(navigation image)
Home American Libraries | Canadian Libraries | Universal Library | Community Texts | Project Gutenberg | Children's Library | Biodiversity Heritage Library | Additional Collections
Search: Advanced Search
Anonymous User (login or join us)
Upload
See other formats

Full text of "Finite-amplitude standing waves in a cavity with boundary perturbations"

—• ■ 



LIBRARY 

RESEARCH REPORTS DIVISION 
NAVAL POSTGRADUATE SCHOOL 
MONTEREY, CALIFORNIA 93940 



NPS61-82-005 



HAVAL POSTGRADUATE SCHOJL 
MONTEREY. CALIFORNIA 93940 



NAVAL POSTGRADUATE SCHOOL 



'i 



Monterey, California 




FINITE-AMPLITUDE STANDING WAVES 
IN A CAVITY WITH BOUNDARY PERTURBATIONS 

BY 

A.B. Coppens, J.V. Sanders and I. Joung 

Naval Postgraduate School 
Monterey, CA 93940 

April 1982 



FEDDOCS 
D 208.14/2: 
NPS-61 -82-005 



Approved for public release; distribution unlimited 

Prepared for: 
Chief of Naval Research 
ATTN: Dr. Logan Hargrove 
^00 Quinch Street 
rlington, VA 22217 



DUDLEY KNOX LIBRARY """ 
NAVAL POSTGRADUATE SCHOOL 

MONTEREY, CA 93943-5101 NAV al postgraduate school 

Monterey, California 



Rear Admiral J. J. Ekelund D. A. Schrady 

Superintendent Acting Provost 



The work reported herein was supported in part by the Office of 
Naaval Research, Washington, D.C. 

Reproduction of all or part of this report is authorized. 

This report was prepared by: 



LMCLASSXEIED 



SECURITY CLASSIFICATION OF THIS PAGE (Whan Data Entered) 



REPORT DOCUMENTATION PAGE 



READ INSTRUCTIONS 
BEFORE COMPLETING FORM 



r. REPORT NUMBER 

61-82-005 



,2. GOVT ACCESSION NO 



3. RECIPIENT'S CATALOG NUMBER 



4. TITLE fond Subtitle) 



Finite-Amplitude Standing Waves in a 
Cavity With Boundary Perturbations 



5. TYPE OF REPORT 4 PERIOO COVERED 

Technical Report 



5. PERFORMING ORG. REPORT NUMBER 



7. AUTHORS 

A.B. Coppens , J.V. Sanders and I. Joung 



3. CONTRACT OR GRANT NUMBERS 



9. PERFORMING ORGANIZATION NAME AND AOORESS 

Naval Postgraduate School 
Monterey, CA 93940 



10. PROGRAM ELEMENT, PROJECT, TASK 
AREA 4 *ORK UNIT NUMBERS 

61153n; RR032-01-01 
N0001481WR10171 



II. CONTROLLING OFFICE NAME AND ADDRESS 



12. REPORT DATE 



Chief of Naval Research ONR 

800 Quincy St. ATTN: Dr. Logan Hargrove 

Arlington, VA 22217 



April 1982 



13. NUMBER OF PAGES 

50 



1*. MONITORING AGENCY NAME 4 AOORESSfff al Iterant from Controlling Olttce) 



15. SECURITY CLASS, 'ot thia report) 

unclassified 



15«. DECLASSIFICATION/ DOWNGRADING 
SCHEDULE 



16. DISTRIBUTION STATEMENT (ot thia Report) 

Approved for public release; distribution unlimited 



17. DISTRIBUTION STATEMENT (ot the abatract entered In Block 20, it different from Report) 



18. SUPPLEMENTARY NOTES 



19. KEY WORDS 'Continue on reverae aide It neceeeary and Identity i>y block number) 



Finite amplitude acoustics 
Standing Waves 
Rectangular Cavity 
Boundary Perturbation 



20. ABSTRACT (Continue on reverae aide It neceaaary and identify by block number) 

Finite amplitude acoustic standing waves in a rectangular air- 
filled cavity with various wedge-shape boundary perturbations 
were studied both experimentally and theoretically. The experi- 
mental results show that geometrical perturbations alter the 
finite-amplitude behavior of the cavity and that the nature of 
these changes are in qualitative agreement with the predictions 
of the theory. However, quantitative agreement was not observed, 



do ,; 



FORM 
AN 73 



1473 



EDITION OF 1 NOV 65 IS OBSOLETE 
S/N 0102-014- 6601 i 



UNCLASSIFIED 



SECURITY CLASSIFICATION OF THIS PAGE (Whan Data Sntarad) 



[TMrT.&SSTFTF.D 



LCUW1TY CLASSIFICATION OF THIS PAGEfWiwi Dmtm Entered) 



possibly because the perturbation chosen did not satisfy all 
the assumptions of theory. 



UNCLASSIFIED 
ii 



SECURITY CLASSIFICATION OF THIS PAGE(T«i»n Dmtm Em 



ABSTRACT 

Finite amplitude acoustic standing waves in a rectangular 
air-filled cavity with various wedge-shape boundary perturbations 
were studied both experimentally and theoretically. The experi- 
mental results show that geometrical perturbations alter the 
finite-amplitude behavior of the cavity and that the nature of 
these changes are in qualitative agreement with the predictions of 
the theory. However/ quantitative agreement was not observed, 
possibly because the perturbation chosen did not satisfy all the 
assumptions of theory. 



in 



TABLE OF CONTENTS 

page 

I. INTRODUCTION 1 

II. BACKGROUND AND THEORY 2 

III. SAMPLE CALCULATION OF THE PRESSURE DISTRIBUTION WITH 5 
A WEDGE PERTURBATION 

IV. DEFINITIONS OF SOME PARAMETERS 9 

A. STRENGTH PARAMETER 9 

B. FREQUENCY PARAMETER 9 

C. HARMONICITY COEFFICIENT 10 

V. APPARATUS 11 

A. THE RECTANGULAR CAVITY 11 

B. APPARATUS 12 

VI. DATA COLLECTION PROCEDURE 13 

A. PRE-RUN AND POST-RUN INFINITESIMAL-AMPLITUDE 13 
MEASUREMENTS 

B. THE FINITE-AMPLITUDE MEASUREMENTS 14 

VII. RESULTS 16 

A. THE WEDGE AT THE CORNER OF THE CAVITY 16 

B. THE WEDGE AT THE CENTER OF THE LONG WALL 16 

VIII. CONCLUSIONS 18 
APPENDIX A: Curves 26 
APPENDIX B: Tables 36 
LIST OF REFERENCES 4 7 
INITIAL DISTRIBUTION LIST 48 



I. INTRODUCTION 

Coppens and Sanders [1,2] developed a non-linear acoustic 
model with the dissipative term phenomonologically describing the 
viscous and thermal energy losses actually encountered at the 
walls of a rectangular rigid cavity. Several researchers [3,4] 
have examined the problems and experimental results are in 
excellent agreement with the theory except when a degeneracy 
exists in the cavity. 

The purpose of this research was to examine the 
finite-amplitude behavior within a cavity for which perturbation 
effects can be accurately measured and to compare the results with 
the predictions of the model. 



II . BACKGROUND AND THEORY 

In 1975 Coppens and Sanders [2] formulated a perturbation 
expansion for the non-linear acoustic wave equation with a dissi- 
pative term describing the measured absorption properties and the 
measured resonance frequencies for standing waves within a real, 
fluid filled, rigid-walled cavity. This model predicts that, when 
a near-degeneracy exists, geometrical perturbations provide a 
mechanism whereby a nearly degenerate mode can affect the 
finite-amplitude behavior of the cavity. 

