The Quarterly of Applied Mathematics. Vol. 4, Nr 1

QUARTERLY

OF

APPLIED M A T H E M A T IC S

EDITED BY

H . L. D R Y D E N J. M. L E SSE L L S

T . C. F R Y W . P R A G E R J . L, S Y N G E

W ITH TH E COLLABORATION OF

m; A. B IO T H. W . E M M O N S J, A. G O FF

P, L E C O R B E IL L E R S , A. S C H E L K U N O F F S I R G E O F F R E Y T A Y L O R

L . N . B R IL L O U IN W . F E L L E R j . N . G O O D IE R F . D . M U R N A G H A N W . R . S E A R S

S. P. T IM O S H E N K O

N S S pig

Vo l u m e I V

APRIL • 1946

A u o r ? > ._{v v ;}

T H . v, K A R M A N I. S. S O K O L N IK O F F

J. P. D E N H A R T O G K . O. F R I E D R I C H S G. E . H A Y

E . R E IS S N E R R. V. S O U T H W E L L H . S. T S I E N

---

N u m b e r T

in

Q U A R T E R L Y

OF

a p p l i e d m a t h e m a t i c s

This periodica! is published quarterly under , the sponsorship of Brown Uni­

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?■ i f

1

Q U A R T E R L Y OF A P P L I E D M A T H E M A T I C S

1. Introduction. Although the propagation of sound waves in m oving media has received considerable attention [l, • • • , 9 ] ,1 little inform ation is available concern­

ing the propagation of such disturbances in rotational stream s or concerning the propagation of transient rotational phenomena. It is shown in the present paper that the w ave fronts associated with those parts of a disturbance which are derivable from a potential propagate in a rotational stream according to those laws which th ey are already known to obey in an irrotational stream. I t is further shown that th e rota­

tional disturbances drift with the stream rather than propagate relative to the m oving fluid.

T he analysis consists of an application of conventidnal perturbation procedures to the N avier-Stokes and con tin uity equations. T he equations so derived are treated according to the theory of characteristics. T he results obtained lead to a general ex­

pression for the Mach lines of an arbitrary supersonic flow and also suggest a new m ethod of wind tunnel calibration which elim inates the necessity of placing an ob­

stacle in th at portion of th e stream being calibrated. Finally, predictions are carried out as to the nature of pulses which are formed at a surface and then propagate through a boundary layer into a uniform stream.

2. The equations of motion. In this analysis, we shall consider the propagation of small disturbances in fluid stream s which are characterized by three functions of the space coordinates and the time, nam ely: po (the den sity), p a (the pressure), and Vo (the velo city). N o restrictions will be applied to these functions except th at th ey obey the differential equations im plying the conservation of m om entum , mass, and energy. T hese equations, known fam iliarly as the N avier-Stokes, continuity, and en­

ergy equations, m ay be written in the forms:

V ol. IV A P R IL , 1946 No. 1

ON THE PROPAGATION OF SMALL DISTURBANCES IN A MOVING COMPRESSIBLE FLUID*

BY

G. F. C A R R IE R A N D F. D . C A R LSO N H arvard University

1 i±

( v - g r a d ) v + d v / d t -\ - g r a d p = —L(y)

P P

= -L(v)

p

(1)

div v T d In p / dl + v-grad In p = 0. (2) d U / d t + pd( P~ ' ) / d t = Q + — x- (3)

P

* R eceived Jan. 10, 1946.

1 N um bers in brackets refer to th e bib"

2 G. F . C A R R IE R A N D F . D . C A R L SO N [Vol. IV, N o. 1

In the foregoing equations, p is the viscosity of the fluid, L sym bolizes (A + -J grad div) where A is th e Laplacian operator; Q is the rate of heat accum ulation, U is the internal energy, and x abbreviates th e viscous dissipation terms. D iscussions of these equa­

tions are conveniently found in [7 ] and [8]. T he necessity of m anipulating the energy equation in the investigation may be elim inated by using the following assumption.

