American Institute of Aeronautics and Astronautics

1

Acoustic Modeling of Rocket Payload Bays within Launch

Fairings

Tyler Engberg1, Umesh A. Korde

2

South Dakota School of Mines and Technology, Rapid City, SD, 57701

Acoustic shielding treatments are necessary to prevent damage to the payload during launch. This paper

discusses three modeling approaches used to analyze the efficacy of such treatments. Results from acoustic

modeling studies using a finite element model (FEM), an analytical approach and experimental testing are

presented here. Payload bay pressure distributions and frequency responses are developed using simple two

dimensional scale models. The two dimensional finite element models are developed first to demonstrate

agreement with published results and analytical predictions. They also show the potential for three

dimensional full scale finite element modeling.

I. Introduction

The need for acoustic shielding of payloads has acquired greater importance in recent years since the

introduction of light-weight composite launch fairings. While such fairings successfully protect the payload against

structural loads, their acoustic transmission characteristics are less than ideal. It has been found, that at certain times

during the first minutes of a launch, both structural and acoustic loads are significant and have the potential to

damage the payload1. Much work has been done in the area of acoustic noise shielding. Acoustic blankets are a

passive method that has been used for a long time and are still used. However, acoustic blankets do not perform

well at low frequencies1. Another passive method is that of a composite chamber core method developed by the

U.S. Air Force Research Laboratory. This method was found to perform as well as the passive acoustic blankets2.

A third passive method consists of using tuned passive vibration absorbers to fairing and cavity modes in order to

reduce noise within the payload fairing3,4

. In contrast to passive methods, active noise shielding works well at low

frequencies, but the volume in which noise shielding is effective is often limited. Different methods of active noise

shielding of payloads have been attempted. Some of these include using structural actuators attached to the wall of

the fairing to dampen acoustic noise5, using a loud speaker located within the fairing to actively cancel acoustic

noise6 or using fairing mounted active proof-mass atuators

7. We would like to take a different approach. We would

like to investigate using the geometry of the payload as a design constraint for noise suppression. Acoustic

modeling is being utilized to assess whether different sized payload and payload fairing combinations can reduce the

acoustic pressure within the payload fairing of an Atlas V rocket.

The significance of the acoustic environment within the fairing is not to be underestimated. For instance, the

Atlas V family of rockets can see acoustic pressure levels as high as 140 dB and potentially damaging levels in the

30Hz to 800Hz frequency range1. Currently we are studying noise propagation into cylindrical fairings

encapsulating a cylindrical payload. Much of our early work on this has focused on analytical modeling. The focus

has since shifted to finite element modeling and experimental scale-model testing of a launch vehicle fairing. In

order to develop a suitable scale model for testing, the pressure level had to be scaled. To do this the pressure was

non-dimensionalized using known parameters in the system; namely: length or diameter, air density and gravity.

The non-dimensionalization shows that the RMS pressure varies linearly with how the geometry is scaled. These

scale the sound pressure levels to 114 dB and 106 dB, respectively. However, due to the size of tubing readily

available the first actual scale rocket model is 1/41.3 scale, which scales the sound pressure level down to 107.67

dB. Tables 1-4 in Appendix A show the geometry and pressure scales for the payload fairing on the Atlas V family

of rockets. Along with scaling the pressure, the frequency range had to be scaled. This scales linearly with

geometry and leads to a scale model frequency range of 1249Hz to 33040Hz. Our focus has primarily been on two

dimensional models looking at one plane of the whole rocket. The first goal is to match the pressure distribution

within the region between the fairing and the payload (air cavity) using three different modeling approaches. The

second goal is to then provide frequency response data for varying payload diameters. This will display problem

frequencies as a function of payload diameters. Full understanding of these models will lead to three dimensional

models and their applications to a full scale rocket model.

1 Email: Tyler.Engberg@mines.sdsmt.edu

2 Email: Umesh.Korde@sdsmt.edu; Phone: 1-(605)355-3731

52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference<BR> 19th4 - 7 April 2011, Denver, Colorado

AIAA 2011-2135

Copyright © 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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II. Model Set Up

As stated above the model being used is of 1/41.3 scale. This leads to a 4in fairing diameter, correlating to

dimension „b‟ in Figure 1 below and a payload diameter of dimension „a‟ in Figure 1. The payload diameter will be

varied from 1.5in to 2.5in in increments of 0.25in.