The non-linear wave equation of the viscous fluid [2] is, 

(c o 2 Q 2 +3*/3t) p/p o c o 2 =9 2 /3t 2 [(u/c o ) 2 +^( Y -l) (p/p o c Q 2 ) 2 ] 

where c Q 2 = (3p/9p) (adiabatic), p is the equilibrium 
density of the fluid, p the acoustic pressure, u the magnitude of 
the acoustic velocity, Y the ratio of heat capacities , and <£, an 
operator describing the physical processes for absorption and 
dispersion. The term on the right can be interpreted as a 
distribution of virtual sources created by the self-interaction 
of the standing waves. 

Pressure standing waves in a rectangular cavity of dimension 
L x , Ly, L z have the form 

p cos k x x cos k y y cos k z z cos u)t 
where 

k x = jtt/l x j = 0,1,2,... 

k y = £Tr/L y £ = 0, 1, 2, . . . 

k z = mTr/L z m = 0,1,2,... 

and j+£+rn^0 



If the cavity is being driven near resonance, the contributions of 
non-resonant terms are negligible with respect to the resonant 
terms; the acoustic field has the form 

oo 

P = Z P n (2) 

n=0 n 



where 



2 

P„/P_c =MR cosnk x cosnk y cosnk z sin(nwt-Kf) ) 
n u u n x y" z n 



M is the Mach number | u/c | , R]_=l, and R n is the relative 
amplitude of the n-th standing wave. By substituting (2) into 
(1), a set of coupled, non-linear, transcendental equations is 
obtained , 

r n ~^- 
R (r? 5 }^ -e ) = NMBO cos 6 h l R. R i 003 } (<$> .-$ ■) 
n sin r n n ti n 2 . , j ^i-j sin *j n-y 



COSi 



■] 



j =1 n+j j sin r n+j r j' 

where N = 1 for axial, 2 for tangential, and 3 for oblique 
standing waves. 

If a geometrically-perfect rectangular cavity is driven at 
frequencies near the resonance frequency of the (0,1,0) standing 
wave, only the family members (0,n,0) will contribute 
significantly to the finite amplitude behavior. 

In this research we were interested in this behavior when a 
degeneracy exists between the (0,2,0) and (1,0,0) modes and the 
cavity is not a perfect right parallelepiped. 

If we define a perturbation parameter - as a dimensionless 
measure of the magnitude of any irregularities on the cavity 



(3) 



surfaces compared to the effective dimensions of the cavity, the 
effect of the perturbation on the standing wave p n that would 
exist in the ideal cavity can be expressed as a small correction 
p' so that the true pressure field p n ' is 

Pn' = Pn + £ P* 
For the case of interest, n denotes the (0,2,0) standing wave and 

p' is the perturbation-generated (1,0,0) standing wave of nearly 

identical resonance frequency. 



III. SAMPLE CALCULATION OF THE PRESSURE DISTRIBUTION WITH A 
WEDGE PERTURBATION 

The pressure distribution of the (0,2,0) wave has the form 

p 2 = P cos (27Ty/L y ) cos(2oot+9 2 ) (1) 

where P and 9 2 are constant determined by the driving 
conditions . 

For this example, let the equation for the perturbed boundary 
(Fig. 0) be 

x = L x (1-CA/y [(4/L y ) (y-3Ly4)] [U(y-3L/4)-U(y-L )]) 



= L x [l-t€f(y,z)] (2) 

where 



£ = A/L 
x 



f(y,z) = -(4/L ) (y-3Ly4) [U(y-3L y /4)-U(y-L y ) ] 



if v < 
U(y) = 

1 if y > 



Differentiating p 2 and f(y,z) with respect to y gives 
3p 9 /3y = -(2ttP/L ) sin(27ry/L ) cos (2oot+9 _ ) 

^ y y ^ 



and 



9f/ 9y = -(4/L y ) [U(y-3L y /4)-U(y-L )] 

-(4/L y ) (y-3L y /4) [6 (y-3L y /4 ) -6 (y-L y ) ] (3) 



and 5 (y) is the Dirac delta function. 

Now, it can be shown by standard perturbation analysis [4] that 

the perturbation correction p' must satisfy the boundary 

condition 



OpV3 x ) L = L Of/3y) (3p 2 /3y) 

x x 



Substituting, we obtain 

(3p*/3x) = A sin(2iTy/L v ) cos (2cot+6 ) (u (y-3L /4)-U(y-L ) 

x y y 

+ (y-3L y /4) [5 (y-3L y /4)-5 (y-L y ) ]} (4) 

where 

A = 8ttPL /L 2 
x y 

Equation (4) can be expanded as a Fourier series, 



(3p'/3x)_ = A cos(2cot+9~) I [A cos (nnry/L ] +b m sin(imry/L )J (51 

Li z _ in y in y 

x m=0 ■* 



Only Aq is needed to determine the first order correction term 
due to the degeneracy between the (0,2,0) and (1,0,0) waves: 



A = 1/27T 



The perturbation correction p' is 



2 2 

p' = (Ac o -/ttL x 4o> ) Q 100 sinT 100 cos (ttx/L x ) cos ( 2ait+6 2 +T ioo ) (6) 



where the phase angle T is given by 



tanr l00 = Q 1Q0 [1- ( f 10Q /2f ) 2 ] 



The pressure at the microphone position x=L x can be written as 
follows (with 20)/c o =477/Ly) , 



- -r^ • 



p mic = ? 2 + £ P 

= P[cos(2cot+9 2 ) + (£/2tt ) Q 100 sinT 100 cos(2o)t+9 2 +T 100 l] (7) 

The total pressure amplitude at the microphone will be 

I Pmicl = P\/(l+B cos ^ioo) 2 + (B sin t 100 )2 C8)_ 

where 

B = (=/2tt2) q 1Q0 sin t 100 (9a) 

B can also be expressed in terms of A Q for a wedge anywhere on 
the wall x=L x by 

B - (e A /tt) Q 100 sin t i00 (9b) 



where A is calculated for any arbitrary position and wedge 
dimensions by the same method as developed above. 

[In reference [5], the author made a error in formulating Equation 
(2); he used y instead of (y-aL x ) and got an incorrect value of 



IV. DEFINITION OF SOME PARAMETERS 
Several parameters will be defined for the purpose of 
simplifying the mathematical formulation and elucidating the 
physical content of the equations. 

A. STRENGTH PARAMETERS: S 

S = MflQ 
where 3 = (1/2) (1+y) and Q]_ is the quality factor of the driven 
(0,1,0) standing wave. The strength parameter characterizes the 
strength of the finite-amplitude interaction. It is interesting 
to note that S is one half the Goldberg number. Since the 
microphone sensitivity S m (obtained with a B&K 4220 pistonphone) 
is known, 

S = MgQi = 7.07 x 10" 3 VxQi 
where 

M = "\f2 V 1 /pc 2 S m , 3 = 1.2 for air, 

P = 1.293 kg/m 3 , c Q = 345 m/s 

V}_ = RMS output voltage of the first harmonic component, 
and 

Q^ is the quality factor of the driven (0,1,0) standing 

wave . 

B. FREQUENCY PARAMETER: F n 

F n (f) = Q n [l-(f n /f)2] 
where f is the driving frequency, and f n and Q n are the 
resonance frequency and quality factor for the (0,n,0) wave. 
If F n <l, the n^ harmonic of the driving frequency lies 
within the half-power frequencies of the resonance curve for the 
( 0, n, ) wave . 



