April 2015 

NASA/TM–2015-218706 

Toward a Nonlinear Acoustic Analogy: 

Turbulence as a Source of Sound and Nonlinear 

Propagation

Steven A. E. Miller 

Langley Research Center, Hampton, Virginia 
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April 2015 

NASA/TM–2015-218706 

Toward a Nonlinear Acoustic Analogy: 

Turbulence as Source of Sound and Nonlinear 

Propagation

Steven A. E. Miller 

Langley Research Center, Hampton, Virginia 
Available from: 

NASA STI Program / Mail Stop 148 

NASA Langley Research Center 

Hampton, VA 23681-2199 

Fax: 757-864-6500 

Acknowledgments 

The author is grateful for continuous support from the National Aeronautics and 

Space Administration (NASA), Advanced Air Vehicles Program, Commercial 

Supersonic Technology Project. Brian Howerton of NASA Langley Research Center 

provided measurements from the NASA Langley Liner Lab Normal Incidence Tube. 

Emily Mazur, a 2012 NASA Langley Aerospace Research Student Scholars Program 

participant, is acknowledged for evaluating numerically the Blackstock bridging 

function.

The use of trademarks or names of manufacturers in this report is for accurate reporting and does not constitute an 

official endorsement, either expressed or implied, of such products or manufacturers by the National Aeronautics and 

Space Administration. 
Abstract

An acoustic analogy is proposed that directly includes nonlinear propagation effects. We examine the Lighthill acoustic analogy and replace

the Green’s function of the wave equation with numerical solutions of the

generalized Burgers’ equation. This is justified mathematically by using

similar arguments that are the basis of the solution of the Lighthill acoustic analogy. This approach is superior to alternatives because propagation is accounted for directly from the source to the far-field observer instead of from an arbitrary intermediate point. Validation of a numerical

solver for the generalized Burgers’ equation is performed by comparing

solutions with the Blackstock bridging function and measurement data.

Most importantly, the mathematical relationship between the NavierStokes equations, the acoustic analogy that describes the source, and

canonical nonlinear propagation equations is shown. Example predictions are presented for nonlinear propagation of jet mixing noise at the

sideline angle.

1
Nomenclature

Symbols

A Arbitrary vector quantity

Aijlm Coefficient matrix

Aij Subset of Aijlm

Bn Fourier coefficients of BBF

c Speed of sound

D Nozzle exit diameter

Dj Fully expanded diameter

eo Total energy

F Fourier series dependent on x

g Arbitrary function or Green’s function

Jn Bessel function of the first kind of order n

j Integer index

K Fourier coefficient

k Wavenumber

kmax Maximum turbulent kinetic energy

within jet plume

l Turbulent length scale

M Mach number

Ma Acoustic Mach number

Mc Convective Mach number constant

Md Design Mach number

Mj Fully expanded Mach number

m Integer index

Pf Scaling constant for spectral density

P r Prandtl number

p Pressure

q Heat-flux

q ̃ Square of  ̃p

R Gas constant or propagation distance

Rijlm Two-point cross-correlation of Lighthill

stress tensor

r Vector from source to observer

r Radial coordinate

S Spectral density

St Strouhal number

Sy Equivalent source strength per unit length

T Thermodynamic temperature

Td Thermal diffusion rate

Tij Lighthill stress tensor

t Time

u Velocity

x Observer or position

y Source position or particle displacement

yc Jet potential core length

α Atmospheric absorption coefficient

β Coefficient of nonlinearity

or dispersion coefficient

βs A constant within the two-point

cross-correlation

Γ Group of coefficients involving nonlinearity

γ Ratio of specific heats

δ Dirac delta function or viscous terms

δij Kronecker delta function

 β(ρ∞c3

∞)

−1

η(ξ, η, ζ) Source separation vector

μ Kinematic viscosity

ν Dynamic viscosity

ξ Momentum in Lagrangian coordinates or

source separation in x direction

ρ Density

σ Shock formation distance

or amplification factor

τ Retarded time

τs Turbulent time scale

τij Viscous stress tensor

Φsh Earnshaw phase angle

φ Arbitrary vector quantity or wave

emission time

ω Radial frequency

Abbreviations

BBF Blackstock bridging function

NIT Normal Incidence Tube

PSD Power spectral density

SPL Sound pressure level

TTR Total temperature ratio

2
1 Introduction

The physical mechanism and associated mathematical model to predict acoustic radiation from

high speed compressible fluid turbulence continues to elude investigators after decades of research.

In most fluid flows, the intensity of turbulence gives rise to acoustic radiation that propagates

according to the theory of linear acoustics and dissipative effects dominate nonlinear effects. When

turbulence is highly intense the resultant acoustic waves have magnitudes that result in nonlinear

propagation due to dominance of nonlinear terms over dissipative terms contained in the equations

of motion. The radiating waves contain all non-zero frequency components, are very energetic, and

coalesce into many discontinuities (shock waves). Here, we seek a unified theory of the radiation

source using an acoustic analogy combined with the nonlinear propagation effects approximately

governed by the generalized Burgers’ equation.

High intensity compressible turbulence is present within the exhaust flow created by rocket and

high performance air breathing jet engines. The acoustic radiation created by these exhaust flows

is potentially harmful to the flight vehicle airframe, launchpad, or flight deck through the mechanism of sonic fatigue or sonic failure. It is also potentially harmful or annoying to the surrounding

community and natural environment. A few recent investigations (among many conducted over

decades) illustrate the relevance of this contemporary problem via the noise produced by aircraft.

Recently, Neilsen et al. [1] conducted measurements of the ‘F-22A Raptor’ flight vehicle with afterburning engines. They examined the spectral characteristics spatially and the waveforms’ nonlinear

indicators as described by Gee et al. [2], and showed that nonlinear propagation effects are important. Though many nonlinear indicators have been proposed, Gee et al. [2] created two that are

complimentary to traditional spectral measurement approaches, and successfully separated geometric acoustic effects from nonlinear propagation effects. Petitjean et al. [3] performed experiments to

examine nonlinear distortion of acoustic waves and waveforms in the time and frequency domains.

They proposed that the convective Mach number of turbulent structures is highly correlated with

nonlinear effects, which is certainly the case for high speed rocket and jet exhaust. Highly energetic

jet noise spectra were recently analyzed by Tam and Parrish [4], and they proposed that the divergence of measurements from relatively low intensity similarity spectra might be due to indirect

combustion noise from the afterburner. The peak intensities calculated by Tam and Parrish [4]

are 178 dB. Morfey and Howell [5] conducted flight tests and showed that the inclusion of finiteamplitude noise propagation theory must be used to account for observed aircraft flyover effects.

Thus, prediction approaches for jet noise created by the F-22 or similar aircraft must account for

nonlinear propagation effects.

Contemporary studies of rocket noise with emphasis on nonlinear propagation have recently been

conducted. Rocket noise data was analyzed by McInerny and Olcmen [6], and they observed many

shocks within the waveforms at all angles and distances from the flight vehicle. They concluded

that nonlinear effects must be accounted for to explain the many discontinuities observed in the

measured waveform. Recently, Gee et al. [7] characterized the rocket noise source by performing

measurements of a statically fired rocket engine. These measurements yielded insight into the rocket

noise source strength and spatial distribution based upon near-field microphone measurement.