T he changes in pressure and density accom panying the disturbance are taken to obey the law

P/po = (p/po) 7 (3')

where p, p , v, characterize the disturbed stream ; th at is, the disturbance is a phe­

nomenon such th at the changes in state from undisturbed to disturbed stream are isentropic. N o te th at this in no w ay restricts p oand po. T he appendix indicates briefly the fact th at while this assum ption is by no means rigorously justified, it leads to valid results.

It is convenient a t this point to introduce the small param eter e. Although this may be done in a fairly arbitrary manner, w e shall define it in th e following way in order to avoid any possible am biguities. L et the initial conditions of any particular problem be such th at at tim e zero, p = p q + ep j, where the maximum value of pi/po over the region under consideration is unity. T hus, since w e are considering small disturb­

ances, e is a small number compared to unity. Consistent with this notion, we shall write p = po + ep i+ + e2p2+ • • • , p —po + e p i + • ■ ■ , and v = v0 + e v i+ • • • , at tim e /; and shall require that the series be valid over a range of ^ Since disturbances can usually be expected to attenuate, it is certainly reasonable to expect that the series will converge for sufficiently small values of this parameter.

If we now su bstitute the foregoing forms of p, p , and v, into Eqs. (1) and (2), elim inate the p (except for p o ) by using Eq. (3'), and collect the coefficients of each power of «, we obtain:

1 p

(vo-grad)vo + d v 0/ d t H grad p Q L(v0)

po po

+ e | ( v0-grad)vi + (vrgrad)v0 + dvi/ d t - f grad

^ + • • ■ = 0, (4)

P i , 2 P i P

grad po T do— grad In p0---

P 2 Po Po

and

d In po/dt + div v0 + v0-grad In p0

L( y i) + — L ( v 0)

Po

+ « {d iv Vi - f Vj-grad In p0 + v0-grad — ---( — ) \ + • • • = 0,

1 P o d t \ p o / )

(5) where, a o = y p o / p o -

W hen, in equations (4) and (5), the coefficients of e° are equated to zero, we find two of the necessary conditions th at the functions po, p o , v0, characterize a possible fluid stream. Since these quantities m ust vanish identically, we m ay om it them from Eqs. (4) and (5), and divide the remaining equalities through by e. W hen w e allow e t o approach zero, w e see th at the m athem atically exact solution to the problem is found by setting the coefficients of e in Eqs. (4) and (5) to zero. H ence, we m ay expect

1946] T H E P R O P A G A T IO N OF D I S T U R B A N C E S I N A C O M P R E S S IB L E F L U ID 3

th at the functions pi, Vi, so determined will provide a good first approxim ation to the behaviour of small am plitude disturbances. T his, of course, is the conventional per­

turbation reasoning.

If we had been willing to assume at the ou tset a functional relationship p = p ( p ) applicable both to the stream and the disturbance, the perturbation procedure would have been unnecessary. T he forthcoming techniques could have been applied directly to Eqs. (1) and (2). H owever, th e solution possesses the desired generality only when we refrain from such restrictions on the nature of the stream. T his leads to a choice between working with the energy equation or using the foregoing procedure; the la t­

ter seem s more convenient. A s a m atter of fact, some of the results of this analysis differ from those of previous investigators only in th a t th ey are obtained for any stream wherein the medium behaves as a continuum rather than one of a very re­

stricted character.

Recalling now th at any vector m ay be expressed as the gradient of a scalar plus the curl of a vector and that

(B-grad)C + (C grad )B = grad (B -C ) + (curl B X C) + ‘ (curl C X B), one m ay write

Vi = grad <t> + curl A, and the differential equations defining pi and Vi become

grad (vo-vi) + a^{2 P i} 0 i ^d<p~\ H ^d curl A + a²0 ^pi — grad In p0

po dtJ dt po

Pi M r P i 1

— grad ÿo + ui X v0 + oo X Vi = — Z,(vi) H T (v0)

P„ P o L p o J

(6)

and

d / P i \

A 0 H ( — ) + V i - g r a d In p 0 = 0 (7)

dt \ p o /

whereco; = curl v<, and d / d t = [vo-grad + 3/ch ]. W hen th eop eration “curl” is performed on Eq. (6), the following equality arises:

don i p

= curl < —

d t

I po

— °o — grad In p0 Po

+ — grad i o - u i X v o - u o X v i ^ . Pi (8)

Po

f

It is evident by inspection of Eq. (8) that an identically vanishing initial choice of wi does not im ply th at this function will vanish for all tim e, as for exam ple, is th e case in an irrotational stream. T hus, we cannot om it o i in this investigation.

