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Calhoun: The NPS Institutional Archive

Theses and Dissertations Thesis Collection

1967-12

A Theoretical Investigation of Finite Amplitude

Standing Waves in Rigid Walled Cavities

Ruff, Paul G., III

Monterey, California. Naval Postgraduate School

http://hdl.handle.net/10945/31904

H«AfPOSTGRADUATE SCH^nttJP. 99W0

A THEORETICAL INVESTIGATION OF

FINITE AMPLITUDE STANDING WAVES

IN RIGID WALLED CAVITIES

by

Paul Gray Ruff III

Lieutenant, United States NavyA.B., St. Benedict's College, 1960

Submitted in partial fulfillment of the

requirements for the degree of

MASTER OF SCIENCE IN PHYSICS

from the

NAVAL POSTGRADUATE SCHOOLDecember 1967

Km <W c ,/

ABSTRACT

The Coppens- Sanders perturbation solution for the one- dimensional

nonlinear acoustic wave equation with dissipative term describing

the viscous and thermal energy losses encountered in a rigid walled,

closed tube with large length-to-diameter ratio was extended to include

sixth order terms. The solution was then investigated to determine

the region of validity. Computer programs were written to evaluate

and graph the resulting waveforms. Available experimental results

were compared with the theoretical predictions and good correlation

was found to exist in the region of low Mach numbers. This agreement

was found to gradually deteriorate as the Mach number was increased.

A Fourier synthesis approach is also presented and the leading terms

of the first ten harmonics are derived.

LIE

NAVAL POSTGRADUATE SCHOOrMONTEREY, CALIF. 93940

LIST OF SYMBOLS

".ACVWif)^co

k

a = Lagrangian coordinate measured from piston

A = (peak) acceleration amplitude of piston

B/A = parameter of nonlinearil

c = phase velocity in the tube

- (df/dfy

/z @fô

£ = k - io{

K = m 7[7L

L = Lagrangian coordinate of rigid end of tube

m = normal mode of tube most strongly excited by input frequency

M = U-,/c peak Mach number of first-order solution11 o

p.P = acoustic pressure, equilibrium pressureo

p .

= j-th frequency component of the n-th order perturbation pressure

V ,= (infinitesimal) pressure amplitude at the rigid end of the tube

H = resonance parameter (see Eq. 36)

u = particle velocity

u = particle velocity of the n-th order perturbation solution

u .= j-th frequency component of u

nj n

OC = infinitesimal-amplitude attenuation constant

jS = 1 + \ (B/A)

= c /c = ratio of specific heatsp v

O = dissipation parameter (see Eq. 19)

O-j = value of o for (angular) frequency u)-j

tjy)= phase parameter

^ = particle displacement

f)9Q= instantaneous density, equilibrium density

Wn = forcing term (see Eq. 16)

CO = 27C f = (angular) driving frequency

CO = infinitesimal-amplitude resonance frequency

D = (pc V\~ yD. = D for frequency Cdj

^r = operator for body forces (see Eq. 9)

,~ j~j; ~Z\ = D'Alambertian in Lagrangian coordinates (one

&t spatial dimension)

V. = (peak) amplitude of voltage output of accelerometer

V = (peak) amplitude of voltage output of microphone

S. = V. /A = accelerometer sensitivityA A o '

S™ = V../P = microphone sensitivityMM r J

10

TABLE OF CONTENTS

Section Page

1. INTRODUCTION 13

2. THEORETICAL DEVELOPMENT 19

Perturbation Approach

Fourier Synthesis Approach

Cavity Response Development

3. APPLICATIONS ^X

4. RESULTS AND COMPARISONS 43

5

.

BIBLIOGRAPHY 52

LIST OF TABLES

Table Page

4.1 Tabulated Theoretical and Experimental Values of iythe Harmonic Content in the Waveforms.

4.2 Tabulated Theoretical Values of the Harmonic Phase / QAngles.

B.l Symbology Pertinent to the Computer Program FINAMPI g»

B.2 Symbology Pertinent to the Computer Programs QCURVES, -«QCURC, PHAMP, and PHAMPC

.

LIST OF ILLUSTRATIONS

Figure Page

A.l Definition of Geometrical Relationships for 56Both the Fourier Synthesis Approach and the

Perturbation Approach.

C.l Pressure Waveshapes for M = 0.004, AcO ~ O. 123

C.2 Pressure Waveshapes for M = 0.005, A<^> =0- 124

C.3 Pressure Waveshapes for M = 0.006, A cJ = °"^25

C.4 Pressure Waveshapes for M = 0.009, £±cD~C12£

C.5 Q- curves of the Fundamental, M = 0.004 127

C.6 Q-curves of the Second Harmonic, M = 0.004 128

C.7 Q-curves of the Third Harmonic, M = 0.004 129

C.8 Q-curves of the Fourth Harmonic, M = 0.004 130

C.9 Q-curves of the Fifth Harmonic, M = 0.004 131

CIO Q-curves of the Sixth Harmonic, M = 0.004 132

C.ll Q-curves of the Fundamental, M = 0.005 133



C.14 Q-curves of the Fourth Harmonic, M = 0.005 136

C.15 Q-curves of the Fifth Harmonic, M = 0.005 137

C.16 Q-curves of the Sixth Harmonic, M = 0.005 138

C.17 Q-curves of the Fundamental, M = 0.009 139



C.20 Q-curves of the. Fourth Harmonic, M = 0.009 142

C.21 Q-curves of the Fifth Harmonic, M = 0.009 ^3C.22 Q-curves of the Sixth Harmonic, M = 0.009 -^kk

C.23 Phase Dependence of Various Harmonics on AU? jAe

ACKNOWLEDGEMENTS

The generous aid and encouragement of Professors Alan B. Coppens

and James V. Sanders is gratefully acknowledged.

11

1. INTRODUCTION

The thermodynamic equation of state of a fluid for reversible

acoustic processes can be written as

*' fori *£*(/+)*

or, for an ideal gas,

t^L - (f/P Yft

17'/ (i-2)

where is the ratio of specific heats. The continuity equation for

one-dimensional wave propagation in Lagrangian coordinates is

The force equation in the absence of dissipative mechanisms has the

form

f° ^ u '

(l - 4)

It is possible by straightforward combination to obtain the equation

£.ff .2./. .x\-ir

(1.5)

for a fluid obeying Eq. 1.1 or

—r -Co(

/v j— ) —; * o (i.6)

for an ideal gas.

13

If interest is restricted to acoustic processes for which /&cl -Lj

Eqs . 1.5 and 1.6 may be approximated by

o't - hiWwhich (upon differentiation with respect to time) becomes the non-

dissipative wave equation

£1 fdi )

z

u " - flit (^where

^/ Z (1.9)fi-i+-i(fy

n 2= .i

2J- ^

2

A more general form of Eq. 1.4 is

where ^^ is an operator which generates all forces, other than that

arising from the gradient of the pressure field, which are active in

the system of interest [4], If Eq. 1.10 is used instead of Eq. 1.4,

Eq. 1.7 becomes

îf*^* /£fgf (1.11)

2~

In the Navier-Stokes case

'( csja

where "ft and i(o are the shear and bulk viscosity coefficients, respectively,

jf*(ft+T'Jfr

14

An important contribution to the solution of Eq. 1.11 for the

case of Eq. 1.12 was made by Fay [5]. He investigated the changes

in the steepness of the wavefront in a periodic, finite-amplitude

plane wave of infinite extent in a viscous medium. By means of

Fourier analysis techniques he was able to show that the nonlinearity

of the pressure density relationship results in accumulating distortion

of the waveform with shifts of energy from the lower to the higher

frequency components of the wave. Because the classical Navier-Stokes

absorption coefficient is proportional to the square of the frequency,

the higher frequency components are attenuated more rapidly. Fay's

solution is valid in a region far from the source where the rates of

harmonic growth due to nonlinearity and the rates of attenuation due

to viscosity are tending to balance leading to a waveform of relatively

stable shape over distances corresponding to many wavelengths.