Nonlinear indicators from these measurements implied that the source is intense and necessitates

the use of finite-amplitude acoustic theory for sound propagation. In these investigations, the

far-field noise can only be obtained by propagating the measured signal from the near-field to the

far-field. These contemporary investigations (and those in the past) measure the far-field noise

directly or propagate the measured near-field signal to the far-field from an intermediate point.

Here, we attempt to create a prediction method to propagate acoustic radiation from high intensity

jets to the far-field directly from the source.

3
In this paper, the Navier-Stokes equations are used to derive the generalized Burgers’ equation

and the Lighthill [8] acoustic analogy. The form of the acoustic analogy and associated sources are

modeled for jet mixing noise using the approach described by Miller [9]. We retain the Green’s

function of the wave equation as an argument for the spectral density in the far-field. It is argued

that the modulus squared of the Green’s function of the wave equation, contained explicitly in the

acoustic analogy, can be approximated with solutions of the generalized Burgers’ equation. For low

intensity turbulence, limiting forms of solutions of the generalized Burgers’ equation are equal to

those of the wave equation. When the radiation source is relatively more intense, then nonlinear

effects present within the generalized Burgers’ equation are captured directly within the acoustic

analogy approach. The combination of approximating the Green’s function of the wave equation

with the generalized Burgers’ equation leads to a unique approach that overcomes limitations of

previous linear acoustics approaches. However, some important assumptions must be made. In

particular, for propagation purposes only, the source origin is approximated at a point in space.

The source spectrum is still evaluated with a volumetric integral.

This paper first surveys the mathematical theory of the governing equations of motion and

the subsequent derivation of the source model and propagation model. Particular solutions of

propagation models are developed. The relationship between the source model and the propagation

model is shown. The process of evaluating the closed form models numerically is discussed. Next,

example calculations are presented that illustrate the physics of nonlinear propagation for both the

limiting cases and the model, and are compared with select measurement data. Finally, predictions

are conducted for the power spectral density of high intensity jet mixing noise at various observer

positions to illustrate the effect of nonlinear propagation.

2 Mathematical Theory

This section is heavily based upon the work of Schlicting and Gersten [10], Crighton [11], and

Blackstock [12, 13]. The equations of motion are introduced, a source model and propagation

model are derived independently, and their connection is shown. We begin with the Navier-Stokes

equations as shown by Schlicting and Gersten [10] where the continuity equation is

∂ρ

∂t + ∂ρui∂xi

= 0, (1)

the momentum equation is

∂ρui

∂t + ∂ρuiuj∂xj

= ∂τij

∂xj

, (2)

and the energy equation is

∂ρeo

∂t + ∂ρujeo∂xj

= −∂ujp

∂xj

− ∂qj

∂xj

+

∂uiτij

∂xj

, (3)

where

τij = −pδij + μ

∂ui

∂xj

+

∂uj

∂xi



− 2

3

μ

∂uk

∂xk

δij , (4)

eo is the total energy, p is the pressure, t is time, u is the velocity vector, q is the heat-flux, x is

the spatially independent variable, δij is the Kronecker delta function, ρ is the density, and μ is the

viscosity. We assume that the gas is ideal and p = ρRT, where R is the gas constant and T is the

temperature.

4
2.1 The Propagation of Weakly Nonlinear Waves

We simplify the Navier-Stokes equations by assuming that u = ∇φ + ∇ × A, where φ and A

are used to represent the dependent variable u. Only the first- (linear), second-order (nonlinear),

and dissipative terms that are linear relative to dependent variables and diffusion coefficients are

retained. We also assume that the diffraction coefficients are constant and equal to their reference

(ambient) values. The Prandtl number is P r = νT −1

d , where the kinematic viscosity is ν = μρ−1

∞

and Td is the thermal diffusion rate. We obtain

−c2

∞νPr−1∇4φ +



2 + μv

μ

+ γ

P r

ν∇2 ∂2φ

∂t2 +

∂

∂t 

c2

∞∇2φ − ∂2φ

∂t2



= ∂

∂t 

2∇φ · ∇∂φ

∂t + (γ − 1) ∂φ∂t ∇2φ

 (5)

and

∂A

∂t + ν∇×∇× A = 0, (6)

where sub v signifies the dilatational viscosity. If we let A be a constant (the flow is irrotational)

and allow the temperature boundary condition of the fluid domain to vary, we find a simplified

form

c2

∞∇2φ − ∂2φ

∂t2 +



2 + μv

μ

+ γ − 1

P r 

ν∇2 ∂φ

∂t = 2∇∂φ∂t · ∇φ + (γ − 1) ∂φ∂t ∇2φ. (7)

Equation 7 is the basis for a wide range of nonlinear propagation investigations and can be used to

accurately predict ‘weakly’ nonlinear propagation of waves.

Using Eqn. 7 we seek to derive the generalized Burgers’ equation. We assume that the flow

has the properties of a set of symmetries. These symmetries are cylindrical, spherical, and planar.

Also, we assume that k∞r >> 1, where k∞ = ωc−1

∞ is the ‘linear’ wavenumber, ω is the radial

frequency, and c∞ is the ambient speed of sound. Using these assumptions Eqn. 7 simplifies to

∂u

∂t + c∞

∂u

∂r + γ + 12

u

∂u

∂r + jc∞u2r = δ2

∂2u

∂r2 , (8)

where j = 0, 1 or 2 for plane, cylindrical, or spherically symmetric (outgoing waves) solutions

respectively and δ is a group of viscous terms. Crighton [11] wrote this ‘generalized Burgers’

equation’ non-dimensionally and compactly as

∂W

∂Z − W ∂W∂Θ = δω2c2

∞

g(Z)

∂2W

∂Θ2 , (9)

where for j = 0, W = U, Z = R, and g(Z) = 1, for j = 1, W = R1/2U and Z = 2R1/2, and

g(Z) = Z/2, and for j = 2, W = RU, Z = ln R, and g(Z) = exp[Z]. Here, U = (γ + 1)u(2c∞)−1,

R = k∞r,Θ= ωτ , and τ is the retarded time. Unfortunately, the general solution of Eqn. 8

is unknown, let alone a proof of a solution’s existence. Recently some exciting exact bi-soliton

solutions have been proposed by Vladimirov and Maczka [14]. Crighton [11] summarizes some

specific earlier solutions of Eqn. 9. Let us temporarily focus our attention on solutions with planar

symmetry where the wave fronts are perpendicular to the x direction. We write Eqn. 7 or 8, after

integration with respect to time and differentiation with respect to x as

5
∂u

∂t +



c∞ + γ + 1

2

u

 ∂u

∂x = 12

δ

∂2u

∂x2 . (10)

This equation can be written in a more compact form by making a coordinate transform from

x to x − c∞t. Also, for reasons that will become apparent later, we are interested in solutions

of the generalized Burgers’ equation cast as a boundary value problem, that is a solution where

at some point in space the pressure is defined as a function of time. The waveform evolves from

this initial point in the direction of propagation. We also assume c∞φx ≈ −φt, the retarded time

is τ = t − xc−1

∞ , and subsequently obtain an equation shown by Mendousse [15]. We write the

resultant equation of Mendousse [15] as a pressure perturbation using the relation p ≈ ρ∞c∞u and

find

∂p

∂x − 
p

∂p

∂τ = δ2c3

∞

∂2p

∂τ 2 , (11)

where β = (γ + 1)/2 is the coefficient of nonlinearity, 
 = β/ρ∞c3

∞, and

δ = ν

4

3 + μvμ + (γ − 1) P r−1



. (12)

