1

Nonlinear and dissipative Nearfield Acoustical Holography

algorithms based on Westervelt Wave Equation

Yaying Niua)

Yong-Joe Kimb)

Acoustics and Signal Processing Laboratory, Department of Mechanical Engineering, Texas A&M

University, 3123 TAMU, College Station, TX 77843-3123, USA

When a conventional, linear, lossless Nearfield Acoustical Holography (NAH) procedure is

applied to reconstruct the three-dimensional (3-D) sound fields that are radiated from a

high-level noise source and include significant nonlinear components, it can result in

significant reconstruction errors. Here, a nonlinear, dissipative, planar NAH procedure is

introduced that can be used to identify nonlinear noise characteristics in the 3-D nearfield

of the high-level noise source from two-dimensional sound pressure data measured on a

hologram surface. The proposed NAH procedure is derived by applying perturbation and

renormalization methods to the nonlinear, dissipative Westervelt Wave Equation. In order

to validate the proposed procedure, the nonlinear and dissipative sound pressure field

radiated from a high-level pulsating sphere at a single frequency is calculated from the

spherical Burgers Equation. An improved SONAH procedure is applied to reconstruct the

source sound pressure field that is input to the proposed nonlinear projection

procedure. Within 2.5 m nearfield reconstruction distance from the pulsating sphere, the

nonlinear sound pressure field reconstructed by applying the proposed NAH procedure

matches well with the directly-calculated field at the maximum reconstruction error of 0.5

dB.

1 INTRODUCTION

The sound fields radiated from high-level noise sources include significant nonlinear

components that can lead to shock wave propagations. Thus, it is critical to identify noise

propagation characteristics in the nearfield of the sources in order to design optimal noise control

schemes to suppress the nonlinear noise components before they turn into shock waves.

However, the conventional, linear, lossless NAH procedures1-9

cannot be used to visualize the

highly nonlinear and dissipative sound fields correctly. Therefore, it is here proposed to develop

a novel, nonlinear, dissipative, NAH procedure in an open 3-D space to consider the nonlinear

and thermoviscous effects. Combined with a linear NAH procedure, the proposed NAH

procedure can be used to nonlinearly project the sound pressure data measured on a 2-D planar

a)

y-niu@tamu.edu b)

joekim@tamu.edu

2

measurement surface close to the noise sources into a 3-D space to visualize both linear and

nonlinear acoustic fields.

Hamilton and Blackstock10

summarized nonlinear acoustics research work done by many

researchers11-17

. In particular, perturbation and renormalization methods are used by Ginsberg13-

16 and Nayfeh

17 to investigate nonlinear acoustic wave propagations in planar, cylindrical, and

spherical cases. According to Ginsberg’s investigation on 2-D planar and 3-D cylindrical

nonlinear wave propagations based on renormalization procedures13-16

, the particle velocity

along a wave propagating direction can advance or retard wavefronts, while the transverse

particle velocity can bend the rays of the nonlinear waves. For a spherical wave, however, the

transverse particle velocity decays in the factor of 1/r compared to the radial velocity17

. Thus,

the bending of the rays is negligible and the rays remain straight in this case.

Among the nonlinear and thermoviscous acoustic equations for perfect gases10

, the

Westervelt Wave Equation10,11

(WWE) is selected to derive the proposed nonlinear and

dissipative NAH procedure. The WWE is solved by using perturbation and modified

renormalization methods to give linear and nonlinear NAH reconstructed sound pressure fields,

which include the nonlinearity-induced steepening effects of wavefronts but do not include the

transverse particle velocity-induced bending of rays. Thus, the proposed nonlinear NAH

procedure can be applied to nonlinear wave propagation problems only with straight rays: e.g.,

spherical and 1-D planar wave cases. In addition, local nonlinearity caused by source surface

displacements is not included in the WWE. That is, the WWE is appropriate for sound waves

only with dominant cumulative nonlinearity away from a noise source surface.

The proposed nonlinear and dissipative NAH procedure is valid for monofrequency source

cases. Thus, the first-order, linear sound pressure at a fundamental frequency is used to calculate

the second-order, nonlinear sound pressure at twice of the fundamental frequency. In multi-

frequency source cases, however, the Fenlon’s solution10,12

indicates that each frequency

component consists of the harmonics, summations, or differences of other frequency components.