Fox and Wallace [l] investigated the same problem as Fay using a

graphical analysis technique. Their result is essentially the same

as Fay's. Keck and Beyer [6] developed a perturbation analysis technique

for periodic plane progressive waves of infinite extent in a viscous

medium with which they generate terms through sixth order. The wave

equation is the same as that treated by Fay, but the solution is valid

only near the source.

Weston [20] has presented a linearized wave equation for the

propagation of monofrequency sound in tubes. He assumes that the wave-

fronts are planar except near the walls. The viscous and heat-conduction

losses in the boundary layer are assumed to be the dominant dissipation

mechanisms in the fluid. Except in this boundary layer, the wavefronts

are independent of a radial coordinate. This suggests that the boundary

15

layer loss can be replaced by an equivalent absorptive process active

throughout the volume of the cavity [4], More will be said about this

later.

Saenger and Hudson [14] present a simple description of periodic

shocks at resonance using a two part solution. Their solution is

based on the assumption of the applicability of the linearized acoustic

equations everywhere except in the region of the shock where the Rankine-

Hugoniot shock conditions are assumed. They consider the case in which

the piston oscillates at the fundamental frequency. Compressive viscosity

is ignored but the effects of shear viscosity and heat conduction in the

boundary layer are considered. Their solution remains finite only because

of these considerations.

Betchov [13] also postulates the existence of the shock wave and

constructs a solution at resonance based on a continuous and a discontinuous

part. For an inviscid fluid the amplitude at resonance is found to be

finite and to be determined by nonlinear effects. The effect of wall

friction is discussed and it is suggested that this could modify the

solution significantly.

Weiss [7] has applied finite-difference techniques to the nonlinear

inviscid acoustic equations and has shown that repeated reflections at

the rigid boundaries tend to promote the development of a discontinuity

in the velocity profile. The absence of any dissipative mechanisms,

however, eliminates the possibility of any steady-state standing wave

patterns

.

While most investigations have been confined to traveling waves,

a few have considered standing waves.

16

Solutions of the nonlinear wave equation have been obtained for

the case of finite-amplitude standing waves in a closed tube, where

the tube is driven at one end by a piston source. This work was done

by Keller [21] using the Lagrangian formulation and assigning an

unrealistic value of -1 to the adiabatic exponent Y . Due to neglect

of any dissipative mechanisms, his expression for the particle velocity

becomes infinite if the piston frequency equals a natural frequency

of the tube.

A detailed theoretical analysis based on the nonlinear acoustic

equations including the effects of compressive viscosity and shear

viscosity in the boundary layer has been done by Chester [8]. He

successfully predicted asymmetries in the finite-amplitude standing

waves. Unfortunately his results are not easily compared to experi-

mental results.

Coppens and Sanders [4] developed an extension of the Keek-Beyer

perturbation approach wherein wall losses as well as bulk losses are

included. Their solution is applied to the case of finite-amplitudes

standing waves in rigid walled cavities. The results yield information

concerning the amplitudes and phases of the Fourier components of the

waveform, and indicate the importance of the type of absorptive process

on the resulting waveform.

The purpose of this thesis is to extend the Coppens-Sanders

perturbation approach. In part this is accomplished by a direct

application of their method to obtain sixth-order terms. Information

concerning the response of the cylindrical cavity to the sinusoidal

excitation is extracted from this solution by some algebraic manipulations

17

A Fourier synthesis approach to the problem is presented and from It

the leading terms of the first ten harmonics are derived. Information

similar to that obtained from the perturbation approach is acquired

and both results are compared to the experimental results obtained

by Beech [19].

IS

2. THEORETICAL DEVELOPMENT

In this section both a perturbation approach and a Fourier-synthesis

approach to the solution of the one- dimensional nonlinear, dissipative

acoustic wave equation are presented. These approaches are formulated

for a rigid-walled, closed, cylindrical cavity with large length-to-

diameter ratio. The solutions obtained are for finite-amplitude standing

waves. Manipulations are also presented which enable each approach to

yield resonance response information which is convenient for comparison

with experimental results.

Perturbation Approach

Assume a perturbation series ,5" 2± $y\ such that the n-th

term is of order n-1 in the Mach number. Substituting this into Eq. 1.8

and collecting terms of equal order yields the set of equations

where

(2.2)

-n<l

If Uy) - ^~Z tty)J , where the summation over j is understood to

encompass only those frequencies included in each Wy) , Eq. 2.1 becomes

^(Ot+ Z)

;

- V- = fa(2.3)

i

19

In order to utilize this formulation, it is first necessary

to obtain the form of D j . For the case of finite amplitude sound

in a duct, Weston [20] has presented a wave equation valid for the

propagation of monofrequency sound. Weston's equation may be written

in Eulerian form as

where

I—If~

S-$)(**f*['**('-?»'*]

(2.4)

(2.5)

and

jO (uniform) cross-sectional area

\J( = perimeter of area S

C«J = (angular) frequency of the propagating waveform

V = kinematic shear viscosity of the fluid

V thermometric conductivity of the walls

This may be rewritten as

(2.6)

When dissipation processes are included, it is usually assumed, explicitly

or tacitly, that they are weak. Under this assumption, and for small

Mach number, the derivatives (yQ&H and ( vdX, ) t maY be interchanged.

This also holds for the pair (%£)a and (/ofc)x. Thus Eq. 2.3 must be

solved with2.

(2.7)

where o -t represents the value of 6 for a frequency CuJ.

20

SCDÎ Xr.~Si&)u^(2.8)

2. »-/

The final result is the approximate wave equation

Eq. 2.8 was solved [4] for a uniform tube of length L excited

at end a = by a piston driven with constant acceleration A at aJ o

frequency C*s, and terminated at a = L by a rigid cap. In this

solution, the boundary conditions were given by

ujo^t) = (Jo/a) AUu(ajt - e,)

(2.9)

where t/ will be defined later.

The lowest order solution of Eq. 2.8 is

Ui ~ un ~ (^y(^n) (2.10)

where

Ull . A ,,<(**-*)_e ê~ '*> UL -lit

e ~ -aand k = k -io^ is the complex propagation constant.

The solution for UL , may be rewritten as

Ultm U„

£ I Co*fi °((l- *.) M*v k U' o-jjOc^ cot

+[M*/>*(L-Cl) Cc^l. kU-*.)]^ cotf

where

(2.12)

n*@)(^*''t + ji.*ti)~yz

21

and

// M* kl>

Eqs. 2.12 and 2.13 will be rewritten in terms of (jj^ , the

resonant frequency which maximizes the pressure amplitude -^ ,

associated with IL , , at the rigid end of the cavity.

c l >* J

(2>15)

a) = a)u + Aa)

The integer m represents the normal mode most strongly excited by the

input frequency. For o(L«l and Atu«7~>(J'and T// are

U, * 7 Cc^ e' (2.16)

(2.17)

From this point on, higher-order solutions are obtained by first

making the approximation

(J-u

JJ= JAv/<(l-*.) W>t (2.18)

which is Eq. 2.12 with terms of //£ omitted. The procedure (presented

in Appendix A), is then merely an iteration process.

In view of later discussions it is advantageous to point out that

Eq. 2.18 may be rewritten in terms of the associated acoustic pressure

by merely making the substitutions

(2.19)

22

This is equivalent to ignoring terms of order /^ in Eq. 1.10, and

converts the calculated variable from velocity to pressure, a more

commonly measured quantity.

In terms of the pressure, the solution presented by Coppens and

Sanders is

(2.20)

+ • « •

23

where

(<<ijf' lurir^tr***^*^ (2 - 21)

M * V,,/ (2.24)

and M is the peak Mach number of the first order solution.

Using the same scheme, the sixth order terms were derived,

The solution for x)g which is to be added to Eq. 2.20 is

% = ft* 2k(L-».){2$f[(WSk

+i(-'>2> -i

2-3h *i(Hwfy}

24

- <** 4 KCL-i,) UqZ) (2. 1 4,4, 5)4

i

+(2AlU)i+(/,22,S4)*

^frMVh ^(22SfQL

25

The entire perturbation solution herein developed, which is the

sum of Eqs. 2.20 and 2.25, will be discussed in some detail after

the Fourier synthesis approach has been presented.