It will prove useful to define an (artificial) auxiliary potential ^ in the following manner2 (Eq. (6) im plies the existance of this quantity).

1 T h is will allow us to elim in ate p i/poand thu s ob tain eq u ation s in which each unknown has the d i­

m ensions of a v elo city potential.

4 G. F . C A R R IE R A N D F . D . C A R LSO N [Vol. IV, N o . 1

dp d A ix

r

— grad —• + cu rl = — .

dt dt p o L

L(y i) + — L ( v 0)

P o

— al — grad In p0

Po

, P1

+ — grad po Ml X Vo - oo X Vi = 0. (9) Upon substitution of Eq. (9) into Eq. (6), the latter becomes

grad Vo-Vi + i do2 pl ib Po

dp dp~\

— + — = 0. dt dt A

T his however, we m ay solve for pi/po arbitrarily choosing the “constant of integra­

tion ” to be zero.3 W e obtain

P i / p o =

T

d A p

dt

/ 1

r l

( — — + Vi grad In po = — — (-b v 0-curl A

\ a - dt J dt \ d t / .

( 10)

(ID It will be shown directly th at the w ave front propagation can be derived from Eqs. (8), (9), and (11), provided we can ju stify the omission of the term (p /p0)L(vi) from Eqs. (8) and (9). W e note, considering Eq. (8), th at if only the terms dcoi( dt and curl (pt/po)T(vi) were non-vanishing, we would have virtually the equation for the conduction of heat, th at is d<oi/d/=(p/po)Aoi. T he “conduction coefficient” is very small (in air the spreading of vorticity is known to be very slow) so th at the term in question may be thought of as one which causes a small dispersive effect. It is to be understood, then, th at this effect is to be superimposed on any results which are ob­

tained by treating the equations from which this term has been om itted. W ith this omission we are now ready to apply the method of characteristics.

Hadamard [ l ] has shown the following facts concerning second order differential, equations which will be useful in the analysis of the foregoing equations. H e con­

siders the equation

n

E dikpik + h = 0

in the n independent variables x\, ■ • • , x n, where the a,-* and h are functions of the unknown quantity z, th e and the first partial derivatives of z with regard to the x,-;

pik — d2z / dx i d xk ■ T he differential equation which defines the characteristic surfaces (wave fronts) of this equation is given by

n—1 n—l

B = £ aikPiPk - E atnPt + = 0 (12)

i.k-l i- 1

where P , = 3.r„/3x,- when the “surface” is written in the form

Xn ~ ^n(Xi, • ' * , Xn— l)* (12a)

3 A ny such (actu ally tim e depend en t) con stan t could be absorbed in d<p/dt and would contribute nothing to V i.

1946] T H E P R O P A G A T IO N OF D IS T U R B A N C E S IN A C O M P R E S S IB L E F L U ID 5

Furthermore, let there be 5 unknown functions Zi, • • • , z „ and s equations of the form y*. 0,ikpik + bikQik + ‘ ‘ • + Cikgik + h = 0.

i,k

Here the a n , bik, ■ ■ • are respectively the coefficients of the second derivatives of Zi, z2> • • • . T he characteristic surfaces of this system of equations are determined by the relation

-Bui -B12, • • • , B u

B, i B „

T he B aßare analogous to th e q uan tity B of Eq. (12). In fact, when a takes the values 1, 2, • ■ • , s, B is derived respectively from the first, second, • • • sth, equations.

When ß takes these values, the ^{B aß} are obtained from the a ik, bik, • • • , c« , respec­

tively. T he present problem deals with the five unknown quantities 0 , 0 , and th e three com ponents of A. Eq. (9) is equivalent to four scalar equations4 if we specify (for exam ple) th at \p and A are to be those solutions for which d iv A = 0. T his is no restric­

tion since only curl A appears in Vj. W e m ay, then, apply the foregoing typ e of analy­

sis to Eqs. (9) and (11) (with po/pi replaced by the expression given in Eq. (10)).