For reasonably large values of Q]_, 
Fl (f) = 2 Q L (f-fxJ/fx 

C. HARMONICITY COEFFICIENT: E(n) 
E(n) = (f n -nf 1 )/nf 1 

E(n) characterizes how well the modes of a given family are 
tuned (harmonic). If | E(n)| < 4, the corresponding harmonic will be 
strongly excited. 



10 



V. APPARATUS 

A. THE RECTANGULAR CAVITY 

The cavity (Fig.l), constructed from 0.75-in. aluminum, "has 
interior dimensions 12.00 in. long, 2.50 in. high, and a width 
that can be varied between 5.50 in. to 7.00 in. in 0.25 in. 
increments. All joints were sealed with a thin layer of silicon 
grease . 

Figure 2 shows the pressure distribution for several of the 
lower modes of this cavity, and Table presents the theoretical 
eigen frequencies calculated for a cavity 12 x 6 x 2.5 in. Note 
that the (0,2,0) and (1,0,0,) modes are predicted to be 
degenerate. Experimentally, the resonance frequency of the 
(0,2,0) standing wave was about 3 Hz higher than that of the 
(1,0,0) standing wave. The various configurations of wedge- 
perturbations are shown in Fig. 3. 

The source piston is set flush with the bottom of the cavity 
as near to the wall y=0 as practical, and halfway between the wall 
at x=0 and x=L x . In this position it can efficiently excite the 
(0,n,0) family of waves without appreciably exciting the (1,0,0) 
wave. To determine the proportion of the (1,0,0) mode, an 
auxilliary driver, an (ID-30), can be inserted at Position A. 

A microphone at Position B will sense the pressure of all 
standing waves. If the microphone is placed at Position A, it 
senses the (1,0,0) wave with minimum contamination from the 
(0,2,0) wave and at Position C it senses the (0,2,0) wave with 
minimum contamination from the (1,0,0) wave. 

11 



B. APPARATUS 

A block diagram of the apparatus is shown in Fig. 4. A GR 
1161 -A coherent decade frequency synthesizer is used to produce a 
driving signal precise to within +0.001 Hz. This signal is 
applied to a 2120 MB power amplifier which, in turn, drives the 
shaker. The motion of the piston was continuously sensed by an 
Endevco Model 2215 Accelerometer mounted within the piston, and 
the output of the accelerometer was observed on a Model 130BR HP 
oscilloscope and measured on a HP 400D Voltmeter. A Schlumberger 
spectrum analyzer was used to measure the harmonic distortion in 
the piston motion. 

The output of the B&K 1/4-in. microphone (with matching 
preamplifier B&K 2810) was fed into three devices; (1) an HP 400D 
VTVM to measure overall voltage level, (2) a Schlumberger spectrum 
analyzer to display the spectrum of the waveform, and (3) two HP 
302A wave analyzers to measure the amplitudes of the first two 
harmonics of the pressure waveform. 



12 



VI . DATA COLLECTION PROCEDURE 

Since the resonance frequencies of the cavity were observed 
to vary with time, the system was allowed to warm-up for at least 
one hour prior to data collection. The piston was then driven at 
the maximum amplitude to be expected during the run, and the 
harmonic content of the accelerometer output was analyzed and the 
piston was adjusted until the second harmonic of the accelerometer 
output was at least 50 dB below that of the fundamental. Figure 5 
shows typical results for the percent second harmonic in the 
acceleration with the piston driven at a rather large amplitude. 
Finite-amplitude measurements were limited to those frequencies 
for which V 2 /V 1 <0.01. 

There are three steps necessary for collecting accurate data; 
(1) pre-run infinitesimal-amplitude measurements, (2) finite- 
amplitude measurements, and (3) post-run infinitesimal-amplitude 
measurements . 
A. PRE-RUN AND POST-RUN INFINITESIMAL-AMPLITUDE MEASUREMENTS 

During these measurements, the piston was driven with the 
accelerometer output less than 0.1 V. Observation of spectra of 
the pressure waveforms obtained at these low amplitudes showed 
that the amplitudes of all overtones were at least 60 dB below 
that of the fundamental. The resonant frequencies f n and the 
quality factors Q n for the first few (0,n,0) waves were 
determined from 



13 



ff n = (f u +f e )/2 

Q n = f n /(f u -f e ) 
where f u and f^ are the upper and lower half-power frequencies. 
The resonance frequency and Q of the (1,0,0) wave were determined 
by the same procedure but with the cavity driven at port A. 
The harmonicity coefficients were then calculated from 

E n = (f n - nf l)/ n fl' f° r (0,n,0) wave 

= ( f ioo~f 2 ) /^2' f° r t ^ rie (1/0,0) wave 

where fioo ^ s t ^ ie resonance frequency of the (1,0,0) wave. 
The time at which each f u and f . were taken was recorded. 

B. THE FINITE AMPLITUDE MEASUREMENTS 

Throughout this portion of the run, the strength parameter S 
was maintained constant by adjusting the driving voltage applied 
to the piston; with the microphone in Position B, the HP 302A wave 
analyzer was set to the driving frequency and the driving voltage 
adjusted to keep the amplitude of the fundamental of the pressure 
waveform V]_ constant as the frequency was changed. The 
frequency was increased 0.5 Hz steps through +6 Hz about the 
resonance frequency of the fundamental. At each driving 
frequency, the harmonic content of the microphone output (usually 
up to the fourth harmonic) was measured with the second HP 302A 
wave analyzer set on AFC mode. The time of each measurement was 
recorded . 



14 



To determine the frequency parameter at the time the harmonic 
content was measured, f n at the instant of the measurement of 
V n was estimated by interpolation of the f n found in the pre- 
and post-run procedures. Figure 6 shows the drift in f n is 
approximately linear in time. The results were presented as 
V n /V]_ vs F]_(f) for a given strength parameter and 
perturbation. 



15 



VII. RESULTS 

The results are presented in graphical form in Appendix A and 
in tabular form in Appendix 3. 

Figure 7 shows the excellent agreement between theory and 
experiment obtained when there are no perturbations so that only 
one family of waves is excited. Some discrepancies were always 
observed when the driving frequency was far from the fundamental 
resonance frequency (for | F^j >2). In this region, the piston 
had to be driven hard to keep V-]_ constant, causing significant 
second harmonic to appear in the piston waveform (as shown by 
Fig. 5 ) , thereby introducing a linear ly -gener ated second harmonic 
into the cavity. 

For all cases studied, the pressure was calculated following 
the methodology of Sect. III. 

For the boundary perturbation of Fig. 3, the results are 
plotted in Figs. 8 through 15. The thin line is the theoretical 
prediction in the absence of any perturbation and the thick line 
includes the perturbation correction. The circles are the 
experimental results. Along the edge of each figure is the 
information about the cavity configuration. 

A. THE WEDGE AT THE CORNER OF THE CAVITY 

Figures 3 through 11 show the results for a wedge at the 
corner of the cavity. The wedge used are those of Fig. 3(a)-(d) 
respectively. 

3. THE WEDGE AT THE CENTER OF THE LONG WALL 

Figures 12 through 15 show the results for the same wedges 
[Fig. 3(e) -(h)] but now located at the center of the long wall. 



16 



For Fig. 12 and 13, A Q = 0.177 with & for Fig. 13 twice that for 
Fig. 12. For Fig. 14 and 15, A Q = 0.500 with A the same as for 
Fig. 12 and 13. 



