size depends on the excitation frequency of the 

incident sound, and on the shape/surface area of the scatterer. In general, for moderate to high 

frequencies and relatively large, thick bodies, srcalpha ~ 0.9 to 0.95 gives excellent results; the 

thinner the body and the lower the frequency, the smaller srcalpha should be. For example, 

srcalpha ~ 0.75 to 0.8 works well for nacelles at most frequencies of interest; for very thin 

shapes, srcalpha ≤ 0.1. Source surface sizes for the spheroids used in the study are included in 

Table 1. 

Source surface shape – Ideally, a constant separation between source and scattering surfaces 

should be maintained. This is precluded in the regions where the source surface is anchored to 

the scattering body. Most geometries of interest are composed of wing-like structures and/or 

fuselages; for these, the effects of non-constant separation at the edges are small. For canonical 

shapes like spheres, spheroids, cylinders, and cubes, the effect is more pronounced. In these 

cases, the geometry module can still be used advantageously by employing separate, properly 

constructed scattering and source surfaces as inputs. The procedure is as follows. First, using the 

scattering surface as input, obtain the collocation points (Nc1). Run the module again, using a 

smaller replica of the scattering surface (scaled so as to meet source surface size requirements) as 

input, to obtain a second set of collocation points (Nc2). Note that by adjusting the number of 

points per wavelength (Nw) on the second run, Nc2 ~ 1/3 Nc1. Thus, Nc2 will be used as equivalent 

sources in a subsequent ESM module run (the two sets of equivalent sources generated by the 

geometry module are discarded). This approach was followed to discretize the sphere surfaces; 

for consistency, it was also employed with the oblate spheroids – the geometric definitions of the 

source surfaces were created by adjusting the scale in every dimension to maintain an 

approximately constant separation to the parent scattering surface. 

To correctly solve exterior BVPs, the ESM requires sufficient space inside the scattering body 

for the placement of a suitable source surface – a certain minimum thickness for the scatterer is 

necessary. It was found during the present study that, as expected, it is quite difficult to obtain 

well behaved solutions for very thin oblate spheroids (maximum thickness ratio (t/D)max < 0.04), 

the reason being extreme thinness toward the edge. This problem was eliminated by considering 

a constant thickness, flat disk with a rounded edge. The resulting geometry (see Fig. 3 and Table 

1) had a smaller (t/D)max than that achievable for oblate spheroids. 
8 

Because all the configurations or systems (scatterer plus incident sound generator) considered 

here are symmetric with respect to both the x-z and y-z planes, only one quarter of each surface 

was discretized, and two levels of symmetry were used to obtain the scattered acoustic fields. 

2.3.2. Flat Plate

The last configuration to be considered in the study was a relatively small flat plate with different 

edge geometries (square, rounded, sharp). The dimensions of the plate were 0.5 m x 2.0 m. The 

acoustic source, a stationary monopole of unit strength, was placed 0.25 m from the center of the 

surface. The plate dimensions, the type, location, and frequencies of the sound generator, as well 

as the observer field location, were chosen to represent a subset of the configurations tested in 

reference 14. 

2.3.2.1 Source Surface Construction 

For a given scatterer, Nc, Ns, and the relative size of the source surface are frequency dependent. 

In general, the size of the source surface necessary to obtain a good solution decreases with 

frequency, scattering body thickness, and distance to the acoustic source. Thus, it was decided to 

select the plate thickness so that a correct solution for the lowest frequency of interest could be 

obtained, and use the same geometry for the higher frequencies. This resulted in a plate with a 

thickness of 0.07 m, and a 0.006 m thick source surface. In order to maintain the slope 

discontinuities at the edges inherent in the geometry, the surfaces were defined as multicomponent entities; as was done with the spheroids, each surface (scattering and source) was 

discretized separately using only one quadrant, and two levels of symmetry were used during 

execution of the ESM module.

3. RESULTS AND DISCUSSION 

All results presented in this report were obtained using two levels of symmetry*, and stationary 

monopoles as equivalent sources. The configurations were immersed in a quiescent flow (M∞ = 

0.0). The simulations were conducted in the NAS Columbia super computer using FSC v3.1 for 

most of the frequency range; because it has better scalability, FSC v3.2 was used for the highest 

frequency cases. The runs were stopped after reaching either the user supplied tolerances for two 

of the convergence metrics employed in the code (L2 norms of the solution vector residual and 

the source strength vector, set to 1.0 x 10-5 and 1.0 x 10-6, respectively), or enough iterations to 

properly resolve the shadow zone. 

3.1. Sphere 

The excitation frequencies, f, considered in the exercise covered the range of Helmholtz number 

1 < ka < 500. These frequencies are listed in Table 2. Also included in the table are grid (matrix) 

sizes and metrics at each frequency of interest, as well as computer resource utilization statistics. 

FSC versions 3.1/3.2 have been optimized for high frequency/large scale applications. The use of 

v3.1 to solve the low frequency scattering problems presented here (f < 1000 Hz) was done for 

consistency only, as the computer resources (processors and time) required to obtain a converged 

 

*

Superposition of the acoustic fields is assumed when using symmetry. If the acoustic source lies on a symmetry 

plane, its strength is doubled. Thus, 6 dB per plane of symmetry must be removed from the solution to account for 

this fact. 
9 

solution for such small cases will certainly exceed those of FSC v2.0 – results from a comparison 

test (not shown) using a matrix of size Nc x Ns ~ 3500 x 1200 indicated that the single-processor 

v2.0 is approximately 17 times faster than v3.1 with 10 processors. Thus, the execution times 

presented in Table 2 for the two lowest frequencies should not be used to assess code 

performance. 

The total acoustic field was sampled at 1) a plane bisecting the configuration, 2) a straight line 60 

meters long and located at z = -30 m (8a away from the source), and 3) a ring placed around the 

sphere, centered at the origin, with a radius r = 7.5 m (1.5a); on this ring,  = 0o

 corresponds to 

the point closest to the source (see Fig. 2a). An analytical approximation for the noise scattered 

by a single sphere with an acoustically hard or soft surface, based on spherical harmonic 

expansions (see section 2.2), was used to validate the simulations for the straight line and ring 

observer fields; the number of terms included in the spherical harmonic expansions, nn, is also 

provided in Table 2. Two surface impedance cases have been considered, namely, a hard-wall 

sphere with admittance A = 0.0 + 0.0i rayl-1, and an acoustically soft sphere with A = 9.0 x 10-4 + 

9.0 x 10-4i rayl-1. The latter admittance is within the range of values used in engine liner design. 

Table 2. – Acoustic source excitation frequencies, grid metrics, and execution time statistics for 

sphere scattering. 