Fourier Synthesis

Beginning with the approximate wave equation Eq. 2.8 and assuming

the total solution is composed of the known classical solution £-

satisfying

3^ ,t

rV*> , k <> & f- Alt - O

and some other portion J defined by

(2.26)

f*ft*f (2.27)

then

~~ 2 Sau \Jcu Jo. J

Define the phase velocity Cj to be Cj - C \ /~dj ) anc* Eq. 2.28 may

(2.28)

be written

2^0

[cj ** c ** LjJ+w(2 . 29)

hi i /Hi + ifV

Assume \ may be written as a Fourier series

— =T)C Cc^K(L'&) C&4-.faojt+<jfc ) (2.30)

^ >,-0

26

and

i^ = Xc Co*- #&-*J Co* a)t (2.31)

and let

CO

(2.33)

(J£f=23^ Oo^lcSt tR,) (2.32)

where /iy^ and />) will be obtained later. Combination of Eqs. 2.29

through 2.32 yields

[(cVz-(k)j^^*^ *~ fat * &> ')

- X) X»° J<^ °1 £(L~*) jl*J (foot + 0y,°)

We impose the conditions that

hM = 3-* Co* T* *(£-*>)(2.34)

and

so that Eq. 2.33 becomes

= &&&*(»»*+%)Introduction of the phase angles "C/Vj of Eq. 2.17 allows Eq. 2.35 to be

o

(2.35)

written

^Vp^'-«»). §-' &&- 3*c~fa4+Z) <2 - 36

27

In order to simplify the manipulations to obtain Jj^ and /->,

define

+ Xc co* XU-*.) e<^cot < 2 - 37 >

and

^tic^ML'^cruiyuol+fa ) ,yi*l (2 * 38)

Thus

and

oo

rCombining Eqs. 2.40 and 2.36 gives

iL. 4l! V ° / J J /I \

0+1 C<H &-*

jr/7 X-jX-h+j C<*(hart +<p-hij ~4-j)

= l -h-l

28

(2.39)

(2.40)

oo (2.41)

It is convenient to shift to complex notation at this time. Let

X, Co* 4, - X° c<* 4>,° + Xc

(2.42)

A/ a^ 4 » * X,e

J^ 4>,

°

so that

X,e' ' - X° e y- Jfc (2.43)

and

X„ e'"^ - )f„

6«•

<V\ -*>i (*•«*)

Then Eq. 2.41 becomes

-£**1 ^'V+iSxjKje^***(2 ' 45>

If the equivalent pressure equation is desired, make the

substitution

x~n = - ?/F - zfc^c (2.46)

where M is the amplitude of the Mach number for the particular harmonic

Define

so that Eq. 2.52 becomes

CO

= a^v'^y^"^(2.48)

}

r 1

29

Examination of Eq. 2.48 and Eqs . 2.20 and 2.25 shows that the two

methods are equivalent since expansion of the term

iZLî^-i etc**-i+*i)

yields the leading terms of each harmonic derived in the perturbation

scheme, while the other portion of Eq. 2.48 contains the correction

terms

.

Recall that in both the perturbation approach and the Fourier

synthesis approach the leading term of the fundamental was calculated

to be t].J^ K(L-cl) Co*- &> v . This is of order one in the Mach number.

It is also a known experimental fact that the fundamental remains

essentially unchanged with respect to Mach number. This suggests that

the phase angles associated with the corrective terms might cause the

corrective terms to sum to approximately zero. It then becomes

conceivable that a solution composed of only leading terms could

yield significant results. In order to investigate this possibility,

Eq. 2.48 was used to obtain the leading terms of the first ten harmonics

The result is

(2.49)

+j**

30

+iM+W]\

IV

31

+1

+2<2M1iZ*h*£(*2*i<Z*)i

32

, 9

33

+j e* tow-*) kffi[(wsAV,%»),.

34

/c

fd+£W,3frA%«h +i (22234589,01

4- v , , >> , >j ? Jfc •+ ^ {223W6 7/oJfo

/iC// , > , / ,r/M +£(2222*4 SfroL

+ 4 (222S</S-^?/ci 4 ~ (221 Sii-i fill

to

35

> i ( 2/A 2*fI8,'<J* * 8 (&*Wl$<4i (<WZ*mJu 4 it <WMPt

*/c)lt

36

+ ^r(2222 3</SX/o) 1}/4 v

' > ' ' ' ' ' ' 'h If

J- m

where (a,b)g has been given in Eq. 2.21

37

Cavity Resonance Response

Investigation of the behavior near resonance of each approach

leads to the desirability of obtaining the response of the cavity

for each harmonic as the fundamental is swept through resonance.

It is obvious from the form of Eqs . 2.27, 2.32, and 2.55 that both

approaches yield results which may be written as sums of terms

having like factors of K(L-a) ando) t . It is also evident that each

of these harmonics is heavily dependent upon M, the Mach number, and

Au).

The algebraic process necessary to bring each of the two solutions

to a form which conveniently yields the desired information will be

derived in a general form and then applied to the perturbation

solution. It is then easily extended to the Fourier synthesis solution,

We begin with the relationship

"V?

£& ^ (j"* *4<~) ~ /4J*o Q~OJ?* )• (2.50)

I =/

At t = Eq. 2.50 becomes

V\

y*etc j^ 4c - a m^ <f.

(2.51)

The derivative of Eq. 2.50 evaluated at t = yields

V\

/^ ai cot 4c - /Co± 4 (2= 52 )

which when combined with Eq . 2.51, yields

2. / * . W

If Eq. 2.51 is divided by Eq. 2.52 the result is

(2.53)

(2.54)

€'»

38

The sum of Eqs. 2.20 and 2.25 may be written as

r" < '<=, f (2.55)

„ A Cc+J #£-*.) J^ Jtut + ^ .

for each harmonic, where j indicates the harmonic number and n

indicates the number of terms in that harmonic. It is evident

that the amplitude of each harmonic may be obtained by applying

Eq. 2.53.

In applying Eq. 2.53 to the perturbation approach the magnitude

of each harmonic was normalized to approximately one by the maximum

magnitude of a factor chosen from the leading term of that harmonic.

This normalization was done to yield a set of curves whose maxima are

of order unity. For the fundamental this normalization factor is one,

The method of choosing the normalization factor fbr higher harmonics

will be demonstrated for the second harmonic. The leading term of

the second harmonic is

(¥H -ifi * &* '«) (2.56)

Now

fl - Hk —

so that

(2.57)

2 ^£»r«>& H, < 2 - 58 >

and the leading term of the second harmonic may be written

_ / . , .

-—, Mp^ (J?ajt -/- &Z J (2.59)

39

Since

4 - V***. (2 - 60)

Eq. 2.59 may be written

M- &4- £t±* AJU***) (2 . 61)

H has as its maximumn

V^ ^ < 2 - 62 >

and the normalization factor for the second harmonic is then chosen as

(2.63)

It should be noted that these normalization factors are not the maximum

amplitudes of the various harmonics, neither are they the maximum

amplitudes of the leading terms of each harmonic; rather, they are

factors chosen to yield maxima of order unity.

These factors are given the symbol MAXA where n refers to the n-th

harmonic so that

Mm z = t~t 77 (2 - 64)

* *TnlCtyCc dot

By combining Eqs . 2.53 and 2.64 it is possible to define Q as the

normalized response of the n-th harmonic. It should be noted that A

defined in Eq. 2.53 is a function of the Mach number and &oJ.

Thus

fy(H,A*)) - SMXAti (2.65)

where n is the harmonic number.

4o

3. APPLICATIONS

Associated with each of the solutions formulated in the previous

section are three computer programs. These programs are concerned

with the pressure waveform, the Q-curves, and the phase information.

The perturbation solution has the programs FINAMPI, QCURVES, and PHAMP

connected with it while FINAMPIZ, QCURC and PHAMPC are related to the

Fourier synthesis.

The first attempted utilization of the perturbation solution

consisted of writing a computer program involving all terms through

sixth order. This program, entitled FINAMPI, was designed to compute

and graph both velocity and pressure waveforms. FINAMPI was first used

to graph the waveforms contained in Ref. [4] and has been used since

in determining for what Mach numbers the theory begins to deviate

significantly from the experimental results. The graphical output

from the computer is contained in Figs. C.l through C.4.

