*A *SINGLE VORTEX WITH *A *SHOCK WAVE

BY

H. S. RIBNER

\

THE SOUND GENERATED BY INTEHACTION . .- OF A SING LE VOR T EX WITH A SHOC K WA VE

BY

H. S. RIBNER

This paper was presented at the 1957 Heat Transfer and Fluid Mechanics Institute. California Institute of Teclmology,

Pasadena. June 19-21, 1951. The present content is that of the Stanford University Press preprint with minor cox-rections and the addition of a list of symbols.

I am indebted to Mr. G.S. Ram for his considerable assistance with the calculation of the two-dimensional Fourier transforms of the velocity components of {he columnar vortex; to Professor B. Etkin for many discussions on the ideas; and to Dr. G.N. Pattersol1. for his continued interest.

Support was provided by the Defence Research Board of
Canada under DRB Grant Number 9551-02 and by the United States
Air Force under Contract Number *AF *49~638)-249, the latter
monitored by AF Ofîice of Scientiîic Research of the Air Research
and Development Commando

### .

: .SUMMARY

The passage of a columnar vortex 'broadside' through a shock is investigated. The vortex is decomposed (by Fourier trans-form) into plane~sinusoidal shear waves disposed with radial

symmetry. The plane soUnd waves produced byeach shear wave-shock interaction~ known from previous workl" are recombined in the Fourier integral. The-waves possess an envelope that is

essentially a growing cylindrical sound wave, partly cut off by the shock. The sound wave is centered at the transmitted (and modified) vortex and the peak pressure attenuates inversely as the square root of the growing radius. The strength varies smoothly around the arc J- from compression at one shock intersection to rarefaction at the other shock intersection . Comparison is made w ith results of. a shóck-tube investigation and heuristic theory by Hollingsworth and Richards in England .

TAB LE OF CONTENTS

Page

NOTATION ii

1. INTRODUCTION 1

II. QUA LIT A TIVE DISC USSION 1

lIl. QUANTITATIVE DISCUSSION 2

3.1 Resolution of Vortex into Shear Waves 2 3.2 Transforrnation of Origin to Shock Front 3 3.3 Shear Wave -SrlOck Interact:lon 3 3. 4 Transformationl~of Origin to Core of

Trans-rnitted Shock 4

3.5 Pressure in Cylindrical Sound Wave: Upper

and Lower Bounds 5

3.5.1 Basic Forrnulation 3.5.2

3.5.3 3.5.4 3.5.5

A pproximations in the Vicinity ril ~ R. for R 7";> ro

Forms of the Inte~rand Limits of lntegration

Final Results for Pressure Field

5
*'1 *
7
8
8
IV. COMPARISON WITH HOLLINGSWORTH AND RrCHARDS 9
4. 1 Comparison v/ith Theory of Reference 9 9
4.2 Cornparison wJ:th Experiment of R eference 8 9

REFERENCES 11

c
c:A
*f(ot') *
*GLo-) *
g (~; r 0)

### g

i J*(0<.") *

J 1 (k ~ 0)
k
MA
ril
"-P ### Co<.

*liJ *

P
q
R
r
ro
*1\*r s A s { > t U UA ( ii ) NOTATION

velocity of sound downstream of shock velocity of sound upstrearn of shock

*4:[4 *

*P{«'I) J{rxV *

function describing radial distribution of pressure in

cylindrical sound wave (Eq. 24 and Fig. 11)

elementary wave profile (Eq. 10) upper and lower bounds to g (Eq. 16)

*·R *

Bessel function of first kind anç:l first order

wave number (2

*TT *

Iwave length)
Mach number upstream of shock (U *AI GA) *

velocity ratio across shock (UrA/U)

( 2 '( /7)

### 7T~ec ~/)

transfer function - lr~l)m

### _

*Cr-I) *

pressure

resultant vortex velocity

*(Iu"-t-*

*Vi) *

nominal radius of cylindrical sound wave (ct)

radial distance from centre of vortex

### (~x,

**2.-r **

'X,. ### '2.)

vortex core radiuscomponent of r parallel to wave-normal

### !i

(r cos*(o<.-r(J) ) *

radial distance from origin 0 on shock component of sparallel to wave-normal ~

---

-time rneasured from instant vortex centre ~l':'~sses;shQ:ck

stream velocity downstream of shock

*TT *

*p *

perturbation velocity components of vortex

; . . ' *I * "0

factor having the respective values 2 and 1 to yield upper and lower bounds

angle of wave -number vector with horizontal

limits of integration (Fig. 10)

I'W

values of

### (3

associated with*f3t) *

**t3z . **

respectively
**t3z .**

angle of elementary wave with horizontal .

angle ot"transmitted shear wave with horizontal (Tabulated

v.