3.1.1 Acoustically Hard Surface

In general, surface discretization using 5 < Nw < 12 is sufficient to properly capture the physics 

in the vast majority of scattering problems. However, this rationale does not apply to very low 

frequencies, where the wavelength of the sound is larger than the dimensions of the body. In 

such cases, a minimum of a few hundred points is necessary for an adequate definition of the 

scattering surface of interest. For the lowest excitation frequency considered in the study, 11 Hz 

(ka = 1), surface discretization was performed using Nw = 50, resulting in Nc ~ 200. The total 

(incident plus scattered) acoustic field at this frequency, sampled at the three observer groups 

described previously, is presented in Figure 4. Sound pressure level (SPL) contours for a plane 

bisecting the configuration are presented in Fig. 4a – note that, as expected, the omni-directional 

incident sound field is only minimally disturbed by the presence of the sphere. On a ring 

surrounding the body (Fig. 4c), the spherically radiating nature of the acoustic field is apparent, 

with minor differences between incident and total components. Plane wave propagation is almost 

fully established by the time the sound reaches a line 40 meters away from the source (Fig. 4d); 

at this distance, the differences between incident and total acoustic fields are very small, 

Frequency,

Hz ka

Grid size,

( Nc x Ns)

Points per

wavelength,

Nw

Processors 

used,

Nproc

Number 

of 

iterations,

Niter

Execution 

time, texec

hours/job

Terms in 

analytic 

expression,

nn

11 1.0 196 x 65 50.00 4 19 9.478 x 10-550

541 50.0 4940 x 1688 5.00 16 114 4.067 x 10-2100

1000 92.4 16997 x 5548 5.00 64 354 1.964 x 10-1100

1623 150.0 33346 x 9858 4.32 64 305 8.470 x 10-1175

2164 200.0 67906 x 22150 4.62 64 1800 2.035 x 101250

3247 300.0 180192 x 58570 5.00 64 1500 1.343 x 102325

4329 400.0 500669 x 162850 6.00 64 1113 7.011 x 102 N/A

5411 500.0 720890 x 162224 6.00 128 680 3.355 x 102 N/A
10

indicating negligible scattering by the sphere. Note from the latter two figures that agreement 

between FSC and analytic results is excellent. Also included as Figure 4b are iteration history 

plots for the convergence metrics – observe that the solution reached user specified tolerances 

very rapidly. 

 

 

Figure 4. Total acoustic field for sphere/point source system, f = 11 Hz (ka = 1), M∞ = 0.0.

Observer field results for f = 541 Hz are presented in Figure 5. Note from Fig. 5b that the FSC 

solution reached steady levels very quickly, converging to specified tolerances after about 100 

iterations. SPL contours on a plane bisecting the system (Fig. 5a) indicate that there is substantial 

scattering of sound by the sphere, giving rise to a well defined shadow zone behind the body. 

The symmetric and highly oscillatory nature of the total acoustic field is clearly observed in 

Figures 5c and 5d, where the incident field (in the absence of scattering surfaces) is provided for 

reference purposes. On a ring surrounding the sphere (Fig. 5c), shielding effects are obvious for 

(a) Plane bisecting the configuration (b) Convergence history

(c) Ring surrounding sphere (d) Observer line at z = -30 m
11

90o<  < 270o, indicating the boundaries of the shadow zone. Within this region, the total sound 

decreases and then increases as it reaches the point on the ring farthest from the source ( = 

180o), where its amplitude is still several dB lower than that of the incident field. On a line 40 m 

away from the source (Fig. 5d), shielding causes the total noise to decrease and then increase as 

the observer moves toward the sphere, peaking at the point closest to the body. At this point, the 

total noise level is slightly higher than that of the incident field. Note from the line plots that 

agreement between FSC and analytic results is excellent. It is not known at present if the 

“singularity” at points in the shadow zone directly opposite the source has a physical meaning. 

 

 

Figure 5. Total acoustic field for sphere/point source system, f = 541 Hz (ka = 50), M∞ = 0.0. 

Observer field results for f = 1000 Hz (ka = 92.4) are presented in Figure 6. Note from Figure 6b 

that the FSC solution is well converged after a few hundred iterations. SPL contours on a plane 

bisecting the configuration are given in Fig. 6a; note that a clearly defined shadow zone behind 

(a) Plane bisecting the configuration (b) Convergence history

(c) Ring surrounding sphere (d) Observer line at z = -30 m
12

the sphere is present. Comparisons between FSC results and those obtained from the analytical 

solution are given in Figures 6c and 6d for the ring and straight line observer fields, respectively. 

Note from the figures that the solutions are symmetric and highly oscillatory, as expected, and 

practically identical – the SPL values at the ring of observers (Fig. 6c) clearly indicate that the 

shielding effect is substantial, manifested by a 52 dB drop in noise between highest and lowest 

levels. The line of observers (Fig. 6d) indicates that, at 40 meters away from the source, the total 

noise is highest at the points most distant from the sphere, decreases as shielding by the sphere 

increases, and peaks again at the point closest to the sphere, where the level is comparable to that 

at both ends of the line. 

 

 

Figure 6. Total acoustic field for sphere/point source system, f = 1000 Hz (ka = 92.4), M∞ = 0.0. 

Solution convergence histories and observer field comparisons for an excitation frequency of 

1623 Hz are depicted in Figure 7. As before, excellent FSC convergence is achieved within a few 

hundred iterations (Fig. 7b). The trends are very similar to those observed at the lower 

(a) Plane bisecting the configuration (b) Convergence history

(c) Ring surrounding sphere (d) Observer line at z = -30 m
13 

frequencies (once scattering becomes evident) – the oscillatory nature of the total field is 

apparent, with the added effect of improved shielding as a result of shorter wavelengths (Figs. 7c 

and 7d). The latter is manifested as a larger noise differential within the shadow zone. Also note 

that, as before, agreement between FSC and analytical results is remarkable. 

 

 

Figure 7. Total acoustic field for sphere/point source system, f = 1623 Hz (ka = 150), M∞ = 0.0. 

The amount of noise shielding by the sphere increases with frequency, as can be observed in 

Figures 8 and 9, which present the acoustic fields for f = 2164 Hz (ka = 200) and f = 3247 Hz (ka

= 300), respectively. It was also noted during this exercise that the portion of the total acoustic 

field where noise amplification occurs due to reflections from the sphere surface converges very 

rapidly, independent of frequency; however, as the frequency increases, substantially more 

iterations and/or denser grids (increased Nw during surface discretization) may be necessary to 

fully converge the portion of the field in the shadow zone. Observe from Table 2 that, in general, 

(c) Ring surrounding sphere (d) Observer line at z = -30 m

(a) Plane bisecting the configuration (b) Convergence history
14

Nw ~ 5 sufficed to obtain very good agreement between FSC and analytical results, but 

noticeably more iterations were required for the higher frequencies, as seen in Figures 8b and 9b. 

 

 

Figure 8. Total acoustic field for sphere/point source system, f = 2164 Hz (ka = 200), M∞ = 0.0. 

FSC interim solution convergence histories and observer field plots for f = 4329 Hz (ka = 400) 

are presented in Figure 10; no analytical solution could be obtained at this frequency because of 

convergence problems†

. Although the trends observed previously are already well established, it 

is apparent that more iterations are necessary if a better resolution of the acoustic behavior in the 

shadow zone is desired. The line fields are expected eventually to collapse to patterns similar to 

those obtained for the lower frequencies; however, because of the large size of the problem and 

 

†

 Convergence problems with the analytical expression derived in section 2.2 were encountered for the higher 

frequency cases ka = 400 and 500. The problems were traced to the Legendre function calculations in the reference 

2 software. Alternate Legendre function software was not considered for this exercise.