The program designed to compute and graph the cavity resonance

response for the perturbation solution is entitled QCURVES. This program

is a direct application of Eq. 2.65 and has been used to obtain the

curves contained in Figs. C.5 through C.22.

PHAMP, the third program associated with the perturbation solution,

was designed to compute the phase angles of each harmonic as the

fundamental was swept through resonance. It is a direct application

of Eq. 2.54 and resulted in the data contained in Table 4.2.

In an attempt both to determine the importance of the corrective

factors to each harmonic and to test the validity of the Fourier synthesis,

computer programs involving only the leading terms of the various harmonics

were written. Unfortunately a program involving all ten harmonics could

4i

not be brought to fruition because of time limitations. A second

program, FINAMPIZ, which uses the leading terms of the first six

harmonics has been used to graph the pressure waveforms also contained

in Figs. C.l through C.4.

Eqs . 2.65 and 2.54 were also applied to the Fourier synthesis

yielding the programs QCURC and PHAMPC . QCURC computes and graphs

the Q-curves for each of the harmonics while PHAMPC computes and graphs

the phase dependence of the individual harmonics on A cd .

It should be noted that of the two programs concerned with phase

dependence, only PHAMPC graphs its output. This was done because the

Q-curve programs indicated the Fourier synthesis approach produced

curves which were in much better agreement with the experimental results

than did the perturbation approach. The graphical output of PHAMPC is

contained in Fig. C.23.

In this section only a superficial description of each program

was given. A detailed description of all programs is contained in

Appendix B, and a more complete discussion of results follows in

Section 4.

42

4. RESULTS AND COMPARISONS

In this section the theoretical predictions obtained from the

applications described in Section 3 will be compared to the experimental

results obtained by Beech [19].

Before proceeding to a discussion of these results it is necessary

to attempt some qualitative evaluation of the apparent properties of

the perturbation solution. The amplitudes of the (n + l)th order terms

tend to behave as [4].

It is clear that this expression can diverge for increasing n. This

divergence, which occurs because of the dependence aiy r/-Jy^ > would

not occur if O^^Cf since the left hand side of Eq. 4.1 would then

be bounded by ~Y\(</2<f ) an<^ a ll series would converge for Mp< 2-V}

The Mach number which satisfies the expression l/pfT ~I an^ w°uld

therefore appear to be related to shock formation is M = 0.02. This is

within the same order of magnitude as the experimentally measured Mach

number at shock formation, M = 0.01.

The divergence may be remedied if it is recalled that the bulk

absorptive processes have been neglected up to this point. If the

viscous terms are retained and assumed to be additive to the wall losses

then

~>£. , 1.A-. r.

<*>

'

w- (4 - 2)

where

43

The analysis and results are the same as before with the H's and O's

redefined as

n% ~ ~2 (4.4)

^<& + -**>€

With these modifications the amplitudes of the (n + l)th order terms

now behave, for sufficiently large harmonic, as

which tends to zero with increasing n.

The modifications introduced by Eqs . 4.2 and 4.3 are important

in the perturbation expansion for those harmonics whose indices

satisfy the inequality n yS /&)£ . For the problem dealt with

in this research the theory developed without the bulk absorptive

terms is adequate as long as attention is restricted to frequencies

below about the thousandth overtone.

In comparing the theoretical and the experimental results it is

convenient to divide the discussion into three areas: the low Mach

number region, the intermediate Mach number region and the high Mach

number region.

In the low Mach number region, below M ^2 0.006, it is obvious

from Figs. C.l and C.2 that theory and experiment are in very close

agreement. In these figures of pressure waveforms the large drawing

contains the waveforms predicted from the computer programs and the

smaller inset is a drawing from an oscilloscope photograph of the

waveform observed experimentally. The dashed line in both drawings

is a sine wave which is included for reference purposes. One of the

44

main reasons for the good agreement in this region is due to the

fact that the ( //2J~, ) terms are as yet relatively small.

It should be noted here that in all the figures of pressure

waveforms, the major difference between the FINAMPI and FINAMPIZ

predictions is in the phase dependence. Recall that the development

of the reasoning behind the FINAMPIZ program rested on the possibility

that corrective terms to the generated harmonics could be ignored.

These curves lend support to that suggestion in the low Mach number

region.

The low Mach number region being discussed also includes the

first two sets of Q-curves, Figs. C.5 through C.16. It is evident

in both sets that the agreement of the experimental results with the

predictions of the Fourier synthesis approach is excellent in the

lower harmonics and deviates only slightly in the higher harmonics.

When the agreement does deviate, the experimental results in general

fall between the two theoretical predictions. This tends to indicate

that the two solutions have the exact solution bracketed. It is

evident in the QCURVES predictions for M = 0.005 that the third

harmonic curve is no longer a smooth bell-shaped curve. The reason

for this is the growth in the third harmonic of the correction terms

arising from the fifth-order solution. For instance, the ratio of

the approximate magnitude of the first corrective term to the leading

term in the third harmonic is about 0.5 at M = 0.005. This is no

longer insignificant so that the predictions based on the perturbation

approach begin to show unrealistic fluctuations.

45

Theoretically predicted amplitude information is obtained from

the two programs, PHAMP and PHAMPC . The amplitude results from these

programs together with the experimental results are contained in

Table 4.1. It should be noted that Beech's uncertainty of + 0.1 Hz

in frequency results in an uncertainty of + 0.628 radians in Figs. C.5

through C.23. A comparison of the uncertainty in amplitude (based

on the uncertainty in A<0) with the observed discrepancy between

measurement and theory for M = 0.004 and AoJ = yields

n uncertainty in discrepancyharmonic content between PHAMPC

and experiment

2 0.5 0.4

3 0.3 0.17

4 0.15 0.12

5 0.07 0.08

6 0.03 a. 07

It is evident that the uncertainties in harmonic content are consistent

with the discrepancies in harmonic content noted between PHAMPC and

experiment

.

As was stated in Section 3, PHAMPC also predicts the phase

dependence of the various harmonics. The results of this program

are contained in Table 4.2 and Fig. C.23. One thing should be noted.

The sets of phase angles predicted by PHAMPC are identical for all

Mach numbers, which is not surprising if the reader recalls that the

normalization process destroyed the Mach-number dependence of the

leading terms of each harmonic.

46

4-1

cOn 

1-4

COinco

COo O 00 <* r—

1

mB3 o

CO

pa r^ vD co CM iH r^ m CM r-l O2X

4-1

c

1-4 i—i

CJ CD

03

8 oooUcu o m

COr-- CJ>

COON CO co CM

CNin

oCM

oPm

CMi—

1

CO 1-4 !-< oi—i

CO r-l o o

u•F-l UC 1 P

cf < CU WS-i 55 5 -U

60 < •1-4 T-4

O 3 S-i 3Vj o cu

o 4-1 m coo a CM vO m r^ CTs <f m o r~- o>

• o • • • • • • * • •

o o o <r CO CO CO l^~ r>» m <* mCM i—i t—

i

1—1 CM i—i .—1 i—i i—i

CJ

•r-l

cu in o in<u o g 00 CO r^- 1—4 r^ <tXI o S-i

CO

vO CM CO co CO CM <t- m «tf 00

3 o X CO <t 1—1 i—i o in m CM 1-4 oZ r-J 1—4

4-1

X Cu <}• CU r^ Csl

m o u 00 ô t0 r^ r-» CM r^-

8 o 5-1

CO

12

01

X

cu

4-1

cou

coEuCO

cd



ccu

J!-4

CUaxw

CCO

CU

S-i

oa>

xH

CU4-1

COi—i

3XCO

H

<t

OJi—i

XCO

H

47

One other program exists for the investigation of phase and

amplitude characteristics. The computer program FOUANAL is being

introduced here since it was not derived from either theoretical

approach. FOUANAL takes data from an oscilloscope photograph and

computes the Fourier coefficients of the various terms of a Fourier

coefficients of the various terms of a Fourier expansion of the

waveform. The results from this program are also contained in

Tables 4.1 and 4.2.