Figure 1: Scale Model

Three different modeling approaches are used. The first is an analytical approach that has the capability of finding

the frequency response peaks of the air cavity due to its axial resonances. The second modeling approach is

physical scale model testing. Due to the scale of the model and the equipment available this model can only produce

the frequency response peaks due to the first two radial resonances. Therefore, this model is unable to verify the

validity of the analytical approach. The third modeling approach is a finite element approach. This approach will

bridge the gap between the other two models as it has the capability to produce frequency response peaks due to any

of the resonances of the air cavity. Once the three models have validated each other the finite element approach will

be used to look at how the frequency response changes as a function of payload diameter.

All of the model material properties match that of physical scale model. It has a fairing built from acrylic tubing,

payloads built from 6061 aluminum and dry air at atmospheric pressure. The material properties used are

summarized in the table below.

Table 1: Material properties

Before any modeling was performed, the two dimensional resonant frequencies of the air cavity were found

using published analytical methods by Blevins2. This gave us an idea of where to look for frequency response peaks

and also helps validate the methods. The resonant frequencies as a function of payload diameter can be seen in the

table below:

Density (kg/m^3) 1180 Density (kg/m^3) 2700

Young's Modulus (Gpa) 2.88 Young's Modulus (Gpa) 68.9

Poisson's Ratio 0.402 Poisson's Ratio 0.33

Air

Density (kg/m^3) 1.225

Bulk Modulus (Pa) 141610

Speed of Sound (m/s) 340

Acrylic Aluminum

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Table 2: Analytical Resonant Frequencies

III. Acoustic Field within the Fairing: Analytical Approach

Analytical modeling of the sound pressure levels within a launch fairing can be approached in a number of ways

to various ends. For instance, Farinholt and Leo analyze oblique propagation into a conical fairing and use it for

active control design9. Our goal here is to use analytical modeling to understand the effect of the fairing/payload

relative sizes. In particular, it would be necessary to avoid any type of resonant response within the payload/fairing

gap. Here we focus on just the response in the radial direction through a 2-dimensional model. We study a

cylindrical fairing of radius b enclosing a concentric cylindrical payload of radius a., with a < b. A radially

propagating horizontal incident wave field is assumed. The excitation force f(t) due to this field on the fairing wall

causes an acoustic pressure variation p(t) within the cavity. Assuming linearity, the motion within the region a<r<b

can be described as with the partial differential equation , with a forcing function10

−𝒄𝟐𝛁𝟐𝒑 +𝛛𝟐𝒑

𝛛𝒕𝟐= 𝒇(𝒕) (1)

Where c denotes the acoustic speed in air, and for radial symmetry the gradient operator is,

𝛁𝟐 =𝝏𝟐

𝝏𝒓𝟐+

𝟏

𝒓

𝝏

𝝏𝒓 (2)

Now consider an eigenfunction expansion for the solution with time-dependent modal coefficients,

𝒑 𝒓, 𝒕 = 𝚼𝒎 𝒕 𝝓𝒎(𝒓)∞𝒎=𝟏 (3)

The eigenfunctions for the axisymmetric response within the cavity can be shown to be:

𝝓𝒎 𝒓 = 𝑱𝒐 𝜷𝒎𝒓 −𝑱𝟏 𝜷𝒎𝒂

𝒀𝟏 𝜷𝒎𝒂 𝒀𝒐(𝜷𝒎𝒓) (4)

Substituting equation (3) into equation (1) we get:

−𝒄𝟐 𝚼𝒎 𝒕 𝒅𝟐𝝓𝒎

𝒅𝒓𝟐+

𝟏

𝒓

𝒅𝝓𝒎

𝒅𝒓 + 𝚼 𝒎 𝒕 𝝓𝒎(𝒓) = 𝒇(𝒕)∞

𝒎=𝟏∞𝒎=𝟏 (5)

By definition, the eigenfunctions satisfy the relation,

𝒅𝟐𝝓𝒎

𝒅𝒓𝟐+

𝟏

𝒓

𝒅𝝓𝒎

𝒅𝒓 = −𝜷𝒎

𝟐 𝝓𝒎 (6)