17 



VIII. CONCLUSIONS 

A. The experimental apparatus and procedures are capable of 
providing data sufficiently precise to verify the prediction of 
the theory in the absence of geometrical perturbation if the range 
of driving frequencies is restricted so that the absolute value of 
the frequency parameter is less than 2. 

B. To verify the correction for a geometrical perturbation, 
it is useful (1) to have a sufficiently large correction to the 
unperturbed prediction and (2) that the sign of the correction 
change for an absolute value of the frequency parameter less than 
2. 

C. The experimental results show that geometrical 
perturbation alters the finite amplitude behavior of the cavity, 
and that the nature these changes are in qualitative agreement 
with the predictions of the theory. However, quantitative 
agreement was not observed. 

D. Sources of the difficulties in obtaining good agreement 
might be (1) the inability to experimentally satisfy conditions of 
B above, and (2) the higher order terms neglected in developing 
the theory may not all be small. For example, terms of order 
higher than first order in Eq. ( 4) of Sect. II and terms of order 
higher than Aq and all b m in Eq. (5) of Sect. IV. 



18 



V 

\ 



V 



3L 




x 



Fig. 0. Geometry of the wedge perturbation. 



19 



cc 



PS 

o 



H 



o 



E- 

o 

Cm 



U 
H 

s 




on 



6" 



Piston Position-*0 



(a) 



12" 



-=> v 





b) 



O 



(d) 



>Fig. 2 



(a) Cavitv Orientation, (b) Pressure Nodes for 
(c) For (0,2,0), (d) For (1,0,0). 



0,1,0) , 



21 



a) 



3" 



1/8" 





b) 



3" 



1/4" 





(c) 



6" 



1/8" 





(d: 



6" 



1/4" 




(h: 



Fig. 3. Perturbation Configuration 



22 






Eh 


H 


Eh 


2 


'-O 


Z 


>n 


rH 


u 


Cfl 


r-H 


ex 






w 


• 


'X 




CN u 


! ° 


(X 




1 u 


fa 





M 

en 

■H 

c 

_^ 
u 



rH 



•H 

fa 





OS 


fa 


fa 1 


Q 


CS3 


< 


H 


1 u 

' fa 


fa 



23 



r 



; .:- ; 



I...! - I? : " 
t 

. ; ; 1 a — 



-I 



! 



! i 



I I 



I : I 



J 1 : l. 

L- I 



i i 



4~ 



.... ,. 



i t-j- 



— _ — , — «. 



j u 



. i 



■U 



I_L 









U... 



— <: — - 



•(- 



»•« 






<*? 



Figure 5 
Typical harmonic distortion in output of the accelerometer . 



24 



i i 



I • 






1 

L 






! i 






H 



i 
I 



I 






r. 



: 



• 



\ ■ ■ ; - •, 



• 



--1 






pimtf 



• •> 



I i.J 



1_: 



"U{- !-l--i-p 



TTi tttttt 

■■4— LJii. 



..4. 



i__ 



— &- 



Figure 6 

Typical time dependence of the resonance frequency of 
the lowest standing wave (0,1,0) . 



25 



APPENDIX A 



2" 




•H 
fa 



29 



CO 

•H 
fa 




pc\sc 



30 




tm 



31 



---7-t-r- 



li 



i I 



I ! 



— r— , 



1 r*N's 

!*H 11: 







v^ 



32 




wife 



33 






en 

■H 

&4 




O -. 1 £• 

u » ■ "i m ; 

c <t,^,' 8' 



p4* 



34 



! ! 
! : 



I ! ■ 



rJ 



. : 



—L— ^j ._ 



::.:. *■: ' :j :: . : ::. : | 



, -j - -■- j- 



' iz — r.- 




a 



::::. t :: 



4- 



— i 



r 



i 



! i 



i- 



<» .. o 
• : ml u I 

IP 



V-7* 

• C1J.1? 






n 






::•!;::::. 



I -, .-,. ,ti . 



-T— 



:::::::. 



5SB 



-4 



X<* •»'•« i 



: 

III 






35 



•H 



T" 




1 

! 
1 

i 


? • 




^ 



36 




idr 



37 



APPENDIX B 



39 



TABLE 



ode 


Frequency 


010 


565 


020 


1131 


030 


1697 


100 


1131 


110 


1205 


120 


1599.5 



Hz) 



Mode 



Freauencv (Hz) 



130 


2039.4 


140 


2529.0 


150 


3032.0 


111 


2980.5 


101 


2 7 5 7.5 


Oil 


2926.4 



41 



TABLE I (Fig. 7) 

mode time f f f 
u "-L r 



010 565.77 562.38 564.33 195.40 

020 3 1131.84 1127.79 1129.81 278.69 O.lOxlO -2 

°30 6 1698.27 1693.52 1695.90 356.51 0.17xl0 _2 

100 9 1131.87 1127.59 1129.73 264.20 -0.74xl0 -4 

v^lldB time f v 2 y 3 y 4 p 2/Pi p 3/Pi p 4/P> Fp 

12 558.5 -50.0 -70.5 -70.5 0.0113 0.0011 0.0011 -4.19 

13 559.0 -47.9 -72.0 -69.0 0.0143 0.0009 0.0013 -3.85 

14 559.5 -46.9 -73.8 -67.8 0.0160 0.0014 -3.50 

15 560.0 -46.0 -72.6 -67.5 0.0179 0.0015 -3.15 

16 560.5 -44.8 -71.2 -69.7 0.0204 0.0011 0.0012 -2.81 

17 561.0 -43.7 -69.7 -73.5 0.0233 0.0012 -2.47 

18 561.5 -42.4 -67.8 -76.5 0.0269 0.0014 -2.12 

19 562.0 -41.0 -65.7 -75.9 0.0316 0.0018 -1.78 

20 562.5 -39.4 -63.2 -82.5 0.0380 0.0025 -1.43 

21 565.0 -37.8 -60.3 -85.0 0.0457 0.0034 -1.09 

22 563.5 -35.9 -56.6 -74.9 0.0569 0.0052 — -0.74 

23 564.0 -34.0 -52.2 -67.5 0.0708 0.0007 0.0015 -0.40 

24 564.5 -32.2 -47.6 -60.6 0.0876 0.0148 0.0033 -0.05 

25 565.0 -31.1 -45.0 -54.5 0.0994 0.0200 0.0067 0.29 

26 565.5 -31.6 -43.9 -52.9 0.0933 0.0226 0.0081 0.64 

27 566.0 -33.3 -46.7 -56.0 0.0772 0.0164 0.0056 0.98 

28 566.5 -35.3 -51.0 -63.2 0.0610 -0.0100 0.0025 1.33 

29 567.0 -37.6 -55.5 -70.8 0.0484 0.0060 0.0010 1.67 

30 567.5 -39.8 -60.0 -73.6 0.0365 0.0036 2.02 

31 568.0 -42.1 -64.3 -74.7 0.0279 0.0022 2.36 

32 568.5 -44.1 -68.5 -74.2 0.0221 0.0013 2.71 

33 569.0 -46.2 -69.9 0.0174 0.0011 3.05 

34 569.5 -47.2 -68.7 -67.0 0.0156 0.0013 0.0016 3.40 

35 570.0 -48.0 -65.4 -67.0 0.0141 0.C019 0.0010 3.14 

36 570.5 -48.9 -65.1 -67.2 0.0127 0.0020 0.0016 4.09 

37 571.0 -49.8 -65.2 -67.5 0.0115 0.0019 0.0015 4.43 

38 571.5 -50.8 -65.4 -68.2 0.0103 0.0019 0.0014 4.78 

39 572.0 -51.7 -65.8 -68.6 0.0093 0.0018 0.0013 5.12 

40 572.5 -52.5 -66.3 -67.4 0.0084 0.0017 0.0015 3.47 

41 573.0 -54.7 -65.1 -66.7 0.0065 0.0020 0.0016 5.31 

mode time f f „ f Q E 
u L r 

010 44 566.28 563.37 564.32 194.03 

020 47 1132.74 1128.67 L130.71 277.82 0.94xl0" 3 

030 50 1699.77 1694.97 1697.37 353.84 0.17xl0 -2 

100 53 1132.80 1128.48 1130.64 262.03 -0.61xl0" 4 



43 



TABLE II (Fig. 8) 