In fact, in order to determine the characteristic surfaces which define a motion involv­

ing the function 0 , we need only a brief inspection of Eq. (9). I t is evident that no sec­

ond derivatives of 0 appear in this equation. T hus, when it is split into its four subdivisions, we find that the four quantities Bn , • ■ ■ , B n , which appear in the left column of Eq. (13), vanish. T his implies th at Eq. (13) is satisfied when either Bu or the minor associated w ith this quantity vanishes. Since the vanishing of the former involves only the coefficients of derivatives of 0, we m ay assume th at this surface will be associated with th e potential ty p e of disturbance. T he vanishing of the minor will correspond to the propagation of disturbances of the rotational type.

If we now com pute B n using, of course, the a « of Eq. (11), we find the same w ave front equation which w as found by Hadamard for the isentropic stream. T h at is, the tim e-position correlation of a w ave front does not depend on the character of the stream but only (as the following equation will show) on the local values of the quantities no, Vo, wo, and do- T he first three of these are the com ponents of v0. This wave front equation, in a form som ewhat more convenient for our purposes than Hadam ard’s, is shown below.

d y / d t + Uody/dx + Wady/dz — v0 ± a0[l + (d y / d x) 2 + ( d y / d z) 2] 112 = 0. (14) Eq. (8) indicates th at whenever o0 and 0 are each non-vanishing in a given region, a rotational motion oi, is generated continuously. T his being so, there is always a possible “vorticity wave front” coincident with the w ave front associated with 0. Hence, if we treat Eq. (8) according to the foregoing m ethod, using the com ponents of curl A as the unknown function, w e find th at the determ inant vanishes identically.

T his is to be expected since the operation which led to Eq. (8) elim inated the higher derivatives of 0 while retaining the higher derivatives of A. Hence, formally, the char­

* A n y equation of the form curl M + g r a d Q = C can be reduced to the forms grad Q = P and curl M = N if one is ingenious enough to separate C in to the required parts.

6 G. F . C A R R IE R A N D F . D . C A R L SO N [Vol. IV, N o . 1

acteristics method fails to give the desired information. W e note th a t in this method, however, th e only terms which affect the positional nature of the propagation are those containing second derivatives of the unknown functions. If, in Eq. (8), we segragate these terms of the required order w e find th at they comprise exactly the single term dan/dt. Therefore, in so far as the position of the disturbance is concerned, w e h aved w i/d L =0; th at is, th e tim e rate of change of vorticity, relative to an observer m oving w ith th e particle, vanishes. In other words, the rotational disturbance drifts w ith the stream instead of propagating relative to it. T his statem en t m ust be modi­

fied, of course, by the results of the diffusion-implying viscous term which was om itted in this analysis.

3. The tw o-dim ensional problem. Since, in general, the functions p0, po, v0 asso­

ciated with any given stream are not known (even approxim ately in m any cases), it seem s of interest to describe a method of wind tunnel calibration based on the fore­

going analysis (in particular on Eq. 14). T his proposed procedure will be seen to have the advantage th at it does not require the insertion of an obstacle into th at portion of th e stream being calibrated. L et us consider only tunnels which are bounded by the side walls z = ± b , where b is some constant. In this two-dimensional wind tunnel, the flow in the neighborhood of z = 0 is essentially independent of z. L et us also restrict our consideration to disturbances having reflective sym m etry about th e plane z = 0. Then a t z = 0, E q. (14) reduces to

d y / d t ± a0[l + { d y / d x y ] lli — z0 + u^dy/dx —0. (15) We now have an equation, linear in the three quantities which we wish to deter­

mine; Mo, Vo, and a 0. Suppose we generate pulses a t several points along some bound­

ary of the stream, say by the use of an electric spark. T h e w ave fronts of these pulses m ay be observed (photographed) at successive tim e intervals. T he values of d y / d x and d y / d t can be determined from the photographs for each pulse throughout the region it traverses. For each point traversed by at least three pulses, we m ay form three sim ultaneous equations in the unknown quantities by using these experimen­

tally determ ined values as coefficients in Eq. (15). Figui-es 1 to 4 illustrate such p h oto­

graphs of sound pulses in a fairly uniform stream of air. T he developm ent of the techniques used in obtaining these Schlieren photographs should be credited to the authors of [9]. In [9] the details of th e experimental procedure are explained quite fully.