(a) Plane bisecting the configuration (b) Convergence history

(c) Ring surrounding sphere (d) Observer line at z = -30 m
15

the fact that a considerable noise differential already exists in the shadow zone, the use of 

additional computer resources for this simulation cannot be justified. A solution using Nw = 5 

was also obtained for this frequency; those field results (not shown) indicated that the shadow 

zone was slightly less resolved at the same stage in the solution process. In hindsight, it may 

have been more efficient to continue that solution than to start a new one on a denser (Nw = 6) 

grid. Similar observations apply to the field plots for f = 5411 Hz (ka = 500), depicted in Figure 

11.

 

 

Figure 9. Total acoustic field for sphere/point source system, f = 3247 Hz (ka = 300), M∞ = 0.0. 

(a) Plane bisecting the configuration (b) Convergence history

(c) Ring surrounding sphere (d) Observer line at z = -30 m
16

 

 

Figure 10. Total acoustic field for sphere/point source system, f = 4329 Hz (ka = 400), M∞ = 0.0. 

(a) Plane bisecting the configuration (b) Convergence history

(c) Ring surrounding sphere (d) Observer line at z = -30 m
17

 

 

Figure 11. Total acoustic field for sphere/point source system, f = 5411 Hz (ka = 500), M∞ = 0.0. 

3.1.2. Acoustically Soft Surface

The total acoustic field on a ring (r = 7.5 m) placed symmetrically around a lined sphere is 

provided in Figures 12 and 13 for f = 1000 Hz and f = 1623 Hz, respectively. The entire surface 

of the body has been acoustically treated (A = 9.0e-04 + 9.0e-04i rayl-1). Note from Figs. 12a and 

13a that the treated surface substantially absorbs the noise radiated by the source, as manifested 

by an enhanced noise differential in the shadow zone‡. As observed previously for the hard-wall 

case, the “depth” of the shadow zone increases with frequency. A comparison between FSC and 

analytic results for f = 1000 Hz is presented in Fig. 12b. Note that the agreement between the two 

 

‡ Although negative SPL values do not have practical significance, they represent acoustic pressures below the 

reference value, 2.0 x 10-5 Pa.

(a) Plane bisecting the configuration (b) Convergence history

(c) Ring surrounding sphere (d) Observer line at z = -30 m
18

solutions is excellent, with only minor differences in the “zone of silence” below the threshold of 

human hearing. A converged analytic solution for f = 1623 Hz could not be obtained. 

 

Figure 12. Total acoustic field on a 7.5 m radius ring surrounding a lined sphere; 

f = 1000 Hz, M∞ = 0.0, A = 9e-04 +9e-04i. 

 

Figure 13. Total acoustic field on a 7.5 m radius ring surrounding a lined sphere; 

f = 1623 Hz, M∞ = 0.0, A = 9.0 x 10-4 + 9.0 x 10-4i rayl-1

.

3.2 Oblate Spheroids 

The source excitation frequencies considered for the spheroids and flat disk were similar to those 

used for the sphere cases; these frequencies, along with other grid metrics, are given in Table 3.

Execution time statistics are provided in Table 4; the observations made for the sphere cases with 

regard to low frequency resource allocation (see section 3.1) apply to the oblate spheroids as 

well. The total acoustic field was sampled at 1) a 60 m x 60 m plane bisecting the configuration, 

(a) Liner effects (b) Comparison with analytic results

(a) Liner effects (b) Comparison with analytic results
19

2) a 60 m x 60 m plane located at z = -30 m, 3) a line formed by the intersection of the two 

planes, and 4) a ring placed around the configurations, centered at the origin, with a radius r = 

7.5 m; on this ring,  = 0o

 corresponds to the point closest to the source. Only acoustically hard 

surfaces were considered. 

Table 3. – Acoustic source excitation frequencies and grid metrics for oblate spheroids. 

Table 4. – Execution time statistics for oblate spheroids. 

Total acoustic fields at two intersecting planes are presented in Figure 14 for the sphere and OS1, 

and in Figure 15 for OS2 and the flat disk. The excitation frequency is 1000 Hz. Several general 

observations can be made regarding the sound scattering characteristics of each configuration. 

Note from the figures that the oblate spheroids generate larger, stronger shadow zones: they are 

more effective at shielding monopole noise than the sphere or flat disk. Of the two, OS1 is 

slightly better. Because of enhanced diffraction as the sound waves encounter the surface, the 

sphere projects the smallest shadow zone. Also note that the sound in the amplification zone has 

shorter wavelength for the spheroids and disk than for the sphere; this is most likely caused by 

enhanced reflection from the flatter surfaces. A region of relatively high sound amplitude can be 

observed in the shadow zone opposite to the acoustic source. This behavior is intrinsic to the 

solution of electromagnetic and acoustic radiation scattering by spheres, spheroids and disks. 

Frequency,

Hz ka

OS1 OS2 Flat Disk

Grid size, 

( Nc x Ns)

Nw

Grid size

( Nc x Ns) Nw

Grid size

( Nc x Ns) Nw

11 1.0 174 x 56 55.00 161 x 52 55.00 156 x 52 70.00

541 50.0 3322 x 1173 5.00 3360 x 1089 5.00 1954 x 620 5.00

1000 92.4 12443 x 4042 5.00 11580 x 3860 5.00 6718 x 2104 5.00

1623 150.0 32870 x 10655 5.00 30504 x 9838 5.00 17710 x 5789 5.00

2164 200.0 58364 x 19527 5.00 54315 x 18068 5.00 31430 x 10480 5.00

3247 300.0 131344 x 42504 5.00 122100 x 39234 5.00 70882 x 23156 5.00

4329 400.0 233914 x 78212 5.00 217364 x 72195 5.00 126160 x 42372 5.00

Frequency,

Hz ka

OS1 OS2 Flat Disk

Nproc Niter

texec,

hours/job Nproc Niter

texec, 

hours/job Nproc Niter

texec,

hours/job

11 1.0 32 19 9.828 x 10-516 44 1.575 x 10-416 75 2.415 x 10-4

541 50.0 32 89 1.310 x 10-216 70 1.103 x 10-216 209 1.521 x 10-2

1000 92.4 64 626 2.971 x 10-132 166 1.497 x 10-116 278 1.753 x 10-1

1623 150.0 64 932 2.452 64 508 1.291 32 1000 1.537

2164 200.0 64 1812 1.471 x 10164 1000 6.797 32 1000 4.820

3247 300.0 64 1500 6.272 x 10164 1000 3.831 x 10164 1000 1.206 x 101

4329 400.0 128 1000 7.114 x 101128 1000 5.948 x 101128 1000 1.952 x 10-1
20

 

Figure 14.- Total acoustic field, in dB, for sphere and OS1; f = 1000 Hz, M∞ = 0.0. 

 

Figure 15. – Total acoustic field, in dB, for OS2 and flat disk; f = 1000 Hz, M∞ = 0.0. 