mode time 



u 



Q 



010 
020 

030 

100 



568.27 

1132.37 

1701.89 
1132.44 



565.28 
1127.62 

1696.79 

1127.71 



566.77 
1129.99 

1699.34 

1130.07 



139.81 
238.25 

333.14 

238.76 




-0.31x10-2 

-0.57x10-3 

0.69xl0~ 4 



V 1 =-lldB tire 



V, 



V. 



V, 



V P i 



V P 1 



V p i 



FP 



12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 



561.3 
561.8 
562.3 
562.8 
563.3 
563.8 
564.3 
564.8 
565.3 
565.8 
566.3 
566.8 
567.3 
567.8 
568.3 
568.8 
569.3 
569.8 
570.3 
570.8 
571.3 
571.8 
572.3 
572.8 
573.3 
573.8 
574.3 
574.8 



-44.3 
-43.2 
-41.7 
-40.2 
-38.5 
-36.4 
-34.3 
-32.8 
-32.5 
-33.5 
-35.1 
-36.9 
-37.9 
-38.6 
-39.2 
-40.3 
-41.5 
-42.4 
-44.3 
-45.5 
-46.9 
-48.1 
-49.3 
-50.6 
-51.0 
-51.8 
-52.5 
053.8 



-69.9 
-69.2 
-68.3 
-66.8 
-65.2 
-62.9 
-60.3 
-57.2 
-55.0 
-53.5 
-51.8 
-50.9 
-51.2 
-52.8 
-55.3 
-58.7 
-61.7 
-64.3 
-67.3 
-70.8 
-75.0 
-76.9 
-75.2 
-73.5 
-71.7 
-71.1 
-70.3 
-72.3 



-71.3 
-72.2 
-74.0 
-80.0 



•72.0 
-67.8 
-62.5 
-60.9 
-60.8 
-64.6 
-69.2 
-75.0 
-78.5 



-69.1 
-68.0 
-66.5 
-66.0 
-65.7 
-66.7 



0.0216 
0.0247 
0.0293 
0.0347 
0.0422 
0.0537 
0.0684 
0.0813 
0.0841 
0.0750 
0.0624 
0.0507 
0.0452 
0.0419 
0.0391 
0.0343 
0.0300 
0.0271 
0.0218 
0.0188 
0.0160 
0.0140 
0.0122 
0.0105 
0.0100 
0.0092 
0.0084 
0.0073 



0.0011 
0.0012 
0.0014 
0.0016 
0.0020 
0.0025 
0.0034 
0.0049 
0.0063 
0.0075 
0.0092 
0.1002 
0.0098 
0.0082 
0.0061 
0.0041 
0.0029 
0.0022 
0.0015 
0.0010 
0.0006 
0.0005 
0.0006 
0.0007 
0.0009 
0.0010 
0.0011 
0.009 



0.0009 
0.0014 
0.0024 
0.0032 
0.0032 
0.0021 
0.0072 



0.0012 
0.0014 
0.0017 
0.0018 
0.0019 
0.0016 



-3.95 

-3.61 

-3.26 

-2.92 

-2.58 

-2.24 

-1.90 

-1.56 

-1.22 

-0.87 

-0.53 

-0.19 

0.15 

0.49 

0.83 

1.78 

1.52 

1.86 

2.20 

2.54 

2.88 

3.22 

3.57 

3.91 

4.25 

4.59 

4.93 

5.27 



mode 



time 



u 



Q 



010 


42 


020 


45 


030 


48 


100 


51 



568.82 565.95 567.38 

1133.50 1128.70 1131.10 

1703.98 1698.69 1701.34 

1133.85 1129.10 1131.47 



197.49 





235.40 


-0.32x10 


321.25 


-0.48x10 


238.00 


0.33x10 



-2 
-3 
-3 



44 



TABLE III (Fig. 9) 



mode 



tune 



u 



'J 



010 
020 
030 
100 



567.85 
1128.61 
1697.95 
1128.77 



565.01 
1124.81 
1693.16 
1124.87 



566.43 
1126.71 
1695.39 
1126.82 



199.52 
296.58 
354.31 
288.63 




-0.54x10 
-0.23x10 
-0.96x10 



-2 
-2 
-4 



V^-lldB 



time 



V. 



v. 



v. 



p /p 

2 / 1 



V p i 



V p i 



r ~v 



mode 



010 
020 
030 
100 



12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 

time 



561.0 
561.5 
562.0 
562.5 
563.0 
563.5 
564.0 
564.5 
565.0 
565.5 
566.0 
566.5 
567.0 
567.5 
568.0 
568.5 
569.0 
569.5 
570.0 
570.5 
571.0 
571.5 
572.0 
572.5 
573.0 
573.5 

£ 
u 



-37. 
-36. 
■34. 
-32. 
-30. 
-29. 



•30.8 
-33.3 
-35.5 
-37.3 
-38.9 
-40.3 
-41.7 
-42.5 
-42.6 
-43.1 
-43.3 
-43.8 
-44.5 
-45.2 
-45.6 
-46.2 
-46.8 
-47.5 
-47.5 
-48.2 



-66. 

-64. 

-62. 

-60. 

-57. 

-55. 

-55. 

-55. 

-55.8 

-58.0 

-60.5 

-61.5 

-61.7 

-61.3 

-60.8 

-62.2 

-63.7 

-65.9 

-68.0 

-69.6 

-70.9 

-71.7 

-72.7 

-73.3 

-73.5 

-77.0 



40 

43 
46 
49 



567.99 
1128.94 
1698.43 
1129.10 



565.11 
1125.12 
1693.67 
1125.21 



-76.0 
-73.4 
-74.0 
-74.7 
-71.8 
-70.2 
-71.4 
-76.0 
-80.0 
-81.3 
-80.3 



566.55 

1127.03 
1696.05 
1127.15 



0.0468 

0.0550 

0.0684 

0.0876 

0.1096 

0.1189 

0.1023 

0.0772 

0.0599 

0.0484 

0.0403 

0.0345 

0.0293 

0.0266 

0.0263 

0.0250 

0.0243 

0.0230 

0.0213 

0.0196 

0.0186 

0.0174 

0.0162 

0.0150 

0.0150 

0.0139 



0.0017 — 

0.0021 

0.0026 

0.0034 
0.0047 
0.0059 
0.0062 
0.0060 
0.0058 
0.0045 
0.0033 
0.0030 
0.0029 
0.0031 
0.0032 
0.0028 
0.0023 
0.0018 
0.0014 
0.0012 
0.0010 
0.0009 



0.0006 
0.0008 
0.0007 
0.0007 
0.0009 
0.0011 
0.0010 
0.0006 



-3.84 

-3.49 

-3.14 

-2.79 

-2.44 

-2.09 

-1.74 

-1.39 

-1.04 

-0.67 

-0.34 

0.00 

0.36 

0.71 

1.06 

1.41 

1.76 

2.11 

2.46 

2.81 

3.16 

3.51 

3.86 

4.21 

4.56 

4.91 



196.79 

295.42 
355.76 
289.76 




-0.54x10 
-0.21x10 
0.11x10 



-2 

-2 
-3 



45 



TABLE IV (Fig. 10) 



mode 



time 



Q 



010 
020 
030 
100 



566.19 
1128.86 
1696.18 
1128.46 



563.20 
1124.95 
1691.38 
1124.62 



564.70 
1126.91 
1693.78 
1126.54 



189.30 
287.70 
353.24 
293.29 




-0.2x10 



-2 



-0.18x10 
-0.32x10" 



-3 



_-} 



V 1 =-lldB 



time 



V, 



V, 



V. 