Thus, the fundamental frequency component can include nonlinear components and cannot be

directly applied for the calculation of its second-order nonlinear component.

A nonlinear and dissipative pulsating sphere simulation is performed to validate the

proposed NAH procedure applicable to spherical sound fields. The generalized Burgers

Equation in the spherical coordinates10

is solved by using the aforementioned perturbation and

renormalization procedures to obtain the nonlinear and dissipative sound pressure field radiated

from the high-level pulsating sphere. The calculated hologram data is backward projected to the

source surface, and then the reconstructed source data is input to the proposed, nonlinear,

forward, NAH projection procedure. The sound pressure field reconstructed by using the

proposed NAH procedure is compared with the directly-calculated sound pressure field.

2 THEORY

2.1 Perturbation Procedure for Westervelt Wave Equation

The nonlinear and dissipative Westervelt Wave Equation10,12

(WWE) is represented as

2 3 2 22

2 2 4 3 4 2

0 0 0 0

1 p b p pp

c t c t c t

, (1)

where p is the acoustic pressure, 0 is the ambient fluid medium density, c0 is the speed of sound,

β = (γ + 1)/2 is the nonlinearity coefficient for an ideal gas, and b is the sound diffusivity that

3

includes viscous and thermal conduction effects10

. The two terms on the right-hand-side (RHS)

of Eq. (1) thus represent thermoviscous and cumulative nonlinear effects, respectively.

By using the perturbation method13-17

, the acoustic pressure, p in the Cartesian coordinates,

(x, y, z) can be expanded as

2

1 2, , , , , , , , ,p x y z t O O p x y z t p x y z t , (2)

where ε is the small perturbation parameter (e.g., acoustic Mach number, ε = u0/c0 where u0 is the

particle velocity magnitude) and pn is the n-th order acoustic pressure (n = 1,2,3,). The sound

diffusivity, b is a small number in the order of ε in air10

. By substituting p in Eq. (2) into Eq. (1)

and neglecting the O(ε3) and higher-order small terms, the first- and second-order equations can

be associated with the O(ε) and O(ε2), respectively: i.e.,

22 1

1 2 2

0

10

pp

c t

, (3a)

2 3 2 22 2 1 1

2 2 2 4 3 4 2

0 0 0 0

1 p p pbp

c t c t c t

. (3b)

Eq. (3a) is a linear, lossless, homogeneous wave equation and Eq. (3b) is a nonlinear, dissipative,

inhomogeneous wave equation. The left-hand-side (LHS) of Eq. (3b) is a linear wave equation

with the unknown of the second-order pressure, p2, while the RHS consists of the dissipative and

nonlinear source terms that are composed of the first-order acoustic pressure, p1. Thus, once the

first-order sound pressure solution is obtained from Eq. (3a), the second-order equation becomes

a linear, inhomogeneous partial differential equation of which solution can be obtained

analytically for certain cases.

2.2 Nonlinear, Dissipative, Planar NAH Projection

In practical measurements of the sound pressure field generated by a monofrequency sound

source, the measured sound pressure data at the fundamental frequency contains the only first-

order, linear sound pressure components. The linear NAH algorithms1-9

can be thus applied to

the backward projection from the hologram surface to the sound source surface at the

fundamental frequency. Here, an improved Statistically Optimal Nearfield Acoustical

Holography (SONAH) algorithm9 is applied to backward project the hologram sound pressure

field at z = zh to the source surface at z = z0. As described in Ref. 9, this improved SONAH

procedure allows accurately reconstructing a source sound pressure field even with the hologram

data measured by using a microphone array with a small measurement aperture. The linearly

reconstructed sound pressure field on the source plane is then input to the nonlinear and

dissipative NAH forward projection procedure.