A more detailed assessment of the interactions between the sound emitted by the source and the 

nearby surfaces can be obtained by sampling regions in the near and far acoustic fields. Sound 

pressure levels (SPL) in the near field – sampled at a ring with radius r = 7.5 m surrounding the 

scattering body (see Fig. 2a) – and in the far field – given at a line formed by the intersection of 

the two observer planes – are presented in Figures 16a and 16b, respectively, for an excitation 

frequency of 11 Hz. At this frequency, the wavelength of the sound is several times larger than 

the reference dimension for the surfaces (r or a). Thus, the incident field is only slightly affected 

by the presence of the bodies, with minor differences being caused by the geometry of the 

scatterers (see Fig. 16a). Plane wave propagation is almost fully established by the time the 

sound reaches the observing line; at this distance, scattering by any of the surfaces is negligible. 

(a) Sphere (b) Oblate spheroid 1

(a) Oblate spheroid 2 (b) Flat disk
21

 

Figure 16. – Total acoustic field for oblate spheroids; f = 11 Hz (ka = 1), M∞ = 0.0. 

Observer ring and line results for f = 541 Hz are presented in Figures 17a and 17b, respectively. 

It is apparent from Fig. 17a that both sound amplification and shadow zones are different for all 

geometries considered. In the former, the interaction field for the spheroids and disk has shorter 

wavelength and larger amplitude than that for the sphere; the latter indicates that the spheroids, 

especially OS2, are more effective at attenuating sound than the sphere and disk. It can also be 

observed that the sphere is a slightly better noise shield than the disk in the fringes of the shadow 

zone (note that this may be a consequence of the relative position of the disk with respect to the 

observer ring). At a line 40 m from the source (Fig. 17b), the spheroids and disk have similar 

shielding characteristics, all better than the sphere. 

Figure 17. – Total acoustic field for oblate spheroids; f = 541 Hz (ka = 50), M∞ = 0.0. 

(a) Observer ring (b) Observer line

(a) Observer ring (b) Observer line
22

Similar trends can be observed at an excitation frequency f = 1000 Hz (Fig. 18), except that now 

OS1 has slightly better noise shielding characteristics than OS2. At f = 1623 Hz (Fig. 19), the 

sphere produces a larger, stronger shadow zone than the disk in the near field (Fig. 19a); in the 

far field, OS1 is clearly superior (Fig. 19b). These trends continue as the frequency increases (see 

Figures 20 – 22): OS1 has the best overall shielding characteristics; in the near field, the sphere 

is superior to the disk, whereas the opposite is generally noted in the far field. 

 

Figure 18. – Total acoustic field for oblate spheroids; f = 1000 Hz (ka = 92.4), M∞ = 0.0.

 

Figure 19. – Total acoustic field for oblate spheroids; f = 1623 Hz, (ka = 150), M∞ = 0.0.

(a) Observer ring (b) Observer line

(a) Observer ring (b) Observer line
23

 

Figure 20. – Total acoustic field for oblate spheroids; f = 2164 Hz (ka = 200), M∞ = 0.0.

Figure 21. – Total acoustic field for oblate spheroids; f = 3247 Hz (ka = 300), M∞ = 0.0.

(a) Observer ring (b) Observer line

(a) Observer ring (b) Observer line
24

 

Figure 22. – Total acoustic fields for oblate spheroids; f = 4329 Hz (ka = 400), M∞ = 0.0.

3.3 Flat Plate 

The acoustic source used during this exercise consisted of a stationary monopole of unit strength 

located 0.25 m from the center of the surface; excitation frequencies were 1.0 kHz, 2.0 kHz, 4.0 

kHz, and 8.0 kHz; background flow was quiescent. Grid sizes for the frequencies of interest and 

other metrics are provided in Table 5. Execution time statistics are given in Table 6. The total 

acoustic field was sampled at 1) a 2.0 m x 2.0 m plane bisecting the configuration along the plate 

width, and 2) a 20.0 m long line of observers located 5.0 m from the surface, opposite the sound 

source, traversing the width of the plate at its center. 

Table 5. – Acoustic source excitation frequencies and grid metrics for flat plates. 

Table 6. – Execution time statistics for flat plates. 

3.3.1 Flat Plate with Square Edges

The total acoustic field at a plane bisecting a plate with square edges is presented in Figures 23a 

– 23d for excitation frequencies of 1.0 kHz - 8.0 kHz. Note from the figures that 1) a wellFrequency,

Hz

Square Edges Rounded Edges Sharp Edges

Grid size, 

( Nc x Ns)

Nw

Grid size

( Nc x Ns) Nw

Grid size

( Nc x Ns) Nw

1000 632 x 210 10.00 634 x 210 10.00 550 x 200 10.00

2000 2140 x 702 10.00 2297 x 820 10.00 1935 x 720 10.00

4000 8412 x 2808 10.00 8810 x 2926 10.00 7574 x 2508 10.00

8000 32728 x 10780 10.00 35094 x 12472 10.00 29926 x 9848 10.00

Frequency,

Hz

Square Edges Rounded Edges Sharp Edges

Nproc Niter texec

, Nproc Niter

texec, Nproc Nitertexec,

1000 4 139 3.649 x 10-34 151 3.632 x 10-34 153 3.117 x 10-3

2000 4 204 3.811 x 10-24 216 4.908 x 10-24 290 4.722 x 10-2

4000 8 1000 1.518 16 1000 7.432 x 10-18 1000 1.126

8000 64 1600 3.987 64 1000 2.955 64 500 1.225

(a) Observer ring (b) Observer line
25

defined shadow zone is created by the plate, and 2) diffraction of sound by the edges increases 

with frequency, as manifested by “bending” of the lobes in the vicinity of the plate surface. 

Figure 23.- Total acoustic field, in dB, for a flat plate with square edges, M∞ = 0.0. 

The predicted total acoustic field at a line of observers crossing the shadow zone 5.0 m from the 

surface of the plate is compared to experimental data in Figure 24. The measurements were 

obtained during a series of tests described in reference 14. The data are not available in 

electronic form; thus, they were digitized from several figures found in the reference. As such, 

small inaccuracies in both amplitude and directivity are to be expected, especially at the higher 

frequencies. Note from Fig. 24 that the nature of the sound and its directivity are captured very 

well by the simulations – the peaks and valleys of the waves are generally coincident. However, 

the simulations tend to overpredict the benefits of shielding by 3 – 5 dB at the lower frequencies; 

at the highest frequency presented here (8.0 kHz), the overprediction is about 1 – 2 dB. The 

potential reasons for the discrepancies are numerous, notable among them: 

(a) f = 1000 Hz (b) f = 2000 Hz

(c) f = 4000 Hz (d) f = 8000 Hz
26

 Limitations within the FSC methodology that prevent an accurate prediction of edge 

diffraction effects. 

 Differences in plate thickness – The aluminum plate used in the experiment had a thickness 

of 0.0032 m, being almost 22 times thinner than that employed for the simulations. The close 

proximity of the sound source to the plate, and the low frequencies involved in the 

comparisons, precluded using a thin plate for the simulations. However, limited calculations 

using plates of different thickness (not shown) indicate that this factor may be secondary. 