V s ! 



V p i 



V p i 



FP 



mode 



12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 



time 



558.7 
559.2 
559.7 
560.2 
560.7 
561.2 
561.7 
562.2 
562.7 
563.2 
563.7 
564.2 
564.7 
565.2 
565.7 
566.2 
566.7 
567.2 
567.7 
568.2 
568.7 
569.2 
569.7 
570.2 
570.7 
571.2 
571.7 
572.2 

f 
u 



-42.8 
-41.8 
-40.8 
-39.7 
-38.3 
-36.7 
-34.9 
-33.0 
-30.7 
-28,9 
-28.5 
-30.4 
-32.2 
-33.7 
-35.6 
-37.0 
-39.6 
-42.1 
-43.7 
-44.5 
-45.1 
-45.6 
-46.1 
-46.6 
-47.0 
-47.2 
-47.7 
-48.6 



-69.8 
-68.7 
-67.9 
-66.9 
-65.5 
-63.5 
-61.0 
-57.8 
-53.7 
-50.2 
-47.5 
047.2 
-48.5 
-52.2 
-56.4 
-58.4 
-59.7 
-61.5 
-63.7 
-65.5 
-66.9 
-68.0 
-69.0 
-69.8 
-70.4 
-70.6 
-71.3 
-63.9 



0. 
0. 
0. 
0. 
0. 
0. 
0. 
0. 
0. 
0. 
0. 
0. 
0. 
0. 
0. 
0. 
0. 
0. 
0. 
0. 
0. 
0. 
0. 
0. 
0. 
0. 
0. 
0. 



0257 
0288 
0324 
0367 
0434 
0519 
0638 
9794 
1041 
1274 
1334 
1078 
0873 
0733 
0592 
0501 
0314 
0279 
0232 
0221 
0197 
0186 
0176 
0167 
0158 
0155 
0146 
0132 



0.0011 
0.0013 
0.0014 
0.0016 
0.0019 
0.0024 
0.0032 
0.0046 
0.0074 
0.0110 
0.0150 
0.0155 
0.0133 
0.0087 
0.0054 
0.0043 
0.0037 
0.0030 
0.0023 
0.0019 
0.0016 
0.0014 
0.0013 
0.0012 
0.0011 
0.0011 
0.0010 
0.0023 



0.0010 
0.0019 
0.0030 
0.0040 
0.0042 
0.0026 
0.0010 



010 
020 
030 
100 



42 
45 
48 
51 



566.35 
1129.43 
1697.04 
1129.12 



563.33 
1125.5:. 
1692.26 
1125.28 



564.84 
1127.48 
1694.65 
1127.20 



186.79 

1287.84 

353.94 

294.16 




-0.19x10 
-0.78x10 
-0.24x10 



-2 

-4 
-3 



46 



TABLE V (Fia. 11) 



mode 



time 



010 
020 
030 
100 




3 
6 
9 



u 



566.90 
1127.19 
1695.31 
1126.76 



"L 



563.86 
1123.28 
1690.47 
1122.90 



565.38 
1125.23 
1692.89 
1124.83 



2 



185.80 
288.08 
349.27 
291.41 




-0.49x10 
-0.19x10 
-0.36x10 



-2 
-2 
-3 



V^-lldB 



time 



V. 



V- 



V. 



V*i 



V p i 



V*i 



FP 



12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 



559.4 

559.9 

560.4 

560.9 

561.4 

561.9 

562.4 

562.9 

563.4 

563.9 

564.4 

564.9 

565.4 

565.9 

566, 

566, 

567, 

567, 

568.4 

568.9 

569.4 

569.9 

570.4 

570.9 

571.4 

571.9 

572.4 

572.9 



.4 
,9 

.4 

,9 



-39.5 

-37.9 

-36.4 

-34.5 

-32.4 

-30.3 

-28.7 

-28.9 

-30. 

-32. 

-34. 

-36. 

-37. 

-38.4 

-39.3 

-39.7 

-40.6 

-40.7 

-41.6 

-44.0 

-47.3 

-49.1 

-49.9 

-50.0 

-50.3 

-50.4 

-50.4 

-51.1 



-66.9 
-65.3 
-63.9 
-61.8 
-59.1 
-55.7 
-55.6 
-50.8 
-50.0 
-49.7 
-50.9 
-54.3 
-59.3 
-65.0 
-69.7 
-72.2 
-74.5 
-75.4 
-73.5 
-71.5 
-70.9 
-71.1 
■71.6 
-71.8 
-72.3 
-72.4 
-72.4 
-71.6 



-78.0 
-72.7 
-68.2 
-65.3 
-62.5 
-60.3 
-61.4 
-67.0 
-75.2 



0.0376 

0.045 

0.054 

0.0672 

0.0856 

0.109 

0.131 

0.127 

0.101 

0.081 

0.065 

0.055 

0.048 

0.043 

0.039 

0.037 

0.033 

0.030 

0.030 

0.022 

0.015 

0.012 

0.011 

0.011 

0.011 

0.011 

0.011 

0.010 



0.0016 
0.0019 
0.0023 
0.0029 
0.0039 
0.0058 
0.0059 
0.0103 
0.0112 
0.0116 
0.0101 
0.0068 
0.0038 
0.0020 
0.0012 
0.0009 



0.0009 

0.0010 

0.0010 



0.0008 
0.0014 
0.0017 
0.0027 
0.0034 
0.0030 
0.0016 



-3.99 

-3.66 

-3.33 

-3.00 

-2. 66 

-2.33 

-2.00 

-1.67 

-1.34 

-1.01 

-0.67 

-0.34 

-0.01 

-0.32 

-0.65 

0.98 

1.32 

1.65 

1.98 

2.31 

2.64 

2.97 

3.31 

3.64 

3.97 

4.30 

4.63 

4.96 



mode 



time 



u 



010 
020 
030 
100 



42 
45 
48 

51 



566.95 
1127.31 
1695.54 
1126.98 



563.96 
1123.40 
1690.73 

1123.12 



565.46 

1125.37 
1093.14 
1125.05 



189.24 
288.18 
351.57 
291.39 





-0.49x10 
-0.27x10 
-0.27x10 



-2 
-3 
-3 



TABLE VI (Fig. 12) 



node 



time 



u 



~L 







010 
020 
030 
100 



564.94 
1137.75 
1699.31 
1137.85 



562.04 
1133.55 
1694.34 
1133.56 



563.49 
1135.65 
1696.82 
1135.71 



194.44 
270.78 
341.28 
264.96 





0.77x10 
0.28x10 
0.48x10 



-2 
-2 
-4 



V 1 =-lldB tine 



V, 



V- 



V. 