*e *

inRef. 3)
function defined in Ref. 2 and tabulated in Ref. 3

*(pIl _ R) Ir /1 *

o

nondimensional deviation from radius R ( *(rIl - R) *Ir~)
angle of radius vector r with horizontal

Where symbols appear w ith or without 11 the unprimed symbols refer to the geometry or properties of the initial shear wave.

tne primed symbols refer to the geometry or properties of the sound

wave.

( 1 )

1. INTRODUCTION

In recent years considerable effort has been directed

toward the understanding of the noise produced by jet airplanes. One

by-product has been the recognition of the role played by shock waves.

e. g., in the case of choked jets (Refs. 1 to 6). It is now believed that

either vorticity (in the form of turbulence) or temperature spottiness

in a jet will interact with any shock waves that are present and generate

intense noise.

Until recently the experimental evidence (Refs. 1, 7) has

been indirect. N ow. however, Hollingsworth and Richards (Ref. 8)

have reported on a schlieren study of the interaction between a single

vortex and a shock wave in a shock tube. A single cylindrical sound

wave is generated in the experiment, and such a wave seems eminently

suited to theoretical treatment for comparison. An heuristic theory

has been given in a second paper by the same authors (Ref. 9). and

the present paper is an attempt at a quantitative theory.

The starting point of the analyses is the known result for

the interaction of an inclined sinusoidal shear-flow w ith a shock w"ave

(Refs. 2, 6). Both discussions exploit the-concept tha~ a vortex flow

can be synthesized by a radially symmetrie distribution of such shear

waves in a Fourier integral. In Reference 9 this concept forms the

basis for a qualitative argument; it leads to a foqnula for the estimated

sound pressure as a function of arc length around the cylindrical sound

wave. In the present paper the Fourier integral is explicitly formulated

and the derivation of the sound press ure level is carried through

analytically.

The physical picture is c arried along in the analysis with

. tl;1e aid of a number of simple sketches that portray a 'model' of the vortex-shock interaction process.

**II. ** QUALITATIVE DISCUSSION

The interaction of an inclined sinusoidal shear wave with

a shock gives rise to a refracted shear-entropy wave and a sound wave.

also sinusoidal (Refs. 2, 5, 6). (See Fig. 1). These sine waves may be superposed in a Fourier integral to yield the interaction of waves of arbitrary profile with a shock.

*A *particular example is shown in F ig. 2. Suppose now that

infinitely many weak shear waves of this profile are uniform ly dis-tributed radially like the spokes of a wheel; then the resultant velocity field is that of a vortex of core radius ro (Fig .. 3). This can be dem-onstrated formally by reduction of a two-dimensional Fourier integral

If the ,vortex is convected the constituent shear waves are

convected. The situation before and after encountering the shock,

omitting the sound waves~ is shown in Fig. 4. The angles of refraction

or" the shear waves are such as to bring them to a focus in the "af ter" sketch. This focus can be interpreted as the core of the transmitted, m odified vortex.

The sound waves are added to the llafter" -interaction

view in Fig. 5. These plane waves emanating from the s.b~k come

off at the proper positions and angles so that they possess a

cylin-drical envelope of radius ct.

The detail insert Glarifies the formation of the enveJ.ope.

Let U be the subsonic stream velocity downstream of the shock and t

be the time. Then W is a velocity determined by the inclination of

the shear wave and

*JA *

is the Mach angle associated with that velocity ,
if supersonic*, according to Reference 2. The relation sin / - ### =

c/Wthen requires the distance from the vort ex core to the sound wave to

equal ct. This distance remains constant at ct for all the sound

waves, even though W changes with the shear wave inclination. Thus

the sound waves are tangent to a circle of radius R

### =

ct.The strength of the sound pressure field maximizes

rather sharply .at this circle (or *i * rather , circular cylinder) of

tangency. The effect is that of a cylindrical sound wave. The

pres-sure varies around the circumference of the wave. being dominated

at any point by the contribution of the plane waves tangent at that

point. Since the plane waves are spread more thinly as the radius

growsi the pressure diminishes; it turns out that the peak pressure

varies inversely as the square root of the radius.