In the Cartesian coordinates, (x,y,z), the acoustic pressure solution of Eq. (3a) at the

fundamental frequency of ω can be obtained from the conventional, linear, planar NAH

projection1-3

in a discretized form as

0

11

1 1 0

0 0

1, , , Re , , ,

yx

y zmnx

NNin k y ik z zim k x i t

xm yn

m nx y

p x y z t P k k z e e e eN N

, (4)

where z0 denotes the sound source surface location, z is a reconstruction surface location (z ≥ z0),

Nx and Ny are the number of measurement points along the x- and y-directions, respectively, and

4

(kxm,kyn) = (mkx,nky) are the discrete wave numbers. In Eq. (4), the wave number spectrum,

P1(kxm,kyn,z0,ω) = P1mn is obtained by applying the spatial Fast Fourier Transform (FFT) to the

source sound pressure data, p1(x,y,z0,ω) that is obtained by applying the linear, backward

SONAH procedure to the measured sound pressure data on the hologram plane at z = zh. The

projection relation between the reconstruction and source surfaces is

0 0

1 1 0 1, , , , , , zmn zmnik z z ik z z

mnP m n z P m n z e P e

. (5)

In Eq. (5), the z-direction wave number, kzmn is the function of ω, kxm, and kyn: i.e.,

2 2 2 2( ) ( )zmn x yk k m k n k , (6)

where k = ω/c0 is the acoustic wave number. By substituting Eq. (4) into Eq. (3b) and writing

the second-order acoustic pressure solution as p2 = Re(p21e-it

+ p22e-i2t

), Eq. (3b) can be

decomposed into two inhomogeneous Helmholtz equations: i.e.,

0

1132 2

21 140 00

, ,yx

y zmnx

NNin k y ik z zim k x

mn

m nx y

ibk p x y z e e P e

N N c

, (7)

( )( ) 0

2 2

22

1 11 12

1 1( )( )2 2 40 0 0 00 0

4 , ,

2

y yx x

zmn z l m q nyx

N NN Ni k k z ziq k yil k x

mn l m q n

l q m nx y

k p x y z

e e P P eN N c

. (8)

In Eqs. (7) and (8), p21 is associated with the “dissipative” sound pressure component at the

frequency of ω, while p22 corresponds to the cumulative “nonlinear” sound pressure component

at the frequency of 2ω. In Eq. (8), the high wave number components (i.e., l ≥ Nx and q ≥ Ny) are

set to zero since they are the subsonic components decaying out exponentially during the forward

projections along the z-direction. Similarly, the P1 and kz terms with the subscripts of (l-m)(q-n)

have only non-zero values when the subscripts, (l-m) and (q-n) are within [0, Nx-1] and [0, Ny-1],

respectively.

The solution of p21(x, y, z) in Eq. (7) can be written as

11

21 21

0 0

, ,yx

yx

NNin k yim k x

mn

m n

p x y z P z e e

. (9a)

By substituting Eq. (9a) into Eq. (7), the wave number spectrum, P21mn can be represented as

0

3( )0 1

21 4

0

( )

2zmnik z zmn

mn

x y zmn

ib z z PP z e

iN N c k

. (9b)

In Eq. (9), homogeneous solutions are neglected to consider the only particular solution since the

transient responses associated with the homogeneous solutions decay out quickly. Reflective

waves propagating in the negative z-direction are also neglected in Eq. (9) since no reflection

occurs at the infinite boundary of the open 3-D space. Eq. (9) also satisfies the assumption that

there is negligible “dissipation” at the source surface: i.e., P21mn(z0) = 0. The dissipative sound

pressure spectrum, P21mn(z) increases as z increases in the supersonic region, i.e., (kxm)2 + (kyn)

2 <

k2. Therefore, a renormalization procedure is required to remove this secular term to obtain a

uniformly valid solution.

The solution of p22(x, y, z) in Eq. (8) can be written as

5

11

22 22

0 0

, ,yx

yx

NNiq k yil k x

lq

l q

p x y z P z e e

, (10a)

where the nonlinear wave number spectrum, P22lq is represented as

2 0

( )( ) 0 2 0

( )20 2

1( /2)( /2)2 2 4

0 0 2

22 ( )1 ( )12

1 1( )( ) 2 22 2 40 00 0 2 ( )( )

2,when 2 and 2

2

2,

zlq

zmn z l m q ny zlqx

ik z z

l q

x y zlq

lq i k k z zN ik z zN

mn l m q n

m nx y zlq zmn z l m q n

z z eP l m q n

N N c ik

P ze e

P PN N c k k k

otherwise

, (10b)

where

222 2

2 4zlq x yk k l k q k . (10c)

Similar to Eq. (9), homogeneous solutions are neglected in Eq. (10). Eq. (10) also satisfies the

assumption that there is neither a reflective wave generated at the infinite boundary of the open

3-D space nor a nonlinear component at the source surface, i.e., P22lq(z0) = 0. As in the

dissipative sound pressure spectrum shown in Eq. (9b), the nonlinear sound spectrum, P22lq(z)

includes the secular term that increases as z increases in the supersonic region.