It is clear that the results from FOUANAL are fairly accurate

in the lower harmonics (first, second and third) but tend to become

much worse in the higher harmonics (fourth, fifth and sixth). Judging

from the amplitude results it does not seem likely that FOUANAL would

yield accurate results in the phase angles of these higher harmonics.

Examination of Table 4.2 verifies this supposition. These erratic

results in the FOUANAL calculations may possibly be explained in the

following manner. The input data for the program come from extracting

sixty- four points from a standard oscilloscope photograph. This was

done by hand and was subject to many errors. It is possible that the

data were not given to sufficient accuracy to enable the computer to

predict accurately past the third harmonic. The most feasible suggestion

for improvement in this respect is to use a digital counting procedure

to obtain the data. It should be noted that the data in Fig. C.23

are the FOUANAL results for M = 0.004. Clearly these points are

extremely random and the uncertainty in the experimental &(*J cannot

possibly account for the entire discrepancy.

Shifting now to the medium Mach number region (COOG^M^ 0.009)

we see from Fig. C.3, for M = 0.007, that although the general agreement

is still good, deviations from the previous good agreement are becoming

48

^DON m ^D 00 m vO i—I cn i—i oo o <t r^ St <r 00 cn o m rô o 1—1 <t I

s". i-( o o r—

1

<t ^D 00 mo

CO

CCO

o o O o r—

1

-J o o o O © .—

H"3

u CO ^DCD m fa 00 \£> i-4

.

m i—

i

oX o r-~ cl- m T—1 1**. 00 en o m r~~

ou

o i—i in i—i r-~. CM o i—i <t vO 00 m3 o 1—1

60o O o i—i ,—

t

CN o o o o o i—

Xi ccj

CO

<2

S CD

co

<f

CO

X5 i—i cn i—i oo 00 vD en cn 00 IT) o m r>o o i—i m I-. as m o i—

(

<t vO 00 ir

»

o o o o O o i—i o o o o o i—

c 1

•n I uc : xi

1n

iE 1—1 CN m -^ m vO 1— CN cn o- m ^D

i isS i

B XIcd < o00 Iou § afa fa fa

co

<u1—1

00C<cd

CO

cd

jfl

fa

co6

CO

fa*

cd

CO

CD

flr-l

CO

>

CO

o•H4-1

CD

S-i

oCD

XH"3CD4-J

COi—

'

XcO

H

CN

<f

CDi—i

XCO

H

49

evident in the computer predictions. Unfortunately experimental

data for Q-curve and phase angle comparisons were not obtained

but it is not unreasonable to assume that these comparisons would

show essentially the same result as stated for the pressure waveform.

Moving into the region of high Mach numbers, M = 0.009, we find

that the pressure waveforms predicted by each program are generally

invalid. This is nearing the experimentally observed Mach number for

shock formation (M = 0.01), and since the theory is predicted on small

Mach number, pre-shock conditions it is not surprising to see these

deviations

.

Essentially the same results are observed in the Q-curve comparisons,

Figs. C.17 through C.22. The reason for the experimental amplitudes

falling off so rapidly is at present unknown. Tables 4.1 and 4.2

again reflect the serious discrepancies in the high Mach number region.

It is now possible to make some general comments concerning these

two approaches. Both the perturbation approach and the Fourier synthesis

approach have adequately described the effects of finite-amplitude standing

waves in rigid-walled cavities provided the Mach number is relatively low.

This precludes that area in the pre-shock region where significant

distortion exists. It has been demonstrated that in the region where

the approaches are fairly accurate, Q-curve information can be easily

obtained. This information is convenient for comparisons with experi-

mental results. It is also possible to obtain some information on the

phase angles of the harmonics.

50

It seems plausible now to suggest the following: Since the

Fourier synthesis approach gave the most accurate information, the

computer program involving all ten harmonics should be reviewed in

the hope that a usable program will result. This program could be

used to investigate the effects of the higher order harmonics on

the existing solution and possibly give an indication of whether

extending the present developments past six harmonics is beneficial.

Finally, after some exposure to basic communications theory, it

appears that a good grounding in this subject is mandatory if the

present theoretical line is to be pursued further. It would not be

surprising to find that a more exact solution may be obtained as a

result of the techniques used in communications theory.

"H

BIBLIOGRAPHY

1. F. E. Fox and W. A. Wallace, "Absorption of Finite AmplitudeSound Waves," J. Acoust. Soc . Am. 26, 994-1006 (1954).

2. W. W. Lester, "On The Theory of the Propagation of Plane FiniteAmplitude Waves in a Dissipative Fluid," J. Acoust, Soc. Am, 33 ,

1196-1199 (1961).

3. A. L. Thuras, R. T. Jenkins and H. T. O'Neil, "ExtraneousFrequencies Generated in Air Carrying Intense Sound Waves,"J. Acoust. Soc. Am. 6, 173-180 (1935).

4. A. B. Coppens and J. V. Sanders, "Finite-Amplitude StandingWaves in Rigid-Walled Tubes," Scheduled to be Published in

J. Acoust. Soc. Am. (1968).

5. R. D. Fay, "Plane Sound Waves of Finite Amplitude," J. Acoust.Soc. Am. 3, 222-241 (1931).

6. W. Keck and R. T. Beyer, "Frequency Spectrum of Finite AmplitudeUltrasonic Waves in Liquids," Phys . Fluids 3, 346-352 (1960).

7. N. 0. Weiss, "The Development of a Shock From Standing Wavesof Finite Amplitude in an Isentropic Fluid," Proc. Camb . Phil.

Soc. 60, 129-135 (1964).

8. W. Chester, "Resonant Oscillations in Closed Tubes," GraduateAeronautical Laboratories, California Institute of Technology(N. D.).

9. Yen Fu Bow, "Propagation of Plane Compressional Waves of FiniteAmplitude in Real Fluids," Ultrasonics Laboratory, Michigan StateUniversity (1965).

10. W. W. Lester, "A Theoretical and Experimental Study of the Propagationof Plane Finite Amplitude Waves in Real Fluids," Ultrasonics Laboratory,Michigan State University (1965).

11. A. C. Peter and J. W. Cottrell, "Investigation to Define the

Propagation Characteristics of a Finite Amplitude Acoustic PressureWave," NASA (1967).

12. J. S. Mendousse, "Nonlinear Dissipative Distortion of ProgressiveSound Waves at Moderate Amplitudes," J. Acoust. Soc. Am. 15_, 51-54

(1953).

13. R. Betchov, "Nonlinear Oscillations of a Column of Gas," Phys. Fluids

1, 205-212 (1958).

52

14. R. A. Saenger and G. E. Hudson, "Periodic Shock Waves in

Resonating Gas Columns," J. Acoust. Soc . Am. _32, 961-970 (1960).

15. G. E. Hudson, "Periodic Shock Waves in Liquid Filled Tubes,"School of Engineering and Science, New York University (1963).

16. A. Powell, "Distortion of Finite Amplitude Sound Wave," J.

Acoust. Soc. Am. 32, 886 (L) (1960).

17. L. E. Hargrove, "Fourier Series for the Finite Amplitude SoundWaveform in a Dissipationless Medium," J. Acoust. Soc. Am. 32 ,

511-512 (L) (1960).

18. D. T. Blackstock, "Convergence of the Keek-Beyer PerturbationSolution for Plane Waves of Finite Amplitude in a Viscous Fluid,"J. Acoust. Soc. Am. 39, 411-413 (L) (1966).

19. W. L. Beech, "Finite Amplitude Standing Waves in Rigid WalledCavities," Thesis, Naval Postgraduate School, Monterey, California(1967).

20. D. E. Weston, "The Theory of the Propagation of Plane Sound Wavesin Tubes," Phys. Soc. of London, _66B, 695-709 (1953).

21. J. B. Keller, "Finite Amplitude Sound Produced by a Piston in a

Closed Tube," J. Acoust. Soc. Am. 26, 253-254 (L) (1954).