Similar processes can be used to find equations for spherically and cylindrically symmetric

outgoing waves. These can be written in a general form as (containing an extra term relative to

Eqn. 11)

∂p

∂x + m

p

r − 
p

∂p

∂τ = δ2c3

∞

∂2p

∂τ 2 , (13)

where m = 0, 1/2, and 1 for plane, cylindrical, and spherical waves, respectively. Equation 13 can

be found more rigorously as shown by Lighthill [16]. Saxena et al. [17] wrote Eqn. 13 in a form

more amenable to numerical solution in the frequency domain

∂p ̃

∂r + m

p ̃

r + (α + iβ)  ̃p = iω
2 q,  ̃ (14)

where  ̃q =  ̃p2, α is the atmospheric absorption coefficient, and β is the dispersion coefficient. The

tilde represents a Fourier transform. In certain circumstances Eqn. 13 can be solved analytically

for carefully chosen boundary conditions, and a subset of these solutions are described in the next

sections. In most cases of practical interest, Eqn. 13 must be solved numerically. The boundary

condition is typically based on a broadband pressure time history or spectrum. The boundary

condition is assumed to be periodic, and its connection with the source model will be discussed in

the next section.

We survey a numerical method for the solution of Eqn. 13. The approach is heavily based upon

the methods developed by Lee et al. [18] and Saxena [19]. First, the Fourier transform and square of

the Fourier transform of a pressure time history are calculated (boundary condition). It is assumed

that the pressure time history (derived from a ‘source spectrum’ with random phase) is periodic

in time. An explicit second order Runge-Kutta spatial marching technique is used to propagate

the waveform in the radiation direction. The real and imaginary parts of  ̃p and  ̃q are advanced in

space at each discrete step. After each spatial step marched, a filter window is applied to remove

numerical noise, and the complex atmospheric absorption and dispersion as a function of frequency

are applied. Also, a Lanczos filter as described by Duchon [20] is applied at each spatial step to

help minimize the Gibbs phenomenon. The complex coefficients of atmospheric absorption and

6
dispersion are calculated using the method of Bass et al. [21, 22]. When the waveform integration

reaches the observer, the inverse Fourier transform of  ̃p is performed to recover the pressure time

history.

2.2 The Fay Solution

We now examine certain limited solutions for finite-amplitude wave propagation. Fay [23] derived

an equation of motion using physical arguments for the perturbation of a gas in a single spacial

dimension and time. It can easily be shown that the equation of motion that Fay [23] considered

is based upon Eqn. 2 with certain assumptions. These assumptions include the flow varying in

one spatial dimension, varying in time, the dependent variables are small in fluctuation, that

fluctuations in u >> ρ, viscosity is a constant, and additionally that the boundary condition

of temperature is fluctuating. Fay did not explicitly use an assumption regarding temperature

fluctuations at the boundary, but it is required here to obtain Fay’s governing equation from the

Navier-Stokes equations. Multiplying the resultant equation by a differential dx to give a volume

per unit length x, yields Fay’s governing equation

ρ∞

∂u

∂t dx = −∂p∂xdx +

4

3

μ

∂2u

∂x2 dx. (15)

Fay [23] assumes that compression is adiabatic

p

p∞

=

ρ

ρ∞

γ

, (16)

which implies for the one-dimensional problem

p

p∞

+1= ∂y

∂x−γ

, (17)

where y is the distance of a particle from a ‘resting plane’ of reference at time t. Assuming that

c2

∞ = γp∞ρ−1∞ , Fay focuses on a solution of the resulting equation

c2

∞

∂2y

∂x2 = ∂y∂x

γ+1 

∂2y

∂t2 − 4μ3ρ∞

∂

∂t ∂2y∂x2

 . (18)

Fay [23] assumes that the solution is periodic and y = x + F, where F is a Fourier series with

coefficients that are dependent on x. After substituting the relation between y and F into Eqn. 18

and assuming that ∂F/∂x can be represented as a Fourier series, Fay finds the solution for ∂F/∂x

and its Fourier coefficients. The solution of Eqn. 18 in terms of ∂F/∂x is

∂F

∂x = log  16μω3ρ∞(γ + 1)c2

∞K1,1



− 8μω

c2

∞ρ∞ (γ + 1)

n

=∞

n=1

sin n (ωt − ωx/c∞)

sinh n



log  16μω

3c2

∞ρ∞(γ+1)K1,1



+ 2xμω2

3ρ∞c3

∞



(19)

where K1,1 is an arbitrary coefficient that can be set to match a periodic boundary condition. Note

that ∂F/∂x is the ratio of the change in volume relative to specific volume in the undisturbed

medium, thus

p = p∞

	∂y

∂x−γ

− 1




= −p∞γ

	

∂F

∂x −

γ + 1

2!  ∂F∂x 2

+ ...
. (20)

7
Using this relation and Eqn. 19 yields

p

p∞

= 32

3

μω

c2

∞ρ∞

γ

γ + 1 n

=∞

n=1

sin n (ωt − ωx/c∞)

sinh n



log  16μω

3ρ∞(γ+1)c2

∞K1,1



+ 2xμω2

3c3

∞ρ∞

. (21)

Equation 21 represents the Fay [23] solution of Eqn. 15 that is derived after many simplifications

from the Navier-Stokes equations. Blackstock [24] writes Eqn. 21 in a simplified form

p

p∞

=

n

=∞

n=1

2Γ−1

sinh [n(1 + σ)Γ−1]

sin n(ωt − kx), (22)

where Γ groups the coefficients involving nonlinearity relative to dissipation and σ is the nondimensional propagation distance normalized by the shock pressure

σ = x

x = p∞βkxρ∞c2

∞

. (23)

2.3 The Fubini Solution

Another limited solution for finite-amplitude wave propagation is now presented that was proposed

by Fubini [25]. It was brought to the attention of the larger acoustic community through the work

of Westervelt [26]. Fubini used a unique approach and sought a solution of the non-conservative

one-dimensional momentum equation without viscous effects in Lagrangian form

∂2ξ

∂t2 + ∂ξ∂t

∂2ξ

∂x∂t = −1ρ

∂p

∂x, (24)

where a is a function of x only and x = a + ξ. Here, ξ is the displacement of the particle at

position a at time t. The problem is solved with the boundary condition u(0, t) = uo sin[ωt]. The

solution approach is straightforward, and Fubini [25] used the approach of Earnshaw [27] and wrote

a closed-form solution for u as a binomial series. After retaining the first two terms of the binomial

series the solution of Eqn. 24 is

u(x, t) = uo sin 

ωt − ωx

c∞



1 − βuc−1

∞





, (25)

where β = ωu∞(c2

∞Mak)−1 and Ma is the acoustic Mach number. It is desirable to change the

form of Eqn. 25 to something more convenient for our purposes. The term uu−1

∞ is expanded as a

series

u

u∞

= ∞

n=1

Bn sin n(ωt − kx), (26)

where

Bn = 1

n


 2π

0

u

u∞

sin[n(ωt − kx)]d(ωt − kx). (27)

Using conservation of momentum in an Eulerian framework, Fubini [25] showed a simplified

form

Bn = 2

nσ

Jn [nσ] , (28)

8
where Jn are the Bessel functions of the first kind of order n. Using the relation p = ρ∞c∞u and

combining Eqn. 26 and 28, the solution can be written

p

p∞

= ∞

n=1

2

nσ

Jn[nσ] sin n (ωt − kx) (29)

that is also shown by Blackstock [24]. The solutions for weakly nonlinear planar wave propagation of

Fay [23] and Fubini [25] (Eqns. 21 and 29) are very different because they are particular solutions of

the Navier-Stokes equations given greatly different assumptions. In the next section Blackstock [24]

reconciles these differences.