 Experimental difficulties related to (see ref. 14 for a detailed explanation): 

- Simulation of a purely tonal omni-directional sound source, especially at the lower 

frequencies. 

- Reflections from the acoustic wedges lining the anechoic chamber walls. 

Despite these limitations, it is apparent that the FSC can be used to adequately predict the effects 

of noise shielding by a flat plate with square edges in close proximity to a point source. 

 

 

Figure 24.- Comparison between FSC predictions and experimental data (Figure 14 of ref. 14) 

for a flat plate with square edges, M∞ = 0.0; observer line located 5.0 m from plate surface. 

(a) f = 1000 Hz (b) f = 2000 Hz

(c) f = 4000 Hz (d) f = 8000 Hz
27

3.3.2 Flat Plate with Rounded Edges

The total acoustic field at a plane bisecting a plate with rounded edges is presented in Figure 25 

for the excitation frequencies of interest. Although SPL contours are not discernible below 60 

dB, edge diffraction effects appear to be stronger than those observed in Figs. 23a – 23d. This is 

to be expected, as the sound waves are better able to negotiate the gradual changes in surface 

slope associated with rounded edges. 

Figure 25.- Total (incident plus scattered) acoustic field, in dB, for a flat plate with rounded 

edges, M∞ = 0.0.

The simulated total acoustic field at a line of observers 5.0 m from the plate surface is given in 

Figure 26; experimental data, available for 4.0 kHz and 8.0 kHz only, are compared to the 

(a) f = 1000 Hz (b) f = 2000 Hz

(c) f = 4000 Hz (d) ff= = 2000 Hz 8000 Hz
28

simulations in Figs. 26c and 26d, respectively. Note from these figures that, as before, the 

directivity of the sound is well matched, but the magnitude of the waves in the shadow zone is 

under-predicted. At these frequencies, average differences are about 5 – 6 dB. Because the plate 

thicknesses are comparable (0.07 m for the simulations vs. 0.051 m for the measurements), it 

was anticipated that the comparisons would be more favorable. A possible explanation (in 

addition to those mentioned in the previous section) for a portion of the discrepancies is the fact 

that the plate used in the experiments was made of laminated wood, which has a much higher 

surface roughness than that of aluminum. As a result, the reflective/diffractive behavior of the 

waves may have been altered, compounding the effects of edge geometry. The apparent 

superposition of waves in the experimental data (see Figs. 26c and 26d) could be a manifestation 

of this added factor. 

 

 

Figure 26.- Comparison between FSC predictions and experimental data (Figure 23 of ref. 14) 

for a flat plate with rounded edges, M∞ = 0.0; observer line located 5.0 m from plate surface. 

(a) f = 1000 Hz (b) f = 2000 Hz

(c) f = 4000 Hz (d) f = 8000 Hz
29

3.3.3 Flat Plate with Sharp Edges

The total acoustic field at a plane bisecting a flat plate with sharp edges is presented, as it varies 

with frequency, in Figures 27a – 27d. The scattered noise patterns are similar to those observed 

for the previous configurations: a well defined shadow zone whose demarcation increases with 

frequency. 

Figure 27.- Total (incident plus scattered) acoustic field, in dB, for a flat plate with sharp edges, 

M∞ = 0.0. 

The simulated acoustic field at the observer line is given in Figure 28. As was the case with the 

plate with rounded edges, experimental data for the lower frequencies are not available. 

Comparisons between predicted and measured fields are presented in Figs. 28c and 28d for 4.0 

kHz and 8.0 kHz, respectively. Note from these figures that the magnitude of the waves in the 

(a) f = 1000 Hz (b) f = 2000 Hz

(c) f = 4000 Hz (d) f = 8000 Hz
30

shadow zone is reasonably well predicted, the larger differences being on the order of 2 – 3 dB. 

Observe also that the directivity of the simulated sound agrees with that of the measurements in 

the central region of the shadow zone, but has an apparent phase shift toward the edges. This 

may be caused by the differences in plate geometry. The plate used in the experiments had a total 

thickness of 0.0048 m, with 0.0004 m thick sharp edges; the edges had a 20o bevel. Because the 

plate employed in the simulations was substantially thicker, it was decided to maintain the 

sharpness of the edge and increase the bevel angle to about 30o to avoid having sides with 

extensive surface area. Nevertheless, the side surface area is large enough to interact with the 

waves being diffracted by the sharp edge and alter their characteristics. 

 

 

Figure 28.- Comparison between FSC predictions and experimental data (Figure 23 of ref. 14) 

for a flat plate with sharp edges, M∞ = 0.0; observer line located 5.0 m from plate surface. 

(a) f = 1000 Hz (b) f = 2000 Hz

(c) f = 4000 Hz (d) f = 8000 Hz
31

3.3.4 Effect of Edge Geometry on Simulated Noise Scattering

The shape of the edges may begin to affect the noise shielding characteristics of the plate when 

the dimensions of the edge are comparable to 1⁄4 of the wavelength14. For a 0.07 m plate, edge 

effects may be observable at all the frequencies being considered. Simulations for the three edge 

types of interest are presented in Figure 29. Observe that, at the lowest frequency (Fig. 29a), 

variations in the magnitude of the waves toward the center of the shadow zone are very small; 

these differences generally increase with frequency, with the plate with rounded edges providing 

the most attenuation and that with sharp edges the least. Alterations in the diffraction patterns of 

the waves are also observable, causing a phase shift toward the edges of the shadow zone. 

 

 

Figure 29.- Predicted effects of edge geometry on monopole noise scattering by a nearby flat 

plate, M∞ = 0.0; observer line located 5.0 m from plate surface. 

(a) f = 1000 Hz (b) f = 2000 Hz

(c) f = 4000 Hz (d) f = 8000 Hz
32

4. SUMMARY 

A detailed study on the characteristics of monopole noise scattering by several simple shapes 

(sphere, oblong spheroids of different thickness, and a flat disk) has been conducted. The sizes of 

the bodies and their location with respect to the acoustic source were chosen so as to project 

geometrically similar shadow zones. A range of source excitation frequencies 1 ≤ ka ≤ 500 was 

considered, and the total (incident plus scattered) acoustic field was sampled at various observer 

sets. The sphere simulations were compared to analytical results for validation purposes. A 

second study to assess the noise scattering characteristics of flat plates with different edge shapes 

was also performed; these simulations were compared to experimental data available in the 

literature. The following sections summarize the main findings of the investigation. 

4.1 Sphere 

 Advantage must be taken of all possible levels of symmetry within the configuration in order 

to obtain the correct solution and/or to improve computational efficiency. 

 Surface discretization using Nw ~ 5 was sufficient to obtain good solutions. 

 For all frequencies considered, excellent resolution of the portion of the field outside the 

shadow zone was obtained early in the solution process, generally within 100 – 200 

iterations. 

 The number of iterations required to achieve good resolution of the shadow zone increased 

with frequency, possibly as a result of enhanced matrix ill-conditioning due to larger 

numbers of equivalent sources. 