22. I. Rudnick, "On the Attenuation of a Repeated Sawtooth Shock Wave,"J. Acoust. Soc. Am. 25, 1012-1013 (L) (1953).

23. I. Rudnick, "On the Attenuation of Finite Amplitude Waves in a

Liquid," J. Acoust. Soc. Am. 30> 564-567 (1958).

24. R. A. Saenger, "Periodic Shock Waves in Resonating Gas Columns,"Thesis, New York University (1958).

25. F. D. Shields, K. P. Lee, and W. J. Wiley, "Numerical Solution forSound Velocity and Absorption in Cylindrical Tubes," J. Acoust. Soc,

Am. 37., 724-729 (1965).

26. Lord Rayleigh, Theory of Sound (Dover Publications, Inc., New York,1945), 2nd ed., vols. 1 and 2, Topic 350, p. 324.

53

APPENDIX A

Coppens-Sanders Iteration Scheme

From Section 2, the approximate wave equation for finite-amplitude

standing waves in a rigid walled tube is

2 *>f — /

__ y_fL y£< J&.Z _ v^1 (A.l)

f>1

For convenience the right hand side of Eq. A.l will be written

^2

fa -ZY>; -f—t 2Zp*• (A . 2)

7 7Assume

«*, -lij J^ffM* '?"* '"'} (A - 3)

where the ^j and L/^' are complex, and the three dots in the

exponential term represent phase terms to be defined. Work is done

in one n at a time so that the n subscript may be dropped.

Expansion of the left hand side of Eq . A.l yields

where \f~uj ' was previously defined to be (y/co/ «

Thus Eq. A.l becomes

/sf/tu -J- £ & ->£ <*%'mt/.

(A. 4)

(A. 5)

for a given order n. If Eq. A. 5 is generalized in time it becomes

& ±* '&#* % jSk ""^/

54

zc

(A. 7)

(A. 8)

Substitution of Eq. A. 3 into Eq. A. 6 yields

Now each W will be of the form

Substitution yields

Additional substitution of Eq. A. 9 and Eq. A. 3 into Eq. A. 7 gives

£$[-($/+£# -<$]- <'4*//x Ci.10)

Proper manipulation of the bracketed term in Eq. A. 10 will yield the

phase relationships mentioned previously but left undefined.

First recall that

(A. 9)

co - tOp -t Au)

where

K - mr/L >

(A.U)

C^)Z

- -i-tj

and notice that

and

s ± «± J- r / / 4. * r }(A. 13)

55

c * e* (tt joC t % £L rZ )-/(A. 14)

Substitution into Eq . A. 10 yields

It is now possible to write down the now U as

A,,- — 1—*

—

' J~- - r * -2J*J ; r_

7 ' ^~ ld]

Define the following quantities;

&and

</}' ~cC + 2*°Kj,

(A. 15)

(A. 16)

(A. 17)

(A. 18)

(A. 19)

Pictorially these definitions may be represented as in Fig. A.l.

Pictorial representation of the quantitiesdefined in Eqs. A. 17 through A. 19.

Figure A.l

56

With these definitions, Eq. A. 16 becomes

~ 9 4 fl] ' A (A. 20)

and

yielding

&j=Zt;J».fW-<Je'£* (A. 22)

The preceding was a general outline of the procedure to be

followed in the Coppens-Sanders iteration scheme. The remainder of

this section contains a derivation of the terms through third order.

Assume the input

#, * lf„ M» tf(/r*J Oc^-cot (A- 23)

which was shown in Section 2 to be the calculated first-order solution,

Then we have

ft= ^LJ^ £(£'*.)'JUo *>t (A. 24)

£0

fa= -K & <,„ Atf-*) *.*>?

(A ' 25)

da. a)

and since /fc — d

&6U Co (A. 26)

The next step is to obtain j£ *

*-*- Jcl. <*<*- d«_ ^^ (A. 27)c •/

57

Substitution of Eq. A . 26 into Eq. A . 27 yields

%2 - M faz#(£-* ) ****>

2& tr

(A. 28)

where M = ^"/Co

Expansion of the trigonometric term gives

C<v-Z £(l-t) M^ âj~t (A. 29)

- (-i)fc^faMj -"J/**(***) -

1

from which the constants + 1 and - 1 may be discarded since application

of /oac)t wiH remove them. Then

2-/ /

pz = /V ^{"yj C<̂ ^ £fc~*J C™ -?«^ (A. 30)

which implies

/l2t = -//*?^ (A. 31)

and from Eq. A. 20

- J 7/ /M) -I *> **i (A,32)

Recall that ^W"<jf

)

Atif > an^ hence,

"* 'JtifrJA *«**&-*) <?«-(£W +&) (A .33)

which is the second order solution.

The general equation for ' /dOL. is

df* Cl 4,/, i . / -a,)

(A. 34)

Substitution of Eq. A. 34 into Eq. A. 21 results in trigonometric products

such as

(A. 35)

58

For simplicity let

Then Eq. A. 35 may be written

Can AlXM^faT/^J Ok J?X Mv, (fT+fj) (A - 37 >

Expansion of Eq. A. 37 gives

- <*(*-SjX <**ffr-/jT*&-t,J(A. 38)

where terms of the form sin Mk(L-cl) cos JfcO~G have been ignored

in Eq. A. 38 since substitution of these terms into the perturbed wave

equation result in amplitudes of order /£ which have been neglected

throughout. Eq. A. 38 must always be written as (m - 1) where m 7 1,

since a positive frequency has always been assumed.

Before proceding to the derivation of the third harmonic terms,

a brief discussion might be of some use in acquainting the reader with

higher order terms.

In the third harmonic derivation, the user of this iteration

scheme is first exposed to the development of more than one term in

the solution for a particular order. In other words, the iteration

yields both the leading term to the third harmonic plus the first

correction term to the present solution. At the same time, the use

of Eq. A. 38 is demonstrated for the first time. It should also be

pointed out that the methods used in the third harmonic iteration are

identical to those used in the higher order iterations, regardless of

the number of terms encountered.

59

Following the step by step procedure then,

VA «/ ^ ***£ ~£4£ +Pf *^J¥ (A. 39)

Remember

,

4<0

or using Eq . A. 38,

5k ~ 2(f ^J(-j)£~ **«-Je*(r*t+*)

- <?oi_ #(£'«-) Co^-(w/^ &Z) I

6o

(A. 24)

Mi *-{#» c~ *{-*)M»- * & (A ' 25)

&.&(#)£*.****)*#«*+**) (A . 40)

d«. 4atS -2 'Ai / (A. 41)

and

so that

(A. 42)

(A. 43)

Using Eq. A. 8 implies

a . - y & irn" SS * ?o — (A. 44)

/^r/ -2 ^c -^~ (A. 45)fl

and from Eq. A. 10,

'zTTJ^df-Ar ei (e*-f€i)

and

(A. 47)

Now 1*3 - ^(jj * %-3 1 > and when the substitutions are made, the

final form of the third harmonic iteration becomes

I 7r

\ *SMfb(A. 48)

6l

A combination of the first three orders to obtain the total

solution thus far results in

+£%> (ffy£**#*)**&**+**)(A,49)

Notice that the second term of the third order solution is of order

one in j, (with reference to Eq. A. 3), and thus it is termed the

first correction term to the fundamental or first order solution.

Rewriting Eq. A. 49 in order to achieve a form consistent with

this last observation and the form of Eq. 2.20 results in

*?Clh=

Ji* *(£-*)/**•* *> +

/ . • (

J (A. 50)

+ •"]

J, * 4

62

APPENDIX B

Computer Programs

An IBM- 360 digital computer was utilized in performing the

calculations, and obtaining the waveforms predicted by the theory.

Computer programs were written for each of the basic formulations

as well as for each of the variations investigated. Each of these

programs, together with a short description, is presented in this

section. Included with the description is a list of the symbology

peculiar to that particular program.

B-l: Program FINAMPI

This program has written to perform the calculations and plot

the graphs for the perturbation solution. It was originally written

in FORTRAN 60 language and subsequently converted when the IBM- 360

became available.