,lIl. QUANTITATlVE DISCUSSION

3.1 Resolution of Vortex into Shear Waves

The velocity field of a eolumnar vortex of core radius

ro and circulation

*27T *

is given by
r~

### 10

*r *

~r;;
*IA (r; *

*ct) -*

### ~ ~

*/n *

### f/

### .;

### ~

*s" *

*11*

*(J } *

~~ (1)

*1/(r;rf) *

### -

### ça~;

### .

### -~COS~

~ If W is subsonic different relations hold, and the sound waves

decay exponentially wUh distance from the shock. These waves are

neglected herein; they are expécted to be unimportant except near

the shock.

( 3 )

*A *two-dimensional Fourier development can be obtained in the form

### *

*6l *

*(r; *

*rf) *

### =

### ;//?

*J, *

### (A

### r;,)

*.sI" *

*I( *

*r *

*el *

### ~

*s *

### I/;,(

*do< } *

### ,

*-1I/e. * (;)

### ~

*ra *

( 2 )
*tI}-{r; *

*(J) *

### =#

### ,/1}-

*2. J{lrt;) *

*S'/njrP *

*dh *

*COSO(*

*dO<. *

where *-!l'12. * 0

*Ir *

*re, *

*p. *

### =

*r *

*GOS*

### (rx.-~)

. Th

### 2

integrands can be interpreted as the velocitycom-ponents d2u, d v in an elementary. sinusoidal shearing flow (shear

wave) inc lined at an angle

*ot.. *

(See Fig. 6). If the vortex circulation
is generalized from *2?r * to *Î * the resultant velocity

in this wave is

*dtL*

*:. *

*r z *

*J, *

*eh *

*r;) *

### .r

*il'l *

### ,f;'

*dh ti *

### ~

### (

3 )*tT * 2 7T~

*hr;*

*. *

3.2 Transformation of Origin to Shock Front

The shear wave of Eq. 3 is referred to an origin of co-:-ordinates at the vortex centre. The position of this centre is shown

in Fig. 4 af ter an assumed passage through the shock without change

of convection speed (this is called the virtu.al position since the actual

position wil! be'sübstantially upstream of this one). *A * new origin is

taken along the shock as shown in Fig. 7; the shear-wave shock

inter-action formulas take their simplest form with respect to such an origin.

It can be se en that the lin,es of constant phase for the elementary shear

wave of inclination

*e *

### (=

*7flz *

### -

~ Fig. 6) are unchanged if the projection### r

is replaced by its equal §'.3.3 Shear Wave-Shock Interaction

Replacement of sin kt by sin ks in the elementary

sin-usoidal shear wave *I * Eq. *3 _{J }*yields

shear

wave: ( 4 )

The interaction of this wave with the shock gives rise

(Fig. 1) to a pressure wave of different inclination and wave number

(R e f s. 2. 3):

pressure

wave:

....

( 5 )

The 'transfer function' P depends on the wave inclination

*0<"; *the formula is given in "Notation" in terms of functions' .

tabulated in Reference 3.

### *

Mr. G. S. Ram assisted considerably in the calculation of Eqs.### 2

The geometrical relationship between waves (4) and (5) is

exhibited more clearly in the upper part of Fig. 9. Furthermore,

Reference 3 gives

*kIk" *

= cos *e" *

*I *

cos *é3. *

Thus
### ~

### ~/

### _ A-

### ==

### e/JL.

### =

*GOS*6)

*'I*

*r; -*

*k" *

*dk" *

*GOS &*( 6 )

With use of these relations the original wave number k can be

elim-inated from Eq. (5)jwith a little regfouping. /'

### '"

*c/:'lJ= *

*-p L!.... *

*/;//(2J;(*

*Ir*

*I'f;/j*

*.rIa *

*khs"dlr') *

*jo< *

(7)
*r *

~;Lr *r;*

*Ir!;!';*

*'I *

*The integral of Eq. 7 from 0 to *~ in k" and the corresponding

integral of Eq. (4)in k can be evaluated explicitly. giving, af ter

con-siderable reduction. where shear wave: pressure wave: ( 8 )