For an isentropic region of the stream (where th e stagnation condition is known) only two pulses are needed since (see [l 1 ])

2 2 2 2

Oo = ( f ls t ) — ( t — 1 ) ( « 0 + V o ) / 2 .

Finally, for an essentially one-dimensional stream (e.g., a je t or a slowly converg­

ing channel) Eq. (15) becom es

» o / f f .t =

7 - f i

where n ^ a ^ d y / d t .

W hen the stream is supersonic, w e m ay generate stationary disturbances (Mach [ 2 ( t + 1) - 2 ( 7 - 1)m2] 1/2 - 2M

(15a)

1946] T H E P R O P A G A T IO N OF D I S T U R B A N C E S I N A C O M P R E S S IB L E F L U ID 7

lines) by placing very small irregularities in the boundaries of the passage. For this case Eq. (15) reduces to

, , , - ± a \ [ M2 - 1 ] I/2

d y / d x = --- (16)

d2 — u2

o o

where M , the Mach number, is given by M2 = ( n l + v l ) /a%. A mesh of Mach lines pro­

ceeding from both edges of the passage give sufficient information to calibrate any stream known to be isentropic with known stagnation condition. For a one-dim en­

sional stream, we have the familiar formula for the Mach angle 6

d y / d x = tan 0 = ( M2 - l ) 1'2. (17) N o te th at when one wishes to find the characteristics of a given supersonic stream, the classical Charpit procedure will alw ays provide solutions for E q. (16). T he equa­

tion analogous to Eq. (16) in three dim ensions follows directly from Eq. (14) by merely dropping the tim e dependent term.

4. The effect of boundary layers. A problem of considerable interest arises in con­

nection w ith the ideas of the foregoing section when we inquire into the effect of the boundary layer on the form of the w ave front when the pulse is generated a t the

surface of a boundary (or obstacle). W e shall use the Charpit procedure to solve Eq. (15) for this case, using, of course, an idealized group of values for u 0, v 0, and a 0.

T he justification of the steps of this procedure are given in [10] and need n ot be given here, so we shall proceed formally with this method. T he solution obtained will be in closed form and is readily verified (as a solution of Eq. 15) by mere substitution.

W e characterize the stream by the functions

«o = 0, zio — vx/S, for x g 5, Do = » for x ^ 8, a0 = a = const.

T his simplification of an actual situation is som ew hat drastic but useful information results. W e first set £ =x / 5 , i ) = y / 5 , r = a t / d , M = v / a , p = dr)/d%1 q = d r//d r, and Eq.

(15) in the notation of [10] becomes

F(£, v, T , p , q ) = * q ± [1 + p 2] lli - MS = 0 for O ^ ^ l or

F — q ± [1 + p2] 1' 2 - M = 0 for 1 g f. (18) W e proceed by considering the associated ordinary differential equations

dp dq drj d£ dr

F( + pF„ FT + qF1 — pFp — qFq — Fp — Fq (19) and choose any solution which expresses p or q in terms of a parameter a. In our case, (formally)

— dp / p2 \ ~ l

=

= — I t ^ (-

) _dr) =

M

\[1 + p 2]1/2

7

p = [(a - MU) 2 - I}1' 2 and p = [{a - M) 2 - l ] 1'2. (20)

W e now determine 77 by the following integration

V — ¡3 = j ’pdÇ + qdr . (21)

/3 is chosen to suit the initial and boundary conditions; a is determined by the rela­

tion d r j / da = 0, and the sign of p m ust be taken consistent with th at portion of the w ave front under consideration.