The program was designed so that the Mach number, resonance

frequency, length of the tube, position in the tube,P ,A<*J, and d^

are the input parameters. The first half of the program is devoted

to the velocity profiles and the second half to the pressure waveforms

The functions defined at the end of the main program are part of the

conversion from FORTRAN 60 to FORTRAN IV. A list of the symbology

pertinent to this program, together with the corresponding parameter

in the original formulation follows.

63

Table B.l

D(N) '^S = Mach number

WD = A^

WR = Op

B -AF = L

A = a

U = Velocity

P = Pressure

PI = ^L

WT _ cot

q(k) = fy-~/2^

THET(N) = &y,

Vxyz = the expansion of (X,y) for the velocity equation

Yxyz = the expansion of (X,y) for the pressure equationz

64

co

.— ——» —i .—

t

COCOoooc co

oo

O'-'fvjr^^tf' OfÔO C «-, <\l

rvcvrNjf^fMfvrvjrofM cor^mOOCOCCCOO C'C'Cc-eccccccr ©o©

<X

xc

oc

o

u o*- - 3r - z-^ -JO LUO

— >ar aarQl - •

— LL- - LL» -O U_< U_<C XX 31>0 •— ! -Xi— » i— - i—* ax ocx

o • </> «-Zac clo

— 2- - Z- -

ULI

x - XX ^CC

3.CO

—i— I— —1|—

: -J-— OO ^<^

X<T —I

•• >—

u. _I—

CC "J* *

CL —

O •*>»

3EOUJ•"—•_!

l/)X>—

x

cr

I

xcr

Xr\l

mXa

.s

Xrv

Xc

X

x

a

a:

jc<\j

_ l/)Q

CM

O7CM—

"v—Co:

•a

+ #

o-»I

»-l

O-O—i— *.

u_

XX

Xo

ax

i— ujw XO H-

x •>—

x wincm cj-*

oUJ

<xa-3Z

X

^l~n_i || _.-. ||

Z«JI-Q£HQi—

' > UJUTC OC*- X tt-?# * — # X

IJJ_J _l '_? _l <s*

2~<.<l X < oo ——ixx» - UJ* • II

—• a . -J X.^ • tr.

DO- 1 •

~xn< vr;t—lf\Jf—IfNj ».-^

«w »<-» «C—tT>- LfM— <— *—

^,<.w< UJ<r_j 3ra >. ^-X -

<J OKJOC—"X*lLICJUJCJSXLjor. a. at' u_ s a.

-~x — i-w x ^r-«-<ir> U.^—

'

^ O •

C_" U". IX *S> «i X •— ^<• <_£-«»- —^ u.

-~r IS) •< (_j »0—»•—' O

a x—i . ii —x »-<njo >c +fN, •• r-IV. IX »-JH fe«UM a

<'•—"-H w. .— Z3vC«- II -t— 3vT -i— I- II ^lO^^Z— •— >-w I— ^HU. ||

U.KTV'II II |i *»«"-UJ<! MJ<—< »^<•1—2 CL- 1— I— I— >. ro»-5.l—î—O.—Cms ^ ^; luz •— x r- —• ac i: n Q.OJct 'j3i-JLLi~—loacoacuL 1-1 u Z.JE x CJ I— i_5 c*i~o -n

Luh-3a.5Oo

f\) zr>~ — a(M -j »-^

o — i-t j — ocC Qr> » ct

o o<a ou + oo zz^ s—

.

vtJUJ —c—-o^-< -> jo-<-\j

I HU a*-—^"5UJ |l 'X_!I3 •-

• II •— X3 i ii X_jcc>x:0~5>-Z ^c—II II — X——'LLQCUL—m—

«

>— rr_j— t— H-

C -3Z — ZLu^'—^0~- ucji—ox— cl

t— c; i—oo^ oi— a -&

IT XX

oo

65

f^vtip^r-oro-c i-Kvrfi îr setoff o *.. —<(\irr, «j-ir vCr-oca C — csjr^îr vf. MrO'C«-*Nr,>t

COCC < CC CCCCW'CC.CCC'C OCOC COCC'COOCOC CC C C'C'OCCOOOC C C CC CC C CC t. C( r c CC C OC OC GCC.'OC'CCC OC OCCCOOf

+ OJ4-

f\j —— «—-.*- (\J UJ OJw — -—m w x —U! r<^ w r. — — Xwi— — *•» UJ *» •- # — uj —X — l-w — —

•

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B-2: Program FINAMPIZ

This program was designed to duplicate the output of FINAMPI

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Since the final results of the perturbation approach are in the same

form as those of the Fourier synthesis approach the program FINAMPI

was converted to FINAMPIZ by setting the corrective terms to zero.

Because FINAMPIZ is essentially FINAMPI, Table B.l is applicable here,

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B-3: Program QCURVES

This program was designed to compute and graph the normalized

Q-curve for each of the six harmonics using the perturbation solution.

In this program, the input parameters consist of the resonant

frequency, the tube length, J, yO and the piston acceleration. The

symbology pertinent to this program is given in the following table.

Table B.2

AO = piston acceleration

F = length of tube

Delt(N) = <fy

H(N) = hypothenuse of triangle described in Fig. A.l

THET(N) =-Q-n

SCRIPN(M) = unnormalized magnitude of the n-th harmonicat point m

MAXA = the normalizing factor defined for eachharmonic

SMALA = defined subroutine to main program

PHI = defined subprogram to main program

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96

B-5: Program PHAMP

This program was designed to compute the amplitudes and associated

phase angles for each harmonic. Essentially this program is the QCURVES

program with only minor modifications. One new quantity has been defined

as PHASE(IS J). This is the phase angle associated with the i-th harmonic

and the j-th <*-^6J. In all other respects Table B.2 is applicable.

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This program is identical to PHAMP except that it was written for

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m*mm m< (/) •> mmCD *x » mm UJm, xa —

.

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118

•> #—-5 «to

tort -"J** tort to*

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X *x» •-J -»#-J X* ^*—«.

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tort- *tort »..-.

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a uu »»-xooZXOUUH"«0rr>

zai- ii x *u-Otort «-»-# n0«-»

«-n/>zr»» -hz»-ZO—-»îu<a:UUJXH- ||»-»*-X3zixui«- «ai-p3->OXX*<*OUUZULOOt-CL ZSU-OCUJ

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119

B-7: Program FOUANAL

This program is an adaptation of work done by Roy M. Johnson,

Assistant Professor, Naval Postgraduate School. Basically the program

takes a waveform obtained directly from the cavity and performs a

Fourier analysis on it. The output of interest is a set of Fourier

coefficients and their associated phase angles. The program is

written in such a manner that a table of definitions is not appropriate,

Each variable is defined within the program.

120

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oo

oc

vfr

r\i

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—oa.<»t>4

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1O 75 JC I— CLL J. LL-^LLCl IX'-U

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122

CD

>(0

3

CD

c•Hco C1 o1 -H1 4->

- CJ

•U >Hi-l T)3 <1J

co J-i

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CD

I-l

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CO

CD

Ufa

123

I

« cI o- .1-1

4-1 4J.-I o

CO "O0) 0)

o-I—

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CO NJ4-> M<D sa <H ^M H(1) Pn(X •

X •

d) •

BO•i-l

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124

CO

3

CD

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1

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.—' u3 -HCO TJ Uaco n4-1 MC PUa) 26 <3•5 s mU M •

a) tin Oa •

x • -l

3CO 60•H •H

c fe4J OQ) -t-l

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C uM -r4

T3a)

uco CX

ft HO P-i

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a)

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!-!

Pd

125

Den

<D

Ui—

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C0)

e•r-l #*

U c0) oa •r-l

x 4-1

91 u•r-l

en -a•r-l (1) 0)

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en

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126

Q,1.2-

•

1/ .8-•

.6-

• ^V

.4-

1 1 _

•2-

i i i

-4 -2 Lu

Q- curves for the fundamental, M = 0.004,

QCURC prediction, QCURVES predictionExperimental results are indicated by •

Figure C.5

127

Q1.2

•

1.0'

/ /// \ \

•

// ' \ \

I

// \ \

\\\\\\

/ *6 '

\\v\

/ .4"

. ._!_ —1_ __.. A

.2"

* 1 *

-4 -2 4 Acj

Q-curves for the second harmonic, M = 0.004QCURC prediction, QCURVES prediction

Experimental results are indicated by •.