 Compared to the size of the problem (Nc x Ns), convergence performance of FSC v3.1/v3.2 is 

excellent. 

 Agreement between FSC and analytical results is remarkably good. 

 Instability of the analytical approach increased with frequency; analytical solutions for the 

highest frequencies could not be obtained. 

 Shielding effects improve with frequency – increased noise differential in the shadow zone. 

 Improvements in noise attenuation due to a fully lined sphere are substantial – decreased 

sound amplitude and increased shielding. 

4.2 Oblate Spheroids 

 The relative size of the source surface should decrease with scattering body thickness in 

order to obtain the correct solution. 

 A thin flat disk is computationally better than an oblate spheroid with similar (t/D)max

because of increased thickness near the edge. 

 A zone of relatively high amplitude was observed in the shadow zone opposite to the source. 

This behavior is intrinsic to the governing equations. 

 The best overall performer was OS1. 

 In the near field, the sphere was more effective than the disk; in the far field, the opposite 

was true. 

 In general, shielding effects increase with frequency for all configurations. 
33

4.3 Flat Plate 

Comparison with experimental data at several frequencies in the low to mid range indicates that 

the FSC is capable of predicting the correct characteristics and trends of the wave interaction 

patterns in the shadow zone, especially at higher frequencies. 

5. GUIDELINES FOR SUCCESSFUL APPLICATION OF THE FSC TO GENERAL 

SCATTERING PROBLEMS 

There is little guidance in the literature regarding the type and number of source points to be 

used for ESM simulations, their distribution on the source surface, and the location of such a 

surface relative to the scattering body. Extensive numerical experimentation has been performed 

to establish general rules of thumb for using the FSC to solve an arbitrary scattering problem. 

The following sections summarize the lessons learned by the authors during the performance of a 

wide range of scattering simulations, including the present investigation. Where appropriate, 

reference is made to specific portions of the User’s Manual (ref. 2) to supplement the 

information provided here. 

5.1 Solution Quality 

The quality of a solution is affected by several factors: scattering surface definition (shape and 

size of the body), scattering surface discretization (number and distribution of collocation 

points), source surface discretization (type, number, and distribution of equivalent sources),

placement of the source surface inside the scatterer, and configuration symmetry. 

5.1.1 Scattering Surface Definition

The user is responsible for providing a surface definition that is compatible with FSC 

requirements (see section 2.3.1.2 of ref. 2). The input surface grid should 1) be dense enough for 

an accurate definition of the surface of interest; and 2) have uniform spacing, as the surface 

discretization process follows grid topology. 

5.1.2 Scattering Surface Discretization

The geometry module of the FSC produces a uniform§ distribution of collocation points on the 

scattering surface. The number of collocation points (Nc) is proportional to Sk2, where S is 

surface area and k is wave number (see section 1.4.1 of ref. 2). The proportionality factor 

depends on a user selected number of points per wavelength (Nw) and on flow speed. The choice 

of Nw is frequency dependent; a suitable range is 5 < Nw < 12. For higher frequencies, use lower 

values (Nw ~ 5, or even lower, works well); for lower frequencies, user higher values (see section 

2.3.1.6 of ref. 2). For very low frequencies, when the wavelength is larger than the reference 

dimension of the surface, much larger values (Nw ~ 50) should be used. 

5.1.3 Source Surface Discretization

The geometry module generates a uniform distribution of equivalent sources on the source 

surface. The types of equivalent sources that can be used with FSC v2.0 are monopoles 

 

§

 It has been determined that a uniform distribution of collocation points and equivalent sources is necessary to 

minimize matrix ill-conditioning. 
34

(stationary or spinning) and dipoles. Only stationary monopoles can be used with FSC v3.1/3.2; 

however, they are very well suited for the vast majority of applications. For exterior BVPs, the 

number of equivalent sources (Ns) should be about 1/3Nc (see section 2.3.1.6 of ref. 2). For 

interior BVPs, the ratio can be slightly smaller: for room acoustics, Ns = 0.3Nc works well; for 

modal noise propagation within ducts, Ns ~ 0.15Nc is a good choice. 

5.1.4 Source Surface Placement

The FSC generates source surfaces by creating smaller versions of the scatterers based on a 

single user supplied scaling parameter (srcalph). To ensure that the source surface is interior to 

the scattering body (a requirement of the ESM for exterior BVPs), it is anchored at the leading 

and trailing edges for wing-like structures, and at the nose and tip for fuselage-like structures. 

The size and shape of the resulting source surface can substantially affect the quality of a 

solution. 

5.1.4.1 Source surface size

An adequate source surface size (relative to the scattering surface) depends on 1) the excitation 

frequency of the incident sound, 2) the shape/surface area of the scatterer (see section 2.3.1.5 of 

ref. 2), and 3) the distance between the scatterer and the acoustic source. In general, for moderate 

to high frequencies and relatively large, thick bodies, srcalph ~ 0.9 to 0.95 gives excellent 

results; the thinner the surface, the lower the frequency, and/or the closer the acoustic source is to 

the surface, the smaller srcalph should be. For example, srcalph ~ 0.75 to 0.8 works well for 

nacelles at most frequencies of interest; for very thin bodies (thickness ratio of about 0.03 to 

0.05), srcalph can be as low as 0.1. 

5.1.4.2 Source surface shape

A constant separation between source and scattering surfaces is desirable; this cannot be 

maintained in the regions where the source surface is anchored to the scattering body. Most 

geometries of interest are composed of wing-like structures and/or fuselages; for these, the 

effects of non-constant separation at the edges are small. For canonical shapes like spheres, 

cylinders, and cubes, the effect is more pronounced. In these cases, the geometry module can still 

be used advantageously by using separate, properly constructed scattering and source surfaces as 

inputs (see section 2.3.1.1 of this report). 

5.1.5 Use of Symmetry

As indicated in section 2.1.1 of the present report, the ESM is susceptible to matrix illconditioning for highly symmetric configurations. To avoid potential problems, advantage must 

be taken of all degrees of symmetry available in the configuration/system. An added benefit of 

doing so is a substantial reduction in matrix size, as only a portion of the scattering body of 

interest needs to be considered. Thus, for a given computer system, the savings in execution time 

permits consideration of higher excitation frequencies. 

5.2 Solution Accuracy/Convergence 

The accuracy of a solution obtained with the FSC can be assessed in a variety of ways. The 

obvious first choice is to compare FSC results to those obtained from analytical approximations; 
35

however, such expressions do not exist for realistic problems involving bodies of arbitrary shape. 

For these, the following three tests should be performed; if their outcome is satisfactory, the user 

can be confident that the solution is accurate. 

5.2.1 Sound through Solid Surfaces

A simple test to qualitatively determine the correctness of a solution is to sample the total 

acoustic field at a plane bisecting the configuration to determine if 1) sound appears to be going 

through the scattering surfaces, and 2) a well defined shadow zone is present. Consider, for 

example, the system composed of a thin oblate spheroid and a monopole source located 7.5 m 

from the surface presented in Figure 30. The source excitation frequency is 1000 Hz, no 

background flow. Fig. 30a corresponds to a solution obtained with a 90% equivalent source 

surface size. Note from the figure that sound seems to be penetrating the solid surfaces, and that 

a shadow zone – which should be clearly present at this frequency – is missing. This behavior is 

evidence that the relative distance between the scattering and source surfaces is too small. 