For the initial condition 77 = 0 a t £ = r = 0 (a point source), and for £ â l , we have for th at downstream portion of the w ave for which p ^ O ,

77 = ^{o lt} + /3 (a — MÇ) — /3(a) (22) where

[3(a) = a [a2 — l ] I/2 — arc cosh a (23)

8 G . F . C A R R IE R A N D F . D . C A R L SO N [Vol. IV , N o . 1

Mi; M r a = ---+ ---

2 2

4 T / 2

<24) T his solution is valid when £0 ^ £ is min (r, 1);

(0 = [r2 + M~ 2] 1' 2 - 1 / M . W hen p ^ 0, and we must find 77 by writing

v ( r , £) = V(r, {0) + f Pd$

J to

which results in the formula

7?(r, £) = a r — /3(a) — /3(a — M£) (25) where a and /3 are still the quantities defined by Eqs. (2 3 ) and (2 4 ).

When we consider the upstream portion of the w ave front, a m ust be negative.

B y the foregoing procedure we obtain

J7 1 = a ir — /3(— a i ) + |3 (M £ — a/) (26) Ml; M r ~

a i 2 1 +

.2 _ t2

Î2J

1 / 2

(27) or

0i(£> 0 = — ??(£, r) + M£r. (28) T his equation is again valid only when £ — £o*

W e now consider th at portion of the w ave exterior to the boundary layer (i.e.

£ i? l) . W e m ust again extend the integration of Eq. (21), this tim e into the uniform stream. Using now p = — [(a —A f)2] I/2, we obtain for the downstream portion

77 = a r + /3(a — M ) — /3(a) — J ' [(a — M ) 2 — 1 ]1/2d£

= a r + P(a - M ) - /3(a) — [ (a — M) 2 — l]«'2($ - 1). (29) When dr]/da is equated to zero we find

i - 1 = ~ ~ ^ 7 \ t + — [(a - M) 2 - I]»/2 - — [a2 - l ] 1'2} . (30)

a — M { M M J

Fig. 1. Sound w aves proceeding from point source in nose of vane. Tim e delay betw een pulse generation and photography arc 88 and 263 micro­

seconds. T he flow is directed to the left a t a M ach num ber of .423.

Fi g. 2 . (C ontinuation of F i g . 1 ) T im e d elays arc 441 and 700 micro-seconds. N o te refraction and reflection phenomena.

F ig . 3. Sound w aves proceeding from flush sources. B right spots indicate spark gap locations.

M = .33.

Fi g. 4. Sound w aves from flush sources. M = .51.

N o te the poor definition of the w ave front positions in the boundary layer region.

1946] T H E PROPAGATION OF D IST U R BA N C E S IN A COM PRESSIBLE FL U ID 9 E qs. (29) a n d (30) m ay be considered as p a ra m e tric eq u a tio n s fo r £ an d t] in term s of th e p a ra m e te r a for each v alu e of r . In a sim ilar m an n er we o b ta in for th e u p stre a m p ortion

m = a i r - f}(— a i) + ¡3(Af - « 0 + [ (a , - M ) 2 - 1 ]>'*(& - 1) (31) a n d

f( M - a ,)2 - l l 1/s

1 = 1 - 1 + — [a i - 1 ] * " - - [(Ilf - « , ) ' - I f } . (32)

M — a i

In E qs. (29) to (32), 1 + J k T g a < a> an d — 1 S « i > — 00•

T h e solution is now com plete ex cep t for surface reflections. N o te t h a t th e velo city a t w hich th e p o in t of c o n ta c t betw een b o u n d a ry an d w ave fro n t m oves is

dt]

dr £ - 0

or

dy r v-t2nU2

— = a 0 1 + ---

dt L 452_

= a ( r , 0) = T

= aoj^

I V T /2

1 +

(33) F ig u re 5 illu stra te s th e re su lts of th e foregoing section fo r a stre a m w ith M ach n u m b er .50. T h e peculiar b eh av io r of th e solution for r > \ / 5 leads us to in v estig ate th e ra y s of th e p ro p a g atio n . I t has been show n [l, 2, 3] th a t E q . (14) im plies th a t the ra y s be defined b y d x = ( l a0+ u0)dt, dy = (ma0+ v0)dt, an d d z = (na0+Wo)dt, w here I, m, n, a re th e d irec tio n cosines of th e o u tw a rd ly d irec ted w ave fro n t norm al w ith th e co o rd in ates axes. In p a rtic u la r, it h as been show n t h a t for th e co n d itio n s prev ailin g in th e b o u n d a ry lay er specified above, th e ra y s are given by (see [9])