Where 𝜷𝒎 = 𝝎𝑚

𝑐. Substituting equation (6) into equation (5) we get:

𝚼 𝒎 𝒕 𝝓𝒎 𝒓 + 𝒄𝟐 𝚼𝒎 𝒕 𝜷𝒎𝟐 𝝓𝒎 𝒓 = 𝒇(𝒕)∞

𝒎=𝟏∞𝒎=𝟏 (7)

Multiplying both sides by 𝜙𝑛(𝑟) and integrating over r and taking advantage of orthogonality:

𝒓𝝓𝒎 𝒓 𝝓𝒏 𝒓 𝒅𝒓 = 𝜶𝒏,𝒏 = 𝒎= 𝟎,𝒏 ≠ 𝒎

𝒃

𝒂 (8)

From which follows the modal equation for the nth mode:

𝚼 𝒏𝜶𝒏 + 𝒄𝟐𝜷𝒏𝟐𝜶𝒏𝚼𝒏 = 𝒓𝝓𝒏 𝒓 𝒇 𝒕 𝒅𝒓

𝒃

𝒂 (9)

Next we define Φ𝑛 :

Payload Diameter (in) wn1 wn2 wn1 wn2

1.5 1588 3066 5532 10805

1.75 1512 2965 6092 11973

2 1443 2856 6807 13442

2.25 1379 2745 7739 15340

2.5 1322 2637 8993 17876

Radial Axial

Inner Air Cavity Analytical (Blevins) Natural Frequencies

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𝚽𝒏 = 𝒓𝝓𝒏 𝒓 𝒅𝒓𝒃

𝒂 (10)

Substituting Φ𝑛 into equation (9) and dividing through by 𝛼𝑛 :

𝚼 𝒏 + 𝒄𝟐𝜷𝒏𝟐𝚼𝒏 = 𝒇(𝒕)

𝚽𝒏

𝜶𝒏 (11)

With 𝜔𝑛2 such that,

𝒄𝟐𝜷𝒏𝟐 = 𝝎𝒏

𝟐 (12)

absent dissipation, equation (11) describes the dynamics of each mode. If we assume a small amount of dissipation

in the air then:

𝚼 𝒏 + 𝒄𝒅𝒏𝚼 𝒏 + 𝝎𝒏

𝟐𝚼𝒏 = 𝒇(𝒕)𝚽𝒏

𝜶𝒏 (13)

For harmonic forcing 𝑓 𝑡 = 𝐹𝑚𝑎𝑥 𝑒𝑖𝜔𝑡 we can look for a modal solution:

𝚼𝒏 = 𝚪𝒏𝒆𝒊𝝎𝒕 (14)

Substituting equation (14) into equation (15) we get:

−𝝎𝟐𝚪𝒏 + 𝒄𝒅𝒏𝒊𝝎𝚪𝒏 + 𝝎𝒏

𝟐𝚪𝒏 =𝑭𝒎𝒂𝒙𝚽𝒏

𝜶𝒏 (15)

Solving for Γ𝑛 :

𝚪𝒏 =𝑭𝒎𝒂𝒙

𝚽𝒏𝜶𝒏

−𝝎𝟐+𝝎𝒏𝟐 +𝒊𝝎𝒄𝒅𝒏

(16)

Substituting Υ𝑚 (𝑡) and 𝜙𝑚(𝑟) into equation (3):

𝒑 𝒓, 𝒕 = 𝑭𝒎𝒂𝒙

𝚽𝒎𝜶𝒎

−𝝎𝟐+𝝎𝒎𝟐 +𝒊𝝎𝒄𝒅𝒎

∞𝒎=𝟏 𝑱𝒐 𝜷𝒎𝒓 −

𝑱𝟏 𝜷𝒎𝒂

𝒀𝟏 𝜷𝒎𝒂 𝒀𝒐(𝜷𝒎𝒓) 𝒆𝒊𝝎𝒕 (17)

Carrying these calculations out to multiple forcing frequencies we can obtain a frequency response curve based on

the maximum acoustic pressure in the acoustic cavity at each frequency. The figures below show the frequency

responses around the first two axial resonances for a 2in payload

Figure 2. First Axial Resonance, 2in Payload Figure 3. Second Axial Resonance, 2in Payload

The frequency response peaks due to the first two axial resonances as a function of payload diameter is summarized

in Table 3 below.