Reducing the scale of the source surface to 60% (Fig. 30b) and 20% (Fig. 30c) noticeably 

improved the shielding characteristics of the thin spheroid. Although not shown here, additional 

decrements in source surface size did not appreciably change the scattering patterns. Further 

improvements could only be obtained by increasing the thickness of the scattering body, as 

shown in Fig. 30d. 

The process described above was continued until an “optimum” combination of scattering 

surface thickness and source surface size was achieved, resulting in the observation that a thin 

flat disk with a rounded edge is numerically more efficient than an oblate spheroid with similar 

dimensions (see section 2.3.1.1). The total acoustic field for this thin flat disk at a plane bisecting 

the system is presented in Figure 31. Determination of a suitable source surface size using this 

approach can be tedious; however, the authors have observed that once found, the same source 

surface can be used for a wide range of acoustic source excitation frequencies. 

 

(a) b = 0.1 m, srcalpha = 0.9 (b) b = 0.1 m, srcalpha = 0.6
36

 

Figure 30. – Total acoustic field (in dB) on a plane bisecting a thin oblate spheroid (a = 4.33 m), 

f = 1000 Hz, M∞ = 0.0. 

Figure 31. – Total acoustic field (in dB) on a plane bisecting a thin flat disk with rounded edges 

(r = 4.33 m, t/D = 0.035), f = 1000 Hz, M∞ = 0.0. 

5.2.2 Convergence Metrics

The FSC outputs iteration convergence histories for three metrics: the normalized L2 norms of 

the solution (source strength) vector residual, the square root of the solution vector, and the 

residual vector (see section 1.2.1.3 of the FSC v3.1/3.2 user’s manual, ref. 15). The first metric 

gives an indication of solution convergence; the last follows the magnitude of the boundary 

condition residual at the scattering surfaces. Figures 4b - 11b depict the convergence history for 

the first and third metrics, as obtained during the sphere scattering exercise. Final values for 

(c) b = 0.1 m, srcalpha = 0.2 (d) b = 0.125 m, srcalpha = 0.2
37

these two parameters are given in Table 6 for the oblate spheroids. Similar information is 

provided in Table 7 for the flat plate with square edges. 

Table 7. – Final values of normalized L2 norms for oblate spheroids. 

Table 8. – Final values of normalized L2 norms for flat plate with square edges. 

5.2.3 Acoustic Power through a Sphere

The average rate of acoustic energy radiation from a source is known as sound power. Because it 

is spatially and temporally independent, acoustic power can be used to measure the losses in a 

system. For sound scattering simulations using the ESM, the changes in radiated energy may 

result from physical reasons such as the inclusion of sound absorbing material on the surfaces3, 

or from numerical inaccuracies caused by an improper definition/placement of the equivalent 

sources. 

A convenient way to test the correctness of the ESM setup for acoustically hard surfaces is to 

calculate the sound power through a sphere of arbitrary radius surrounding the configuration16. 

The scattering of sound by untreated surfaces is a non-dissipative process; thus, the incident 

acoustic power on the sphere surface should be the same as the total (incident plus scattered) 

power. All versions of the FSC include intensity/power calculations for the sphere observer field 

(iobs = 2). Consider, for example, the monopole/thin flat disk system introduced in section 2.3.1, 

and depicted in Figure 32 as surrounded by a closed sphere of radius 10 m centered at the source. 

The incident power flowing through the sphere is 3.837 x 10-4 W (PWL = 85.84 dB), the total 

power is 3.863 x 10-4 W (PWL = 85.87 dB). The error in PWL is thus 0.035%. 

The outcome from the present test, and from those described in sections 5.2.1 and 5.2.2, indicate 

that the solution obtained with the FSC for the scattering of monopole sound (f = 100 Hz, M∞ = 

0.0) by a thin flat disk is highly accurate. 

f,

Hz ka

Sphere OS1 OS2 Flat Disk

Solution 

vector BC error

Solution 

vector BC error

Solution 

vector BC error

Solution 

vector BC error

11 1.0 7.65 x 10-84.99 x 10-35.07 x 10-61.00 x 10-25.13 x 10-61.61 x 10-31.15 x 10-71.49 x 10-4

541 50.0 9.91 x 10-74.62 x 10-56.59 x 10-65.65 x 10-45.08 x 10-65.63 x 10-44.71 x 10-62.61 x 10-4

1000 92.4 9.55 x 10-64.38 x 10-63.19 x 10-73.55 x 10-47.99 x 10-65.08 x 10-45.69 x 10-61.85 x 10-4

1623 150.0 7.36 x 10-63.77 x 10-69.64 x 10-64.15 x 10-59.81 x 10-62.01 x 10-41.28 x 10-43.10 x 10-4

2164 200.0 4.17 x 10-52.79 x 10-69.80 x 10-62.96 x 10-51.79 x 10-55.35 x 10-36.37 x 10-42.79 x 10-4

3247 300.0 3.52 x 10-58.16 x 10-63.86 x 10-41.77 x 10-45.78 x 10-46.40 x 10-45.52 x 10-43.52 x 10-4

4329 400.0 2.74 x 10-49.59 x 10-56.86 x 10-45.89 x 10-31.87 x 10-32.09 x 10-21.10 x 10-31.90 x 10-2

5411 500.0 9.56 x 10-42.43 x 10-3

Frequency, 

Hz

Solution 

vector BC error

1000 8.24 x 10-61.88 x 10-3

2000 7.75 x 10-65.22 x 10-3

4000 7.03 x 10-43.11 x 10-2

8000 3.68 x 10-42.28 x 10-3
38

Figure 32. – Total acoustic field, in dB, on a sphere (r = 10 m) surrounding the thin flat disk 

/monopole system, f = 1000 Hz, M∞ = 0.0. The sphere is centered at the source. 

5.3 Grid Density Effects on Solution Accuracy 

As can be observed from Tables 2 and 4, the computer resources necessary to achieve 

convergence can be prohibitive for high frequency/large scale scattering simulations. Previous 

investigations by the authors6 suggest that the nominal density (size) of an FSC matrix can be 

decreased substantially for higher excitation frequencies without compromising the accuracy of a 

solution; the reduction factor appeared to increase with frequency. For a given excitation 

frequency, the geometry module can be used to generate FSC grids of any size by adjusting Nw

(see section 5.1.2). 

5.4 Application to Partial Configurations 

For a given configuration, the computational time required to solve scattering problems increases 

substantially with frequency. However, in certain instances, components of interest can be 

isolated without significant loss of accuracy. This can be very useful when performing 

parametric analyses, since execution times are substantially reduced. Results from an exercise 

conducted for a full scale commercial transport (wings, fuselage and sideboard nacelle) similar in 

size and shape to the Boeing 777 are presented in Figure 33. The incident sound was generated 

by the lowest cut-on circumferential mode of rotor-stator interaction tones (m = 10) for an 

engine/nacelle configuration similar to the GE-90; the source excitation frequency was f = 2 x 

BPF (fan), M∞ = 0.2 (uniform). Surface insertion loss** contours calculated using the entire 

configuration are depicted in Fig. 33a in the vicinity of the wing/nacelle area. The fuselage was 

then removed from the configuration and a new acoustic field was obtained; this field is 

presented in Fig. 33b, for the same wing/nacelle region as before. Note that the differences 

between the two approaches are small, and confined mainly to the wing/fuselage junction, as 

expected. 