p /p 



T> /D D /p 

^3 A 1 '4 /F l 



FP 



12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 



557.5 
558.0 
558.5 
559.0 
559.5 
560.0 
560.5 
561.0 
561.5 
562.0 
562.5 
563.0 
563.5 
564.0 
564.5 
565.0 
565.5 
566.0 
566.5 
567.0 
567.5 
568.0 
568.5 
569.0 
569.5 
570.0 
570.5 
571.0 
571.5 



■53.7 
-53.3 
•52.6 
-52.0 
-51.2 
-50.4 
-49.6 
-49.0 
-48.2 
-47.7 
-47.1 
-46.3 
-45.5 
-44.4 
-42.7 
-40.5 
-38.8 
-37.0 
-35.5 
-34.1 
-32.8 
-32.3 
-33.3 
-35.4 
-37.8 
-40.2 
-42.0 
-43.6 
-46.0 



-73.9 
-74.2 
-74.6 
-75.1 
-85.3 
-75.3 
-74.9 
-73.3 
-71.6 
-69.7 
-67.2 
-64.7 
-61.3 
-58.1 
-55.0 
-52.7 
-51.8 
-53.3 
-55.5 
-56.6 
-56.9 
-57.9 
-61.3 
-65.5 
-67.0 
-66.6 
-66.2 
-65.6 
-68.6 



-76.2 
-76.3 
-77.3 
-80.2 
-85.0 



0.0073 
0.0077 
0.0083 
0.0039 
0.0098 
0.0107 
0.0117 
0.0126 
0.0139 
0.0146 
0.0158 
0.0172 
0.0188 
0.0215 
0.0260 
0.0335 
0.0407 
0.0504 
0.0596 
0.0700 
0.0813 
0.0861 
0.0072 
0.0603 
0.0457 
0.0349 
0.0282 
0.0236 
0.0178 



0.0007 



0.0008 

0.0009 

0.0012 

0.0015 

0.0021 

0.0031 

0.0044 

0.0063 

0.0082 

0.0092 

0.0077 0.0015 
0.0060 0.0016 
0.0052 0.0015 
0.0051 0.0024 
0.0045 0.0034 
0.0031 0.0038 
0.0019 0.0035 
0.0016 0.0037 
0.0017 0.0035 
0.0017 0.0034 
0.0019 0.0033 
0.0013 0.0028 



-4.30 

-3.96 

-3.62 

-3.27 

-2.93 

-2.58 

-2.24 

-1.89 

-1.55 

-1.21 

-0.86 

-0.52 

-0.17 

0.17 

0.52 

0.86 

1.21 

1.55 

1.89 

2.24 

2.58 

2.93 

3.27 

3.61 

3.96 

4.30 

4.65 

4.99 

5.34 



mode 



time 



u 



X 



o 



010 
020 
030 
100 



43 
46 
49 
52 



565.47 
1138.89 
1700.94 
1139.00 



562.56 
1134.67 
1695.60 
1134.76 



564.01 
1136.78 
1698.27 
1136.88 



193.75 
269.44 
318.33 
268.20 




0.78x10 
0.37x10" 
0.81x10 



-2 



.-? 



-4 



48 



TABLE VII (Fig. 13 



mode 



time 



010 
020 
030 
100 



u 



561.56 
1137.45 
1694.09 
1137.58 



"L 







558.62 
1133.15 
1689.09 
1133.26 



560.09 

1135.30 
1691.59 
1135.42 



190.57 
264.33 
338.39 
263.01 





0.14x10 
0.67x10 
0.11x10 



-1 
-2 

-3 



V =-HdB 



time 



V. 



V. 



V, 



v> /p 



V P 1 



V p i 



FP 



12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 



554.1 
554.6 
555.1 
555.6 
556.1 
556.6 
557.1 
557.6 
558.1 
558.6 
559.1 
559.6 
560.1 
560.6 
561.1 
561.6 
561.1 
562.6 
563.1 
563.6 
564.1 
564.6 
565.1 
565.6 
566.1 
566.6 
567.1 
567.6 



-53.6 
-53.0 
-52.7 
-52.4 
-52.0 
-51.6 
-51.3 
-50.9 
-50.3 
-49.9 
-49.2 
-48.7 
-48.2 
-47.6 
-47.0 
-46.3 
-45.4 
-44.5 
-43.8 
-43.1 
-42.4 
-41.2 
-40.0 
-38.6 
-37.0 
-35.2 
-33.2 
-31.5 



-74.7 
-74.7 
-74.7 
-75.5 
-76.1 
-76.7 
-77.5 
-77.5 
-76.9 
-75.1 
-72.8 
-70.3 
-67.5 
-65.0 
-62.7 
-60.4 
-57.2 
-55.0 
-54.4 
-55.0 
-57.5 
-60.9 
-64.5 
-67.9 
-70.6 
-73.7 
-71.0 
-66.3 



0.0074 

0.0079 

0.0082 

0.0085 

0.0089 

0.0093 

0.0097 

0.0101 

0.0108 

0.0114 

0.0124 

0.0130 

0.0139 

0.0148 

0.0158 

0.0172 

0.191 

0.0211 

0.0230 

0.0248 

0.0269 

0.0309 

0.0355 

0.0417 

0.0501 

0.0617 

0.0776 

0.0944 



0.0007 
0.0007 
0.0007 
0.0006 
0.0006 
0.0005 
0.0005 
0.0005 
0.0005 
0.0006 
0.0008 
0.0011 
0.0015 
0.0020 
0.0026 
0.0034 
0.0049 
0.0063 
0.0068 
0.0063 
0.0047 
0.0032 
0.0021 
0.0014 
0.0010 
0.0070 
0.0010 
0.0017 



-4.21 

-3.87 

-3.53 

-3.19 

-2.85 

-2.51 

-2.17 

-1.83 

-1.48 

-1.14 

-0.80 

-0.46 

-0.12 

0.22 

0.56 

0.90 

1.24 

1.58 

1.92 

2.26 

2.61 

2.95 

3.29 

3.63 

3.97 

4.31 

4.65 

4.99 



mode 



time 



u 







010 
020 
030 
100 



43 


561.94 


559.01 


560.47 


191.35 





46 


1138.34 


1134.01 


1136.18 


261.97 


0.14x10 


49 


1695.35 


1690.34 


1692.85 


338.37 


0.68x10 


52 


1138.44 


1134.07 


1136.25 


260.07 


0.69x10 



-1 

-2 
-4 



49 



TABLE VIII (Fig. 14) 



mode 



time 



u 



Q 



010 





564.94 


561.90 




563.42 


185.40 





020 


3 


1137.18 


1126.60 




1128.89 


246.64 


0.18xl0" 2 


030 


6 


1703.17 


1698.34 




1700.75 


352.63 


0.62xl0" 2 


100 


9 


1139.73 


1135.13 




1137.43 


247.16 


0.76xl0" 2 


V,—-lldB 


time 


f v 2 


V 3 


V 4 


V P 1 


P /P 


P 4 /P L FP 



12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 



557.5 
558.0 
558.5 
559.0 
559.5 
560.0 
560.5 
561.0 
561.5 
562.0 
562.5 
563.0 
563.5 
564.0 
564.5 
565.0 
565.5 
566.0 
566.5 
567.0 
567.5 
568.0 
568.5 
569.0 
569.5 
570.0 
570.5 