2.4 The Blackstock Bridging Function

Blackstock [24] examined the solutions of Fay [23] and Fubini [25] in an attempt to reconcile their

differences. The Fay and Fubini solutions are not accurate in a range of σ = 1 to σ ≈ 3.5, that

is the transition region from the continuous solution to a discontinuous solution. Blackstock [24]

‘bridged this gap’ between the two solutions by using ‘weak shock theory,’ where progressive wave

relations are used to describe continuous sections of the waveform between shocks (discontinuities

in the solution). Unlike other attempts to find more general solutions, the approach does not

directly make use of the generalized Burgers’ equation or its variations. Using weak shock theory

the governing equations are

u = g(φ), (30)

where u is the particle velocity, g is a function, and φ represents the wave emission time. An

equation is created that relates the difference of wave emission time with source time

τ = φ − (βc−2

∞ )g(φ). (31)

The system of equations is closed by defining an equation that governs the wave path and

amplitude of each shock wave

dt

s

dx = −12

βc−2

∞ (ua + ub), (32)

where subscripts a and b represent quantities just preceding and following the shock front and a

prime denotes evaluation at the retarded time. Equations 30 through 32 are a system of equations

based on weak shock theory, and these equations are solved directly for u after eliminating φ by

substitution. Like Fay [23] and Fubini [25], the solution is sought as a boundary value problem

by specifying u(0, t) = uo sin[ωt] for t >> 0. A transcendental equation for the shock amplitude

results, and the solution process follows that of Fubini almost exactly as shown in the previous

section, where a Fourier series is proposed. After many straightforward mathematical operations

are performed as shown by Blackstock [24], an expression for p is obtained

p(x, t) = po

∞

n=1

Bn sin [nωτ ] , (33)

where po represents the initial wave amplitude at x = 0. Equation 33 is the Blackstock bridging

function (BBF). The coefficients Bn are

9
Bn = 2

n(1 + σ) +

2

nπσ 
π

Φsh

cos [n (Φ − σ sin Φ)] dΦ, (34)

where n is the harmonic number and Φsh is the Earnshaw phase variable. Coefficients Bn are

evaluated with a numerical integration technique. Here, Φsh satisfies

Φsh = σ sin Φsh (35)

and is transcendental. Generally it must be evaluated numerically.

2.5 Acoustic Radiation Source Modeling using the Acoustic Analogy

We have now obtained a general equation for the propagation of weakly nonlinear acoustic radiation

(Eqn. 13) and a highly accurate analytical solution (Eqn. 33) to validate more general numerical

solutions. Here, we seek to predict the broadband source spectrum for use with the developed

propagation theory using the same set of governing equations. We apply the partial derivative

operator on Eqn. 1 with respect to time and apply the divergence operator on Eqn. 2. Subtracting

the modified continuity equation from the divergence of the momentum equation, then adding the

difference of the double divergence of the pressure and the double divergence of the density scaled

by the speed of sound to both sides of the resulting equation and simplifying yields Lighthill’s [8]

acoustic analogy

∂2ρ

∂t2 − c2

∞

∂2ρ

∂xi∂xi

= ∂2Tij

∂xi∂xj

, (36)

where Tij is the Lighthill stress tensor (see Lighthill [8] for details and discussion). The Green’s

function of Eqn. 36 is governed by

∂2g

∂t2 − c2

∞

∂2g

∂xi∂xi

= δ (x − y) δ (t − τ ), (37)

where g(x; y, t; τ ) is the Green’s function, x is an observer location, y is a source location, and δ is

the Dirac delta function. It can easily be shown that the solution of Eqn. 37 in three-dimensional

space is

g(x; y, t; τ ) = δ



t − τ − |x − y|c−1

∞



4π|x − y| (38)

and in the frequency domain

g (x, y, ω) = exp [−ikr]

4πr , (39)

where r is a vector from source to observer. We now can write the closed form solution of Eqn. 36 as

the spectral density of far-field pressure using the theory of Green’s functions following the approach

of Miller [9], but with some important differences. We assume that the observers (from a crossspectral point of view) are at the same position, that the flight stream Mach number is zero, and

that near-field and mid-field terms are negligible in the far-field. The Green’s functions that share

the same observer but differing source positions within the jet exhaust plume are approximated by

a phase change and g(x; y, ω)∗ = g(x; y, −ω), where superscript ∗ denotes the complex conjugate.

10
The density is converted to pressure. We arrive at an equation for the spectral density, S, of

acoustic pressure in the far-field that is similar to Eqn. 1.11 of Ffowcs Williams [28]

S(x, ω) = 
∞

−∞


∞

−∞

rirjr

l

r

m

c4

∞r2r2 g (x, y, ω) g∗ 

x, y

, ω

×

∂4

∂τ 4 Rijlm(y, η, τ ) exp 

−iω τ +

r

c∞

− r

c∞

 dτ dηdy,

(40)

where the prime denotes an alternate source position. The subscripts i, j, l, and m imply summation

from one to three. We have obtained an equation that is a double volumetric integral over the source

and an integral over the retarded time. We require a model for the two-point cross-correlation of

the Lighthill stress tensor Rijlm. Here, we adopt a simplified form from Miller [9]

∂4

∂τ 4 Rijlm(y, η, τ ) = 4Aijlmu4

π1/2l8

s



3l

4

s − 12l

2

s (ξ − uτ )

2 + 4(ξ − uτ )4

× exp 

−|ξ|

uτs



exp 

−(ξ − uτ )2

l2

s



exp 

−η2

l2

sy 

exp 

−ζ2

l2

sz 

,

(41)

where Aijlm is a coefficient matrix, l is the turbulent length scale in the axial (subscript s) and

radial directions (subscript sy and sz), u is the axial averaged velocity component, η = η(ξ, η, ζ)

is a vector from source vectors y to y

, and τs is a turbulent time scale. This model is carefully

chosen to both be integratable analytically and capture trends of measurement of turbulent jets in

the range of 0.5 ≤ Mj ≤ 1.5. Using the proposed model for Rijlm, the integration of τ in Eqn. 40

can be performed. After simplifying we obtain

S(x, ω) = 
∞

−∞


∞

−∞

rirjr

l

r

m

c4

∞r2r2 g (x, y, ω) g∗ 

x, y

, ωAijlm

lsω4

u

exp 

−iξω

u



exp 

−|ξ|

uτs



× exp 

−η2

l2

sy 

exp 

−ζ2

l2

sz 

exp 

−l

2

sω2

4u2



exp 

−iω(r − r

)

c∞



dηdy.