*d*

_{p }

( 9 )
( 10 )
_{p }

" Examination shows that the profile-shape function (ro" *Iro) *

g ( S 11. r Oll) of the press t.!re wave is just a stretched vers ion of the

profile-shape function g *(S * ; rol of the original shear wave; the

stretching. shown in Fig. 9. is in the proportion ro"/ro

### =

cos### e "/

cos

### e .

Eqs. 8 and 9 represent elementary waves of the special

profile sk.etched in Fig. 2. Eqs. 4 and 5 represent the interaction of

a sinusoidal shear wave with a shock (F~g. 1) whereas Eqs. 8 and 9

represent the interaction of ag-profile shear wave w ith a shock (Fig.

*Z). * Since the vortex is synthesized from g-profile shear waves

dis-posed "radially like the spokes of a whee 1" (cf. section TI). the

pressure field generated by the vl9rtex-shock interaction can be synthe-sized from waves of the type of Eq. 9.

3.4 Transformation of Origin to Core of Transmitted Shock

It wi11 be convenient now to transform the origin of

co-ordinates for the elementary pressure wave from the point 0 on the

shock front. Fig. 7 *I * to the centre of the transmitted vortex (actual

position) . The relevant geoIlJetry is shown in Figs 0 8 and 9. Thus the

'.

( 5 )

distance

### .s

### "

is replaced by r### ""

- R, whence the pressure wave reads*cllJ *

= - *P *

### 1:2

### ~

*11*

### a

*( f l ' -*

*R*

*-*

*' *

*r; *

*Ili*

*do<. *

### (

11 )### ,-

### Ut

### 2.

7T~*ro" *

### )

0*j*

Eq. 11 holds only for a r ange of incident shear waves - *o(m *
~ ~ ~ *o{m *; this corresponds to a single travers al of the range

*-ot.'fn *

~ *CJ(II *~

*c<!; *

### .

For this range the construction of the lower part of Ffig. 9, s howing the wave enve lope of radius R. is v alid. The justifi-éation of this construction \\mlS given in the latter part of Section II. in conjunction with Fig. 5. Incident shear waves of larger

### b<1

(smaller### le{

### )

than this range interact with the shock to yield plane pressure waves that attenuate exponentially with distance from the shock (Refs.2. 3). The inclinations are such that these waves do not coelesce to an, envelope. The use of Eq. 11 effectively excludes these "subsonï'c"

waves. Because of the rate of attenuation itis thought the neglected

waves may be important only within distances of the shock ~ r .

o

The angle element d

### IX

can be written.Jo( -

### r~

*dB *

*J&'IJd *

*11 *

*Cl *

### ~

*Let *

& *d&1I ;;&11 *

*0(*( 12 )

The factor in square brackets - the Jacobian of the transformation

from 0( to 0( *11 -* can be evaluated with the aid of the functional relations

connecting

*t9 *

and *e" *

given in References 3 and 2, together wit,h
eX.

### =.f

### -6.)

*o<"=-*

~ ### -

*eH . *

The result may be abbreviated J ( cX")~
the formula is given under "Notation" . Accordingly

. ( 13 )

3.5 Pressure in Cylindrical Sound Wave:- Upper and Lower Bounds

3.5.1 Basic Formulaüon

The resultant sound field

### *

of the vortex-shock interaction .'.is obtained by superposition or integration of the elementary pressure

""" *11 * *..J 11 * *... ./ 1I * , )

waves, Eq. 13,over all angles ... from -~mto

### +

~*m (FIg. 9. To*

this end we abbreviate Eq. 13 further and define the g-function in terms of more convenient variables :

( 14 )

### *

Excluding a contribuUon, localized near the shock, from attenu.ating pressure waves; this was discussed earlier.} where

### .;:.C

0<*11 )*

*-fJ *

*(r"-R) r;) *

=.
### "

_{) }

*ir-*

*I *

~ *I *

( 15 )
### f

_{Y'ö" }

The functions are too complex to invite a generally valid analytical integration. l:Iowever ~ the g-fUnctión approximates a Dirac

### & -

function if R**:» **

ro" (cylindric-al sound wave large
com-pared with vortex core). This behaviour can be exploited to
approximate the press ure integral in the general vicinity of radius R. It can be justified a posteriori that this is the important region~ since the pressure is found to attenuate sharply on both sides of

radius R. .