-50.4 
-50.0 
-49.5 
-48.9 
-48.4 
-47.7 
-47.0 
-46.3 
-45.5 
-44.6 
-43.6 
-42.5 
-41.5 
-40.6 
-40.0 
-40.8 
-42.0 
-42.6 
-41.2 
-39.3 
-37.0 
-35.0 
-33.0 
-32.6 
-33.6 
-35.5 
-37.4 



-74.2 
-74.2 
-74.2 
-74.3 
-74.5 
-74.3 
-73.9 
-73.0 
-71.1 
-68.5 
-65.3 
-62.1 
-59.3 
-57.0 
-54.7 
-54.0 
-54.1 
-54.7 
-54.7 
-55.3 
-57.4 
-58.7 
-58.0 
-57.5 
-58.5 
059.9 
-63.5 



0.0107 
0.0112 
0.0120 
0.0127 
0.0136 
0.0147 
0.0158 
0.0172 
0.0188 
0.0209 
0.0236 
0.0267 
0.0299 
0.0331 
0.0355 
0.0324 
0.0282 
0.0265 
0.0310 
0.0387 
0.0501 
0.0631 
0.0794 
0.0832 
0.0741 
0.0596 
0.0480 



0.0010 
0.0013 
0.0019 
0.0028 
0.0039 
0.0050 
0.0065 
0.0071 
0.0070 
0.0066 
0.0065 
0.0061 
0.0048 
0.0041 
0.0045 
0.0047 
0.0042 
0.0036 
0.0024 



0.0009 
0.0011 
0.0010 
0.0011 
0.0020 
0.0025 
0.0010 
0.0009 



■3.92 

■3.59 

■3.26 

■2.93 

•2.59 

-2.26 

-1.93 

■1.60 

-1.27 

-0.94 

-0.61 

-0.28 

0.05 

0.38 

0.71 

1.04 

1.37 

1.70 

2.03 

2.36 

2.69 

3.02 

3.35 

3.68 

4.02 

4.35 

4.68 



mode 



time 



u 



O 



010 
020 
030 
100 






564.94 


561.93 


563.43 


3 


1131.26 


1126.70 


1128.98 


6 


1703.35 


1698.48 


1700.92 


9 


1139.38 


1135.27 


1137.58 



187.00 
247.64 
349.34 
247.14 





0.19x10 
0.63x10 
0.76x10 



-2 
-2 

-2 



50 



TABLE VIII (Fig. 15 



mode 



time 







E 



010 
020 
030 
100 



561.20 
1125.25 
1700.26 
1143.88 



558.27 
1120.66 
1695.60 
1139.80 



559.73 
1122.96 
1697.93 
1141.84 



190.78 
244.76 
364.21 
280.07 




0.31x10 
0.11x10 
0.17x10 



-2 
-1 

-1 



V^-lldB 



time 



V. 



V. 



V. 



p 2 /p x 



p /p 



V p i 



FP 



12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 



553.7 
554.2 
554.7 
555.2 
555.7 
556.2 
556.7 
557.2 
558.2 
558.7 
559.2 
559.7 
560.2 
560.7 
561.2 
561.7 
562.2 
562.7 
563.2 
563.7 
564.2 
564.7 
565.2 
565.7 
566.2 
566.7 
567.2 



-52.7 
-52.4 
-51.8 
-51.4 
-51.0 
-50.5 
-49.7 
-49.1 
-47.8 
-47.0 
-46.0 
-45.0 
-43.6 
-42.6 
-41.0 
-40.5 
-41.0 
-43.2 
-46.0 
-49.0 
-51.7 
-53.6 
-54.4 
-53.7 
-52.4 
-50.6 
-49.7 



-77.8 
-78.2 
-78.2 
-78.4 
-79.0 
-79.5 
-79.5 
-78.9 
-75.7 
-72.7 
-70.0 
-67.3 
-65.3 
-64.0 
-61.7 
-69.4 
-68.5 
-58.3 
-58.8 
-58.9 
-57.9 
-55.9 
-53.9 
-51.5 
-50.0 
-50.7 
-67.8 




0.0082 

0.0086 

0.0091 

0.0095 

0.0100 

0.0107 

0.0116 

0.0124 

0.0145 

0.0158 

0.0178 

0.0200 

0.0234 

0.0264 

0.0316 

0.0335 

0.0316 

0.0247 

0.0178 

0.0126 

0.0092 

0.0074 

0.0068 

0.0073 

0.-085 

0.0105 

0.0116 



0.0005 - 



0.0011 
0.0015 
0.0019 
0.0022 
0.0029 
0.0012 
0.0013 
0.0043 
0.0041 
0.0041 
0.0045 
0.0057 
0.0072 
0.0094 
0.0085 
0.0105 
0.0115 



,0013 
.0021 
,0027 
.0029 
.0035 
.0056 



-4.17 

-3.83 

-3.49 

-3.15 

-2.81 

-2.47 

-2.13 

-1.79 

-1.12 

-0.78 

-0.44 

-0.10 

0.24 

0.58 

0.92 

1.25 

1.59 

1.93 

2.27 

2.61 

2.95 

3.29 

3.63 

3.97 

4.30 

4.64 

4.98 



mode 

010 
020 

030 
100 



time 



41 
44 
47 
50 



a 



561.45 

1125.52 
1700.77 
1144.24 



f L 



558.48 
1120.92 
1696.08 
1140.17 



559.96 
1123.22 
1698.43 
1142.21 



108.60 
243.91 
362.29 
280.92 




0.29x10 

0.11x10 
0.17x10 



-2 

-I 

-1 



51 



REFERENCES 

1. Coppens, A.B., and Sanders, J.V., "Finite -Amplitude Standing 
Waves in Rigid Walled Tubes," J. Acoust . Soc . Am., 43 , 

pp. 516-529, March 1968. 

2. Coppens, A.B., and Sanders, J.V., "Finite-Amplitude Waves 
Within Real Cavities," J. Acoust. Soc. Am., 5_3, pp. 1133-1140, 
December 1975. 

3. Aydin, M. , "Theoretical Study of Finite-Amplitude Standing 
Waves in Rectangular Cavities With Perturbed Boundaries," 
Thesis, Naval Postgraduate School, Monterey, California, 
1978. 

4. Coppens, A.B., "Finite Amplitude Standing Waves in Rectangular 
Cavities With Degeneracies and Weakly-Perturbed Walls," J. 
Acoust. Soc. Am., 64, Suppl. 1, S34(A), 1978. 

5. Kuntsal, E., "Finite Amplitude Standing Waves in Rectangular 
Cavities with Perturbed Boundaries," Thesis, Naval 
Postgraduate School, Monterey, California, 1978. 



53 



INITIAL DISTRIBUTION LIST 

No. Copies 

1. Defense Technical Information Center 2 
Cameron Station 

Alexandria, VA 22314 

2. Library, Code 0142 2 
Naval Postgraduate School 

Monterey, CA 93940 

3. Department Library, Code 61 1 
Department of Physics 

Naval Postgraduate School 
Monterey, CA 93940 

4. Professor James V. Sanders, Code 61Sd 5 
Department of Physics 

Naval Postgraduate School 
Monterey, CA 93940 

5. Professor Alan B. Coppens , Code 61Cz 5 
Department of Physics 

Naval Postgraduate School 
Monterey, CA 93940 

6. Dr. Logan Hargrove, Code 421 6 
Office of Naval Research 

800 Quincy St. 
Arlington, VA 22 217 

7. CDR II Bok Joung 5 
425 Waupelani Dr #207 

State College, PA 16801 



55 



DUDLEY KNOX LIBRARY 



3 2768 00471965 8