V = [arc cosh (m0 — Af£) + {m0 + M£){ (m0 — M!j)2 — 1} 1/2]cf (34) 2 M

w here m0 is th e v alu e of m ~ l a t th e origin. T h e v a lu e of nip for w hich th e ra y becom es ta n g e n t to th e line £ = 1 is given b y m0 = l + M . H ow ever, a n y ra y associated w ith a uniform strea m w hich is d irected parallel to t h a t stre a m will m a in ta in th is o rie n ta ­ tion. H ence, th is ra y w hich ju s t becom es ta n g e n t to th e uniform strea m b ifu rc ates in to th e cu rves show n in Fig. 5. N o te t h a t no ra y w hich is once reflected b ac k in to th e b o u n d a ry lay er will ever leave th is region. T h is im plies t h a t a sizeable p o rtio n of th e energy of such pulses n ev e r leaves th e b o u n d a ry lay er. F u rth e rm o re , as one can readily see from th e few ra y s p lo tte d in th e figure, th e reflections occur in such a m an n er th a t in terferen ce as well as th e ex tre m ely tu rb u le n t co n d itio n s in such a region m ak e th e o b serv atio n of w aves in such regions im p ro bab le. T h is is bo rn e o u t in F igs. 1 to 4. One large discrep an cy betw een these p ictu re s an d th e th e o ry is easily noticed. T h e lim iting u p strea m ra y given by th e th e o ry does n o t agree to o well w ith th e evidence of Fig. 4. T h is is d u e to th e fa c t t h a t th e pulse used in th e ex p erim en t s ta rte d as one of finite a m p litu d e (p ro b ab ly trav e lin g in itially a t a speed of a b o u t 2o0).

T h is m eans t h a t a t first th e w ave tra v e ls as th o u g h it w ere a sm all a m p litu d e w ave in a slow er s tr e a m ; hence, less d isto rtio n from th e sh ap e w hich w ould be expected w ith no b o u n d a ry la y e r (a fam ily of sem i-circles) is th e logical re su lt to expect. T h is, of course, is co n sisten t w ith th e ob serv atio n s. One o th e r re m a rk is e ss e n tia l in view of

10 G. F . C A R RIER A N D F. D. CARLSON [Vol. IV. N o. 1

th e in itial idealization of th e b o u n d a ry lay er. In th e a c tu a lly occurring physical s itu a ­ tio n , th e b o u n d a ry lay er th ick en s in th e directio n of flow an d th u s th e sh a rp b re ak in w ave fro n t p re d ic ted in th is th eo ry is n o t valid. H ow ever, th e tra n s itio n from large v elocity g ra d ie n t to uniform strea m occurs in a sufficiently sm all region so t h a t little energy tra n sfe r from th e stre a m p a r t of th e w ave to th e b o u n d a ry la y e r is to be ex­

pected.

Fi g. S. P re d ic te d w av e fro n ts for th e s tre a m defined in se ctio n (4 ). W a v e fro n ts a re illu s tra te d fo r t = 1 2, V 5 , 3, S. T h e d o tte d c u rv es a r e ra y s of th e p ro p a g a tio n . T h e so u n d so u rce is a t th e origin.

If one w ishes to ac c o u n t for th e large pulse velo city in a m a th e m a tic a l m an n er, he can replace th e c o n s ta n t v alu e of a0 by a fu n c tio n of th e tim e , large a t tim e zero b u t ra p id ly ap p ro ach in g th e s te a d y v alue. T h is, of course, m akes th e calcu latio n s tedious.