2.3 Renormalization

In order to obtain the uniformly valid solution of Eq. (3) at a large z value, the secular terms

with the growing factor of (z – z0) in Eqs. (9b) and (10b) are eliminated by using a

renormalization procedure. From Eqs. (4), (9), and (10), the total sound pressure is written in the

complex form as

2

1 21 22, , , , , , , , ,i t i t i tp x y z t p x y z e p x y z e p x y z e . (11)

In order to eliminate the secular terms with (z – z0) in Eqs. (9b) and (10b), a strained coordinate,

ζ is introduced as

1 , , , ,z z x y z t , (12)

where ε is the small perturbation parameter (i.e., acoustic Mach number) and the function, z1 is

determined to eliminate all of the second-order small secular terms. When compared to the three

strained coordinates defined in the three directions of a 3-D space in Refs. 15 and 16, the only

one strained coordinate is used in Eq. (12) to renormalize the secular terms in the full, 3-D space.

By substituting Eq. (11) into Eqs. (4), (9), and (10) and retaining up to the second-order terms, a

condition for removing the secular terms is obtained: i.e., the strain coordinate, ζ can be found by

setting the absolute value of the following renormalization function, f to zero for a given set of

(x,y,z,t),

6

0

0

2 0

11

1

0 0

31

04

0

22( )1( /2)( /2) 2

04

0 0 2

1, , , ,

2

2 , when an

2

yx

y zlqx

zlq

zlq

NNiq k y ik zil k x i t

lq zlq

l qx y

ik zlq i t

zlq

ik zl q i t

x y zlq

f x y z t e e z P ik e eN N

Pibz z e e

c ik

Pz z e e l

N N c ik

d are evenq

. (13)

The uniformly valid total sound pressure is then represented as

0

( )( ) 0 2 0

11

1

0 0

( )1 ( )122

1 1( )( ) 2 240 00 0 2 ( )( )

1, , ,

2

yx

y zlqx

zmn z l m q ny zlqx

NNiq k y ik zil k x i t

lq

l qx y

i k k zN ik zNi t

mn l m q n

m nx y zlq zmn z l m q n

p x y z t e e P e eN N

e eP P e

N N c k k k

, (14)

where the subscripts, l and q satisfy l ≠ 2m or q ≠ 2n. The nonlinear and dissipative sound

pressure components are calculated by subtracting the linear NAH forward projected sound

pressure from the total sound pressure in Eq. (14).

2.4 Nonlinear and Dissipative Pulsating Sphere Simulation

In this section, a pulsating sphere, with the radius of rs, that radiates a nonlinear and

dissipative sound pressure field is considered to validate the proposed nonlinear, dissipative,

planar NAH procedure. Note that an acoustic monopole cannot be used to simulate the nonlinear

and dissipative sound field since its infinitely small size compared to the smallest wave length

makes it impossible to effectively radiate nonlinear sound fields10

. Therefore, the radius, rs is

here set to be in the same order of magnitude with the wave length, λ.

The nonlinear and dissipative sound pressure field generated from the pulsating sphere is

thus obtained from the generalized Burgers Equation10

in the spherical coordinates, (r, θ, φ) with

the assumption of the θ- and φ-direction symmetry conditions: i.e.,

2

3 2 3

0 0 02

p p b p p p

r r c c

, (15)

where τ = t-(r-rs)/c0 is the retarded time and r is the radial location (r ≥ rs) of a receiver. Similar

to the WWE, the spherical Burgers Equation also describes cumulative nonlinear effects.