For this purpose the g-function ma~ ~ be expanded in

inverse powers of

*f *

in the range ### tfl

### >

*I-*0 Such a development

has been carried out in unpublished work~ but we will limit 9urselves here to the much simpler procedure of establishing upper and lower

bounds to the pressure 0 These bounds win be introduced by replacing

the g-functionby

### B(

### ~I'-R;

*lfl!:;./ } *

~I>I ( 16)

Comparison with'Eq; 15 shows that

### Ir

e.quals the upper bound ofg-tor ~

### ffr

df!:*land *

asympotîcally approaches the-lower boundl of half the
absolute value, as

*1S'f-" *

*00*; at

*Ifl *

### =

2 the approach is already close 0-It will be convenient now to substitute a new variable

### ($

for### o(/~~/(Fig

0### '

### af.

With this change and the replacement of g by### 'g

the integral of Eq. 14 frorn - ~~ to '" ~ becomes ~### .

~### ..

~"*f *

### =

*f *

*f((/'+(i) *

### f

### 9

*(r'-*

*R; *

*r;') *

*tlfd *

_~_~"

where now ( 17 )

### ..

( 7 )

3.5.2 Approximations in -Ehe Vicinity ril ~ R, for R » ro

*ii2.. *

The approximation Icosf3 ~ / -

### z..

and the definitionmay be used to obtain
*r i l *

### f

### f

( 1~ )

Note that

### cr

is a nondimensional measure of the deviation from radiusR.

To simplify the notation write

*N *

### !.B...

*)IIZ. *

*f3 *

### =

*lZt7,', *

*(3 *

_{( 19 ) }

whençe finally ( 20 )

3.5.3 Forms"oi the Integrand

In the range

*1f/1: *

*Ithe *

integrand of Eq. 17 is
*dA' *

### =

*f{rfJ/I+;3J *

### ~~ ~

*(fl'-R,; *

*fD") *

*df3 *

With use of Eqs. 16' and 20 this is approximately
*dA' *

### =

### ~

### '

### f(~/'+(J)(O-

*-p') *

### d~

### =

### 2

*(J;·R)'h. *

*Cf')'I2.*

_{f }

_{f }

*(rt"11*

*3) *

*(cr *

### -(1&)

*dp *

, Both (r

*"/*

*ro)llz. *

and *f(tI''+*

*(j) *

are functions of ### ~.

The rates of change### wil?

be slow compared with those in*0--*

*fjL , *

if *cr"h *

. is of order unity or smaller. Rence, it will s uffice to expand ({; "/ ### r;,) ,

*-F *

(~/!+ ### (3)

in a Maclaurin series in### (3

### .

and retain the leadingterm. The res uIt is .

*IPI *

~ *J: *

_{( 21 ) }

By a similar process the other form of the integrand is approximated as

*WJ :::./; *

*tin *

### ~

*f( *

### ~

*)112*

*1*

*G..,,)'/z. *

*f'.///J'?*

*7r2.} *

### .

*cI##V *

2
*. *

r~ ### Ll

### roR

*Cr; *

*pil *

TLr *Ij *

*V *

*rr -*

### (3

The .fa.ctor in brackets is independent of

### ~

3.5.4 Limits of Integration

The limits of integration may be established with the aid

of Fig. 10. The integrand changes from dPi to dpo or vice versa at

the crossings:r(->, and

### :i::

*f3J. *

which occur at *r:r *

### =

1 and -i, respectively.This condition yields

*Pi *

### =

*Irr-J *

~

### (32.

### =

### -1

*u-t-I*

For values of

*113 *

*J*

### >

### 1(32.\

the integrand is dpo' For theassumed condition R» ro the very rapid attenuation of dpo as

### [(.31

exceeds

### 115,/

constitutes tne basis of the earlier remarks that theg-function then approxim ates a

### S.-

function . The upper limits on### /1

can thus be removed to

### ±.

00 without noticeable error in the integral -::i.3.5.5 FinalResults for Pressure Field

The results of the integration take different forms for

different ranges of CT, the parameter of deviation from the radius

R. With use of the definition of

*-f( *

*0<.")*in Eq. 1·5 the results may

be written where

*G(er) *

### =

### cr-

### .

### =

### t

### f2.} -'

*s;n *

*-I*

### H

*IR" *

### t-ff}

### ~

*S*

*Inh-*

*I*

### {i-! (

*I *

### ~

*2r:r}/r:r+ *

I ### +

### 1 (;

### -t-

### 20-)

### 1

*lr-I *

### ~fT}.fcr(cosh-/~

### -

*slnh-*

*I*

*1i-) *

### ,

*(rl'- R)/ *

*re *

*,*

*11 *

maximum velocity in vortex

,....;

( 23 )

24 )

The factor

### G

### (0-)

provides the variation in strength inthe radial direction - the profile of the wave; more precisely, two

profiles are provided cönstituting upper and lower bounds to the

actual profile.