(42)

Let us now examine the volumetric integral involving η. The term gg∗ exp[−iω(r − r

)c−1

∞ ] is

approximated as gg∗(y), thus removing its dependence on η and is removed from the integrand of

η. This approximation is valid as long as x is in the far-field. Using the same far-field argument,

we also note that the rirjr

l

r

m term is no longer dependent on η. The prime notation can now be

dropped. The integrals involving the cross-stream variation can be directly evaluated. The term

Aijlm represents coefficients of the fourth order two-point cross-correlation of the stress tensor. We

propose a model for Aijlm that is considerably simplified from Miller [9]

Aijlm = Pfσ1/2A2

ijSy, (43)

where σ is a function that is dependent on the Mach number and frequency of the acoustic radiation

σ = exp 	−



ln[St] − ln  7

100 +

13

100(1 − Mj )

2 

1 + 3

5 (Mj − 1)

σ4

f




. (44)

The function σf is

11
σf =

⎧

⎨

⎩

[(1−McMj )2+(βsMcMj )2]

1/2

[(1−McMj r1/r)2+(βsMcMj )2]

1/2 for McMj < 1

βsMcMj

[(1−McMj r1/r)2+(βsMcMj )2]

1/2 for McMj ≥ 1. (45)

This form is selected to capture fifth power convective amplification effects as proposed by

Ffowcs Williams [28]. The first coefficient of Aij is

a11 =

 

1+(βsMcMj )2

1/2

[(1 − McMjr1/r)2 + (βsMcMj )2]

1/2

5/2

(46)

and all others are approximately 1/3. The form of the coefficient a11 is chosen based on the model

of Ffowcs Williams [28], and βs = 10−1 is a constant. The convective Mach number coefficient is

Mc = 0.70. Recall that the convective Mach number, as discussed by Petitjean et al. [3], plays an

important role in nonlinear propagation of jet noise. The equivalent spatial source distribution is

modeled by the term Sy in Eqn. 43 as

Sy = k2

maxρ2



1 + St−3

200 

exp

⎡

⎣−5

2

ln 	

y1DjSt1/10

yc


2

⎤

⎦ , (47)

where kmax is the maximum effective turbulent kinetic energy in the jet plume. Extensive numerical

simulations suggest that kmax is approximately

kmax = kfM5/2

j TTRP r+P rt(1−erf[2M2

j ]), (48)

where Erf is the error function, kf = 3 × 103 is a constant, P r = 0.72 is the Prandtl number, and

P rt = 0.90 is the turbulent Prandtl number. The total temperature ratio (TTR) is the plenum

stagnation temperature divided by the ambient static temperature. The axial turbulent length scale

is approximated as lsD−1 = 1.07 (0.1028Mj + 0.0654) y1D−1

j . The cross-stream turbulent length

scales, lsy and lsz, are one-third of ls. The turbulent time scale is a function of the local turbulent

length scale, τs = lsu−1. The spatially varying time-averaged stream-wise velocity component and

temperature are approximated using the models of Lau et al. [29] and Lau [30]. These are dependent

on the jet core length and are estimated using the model of Tam [31]. These models are valid for

jets in the range of 0.40 < Mj < 1.5, but we will be exercising this model well outside its range

of validity in the following section. Using these models and assumptions, we simplify the spectral

density of pressure in the far-field as

S(x, ω) = πω4

c4

∞

g (x, ω) g∗ (x, ω)


∞

−∞

Aijlm

rirjrlrm

r4

lslsylsz

u

exp 

−l

2

sω2

4u2



×


∞

−∞

exp 

−iξω

u



exp 

−|ξ|

uτs



dξdy1.

(49)

Equation 49 is used to make predictions of jet mixing noise in the far-field and is in an important form. It shows that the spectral density from jet mixing noise in the far-field is a volumetric

integration that describes a source spectrum that is centered on a point that is approximated at

the nozzle exit. Note that the development of Eqn. 49 is considerably more empirical than other

approaches, but is beneficial to illustrate the proposed concepts. Equation 39 implies that the term

12
gg∗ in Eqn. 49 is (16π2r2)−1 and that the energy decays according to spherical spreading. The remaining part of Eqn. 49 defines the jet mixing noise source spectrum. Now, in Eqn. 49 we discount

the term gg∗ and retain terms that define the source spectrum. In place of gg∗, we use an equivalent expression obtained from the numerical solution of the generalized Burgers’ equation. The

boundary condition of the numerical implementation of the generalized Burgers’ equation is now

the broadband source spectrum defined by Eqn. 49. In essence, we have implemented an equivalent

form for gg∗. For low amplitude sources (source spectrum), the far-field spectral density predicted

using the modified approach is equivalent to that predicted with the more traditional approach.

High amplitude sources will cause the nonlinear term within the generalized Burgers’ equation to

be dominant, and in turn all the characteristics of nonlinear propagation will be apparent in the

predicted jet mixing noise spectrum. Furthermore, the effects of atmospheric absorption and dispersion are directly contained in the solution and not accounted for at a later point. Astute readers

will note that direct approaches to calculate a Green’s function with a Burgers’ like equation, such

as the collapsing sphere approach, will violate the principle of linear superposition. Also, unlike

other approaches to propagate broadband noise that start outside the source region, we propagate

the source spectrum from within its source volume of origin.

The implementation of Eqn. 49 is similar to that of Miller [9] but highly simplified due to lack of

a second observer. Here, the major difference is the integration of the generalized Burgers’ equation

and atmospheric effect numerical solvers. The integrals of Eqn. 49 are approximated numerically.

3 Results

The purpose of this section is to show solutions of equations previously developed graphically . First,

the Fay (Eqn. 21), Fubini (Eqn. 29), and BBF (Eqn. 33) are evaluated for plane wave propagation.

Their respective power spectral density (PSD) is calculated. We then compare solutions of the BBF

to measurement data from the Normal Incidence Tube (NIT). Numerical solutions of the generalized

Burgers’ equation (Eqn. 13) are compared with the BBF (Eqn. 33). The propagation of a jet mixing

noise pressure time history is demonstrated with the numerical solver. Finally, example predictions

for jet mixing noise far-field spectral density with nonlinear propagation effects are performed using

the newly proposed approach.

3.1 Examination of the Fay, Fubini, and Blackstock Bridging Function

We have surveyed mathematically the relation between the Fay (Eqn. 21) and Fubini (Eqn. 29)

solutions and the BBF (Eqn. 33) with governing equations. Here, we will illustrate their physical

significance graphically. Figure 1 shows the amplitudes of the three solutions as a function of nondimensional distance from a sinusoidal boundary condition. The contribution of Fay (Eqn. 21) is

shown as a dashed line with squares, the contribution of Fubini (Eqn. 29) is shown as a dash-dot

line with triangles, and the BBF (Eqn. 33) is shown as a solid line. The amplitudes are normalized

with respect to the amplitude of the source at σ = 0. Recall that σ = xx−1. The amplitude is

calculated numerically by summing the Fourier coefficient magnitudes over all frequencies at each

spatial position. Due to the assumptions and method of solution of Fubini (Eqn. 29), the Fubini

solution is valid from 0 ≤ σ < 1. It immediately decays and approaches zero as the limit σ → ∞.

Fay, who sought stable waveforms, shows a valid solution for σ > 3.5 until viscous effects dominate

the dynamics of wave propagation. Within the region 1 ≤ σ < 3.5 neither the Fay or Fubini solution

are correct. The BBF bridges the two solutions and satisfies (at least to engineering accuracy) the

generalized Burger’s equation.