5000 5500 6000 6500 7000 7500 8000 8500 9000

0

0.01

0.02

0.03

0.04

0.05

Frequency (Hz)

pm

ax/u

nit e

xcita

tio

n a

mp

litu

de

1.25 1.3 1.35 1.4 1.45 1.5

x 104

0

0.2

0.4

0.6

0.8

1

1.2

Frequency (Hz)

pm

ax/u

nit e

xcita

tio

n a

mp

litu

de

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Table 3: Frequency Response Peaks from Analytical Method

IV. Acoustic Field within the Fairing: Scale Model Testing Approach

A scale model of a rocket fairing and payload from the Atlas V family of rockets has been constructed.

Microphones are strategically placed in the space between the payload and the fairing. Monitoring these

microphones provides data to help verify the analytical and finite element analysis results above. The actual 1/41.3

scale model for the Atlas V 400 Extended rocket can be seen in Figure 4.

Figure 4: Scale Model Rocket

Currently, all testing is being done in an acoustics chamber lined with anechoic foam. This foam is designed to

eliminate acoustic wave reflections from the walls and floor in the testing frequency range. Doing this ensures that

acoustic waves exciting the model are planar in nature. The microphones are positioned in two planes of the model.

This allows us to make sure the frequency response peaks do not change in the „z‟ direction and are therefore purely

two-dimensional. The scale model is excited by a speaker, acting as noise from the thrusters, at a range of

frequencies around where frequency peaks are expected to be found. This data is then put into a frequency response

plot where a frequency response peak can be identified. The frequency response plots around the first two radial

modes for a 2in diameter payload can be seen below in Figures 5 and 6.

Payload Diameter (in) Peak 1 (Hz) Peak 2 (Hz)

1.5 5536 10811

1.75 6093 11986

2 6807 13452

2.25 7744 15351

2.5 8999 17889

Small Scale cavity Response Peaks

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Figure 5. First Radial Resonance, 2in Payload Figure 6. Second Radial Resonance, 2in Payload

Frequency response peaks for all the payload diameters are summarized in Table 4 below:

Table 4: Frequency Response Peaks from Physical Testing

During the first round of tests the microphone positioning for each test was not thought through very well.

However, upon looking at the finite element results it was realized that proper microphone positioning could capture

the mode shapes of the peaks being analyzed. With this knowledge the microphones are positioned so that the

mode shapes developing in the air cavity can be mapped out. The general shape of each mode stays the same for

each payload diameter. Therefore, one microphone configuration type can be used to capture the same mode for

each payload configuration. The microphone configuration used to capture the first mode (first peak) for each

payload configuration can be seen in Figure 7. The microphones are overlaid on a contour plot of what the mode

shape should look like produced from the finite element results.

Figure 7 (a): Microphone Overlay on First Radial Mode

With this configuration the microphones at positions 1 and 4 can capture the pressure maximums that oscillate from

high to low and left to right. The microphones at positions 2, 3, 5 and 6 can capture the non-oscillatory intermediate

pressure regions on the top and bottom. A similar plot can be seen in Figure for the second mode shape.

1460 1465 1470 1475 1480 1485 1490 1495 15000.02

0.03

0.04

0.05

0.06

0.07

0.08

Frequency

Vrm

s

First Radial Resonance

2900 2902 2904 2906 2908 2910 2912 2914 2916 2918 2920 29220.01

0.015

0.02

0.025

Frequency

Vrm

s

Second Radial Resonance

Payload Diameter (in) Peak 1 (Hz) Peak 2 (Hz)

1.75 1554.2 3028.3

2 1479 2910.1

2.25 1413.7 2796.9

Small Scale Cavity Response Peaks

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Figure 7 (b): Microphone Overlay on Second Radial Mode

This Configuration puts the microphones in positions 1, 3, 5 and 7 in locations to capture the pressure maximums

that oscillate from side to side to top to bottom. The microphones in positions 2, 4, 6 and 8 are left to capture the

intermediate pressure locations, which do not oscillate. Running the tests again with the microphones in the

positions stated above, contour plots of each mode can be made. Examples of these can be seen below:

Figure 8: Pressure Distribution of the First Radial Mode, 1.75in Diameter Payload

Figure 9: Pressure Distribution of the Second Radial Mode, 1.75in Diameter Payload

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x

y

RMS Pressure of First Mode From Physical Testing, 1.75in Payload

0.2

0.4

0.6

0.8

1

1.2

1.4

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x

y

RMS Pressure of Second Mode From Physical Testing, 1.75in Payload

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

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V. Acoustic Field within the Fairing: Finite Element Approach

Currently we are using finite element modeling as another method to model the acoustic pressure within the

payload fairing of a rocket. Two dimensional scale models have been developed and have been validated using

results from Blevins2. The finite element models provide another modeling method with which to compare the

analytical and experimental results. The models below show the mode shapes associated with the first radial

resonance and the first axial resonance with a 2-in payload.

Figure 10. First Radial Mode Shape, 2-in payload Figure 11. First Axial Mode Shape, 2-in payload

Frequency response characteristics are currently being developed for the air cavity as a function of the payload

diameter. The frequency response peak frequencies for all the different diameter payloads are summarized in the

table below: Table 5: Frequency Response Peaks From Finite Element Modeling

An example of the contour plots correlating to the microphone locations used in the physical scale model testing can

be seen for a 1.75in diameter payload below.

Figure 12: Contour Plot of First Mode Correlating to Physical Scale Model

Payload Diameter (in) Peak 1 (Hz) Peak 2 (Hz) Peak 1 (Hz) Peak 2 (Hz)

1.5 1595.5 3073.5 5560.5 10836.5

1.75 1520.5 2973 6131 12010.5

2 1451.5 2863.5 6851.5 13486

2.25 1389.5 2753.5 7802.5 15396.5

2.5 1333.5 2647 9077 17951

Radial Modes Axial Modes

Small Scale Cavity Response Peaks (Cavity Natural Frequencies)

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x

y

RMS Pressure of First Mode From FEA, 1.75in Payload

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

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Figure 13: Contour Plot of Second Mode Correlating to Physical Scale Model

The finite element results are now plotted to get an idea of how the different peak frequencies vary with payload

diameter. The plot can be seen in Figure

Figure 14: Peak Frequencies as a Function of Payload Diameter

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x

y

RMS Pressure of Second Mode From FEA, 1.75in Payload

0.05

0.1

0.15

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VII. Discussion

Three different types of modeling have been done, the comparison results can be seen below in Tables 6 and 7.

Table 6: Radial Mode Comparison

Table 7: Axial Mode Comparison

The results show a good correlation between the frequency data of the three models. The finite element models

matched within 2 percent of the physical testing models and the finite element models matched within 0.9 percent of

the analytical methods. All of the frequency response peaks fell in the regions of the calculated natural frequencies

of the cavity which helps validate the results even more. Also, the general shape of the mode shape captured using

the physical scale testing matched well with that of the finite element model.

Payload Diameter (in) Type Peak 1 (Hz) Peak 2 (Hz)

FEA 1520.5 2973

Physical Testing 1554.2 3028.3

% Difference 2.19208378 1.84293403

FEA 1451.5 2863.5

Physical Testing 1479 2910.1

% Difference 1.87681283 1.61424415

FEA 1389.5 2753.5

Physical Testing 1413.7 2796.9

% Difference 1.72659817 1.56385125

1.75

2

2.25

Small Scale Cavity Response Peaks (First Radial Modes)

Payload Diameter (in) Type Peak 1 (Hz) Peak 2 (Hz)

FEA 5560.5 10836.5

Analytical 5536 10811

% Difference 0.44158068 0.23559302

FEA 6131 12010.5

Analytical 6093 11986

% Difference 0.62172775 0.20419645

FEA 6851.5 13486

Analytical 6807 13452

% Difference 0.65160889 0.25243151

FEA 7802.5 15396.5

Analytical 7744 15351

% Difference 0.75258097 0.29595902

FEA 9077 17951

Analytical 8999 17889

% Difference 0.86302279 0.34598214

1.5

1.75

2

2.25

2.75

Small Scale Cavity Response Peaks (First Axial Modes)

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VIII. Full Scale Analysis

Now that the frequency data of the finite element model has been verified it will be used to investigate the

frequency response trends of a full scale model. The direct steady state finite element method will be used to look at

how the frequency response peaks vary as a function of payload diameter. Since it has been shown that the natural

frequencies of the cavity correspond to the frequency response peaks, the finite element frequency extraction will be

used to show how the number of natural frequencies varies as a function of payload diameter. The full scale model

utilizes the same material properties as well as the same constraints as the models above. The dimensions used are

from the Atlas V400 rocket which is tabulated in Appendix A. The point of this exercise is to see if the payload size

can be used as a design constraint for reducing the acoustic pressure within the payload cavity.