According to the aforementioned perturbation and renormalization procedures, the sound

pressure, p in Eq. (15) has the uniformly valid solution: i.e.,

1,i

s

ep r Pr

, (16)

where α is the strained coordinate determined by the following two equations, i.e.,

1 , ,r r r , (17)

7

1 1 22, , ( ) ln

ii

s s

s

e rg r Pr r C r r C e

r

. (18)

In the renormalization function Eq. (18), P1 is the acoustic pressure amplitude on the sphere

surface at r = rs and C1 and C2 are the constants defined, respectively, as

2

1 13

02s

bC Pr

c , (19a)

2 2

2 13

0 02sC i P r

c

. (19b)

The linear sound pressure data on the hologram surface (z = zh) at the fundamental frequency, ω

can be obtained from Eq. (16) by replacing α with r = rhn where rhn represents the radial location

of the n-th receiver on the hologram surface. This hologram data is linearly backward projected

to the source surface at z = z0 and the reconstructed source sound pressure data is then used for

the nonlinear, forward NAH projections in the 3-D space. The linear and nonlinear sound

pressure fields directly-calculated from Eq. (16) in the 3-D reconstruction space are compared

with the reconstructed 3-D sound pressure fields to validate the projection performance of the

proposed NAH procedure.

3 PULSATING SPHERE SIMULATION RESULTS

3.1 Simulation Setup

Figure 1 shows the pulsating sphere simulation setup in a free field. The center of the

pulsating sphere with the radius of rs = 0.25 m is placed at (x,y,z) = (0.75,0.75,-0.25) m. The

sound pressure amplitude of the pulsating sphere on its undisturbed surface at r = 0.25 m is 6 kPa

and the excitation frequency is 1 kHz. A 31×31 acoustic pressure transducer array with the

sampling intervals of ∆x = ∆y = 0.05 m is placed at z = 0.05 m to obtain hologram sound

pressure data. Thus, the hologram height is 0.05 m when the plane on the pulsating sphere

surface at z = 0 m is defined as the source surface.

3.2 Directly-Calculated Sound Pressure Fields

The total (i.e., linear, nonlinear, and dissipative) sound pressure field is directly calculated

from Eq. (16), after α is determined by numerically searching for the roots of |g(r,α,τ)| = 0 in Eq.

(18). The linear, lossless component that is obtained by replacing α with r in Eq. (16) is shown

in Fig. 2(a) at the fundamental frequency of 1 kHz. The linear sound pressure amplitude in Fig.

2(a) decreases proportionally to 1/r. The cumulative nonlinear sound pressure field is obtained

by subtracting the linear and dissipative sound pressure from the total sound pressure (see Fig.

2(b)). The nonlinear sound pressure field in Fig. 2(b) has the minimum sound pressure level

(SPL) at (x,y,z) = (0.75,0.75,0) m (i.e., on the sphere surface at r = 0.25 m) where the nonlinear

SPL is approximately 115 dB. The nonlinear SPL is approximately 10 dB less than the linear

SPL at z = 3 m. Although not shown here, the dissipative sound pressure field that can be

obtained by considering only the dissipation source term in Eq. (15) and following the identical

perturbation and renormalization procedures in Sections 2.1 – 2.3 is negligible with low SPLs

when compared to the linear and nonlinear sound pressure fields.

8

3.3 Linearly Backward Projected Sound Pressure Fields

The sound pressure data calculated on the hologram surface at the fundamental frequency of

1 kHz is linearly backward projected onto the source surface by using the improved SONAH

procedure9. The linearly backward projected sound pressure field by using the improved

SONAH procedure is well matched with the directly-calculated sound pressure field as shown in

Figs. 3(a) – 3(c). Although the reconstruction error is relatively large at approximately 0.6 dB

along the measurement aperture edges, a spatial window can be used to suppress this

reconstruction error in the nonlinear forward NAH projections. The reconstructed sound

pressure data on the source surface at z = 0 m is input to the nonlinear NAH projection procedure.

3.4 Linearly and Nonlinearly Forward Projected Sound Pressure Fields

The sound pressures along the four reconstruction lines in Figs. 2(a) and 2(b) are

reconstructed by using the proposed nonlinear NAH procedure. The reconstruction lines are

selected in the “nearfield” of the pulsating sphere since the spatial-FFT-based, linear, forward

NAH procedure is limited to “nearfield” reconstructions due to the farfield reconstruction error

caused by spatial-FFT-periodicity-induced ghost images1-3

.