### *

The two profiles are plotted in Fig. 11; they show### *

This statement is strictly true only for*cr-5=.*

*I *

### ;

because theg-function reverses sign for 0-:> 0 it is not.proved that the integral has

the bounds of ~ln the

*cr> *

*I *

range.
"

I·

( 9 )

that the radial distribution of pressure peaks sharply in the

neighbour-hood of radius R *(r::r *

### =

0) and is essentially zero elsewhere. Thecharacter of the sound pressure field is thus that of a cylindrical sound

wave of nominal radius R (= ct).

The factors P (

*cp *

*11)*and J (

*cp *

*11 )*jointly describing the

variation in press ure Jaround the arc (angle

*.*

*tf *

*/1)*are plotted in Fig.

12 for a shock Mach flumber .25. P (

*tp") *

is the 'transfer funct_{ion' }describing the strength of the plane pressure w ave)w ith normal at angle

~'; generated by interaction of a plane shear wave with the shock.

J ( ~ *11) * is the Jacobian that defines the relative effectiveness of
this plane wave in contributing to the local strength of the cylindrica,l

sound wave at arc position

*tfJll. *

The net variation of pressure with
arc position (at constant

*r:r ) *

is given by the prodl,lct P ( *CfJ/1*)

J ( ~ *If ). *plotted in the same figure. (N ote that I arcs t -

*r;r= *

constant
are not quite true circ ular arcs since r 0" varies with *Pil). *

IV. COMPARISON WITH HOLLThJGSW_{, } OR TH AND R ICHARDS

4.1 Comparison with Theory of Reference 9

The theory of Hollingsworth and Richards largely bypasses

a detailed geometric model. Their assumptions led the~ to

hypothe-size the pressure field. in effect, as

l-J *(r" (IJ *

### I~

**= **

**(Am***10*

### ft

*(*

*rfJ*

*'*

*? *

### ,

### ) r-

### U

*r" *

( 25 )
The differences from the analytical result. Eq. 23 above~ are more

far-reaching than a first inspection would suggest. Space, however ~

does not permit elaborating on them.

4.2 Comparison with Experiment of Reference 8

Figure 13 is a schl;ieren photographof the cyfindrical

sound wave produced by passage of a shock wave over a vort ex in a

shock tube (Ref. 8); it is reproduced from Reference 9. Only qu

ali--tative conclusions can be drawn. The photograph appears to exhibit

the sharp locali~ation of pressure perturbation at radius R .. ct (see Fig. 5) described by the

### ~

*(cr) *

function . T he press ure varies
smoothly around the arc from compression at one shock intersection
( **-** * 0("., * ) to rarefaction at the other (+ (>(,."

### *

### ),

more or less likethe P ( ~ *11 * ) J (

*Cl") *

function (Fig. 12) .
*11 * 0

The pred.icted reversals at

*tp *

~±. 90 • however. do not
appear in the experiment. A possible explanation may be this. The
### *

Fig 12 refers to a c10ckwise vortex '(*r *

positive) ~ whereas Fig.
13 r efers to a counterclockwise vortex with consequent reversal of
'vortex is s ufficiently strong so the press ure reduction at the core is substantial: the schlieren (Fig. 13) suggests this. This pressure reduction is of second order in velocity and is- not allowed for in the

present linear theory. It is hypothesized that the low .pressure zone just

af ter interaction with the shock is in excess of the value for equilibrium

at the core of the stronger transmitted vortex. This excess must then

propagate as a radially symmetrie cylindrical compression wave. The

compression wave will superpose on the wave described by our analysis and reinforce the compression portions and attenuate the rarefaction

portions. The zeros at

*cpll *

*:.t*

### ::i:

900 will be eliminated. All of thisappears compatible with the schlieren photograph.