Figure C.6

128

Q-curves for the third harmonic, M = 0.004, •

QCURC prediction, QCURVES predictionExperimental results are indicated by #4

Figure C.7

129

Q1.2

1.0

.8

v •

•6/

•/

» \\ \\ \

% \

//\\

II

U//

r '

\\

f

.2--

/ •

— - r" i i— __i . i i _i_ J.

-4 -2 4 AuQ- curves for the fourth harmonic, M = 0.004,

QCURC prediction, QCURVES predictionExperimental results are indicated by •»

Figure C.8

130

Q- curves for the fifth harmonic, M = 0.004,QCURC and QCURVES prediction

Experimental results are indicated by ••

Figure C.9

131

Q-curves for the Sixth harmonic, M = 0.004QCURC and QCURVES prediction

Experimental results are indicated by #«

Figure CIO

132

Q•

1.2-

--'-

.8

/.6- •

A-

*

.2-

-4 -2 4 Au

Q- curves for the fundamental, M * 0.005, • *

QCURC prediction, QCURVES predictionExperimental results are indicated by#»

Figure C.ll.

133

Q1.2

1.0 •

'

•

T *.

/ s//

////

/ *

4

>s \\ \\ \

\ \s \\\\\\\\\

•

.2-

t

-4 -2 4 Au

Q-curves for tfce second harmonic, M = 0.005%QCURC prediction, ----QCURVES prediction


Figure C.12.

134

Q-curves for the third harmonic, M = 0.005^QCURC prediction, ---QCURVES prediction

Experimental results are indicated by #•

Figure C.13

135

Q1.2-

k

--. 1.0'

/• \ t

.8- / • \

.6/

Jt//11

It

It

L

'/:V\\ \\ \

^•\\ \\ \\ \\ I

\\\\\\\\\\\\

•J .2-

—i i i 1—. 1 1 1 1 1

-4 -2 4 Au

Q- curves for the fourth harmonic, M = 0.005,QCURC prediction, QCURVES prediction


Figure C.14

136

Q- curve for the fifth harmonic, M = 0.005,

QCURC and QCURVES predictionExperimental results are indicated by ••

V

Figure C.15

137

Q-curve for the sixth harmonic, M = 0.005, .

QCURC and QCURVES predictionExperimental results are indicated by ••

Figure C.16

138

Q1*2-

•

---•'

v^ 1 ' ^x•

/ -8

N XN XN^xs X.

^ X\ x

sX.

.6- •

.4-

.2-

1

-4 -2 4 AuQ- curves for the fundamental, M = 0.009

j QCURC prediction, ---QCURVES predictionExperimental results are indicated by#.

Figure C.17

139

'

«.

•1.2

----1.0' -

g

•>» \\ \\ \i \

/ • ///

/

\ \

• \•

/'•n /r f \ \

\ V\ \\ \

ii

'/•1/ft

\

\s\ \»

// .4N \\ \

\\

//

<7 %

•

1 1 1 1

.2-

' ' 1 I

-4 -2 4 LuQ- curves for the second harmonic, M = 0.009,

QCURC predict ion, ----QCURVES predictionExperimental results are indicated by#»

Figure C.18

i4o

1.2•

1.0-

.8-f x \

/ v \/ /

/ / \ \/ /

% \

/ /\ \\ \

/ /» \

/ / I \/ / " \ \/ / \ \

/ / \ \ •

/ /1 \

/ /» \

/ / •' \i / i 1 » \

i I t \

; A- \ \i I » /"M'/ • v / \

/ / \ / Ax/ / \\

/ /

Xv\\NTs

sS

' . 1 * 1 1 •»««-4 -2 4 ku

Q- curves for the third harmonic, M = 0.009,QCURC prediction, QCURVES prediction,

Experimental results are indicated by# #

Figure C.19

i4i

o1.2

1.0

.8

#.

>*""*\

/ / \

// \

// \

// \

\

\

\

\

• ."A• I

*N\

if i

i

i

// • .2- I

i

i

i

i

i

v// \

VV t i

y* l

i

i 1 1

-4 -2 AuQ-curves for the fourth harmonic, M t 0.009,

QCURC prediction, QCURVES predictionExperimental results are indicated by #•

Figure C.20

142

Q- curve for the fifth harmonic, M = 0.009,

QCURC and QCURVES predictionExperimental results are indicated by #•

Figure C.21

143

1.0

.8-

.6

* *L>

<

•• •

••

•\S^ •

l i teag•f^ i i . i i_

-4 4 Au

Q- curve for the sixth harmonic, M = 0.009,

QCURC and QCURVES predictionExperimental results are indicated by-#>

Figure C.22

144

Phase dependence of the various harmonics r>n /\^)as predicted by PHAMPC. n = harmonic number •- data from FOUANAL for n'-th harmonic

Fi,.

145

INITIAL DISTRIBUTION LIST

No. Copies

1. Defense Documentation Center 20Cameron StationAlexandria, Virginia 22314

2. Library 2

Naval Postgraduate SchoolMonterey, California 93940

3. Commander 1

Naval Ships Systems Command HeadquartersWashington, D. C. 20360

4. Commander 1

Naval Ordnance Systems Command HeadquartersWashington, D. C. 20360

5. Professor Alan B. Coppens 8

Department of PhysicsNaval Postgraduate SchoolMonterey, California 93940

6. Professor James V. Sanders 1

Department of PhysicsNaval Postgraduate SchoolMonterey, California 93940

7. CDR Wayne "L" Beech 1

1108 Leahy RoadMonterey, California 93940

8. LT Paul G. Ruff III 1

USS Dale (DLG-19)FPO San Francisco, California

146

UNCLASSIFIED

Security Classification

DOCUMENT CONTROL DATA - R&D(Security classification ot title, body ot abstract and Indexing annotation must be entered when the overall report ie claeeilied)

GINATIN G ACTIVITY (Corporate author) 2a. REPORT SECURITY CLASSIFICATI1. ORIGIN ATIN G ACTIVITY (Corporate author)

Naval Postgraduate School

Monterey, California 93940

2a. REPORT SECURITY CLASSIFICATION

Unclassified •

2b. CROUP

3. REPORT TITLE

A Theoretical Investigation of Finite Amplitude Standing Waves in Rigid WalledCavities

4. DESCRIPTIVE NOTES (Type ot report and inclusive datea)

Master's Thesis, December 19675. AUTHORfS; (Laat name, tint name, Initial)

RUFF, Paul G. , III

6- REPORT DATE

December 19678a. CONTRACT OR GRANT NO.

b. PROJECT NO.

7a. TOTAL NO. OF PAGES

144

7b. NO. OF REFS

269a. ORIGINATOR'S REPORT NUMBERfS.)

tb. OTHER REPORT UO(S) (A ny other numbers that may be assignedthis report)

10. AVAILABILITY/LIMITATION NOTICES

s»

Etaa

11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY

Naval Postgraduate SchoolMonterey, California 9394°

13. ABSTRACT

The Coppens- Sanders perturbation solution for the one-dimensional non-

linear acoustic wave equation with dissipative term describing the viscous

and thermal energy losses encountered in a rigid walled, closed tube with

large length-to-diameter ratio was extended to include sixth order terms.

The solution was then investigated to determine the region of validity.

Computer programs were written to evaluate and graph the resulting waveforms.

Available experimental results were compared with the theoretical predictions

and good correlation was found to exist in the region of low Mach numbers.

This agreement was found to gradually deteriorate as the Mach number was

increased. A Fourier synthesis approach is also presented and the leading

terms of the first ten harmonics are derived.

DD FORM1 JAN 64 1473 147 UNCLASSIFIED

Security Classification

UNCLASSIFIEDSecurity Classification

KEY WO R OS

Finite-AmplitudeStanding WavesPerturbation ApproachFourier Synthesis ApproachRigid-Walled Cavity

DD ,

FN°o

Rv

M651473 (back)

S/N 0101-807-6821 148 UNCLASSIFIEDSecurity Classification A- 3 1 409

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