Figures 4(a) – 4(d) show the directly-calculated and NAH-projected SPLs along

reconstruction lines 1 – 4. Here, the NAH-projected “linear” SPLs are obtained by applying the

conventional, linear NAH procedure to the source data as in Eq. (4) and the NAH-projected,

“nonlinear” SPLs are calculated by subtracting the NAH-projected “linear” sound pressure

components from the “total” components obtained from the renormalization procedure as in Eq.

(14). The NAH-projected, linear SPLs match well with the directly-calculated, linear SPLs

along the all four reconstruction lines. The NAH-projected, nonlinear SPLs also match well with

the directly-calculated, nonlinear SPLs along the reconstruction lines 3 and 4. Along off-

centered reconstruction lines 1 and 2, the discrepancies between the directly-calculated and

NAH-projected nonlinear components in the region close to the source surface at z = 0 m are

caused by the assumption of no nonlinear component at the planar source surface, although the

pulsating sphere simulation results in nonlinear components at the source surface (see Fig. 2(b)).

Figures 5(a) – 5(d) show the reconstruction errors that are defined as the SPL differences

between the directly-calculated and NAH-projected SPLs in Fig. 4. Along off-centered

reconstruction lines 1 and 2, Figs. 5(a) and 5(b) show that the nonlinear NAH reconstruction

errors decrease quickly around the source surface as z increases. Along reconstruction line 3 at

the center of the measurement aperture, the nonlinear reconstruction error is within 0.2 dB, while

the linear reconstruction error increases oscillatory as z increases. The latter indicates that the

spatial-FFT-periodicity-induced ghost image effects and the measurement aperture truncation

errors become more significant as z increases further from the pulsating sphere1-3

. Along

reconstruction line 4, Fig. 5(d) shows that both the linear and nonlinear NAH reconstruction

errors are approximately 0.1 – 0.3 dB around the center and become larger as the reconstruction

location is further away from the center to the measurement aperture edges where the aperture

edge truncation errors and the ghost image effects are more significant than those at the center.

Note that Eq. (10) can be directly applied for the nonlinear NAH projections without

renormalizations although the renormalization procedure is necessary for long distance farfield

projections. Figures 6(a) and 6(b) show the effects of the renormalization along reconstruction

line 3 extended up to z = 2.5 m (i.e., z 7.3 where is the wave length at 1 kHz). The same

line and marker legends in Figs. 4 and 5 are reused in Fig. 6 except no renormalization case. In

Fig. 6(a), as z increases farther from the source surface at z = 0 m, the nonlinear component

becomes relatively significant compared to the linear component: i.e., the SPL difference

9

between the linear and nonlinear components becomes small as z increases. In Fig. 6(b), the

linear reconstruction error increases oscillatory up to 1.2 dB at z = 2.5 m, due to the spatial-FFT-

periodicity-induced ghost image effects and the measurement aperture truncation errors. The

nonlinear reconstruction error of the no renormalization case in Fig. 6(b) has almost identical

values with that of the renormalization case represented by the solid line with triangle markers.

The maximum nonlinear reconstruction error is approximately 0.5 dB at z = 2.5 m.

For the no renormalization procedure case, the exact and reconstructed nonlinear sound

pressure fields along with the reconstruction errors on x-y plane at z = 1.25 m and x-z plane at y =

0.75 m are shown in Fig. 7. Since both linear and nonlinear NAH reconstructions have

significant edge truncation errors, the x- and y-coordinates are chopped to the range of [0.4, 1.1]

m in Fig. 7. The reconstructed sound pressure fields agree well with the directly-calculated ones:

see Figs 7(a), 7(b), 7(d), and 7(e). Similar to the results shown in Figs. 5(a) to 5(c), the nonlinear

NAH reconstruction errors on the x-z plane decrease as z increases (see Fig. 7(f)).