 

** Insertion loss was calculated by subtracting the nacelle alone contributions from the fuselage/wing/nacelle data. 

Negative values mean a reduction in noise; positive values indicate a gain. 

Incident power = 85.84 dB

Total power = 85.87 dB
39

 

 

Figure 33. Insertion loss for full scale transport component isolation. Incident sound is lowest 

cut-on circumferential mode of rotor/stator interaction; f = 2 x BPF (fan), M∞ = 0.2. 

5.5 Problem Setup – A Roadmap 

Successful application of the FSC to arbitrary scattering problems requires that the user be 

familiar with the code, its workings, and the methodology behind it. The user’s manuals2,15 have 

been written for this purpose. They contain detailed explanations of the various input/output 

parameters and options, as well as several sample cases to guide the learning process. 

The procedure for obtaining an adequate solution with the FSC is the same for all configurations. 

Assuming that the input surfaces are properly defined (see section 5.1.1 of this report), the 

following checks should be performed using the visualization files produced by the geometry 

module (section 2.3.1.4, ref. 2):

a) Spline-fitted shapes for the scattering and source surfaces are as expected (section 2.3.1.7 of 

ref. 2).

b) Collocation points and equivalent sources are uniformly distributed. 

c) Surface normal vectors at the collocation points and equivalent sources are pointing in the 

correct direction (outward for exterior BVPs, inward for interior BVPs. The geometry 

module only supports the former). 

If everything checks, the user can be confident that the collocation point and equivalent source 

files contain the correct information. Input these files into the ESM module: 

Top surfaces Bottom surfaces

(a) Full configuration

(b) Wing plus nacelle only

SPL, dB
40

a) Run the code for a few hundred iterations (more for low frequencies, less for high 

frequencies). 

b) Check the residual history file to determine if the solution is converging. For wellconditioned problems, convergence is fast; for ill-conditioned problems, convergence may be 

slow. 

c) Calculate the acoustic field at a plane bisecting the configuration, including the incident 

sound source: 

 Make sure the incident field is as expected. 

 Examine the total acoustic field. If sound appears to be going through the scattering 

surfaces, and/or an expected shadow zone is not discernible, then 

- The size of the source surface may need to be adjusted (most likely). 

- More iterations are required. 

d) Go back to (a) and repeat the process until the desired convergence/behavior is achieved. 

e) Calculate the acoustic field at other desired observer locations. 

6. REFERENCES 

1. Dunn, M. H., and Tinetti, A. F., “Aeroacoustic Scattering Via the Equivalent Source 

Method,” AIAA 2004-2937, May 2004. 

2. Tinetti, A, F., Dunn, M. H., and Pope, S. D., “Fast Scattering Code (FSC) User’s Manual, 

Version 2.0,” NASA CR 2006-214510, October 2006.

3. Reimann, C. R., Tinetti, A. F., and Dunn, M. H., “Engine Liner Optimization Using the Fast 

Scattering Code,” AIAA 2007-3494, May 2007. 

4. Lee, S., Erwin, J., and Brentner, K., “Acoustic Scattering of Rotorcraft Noise,” AHS 64th

annual forum, April 2008. 

5. Gerhold, C. H., Clark, L. R., Dunn, M. H., and Tweed, J., “Investigation of acoustical 

shielding by a wedge-shaped airframe,” Journal of Sound and Vibration, Vol. 294, pp. 49-63, 

2006. 

6. Tinetti, A. F., and Dunn, M. H, “Scattering of High Frequency Duct Noise by Full Scale 

Hybrid Wing Body Configurations,” AIAA 2009-3400, May 2009. 

7. Posey, J. W., Tinetti, A. F., and Dunn, M. H., “The Low-Noise Potential of Distributed 

Propulsion on a Catamaran Aircraft,” AIAA 2006-2622, May 2006. 

8. Farassat, F., Casper, J. H., Tinetti, A. F., and Dunn, M. H., “Airframe Noise Prediction by 

Acoustic Analogy: Revisited,” AIAA 2006-2564, May 2006. 

9. Tinetti, A. F., and Dunn, M. H., “Acoustic Simulations of an Installed Tandem Cylinder 

Configuration,” AIAA 2009-3158, May 2009. 

10. Dunn, M. H., and Tinetti, A. F., “Application of Fast Multipole Methods to the NASA Fast 

Scattering Code,” AIAA 2008-2875, May 2008. 

11. Ochmann, M., “The Source Simulation Technique for Acoustic Radiation Problems”, 

Acustica, Vol. 81, pp. 512 – 627, 1995.
41

12. Gumerov, N. A., and Duraiswami, R., Fast Multipole Methods for the Helmholtz Equation in 

Three Dimensions, Elsevier Series in Electromagnetism, 2004. 

13. Zhang, S., and Jin, J., Computation of Special Functions, John Wiley & Sons, Inc., 1996. 

14. Ahtye, W. F., and McCulley, G., “Evaluation of Approximate Methods for the Prediction of 

Noise Shielding by Airframe Components,” NASA TP 1004, January 1980. 

15. Tinetti, A. F., and Dunn, M. H., “FSC versions 3.1 and 3.2,” addendum to the FSC v2.0 

manual. 

16. Lee, S., Brentner, K., and Morris, P., “Assessment of Time-Domain Equivalent Source 

Method for Acoustic Scattering,” AIAA 2010-3821, May 2010. 
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The Fast Scattering Code (FSC): Validation Studies and Program 

Guidelines 

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NNL09AA17C

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Tinetti, Ana F.; Dunn, Mark H.

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NASA Langley Research Center 

Hampton, VA 23681-2199 

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14. ABSTRACT

The Fast Scattering Code (FSC) is a frequency domain noise prediction program developed at the NASA Langley Research 

Center (LaRC) to simulate the acoustic field produced by the interaction of known, time harmonic incident sound with bodies 

of arbitrary shape and surface impedance immersed in a potential flow. The code uses the equivalent source method (ESM) to 

solve an exterior 3-D Helmholtz boundary value problem (BVP) by expanding the scattered acoustic pressure field into a 

series of point sources distributed on a fictitious surface placed inside the actual scatterer. This work provides additional code 

validation studies and illustrates the range of code parameters that produce accurate results with minimal computational costs.

Systematic noise prediction studies are presented in which monopole generated incident sound is scattered by simple 

geometric shapes – spheres (acoustically hard and soft surfaces), oblate spheroids, flat disk, and flat plates with various edge 

topologies. Comparisons between FSC simulations and analytical results and experimental data are presented. 

15. SUBJECT TERMS

acoustic scattering, benchmark scattering problems, fast scattering code

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01- 06 - 2011