13
We now examine the waveform of the BBF at various propagation distances. At the boundary

condition (σ = 0) the solution is known and is illustrated in Fig. 2. For example purposes we

select a frequency of 3000 Hz and an initial pressure amplitude of po = 496.07 Pa, which is 144.88

dB. The resultant wavenumber is 54.96 m−1. Ambient conditions are specified as ρ∞ = 1.213

kg m−3, c∞ = 343 m s−1, and β = 1.2. Based on this boundary condition the shock formation

distance is x = 4.362 m. The initial waveform, shown as a function of observer time, is presented

in Fig. 2(a). The PSD of the periodic boundary condition is shown in Fig. 2(b). Unsurprisingly,

the PSD contains only one non-zero discrete component.

The BBF is now evaluated at σ = 1, the shock formation location, and σ = 5.73, which is

well within the validity of the Fay solution. The boundary condition remains the same. These

waveforms are shown in Figs. 3(a) and 4(a). After the waveform is ‘shocked’ there are many

non-zero components in the PSD, as shown in Figs. 3(b) and 4(b).

These solutions are well known to acousticians well versed in the theory of finite amplitude

waves, however this review is conducted to illustrate important points relative to acoustic analogy

theory. Unlike infinitesimal acoustics, multiple non-zero frequency domain Fourier components

appear that are non-negligible that are due to propagation effects. Furthermore, the Gibbs phenomenon appears when we restrict our mathematics to finite numerical approximations. Due to

these nonlinearities we are unable to use the principle of superposition that is central to the evaluation of an acoustic analogy with a volumetric source.

3.2 Comparison of the Blackstock Bridging Function with Measurement Data

of the Normal Incidence Tube

Measurement data of weakly nonlinear propagating plane waves is obtained from the NIT of NASA

Langley Research Center. Schultz et al. [32] discussed the NIT as part of a larger investigation.

The NIT is shown in Fig. 5(a). Six speakers are driven by a function generator and create plane

waves at specific frequencies and amplitudes. These carefully constructed waves propagate down

the circular tube and impinge on a liner sample. Normally, the NIT is used to measure the normal

incidence of various materials for use in aircraft engine liners. Here, a liner is selected that effectively

absorbs the entire wave thereby eliminating reflection. This liner would not normally be placed

in a flight vehicle because of its size and bulk. Figure 5(b) shows a simplified flow chart that

describes the operation of the NIT. A function generator creates a signal that is amplified and

subsequently drives the speaker array. The microphone signals are conditioned and captured with

a digital computer. These signals are then analyzed using digital spectral analysis. High pass filters

are utilized with cut-offs below the planar wave excitation frequency. A microphone is located near

the top of the NIT to measure the source and multiple ‘measurement’ microphones reside near the

liner. The reference and measurement microphones are used to capture near sinusoidal plane waves

for validation purposes.

We evaluate the BBF (Eqn. 33) and compare solutions with various measurements from the NIT.

Figure 6 shows three examples of these comparisons that are representative of a larger investigation.

The BBF is shown as a solid line and the measurement is shown as a dashed-dotted line. Three

comparisons are shown for 140.03 dB at 3000 Hz, 147.94 dB at 2000 Hz, and 144.88 dB at 3000

Hz, in Figs. 6(a), 6(b), and 6(c) respectively. Note the y-axis represents pressure in Pascal and the

x-axis represents retarded time. Generally, this theory and measurement are in agreement both in

magnitude and phase. The relatively short propagation distance within the NIT only allows for

waveforms that are minimally distorted by nonlinear propagation. This is not a major problem

because our focus is on jet mixing noise sources that are approximately 165 dB, that is near the

limit of weak shock theory. Based on these validations, we believe that the implementation of the

14
BBF is correct. There is no reason to expect that larger propagation distances will deviate from

these results.

3.3 Comparison of the Numerical Propagation Solver with the

Blackstock Bridging Function

We have some confidence that the BBF is evaluated correctly using a numerical technique. We

attempt to gain additional confidence in the solution strategy of the generalized Burgers’ equation

by making extensive comparisons with numerical solutions of the BBF. Table 1 summarizes the

conditions examined. At the source position, σ = 0, the boundary condition p(0, t) = po sin [2πft]

is applied. Four values of po are chosen, 2.82843 Pa, 89.4427 Pa, 282.843 Pa, and 2828.43 Pa, that

result in tonal sound pressure level (SPL) of 100 dB, 130 dB, 140 dB, and 160 dB respectively. At

each SPL a frequency of 100 Hz, 1 kHz, 10 kHz, and 100 kHz are applied. For each combination

of frequency and wave amplitude, the planar shock formation distance is calculated using Eqn. 23.

Numerical solutions of plane wave propagation are calculated at three non-dimensional positions,

halfway to shock formation, σ = 1/2, at the shock formation, σ = 1, and three times the shock

formation, σ = 3. Comparisons of these numerical solutions are shown in Figure 7. In each subfigure

the y-axis is pressure in Pascal and the x-axis is retarded time, τ , in seconds. These figures are a

sampling of a larger validation database and are representative of all results. Figures 7(a) and 7(b)

have a boundary condition of 130 dB at 100 Hz and 1000 Hz. Note that the shock formation

distances at the lower frequencies in Figs. 7(a), 7(c), and 7(e) are much larger than the higher

frequencies of Figs. 7(b), 7(d), and 7(f). The numerical solution in the pre-shock region (σ < 1)

agrees very well with the BBF in magnitude and phase. In the post shock region (σ > 1) the Gibbs

phenomenon is apparent. This can be minimized or even almost entirely eliminated by increasing

the number non-zero Fourier coefficients, but doing so increases the amount of computer memory

used by the numerical solver significantly. Unfortunately, in some cases numerical error accumulates

for boundary conditions that are very high in amplitude and frequency. For example, in Fig. 7(f)

near τ ≈ 5 × 10−5 s and σ = 3, small oscillations occur in an ‘odd function fashion,’ centered

about zero. Extensive numerical evaluations using the developed approach have shown that these

spurious oscillations do not grow significantly except at extremely large distances. Overall, the

agreement between the developed numerical method for the evaluation of the generalized Burgers’

equation more than satisfactorily captures the trends of nonlinear propagation of tonal sources.

3.4 Propagation of a Broadband Spectrum

The developed mathematical approach requires that the source spectrum propagate from its location of origin to observer in the far-field. Here, we propagate a measured broadband jet noise

spectrum from the near-field to the far-field to demonstrate the approach. Note that the phase of

each wavenumber component is unknown if only a power spectrum is available from measurement.

A random phase is assigned to each of the frequency components of the measured signal. This is

an important point as the developed prediction method is for spectral density and does not contain

phase. A high speed heated jet at Mj = 1.86 and TTR = 3.20 produces a waveform at the side line

location of one hundred nozzle diameters (R/D = 100). The measured (source) and propagated

(observer) spectrum are shown in Fig. 8(a), where ‘measured’ represents the processed spectrum

at R/D = 100. The propagated spectrum, found by numerical solution of the generalized Burgers’

equation (Eqn. 9), is also shown in Fig. 8(a) as the solid line with circles. The observer spectrum

is at R/D = 200, twice the distance of the measurement location relative to the source. At low

through mid frequencies a decrease of 20 log10 [2] ≈ 6.02 dB of sound power is predicted. This is

15
consistent with the theory of linear acoustics. At high frequencies, simple spherical spreading is

not apparent and only approximately four dB of reduction in SPL is predicted. This is due to

the nonlinear term dominating atmospheric absorption and the transition of energy from lower

wavenumbers to higher wavenumbers. At low through mid frequencies, the same fine ‘peaks’ and

‘troughs’ are present in the spectrum at both field locations. These features are highly distorted

between field locations at high frequencies due to nonlinear propagation effects, atmospheric absorption (dissipation and dispersion), and energy transfer. At very high frequencies additional

energy is present due to the Gibbs phenomenon.