4 CONCLUSIONS

In this article, the novel, nonlinear, dissipative, planar NAH algorithm is introduced that is

based on the first- and second-order perturbation solutions of the Westervelt Wave Equation

(WWE) and the modified renormalization procedure. The proposed NAH procedure is

applicable to sound fields with dominant cumulative nonlinearity and straight wave rays such as

spherical waves and 1-D plane waves. The WWE is decomposed into the first-order,

homogeneous and second-order, inhomogeneous wave equations. The conventional, spatial-

FFT-based, linear, planar NAH technique is applied to solve the first-order wave equation. The

resulting linear sound pressure solution is input to the second-order wave equation as the

inhomogeneous source terms to find the second-order, nonlinear sound pressure solutions. The

secular terms in the second-order solutions are removed by using the modified renormalization

approach to obtain the uniformly valid solution.

By applying the aforementioned perturbation and renormalization procedures to the

spherical Burgers Equation, the sound pressure field radiated from the high-level pulsating

sphere is calculated to validate the proposed NAH algorithm. In the nearfield of the pulsating

sphere up to z = 1.2 m 3.5, the SPLs reconstructed by using the proposed NAH procedure

match well with the directly-calculated SPLs at the maximum reconstruction error of

approximately 0.3 dB for the linear NAH-projections and 0.2 dB for the nonlinear NAH-

projections.

As z value increases farther from the sound source up to z = 2.5 m 7.3, the spatial-FFT-

based linear reconstruction solution suffers from the more significant spatial-FFT-periodicity-

induced ghost image effects and the measurement aperture truncation errors, which increase the

linear reconstruction error up to 1.2 dB. Then, this linear reconstruction error degrades the

performance of the proposed nonlinear NAH procedure with the maximum reconstruction error

of 0.5 dB.

In the proposed nonlinear NAH projections, the secular terms have negligible effects on the

uniformly valid solutions. Thus, it can be concluded that the proposed nonlinear and dissipative

NAH procedure can be applied to reconstruct the sound pressure fields successfully in the

nearfield of the pulsating sphere regardless of the renormalization procedure.

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10

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6 FIGURES

Fig. 1 – Sketch of nonlinear, dissipative pulsating sphere simulation setup.

11

(a)

Reconstruction line 1:

(x,y) = (0.45,0.75) m

2: (x,y) = (0.6,0.75) m

3: (x,y) = (0.75,0.75) m

4: (y,z) = (0.75,1) m

(b)

Reconstruction line 1

2

3 4

Fig. 2 – 3-D sound pressure fields directly-calculated from pulsating sphere on x-y plane at z =

0 m and x-z plane at y = 0.75 m when excited at 1 kHz: (a) Linear, lossless sound pressure field,

and (b) Corresponding nonlinear sound pressure field at 2 kHz.

(c)(b)(a)

Fig. 3 – Linear sound pressure fields on x-y plane at z = 0 m: (a) Directly-calculated sound

pressure field, (b) SONAH-reconstructed sound pressure field, and (c) Reconstruction error (i.e.,

dB differences between reconstructed and directly-calculated sound pressure levels (SPLs)).

(a) (b)

(c) (d)

Fig. 4 – Directly-calculated and NAH-projected SPLs along reconstruction lines 1 – 4: (a)

Reconstruction line 1, (b) Reconstruction line 2, (c) Reconstruction line 3, and (d)

Reconstruction line 4.

12

(a) (b)

(c) (d)

Fig. 5 – NAH reconstruction errors along reconstruction lines 1 – 4: (a) Reconstruction line 1,

(b) Reconstruction line 2, (c) Reconstruction line 3, and (d) Reconstruction line 4. (a)

(b)

Fig. 6 – Reconstruction results with and without renormalization along reconstruction line 3

extended up to z = 2.5 m (The line and marker legends are same as Figs. 4 and 5 except no

renormalization case): (a) Directly-calculated and NAH-projected linear and nonlinear SPLs,

and (b) Reconstruction errors. (a) (b) (c)

(d) (e) (f)

Fig. 7 – Directly-calculated and NAH-projected nonlinear sound pressure fields on x-y plane at

z = 1.25 m and x-z plane at y = 0.75 m: (a) Directly-calculated results on the x-y plane, (b)

Reconstructed results on the x-y plane, (c) Reconstruction errors on the x-y plane, (d) Directly-

calculated results on the x-z plane, (e) Reconstructed results on the x-z plane, and (f)

Reconstruction errors on the x-z plane.