### *

The strong density gradient exhibited in the schlieren photograph is attributable in part to a coexistent entropy gradient (entropy "spot")( 11 )

REFERENCES

1. Powell, A. On the Mechanism and Reduction of Choked

Jet Noise, Part 1., Communicated by Prof. E , J. R ic har ds, A. R . C. 15, 623, F. M. 1856,

Dec. 1952

2. Ribner, H. S. Convection of a Pattern of Vorticity Through

a Shock Wave, NACA Rep. 1164, 1954

(Super-sedes TN 2864, Jan. 1953)

3. Ribner, H.S Shock-Turbulence Interaction and the

Generation of Noise, NACA Rep. 1233, 1955

(Supersedes TN 3255, July 1954)

4. Lighthill, M .J. On the Energy Scattered from the -Interaction

of Turbulence w ith Sound or Shock Waves,

Proc. CaI!lbr. Phil. Soc. 49, Pt. 3, Ju~

195 3, pp. 5 3 1 - 5 5 1

5. Moore, F.K. Unsteady Oblique Interaction of a Shock

Wave with a Plane Disturbance, NACA Rep.

1165, 1954 (Supersedes TN 2879, 1953)

6. Chang, C. T. On the Interaction Between Weak Fluctuating

Fields of Sound, Vorticity, and Temperature

with an Oscilla ting Shoc k, Pt. 1, In An

Infinitely Extended Medium. Dept. of

Aero-nautics , The J ohns Hopkins Univ. {Contract N6ori-105, TaskOrder lIl} Dec. 1,1953

7. Kovasznay, L.S.G. Interaction of a Shock Wave and

Tur-bulence, 1955 Heat Transfer and Fluid

Mechanics Institute, Preprints of papers,

Univ. of Calif., Los Angeles, June 1955

8. Hollingsworth, M. A., A Schlieren Study of the Interaction

Richards, E.J. Between a Vortex and a Shock Wave in a

Shock Tube, A.R.C. 17, 9Jt5, F.M. 2323, 5th Nov., 1955

9. Hollingsworth, M.A., On the Sound Generated by the Inter - .

Richards, E.J. action of a Vortex and a Shock Wave, A.R.è.

Fig. 1. - Interaction of a sinusoidal shear flow with a shock.

shock

### Î

\

Fig. 3. - Synthesis of vortex from radially disposed shear

flows (physical înterpretation of Fourier integraH.

original

vortex

13EFORE shock

vortex, irtual i '

### Sttf°p.

### -,"

### -

/ /"### -

--\ --\ '*---I /*\ \ '"

### ,

--/### '"

/### \ "

### '"

### \ "

### "

\ \ AFTER \ \### \

### "

\### \

\Fig. 4. - Convection of vortex through shock wave, I: focusing of the refracted shear waves.

shock shock

Fig. 5. - Convection of vortex through shock wave, II: formation of envelope by p1ane sound waves generated ~t shock.

{magnitude)

.r

shock

~

Fig. 7. - Transformation of origin from vortex core

(virtual position~ to shock front. ..

shock

## :::r-

*"-R *

core,
actual
. positien
core,
virtual
··}Josition
wave shock \ \ \ \

### \.

.### ,,/"

\ lA.m### ~

\ vortex core kil pressure*pil *

*R *

Fig. 9. - (Upper) The elementary shear wave~pressure wave shock

interaction. (Lower) The pressure wave shown tangent to the envelope

shock

**3-**

2-/'
IV
### ~,

**G(er) **

### 1-0

**I**

**I**

**-8 **

_{-4 }

_{-4 }

### ·

### 0

### 4

### ()

### -

### -

**rl'-**

**rl'-**

**R **

**R**

**1;" **

**1;"**

_{( }

Flg. 11. - Upper and lower bounds to. radial pressure profile of cylindrieal sound

*2.0' *

I'
;A-..
*P, *

*/.0 *

*J, *

*fJJ *

~
### o

### o

~### '1

---... -. -_. I ~*J*

*--0 *

### f'"=

### I~I·

### ..

### fAt

### ~ ~

### I

### I'\: I

*o *

**40**

**40**

**0**"

**80· **

**80·**

**/20**

**/20**

**fJ **

*f *

### '"

Fig. 12. ~ Factors in the product P J that describes the va:;:.'iation of pressure with position around the arc of the cvlindric.aLs_ound wave.