The time history of the spectrum can be compared with the use of the inverse Fourier transform.

The pressure time histories of these spectra are shown in Fig. 8(b), where the y-axis is the pressure

in Pascal and the x-axis represents time. Note that the observer time is not equal to the measured

time, and the propagated signal is translated by t − τ for the purpose of directly comparing the

waveform distortion. Observe wave steepening near certain measured time signals such as t ≈ 0.608

s or 0.6108 s. Figure 8(c) shows the details of the wave coalescence of one particular portion of

the pressure time history. At these times among many, the waves have coalesced into a shock that

exhibits a significant amount of overpressure. Lower amplitude waves, for example near t ≈ 0.6075

s, have only lost energy and not coalesced into a discontinuity. Waves with less energy are dominated

locally through the mechanisms of spherical spreading and atmospheric absorption, that are more

dominant than the nonlinear mechanisms present with the generalized Burgers’ equation. We offer

no further validation of this numerical approach because it is widely used by many investigators

(see for example Saxena et al. [17] or McInerny et al. [6]) to propagate both rocket and jet noise.

3.5 Example Predictions of Jet Mixing Noise Including Nonlinear Propagation

Effects

Now, the numerical method for the solution of the generalized Burgers’ equation with almost no

alteration will be combined with the source spectrum predicted with an acoustic analogy. Figure 9

shows example predictions from the newly developed approach. The jet operates at Mj = 1.86

and TTR = 3.20. Observers are located in the sideline direction of the jet at radial locations

R/D = 100, 140, 200, 240, and 300. The y-axis is SPL per unit St and the x-axis is Strouhal

number, and the former values have been normalized by uj and Dj . The solid line represents the

prediction using the newly developed approach at R/D = 100. This component of jet noise can be

interpreted as the contribution from the relatively incoherent turbulence. Other predictions that

include nonlinear propagation effects are variations of dashed lines and are labeled in the legend of

Fig. 9. These predictions are based on the source spectrum defined by Eqn. 49. To assess the effects

of nonlinear propagation relative to linear propagation, predictions using spherical spreading and

atmospheric absorption of Bass et al. [21,22] are shown in Fig. 9 as symbols. These linear acoustic

predictions are located at the propagation distance R/D = 100. For example, at R/D = 200,

the dash-dot-dash line is the prediction from the jet mixing noise propagated nonlinearly from the

source, and the triangles represent linear propagation using spherical spreading and atmospheric

absorption from the R/D = 100 prediction. Like the previous example shown for purely broadband

noise propagation, we observe a rise in high frequency energy and a generally consistent spherical

spreading rate at low and mid frequencies. Note that the peak SPL per unit St at R/D = 100 is

approximately 164 dB at St ≈ 0.25, that is considerably energetic. Numerical experiments with

this method with less intense acoustic radiation show traditional linear acoustic spreading and

atmospheric absorption, that is generally observed in almost all commercial jet noise measurements

and predictions. At high frequencies some of the energy is due to the Gibbs phenomenon. This

fictitious additional energy can be eliminated by including more non-zero harmonic components in

16
the nonlinear propagation solver. Furthermore, these results are based upon a source model that is

developed for jet Mach numbers up to approximately 1.50, and the overall levels and scaling might

be misleading. To validate and refine this theory, a comprehensive experimental database of high

speed on-design jets must be created, with multiple observer points at various field positions. It is

hopeful that in the future such a database will be available to validate this newly developed theory.

4 Conclusion

An acoustic analogy is proposed that takes into account nonlinear propagation effects from the

source to observer. Propagation effects are captured by approximating the modulus squared of the

Green’s function of the wave equation with a numerical solution in a form of the generalized Burgers’ equation. The numerical solution of the generalized Burgers’ equation is validated with the

use of the Blackstock bridging function and measurement data, that are in turn validated with the

Fay and Fubini solutions. A mathematical survey of the connection between the acoustic analogy

and equations governing propagation is shown based upon the Navier-Stokes equations. The source

model and approximation of the propagation mechanism are consistent with the governing equations. An example of the method based on simplified source models and the nonlinear propagation

numerical algorithm is shown for a high intensity on-design jet.

Validation data is limited for high intensity on-design jets. To fully validate and refine this

theory, a comprehensive experimental database of high speed on-design jet noise must be created,

with various field positions. This data would facilitate the creation of a more advanced source

model for the sound from compressible jet turbulence and help improve the connection between

the turbulent source and nonlinear propagation algorithms. It is hopeful that such a database will

become available to validate this newly developed theory.

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2010, pp. 2586–2595. doi:10.2514/1.J050396.

19. Saxena, S., “A New Algorithm for Nonlinear Propagation of Broadband Jet Noise,” M.S.

Thesis, The Pennsylvania State University, 2008.

20. Duchon, C. E., “Lanczos Filtering in One and Two Dimensions,” Journal of Applied Meteorology, Vol. 18, No. 8, 1979, pp. 1016–1022.

21. Bass, H. E., Sutherland, L. C., Zuckerwar, A. J., Blackstock, T. D., and Hester, D. M., “Atmospheric Absorption of Sound: Further Developments,” Journal of the Acoustical Society of

America, Vol. 97, No. 1, 1995, pp. 680–683. doi:10.1121/1.412989.

18
22. Bass, H. E., Sutherland, L. C., Zuckerwar, A. J., Blackstock, T. D., and Hester, D. M., “Erratum: Atmospheric Absorption of Sound: Further Developments,” Journal of the Acoustical

Society of America, Vol. 99, No. 2, 1995, pp. 1259–1259. doi:10.1121/1.415223.

23. Fay, R. D., “Plane Sound Waves of Finite Amplitude,” Journal of the Acoustical Society of

America, Vol. 3, No. 9, 1931, pp. 222–241. doi:10.1121/1.1901928.

24. Blackstock, D. T., “Connection Between the Fay and Fubini Solutions for Plane Sound Waves

of Finite Amplitude,” Journal of the Acoustical Society of America, Vol. 39, No. 6, 1965,

pp. 1019–1026. doi:10.1121/1.1909986.

25. Fubini-Ghiron, E., “Anomalie nella Propagazione di onde Acustiche di Grande Ampiezza,” Alta

Frequenza, Vol. 4, 1935, pp. 530–581.

26. Westervelt, P. J., “The Mean Pressure and Velocity in a Plane Acoustic Wave in a Gas,”

The Journal of the Acoustical Society of America, Vol. 22, No. 3, 1950, pp. 319–327.

doi:10.1121/1.1906606.

27. Earnshaw, S., “On the Mathematical Theory of Sound,” Phil. Trans. R. Soc. London, Vol. 150,

1860, pp. 133–148. doi:10.1098/rstl.1860.0009.

28. Ffowcs Williams, J. E., “The Noise from Turbulence Convected at High Speed,” Phil. Trans.

R. Soc. Lond. A, Vol. 255, No. 1063, 1963, pp. 469–503. doi:10.1098/rsta.1963.0010.

29. Lau, J. C., Morris, P. J., and Fisher, M. J., “Measurements in Subsonic and Supersonic Free

Jets using a Laser Velocimeter,” Journal of Fluid Mechanics, Vol. 93, No. 1, 1979, pp. 1–27.

doi:10.1017/S0022112079001750.