The direct steady state method is used to look at the frequency response peaks as a function of payload diameter.

The first two modes of the first three mode types are looked at here as an example of how they trend. Contour plots

of the modes, followed by the trend curve are seen below. Starting with the first mode type:

Figure 15: First Two Mode Shapes of Mode Type 1

Figure 16: First Mode Type Trends

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Axial Mode Type:

Figure 17: First Two Mode Shapes of Axial Type

Figure 18: Axial Mode Type Trends

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Second mode type:

Figure 19: First Two Mode Shapes of Mode Type 2

Figure 20: Second Mode Type Trends

The first mode type varies somewhat linearly and the frequencies decrease as the payload fairing diameter is

increased. The other two mode types as well as all higher mode types increase in frequency almost exponentially as

the payload diameter is increased. This could cause the amount number of natural frequencies in the payload cavity

to decrease as the payload diameter is increased. Looking at a plot of number of natural frequencies as a function of

payload diameter one can see that this is indeed the case.

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Figure 21: Number of Modes as a function of Payload Diameter

One can use the knowledge of these trends as a design parameter for reducing the acoustic pressure within the

payload fairing. Varying the payload diameter or geometry could mismatch the natural frequencies of the fairing

and payload cavity which would lessen the excitation of payload cavity modes. One could also design the cavity to

have minimal modes so there are less of them to protect against.

IX. Conclusion

This paper discusses our recent and ongoing studies on analytical, finite element and scale models for the

acoustic pressure within a rocket payload bay during launch. Some frequency response data has been tabulated

using analytic and finite element models. Both methods show a direct correlation between payload diameter and

frequency of peak response. Pressure distribution data have also been obtained from physical scale model and finite

element models. The peak frequencies and mode shapes found using the scale model testing correlate well with the

finite element model. Finally, it has been shown that the payload geometry could play a significant role in the

determining the acoustic pressure inside the fairing. This factor should be considered when determining the size of

the payload and support structure carried within the fairing.

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Appendix A: Scale Rockets

Table 1: Atlas V 400 Large Payload Fairing

Table 2: Atlas V 400 Large Payload Fairing

Table 3: Atlas V 500 Small Payload Fairing

Table 4: Atlas V 500 Medium Payload Fairing

Actual 1/20 Scale 1/50 Scale

Length (m) 12.2 0.61 0.244

Diamter (m) 4.2 0.21 0.084

Pressure (dB) 140 114 106

Atlas V 400 Large (Non Standard)

Actual 1/20 Scale 1/50 Scale 1/41.34 Scale

Length(m) 13.1 0.655 0.262 0.316895238

Diameter(m) 4.2 0.21 0.084 0.1016

Pressure(dB) 140 114 106 107.67

Atlas V 400 Extended (Standard)

Actual 1/20 Scale 1/50 Scale

Length (m) 20.7 1.035 0.414

Diamter (m) 5.4 0.27 0.108

Pressure (dB) 140 114 106

Atlas V 500 Short (Standard)

Actual 1/20 Scale 1/50 Scale

Length (m) 23.4 1.17 0.468

Diamter (m) 5.4 0.27 0.108

Pressure (dB) 140 114 106

Atlas V 500 Medium (Non Standard)

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Acknowledgments

This work is supported by the Air Force Research Laboratory, Space Vehicles Directorate (AFRL/RV). Particular

thanks are due to Jeremy Banik of AFRL/RV for his insight and continued support. Much gratitude goes to Miles

Wickersham and the other co-workers in the Advanced Dynamics lab at the School of Mines. Our thanks also go to

the anonymous reviewer of this paper for his detailed and valuable suggestions.

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