30. Lau, J. C., Morris, P. J., and Fisher, M. J., “Effects of Exit Mach Number and Temperature

on Mean-Flow and Turbulence Characteristics in Round Jets,” Journal of Fluid Mechanics,

Vol. 105, No. 1, 1981, pp. 193–218. doi:10.1017/S0022112081003170.

31. Tam, C. K. W., “Broadband Shock-Associated Noise of Moderately Imperfectly Expanded

Jets,” Journal of Sound and Vibration, Vol. 140, No. 1, 1990, pp. 55–71. doi:10.1016/0022-

460X(90)90906-G.

32. Schultz, T., Liu, F., Cattafesta, L., Sheplak, M., and Jones, M., “A Comparison Study of

Normal-Incidence Acoustic Impedance Measurements of a Perforate Liner,” 15th AIAA/CEAS

Aeroacoustics Conference, AIAA Paper 2009-3301 , 2009. doi:10.2514/6.2009-3301.

19
Table 1. Validation cases for the numerical propagation algorithm.

po (Pa) dB f (kHz) Shock Distance (m) σ1 σ2 σ2

2.82843 100 0.10 22801.5 0.5 1 3

2.82843 100 1.00 2280.15 0.5 1 3

2.82843 100 10.0 228.015 0.5 1 3

2.82843 100 100 22.802 0.5 1 3

89.4427 130 0.10 721.047 0.5 1 3

89.4427 130 1.00 72.105 0.5 1 3

89.4427 130 10.0 7.210 0.5 1 3

89.4427 130 100 0.721 0.5 1 3

282.843 140 0.10 228.015 0.5 1 3

282.843 140 1.00 22.802 0.5 1 3

282.843 140 10.0 2.280 0.5 1 3

282.843 140 100 0.228 0.5 1 3

2828.43 160 0.10 22.802 0.5 1 3

2828.43 160 1.00 2.280 0.5 1 3

2828.43 160 10.0 0.228 0.5 1 3

2828.43 160 100 0.023 0.5 1 3

20
σ

Bn , Harmonic Amplitude

0 2 4 6 8 10 0

0.25

0.5

0.75

1

Shock Amplitude

Shock Amplitude Fay

Shock Amplitude Fubini

Figure 1. Comparison of harmonic amplitudes of the Fay [23], Fubini [25], and Blackstock [24]

bridging function relative to normalized propagation distance.

τ (s)

Pressure (Pa) 0.0E+00 2.0E-04 4.0E-04 6.0E-04 -500

-250

0

250

500

(a) Time history

f (Hz)

PSD (Pa2 / Hz)

.0E+00

1.0E+042.0E+043.0E+044.0E+045.0E+046.0E+047.0E+048.0E+049.0E+041.0E+05

100

102

104

106

108

1010

(b) Power spectral density

Figure 2. Blackstock bridging function at propagation distance σ = 0.

21
τ (s)

Pressure (Pa) 0.0E+00 2.0E-04 4.0E-04 6.0E-04 -500

-250

0

250

500

(a) Time history

f (Hz)

PSD (Pa2 / Hz)

.0E+00

1.0E+042.0E+043.0E+044.0E+045.0E+046.0E+047.0E+048.0E+049.0E+041.0E+05

100

102

104

106

108

1010

(b) Power spectral density

Figure 3. Blackstock bridging function at propagation distance σ = 1.

τ (s)

Pressure (Pa) 0.0E+00 2.0E-04 4.0E-04 6.0E-04 -500

-250

0

250

500

(a) Time history

f (Hz)

PSD (Pa2 / Hz)

.0E+00

1.0E+042.0E+043.0E+044.0E+045.0E+046.0E+047.0E+048.0E+049.0E+041.0E+05

100

102

104

106

108

1010

(b) Power spectral density

Figure 4. Blackstock bridging function at propagation distance σ = 5.73.

22
(a) (b)

Figure 5. a) Photograph and b) diagram of the Normal Incidence Tube. Courtesy of Brian Howerton

of NASA Langley Research Center.

23
τ (s)

Pressure (Pa) 0.0388 0.039 0.0392 0.0394 0.0396 0.0398 -800

-600

-400

-200

0

200

400

600

800

Measurement

BBF

(a) 140.03 dB and 3000 Hz

τ (s)

Pressure (Pa) 0.0388 0.039 0.0392 0.0394 0.0396 0.0398 -800

-600

-400

-200

0

200

400

600

800

Measurement

BBF

(b) 147.94 dB and 2000 Hz

τ (s)

Pressure (Pa) 0.0388 0.039 0.0392 0.0394 0.0396 0.0398 -800

-600

-400

-200

0

200

400

600

800

Measurement

BBF

(c) 144.88 dB and 3000 Hz

Figure 6. Comparisons between the Blackstock bridging function and measurement.

24
τ (s)

p (Pa)

0 0.005 0.01 0.015 0.02 -100

-50

0

50

100

BBF

Prediction

σ=0.5

σ=1.0

σ=3.0

(a) 130 dB and 100 Hz

τ (s)

p (Pa)

0 0.0005 0.001 0.0015 0.002 -100

-50

0

50

100

BBF

Prediction

σ=0.5

σ=1.0

σ=3.0

(b) 130 dB and 1000 Hz

τ (s)

p (Pa)

0 0.005 0.01 0.015 0.02 -300

-200

-100

0

100

200

300

BBF

Prediction

=0.5

σ=1.0

=3.0

(c) 140 dB and 100 Hz

τ (s)

p (Pa)

0 5E-05 0.0001 0.00015 0.0002 -300

-200

-100

0

100

200

300

BBF

Prediction

=0.5

σ=1.0

=3.0

(d) 140 dB and 10000 Hz

τ (s)

p (Pa)

0 0.0005 0.001 0.0015 0.002 -3000

-2000

-1000

0

1000

2000

3000

BBF

Prediction

=0.5

σ=1.0

σ=3.0

(e) 160 dB and 1000 Hz

τ (s)

p (Pa)

0 5E-05 0.0001 0.00015 0.0002 -3000

-2000

-1000

0

1000

2000

3000

BBF

Prediction

=0.5

σ=1.0

σ=3.0

(f) 160 dB and 10000 Hz

Figure 7. Comparisons of the numerical solution of the generalized Burgers’ equation with the

Blackstock bridging function at various propagation distances σ.

25
f (Hz)

SPL per unit Hz

102 103 10 85 4

90

95

100

105

Source

Observer

(a) Sound pressure spectrum

t (s)

p (Pa) 0.606 0.607 0.608 0.609 0.61 0.611 0.612 0.613 -600

-400

-200

0

200

400

600

Source

Observer

(b) Pressure time history

t (s)

p (Pa) 0.6106 0.6108 0.611 0.6112 0.6114 -600

-400

-200

0

200

400

600

Source

Observer

(c) Example of wave coalescence within pressure time

history

Figure 8. Example propagation of broadband jet noise in the sideline direction.

26
St

SPL per unit St

10-2 10-1 10 140 0

150

160

170

R/D = 100

R/D = 140

R/D = 200

R/D = 240

R/D = 300

R/D = 140 Spherical

R/D = 200 Spherical

R/D = 240 Spherical

R/D = 300 Spherical

Figure 9. Predictions of jet mixing noise at the sideline direction at various propagation distances.

27
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