5. In te n s ity d istrib u tio n . In general, it is difficult to o b ta in a solution for <f> from E q . (11). H ow ever, one v ery in te re stin g solu tio n for th e isen tro p ic uniform stre a m has re c e n tly ap p e are d . R o tt [6] has show n t h a t w hen no=wo = 0, u0 = co n st = — M a 0, E q . (11) has a solution of th e form

(¡>1 = —- cos C R

( \

\ O o ( l - M 2) /

o0( l - M 2), (35)

w here R = [#2 + ( l — Af2)(y 2+ s 2) ] 1/2. T h is solution im plies a co n tin u o u s p o in t source a t th e origin an d , of course, assum es no b o u n d a ry la y e r if th e p lan e (say) y = 0 is to be a b o u n d ary .

T h e surfaces of c o n s ta n t p h ase (c h a ra c te ristic surfaces) are given b y

I M x + R

a o ( l - M 2)

= const. = X (36)

a re la tio n sh ip w hich can be show n to sa tisfy E q . (14) for th e given strea m .

1946] T H E PROPAGATION OF D ISTU R BA N C E S IN A COM PRESSIBLE F L U ID 11 In connection w ith o u r pulse problem , we see im m ed iately t h a t we m ay form a new solution as th e integral

f 00 C(m) ( M x + R \

= I cos w ( I --- ) du>. (37)

J o R \ ao(l — M 2) /

If C is p ro p erly chosen, th is solution corresponds to a pulse of a n y desired w ave form o rig in atin g a t th e origin. T h e surfaces of c o n s ta n t phase are again given b y E q . (36) an d , for a given value of t, are circles w ith ce n te rs a t th e p o in ts x = M a0t, y = z = 0, an d radii a0t.

S uppose we now choose tw o valu es of A (say Ai an d X2) to re p resen t tw o c h a ra c te r­

istic su rfaces w hose se p a ra tio n (along a ra d ia l line from th e origin) we can call th e w ave len g th of th e pulse. T h e n th a t u p strea m p o rtio n of th e pulse a t y = z = 0 will h av e a w ave len g th (X2 — Ax)a0( l — M ) a n d th e do w n stream sectio n a t y = z = 0 will h av e a w ave len g th (X2 — X i ) a o ( l + i l i ) . T h a t is, th e ra tio of th e thickn esses of these tw o extrem e p o rtio n s of th e pulse is (1 + A f)/(1 — M ).

In sp e ctio n of E q . (35) also leads to th e conclusion t h a t a t th ese p o rtio n s of th e p ulse th e am p litu d e s v a r y in th e ra tio (1 — M ) / ( \ - \ - M ) . T h u s th e a m p litu d e gradients a t these tw o sections are in th e ra tio (1 —M ) 2/ ( l - t - J l i ) 2. Since th e d en sity g ra d ie n t is essen tially th e q u a n tity observed in th e Schlieren o p tical sy stem , th e foregoing con­

s titu te s an ex p lan a tio n of th e fa r su p erio r c la rity of th e w ave fro n t definition in th e u p strea m p o rtio n s of th e p h o to g rap h s. T h is a rg u m e n t h as assum ed t h a t th e pulse s ta rte d a t tim e zero a n d has a sm all thickness to ra d iu s ra tio a t th e tim e of ob serv a­

tion.

Specifically, th e a m p litu d e ra tio (i.e., th e ra tio of a m p litu d e a t a n y p o in t on th e w ave fro n t d ivided in to th e a m p litu d e a t th e u p strea m ex trem u m ) is given by

M x , v

Amp. ratio = 1 + M --- (38)

a o / ( l — M) APPENDIX

W e wish to ju s tify here th e use of E q . (3 '). W e w rite th e energy e q u a tio n in th e form

R d T d p ~l a n

7- + />— - = — AT + — x (3)

y — 1 dt dt p p

w here T is th e te m p e ra tu re (p = p R T ) a n d x involves a sum of p ro d u c ts of th e form (d u /d y ) (d v /d x ) , (d u / d y ) 2, • • ■ . W hen, as in th e foregoing w ork, th e p e rtu rb a tio n proced ure is applied, th e term s w ith coefficient e can be w ritte n

1 d{T/To) d(p/Po) a n

A T i d X i + x : (3a)

7 — 1 dt dt poRTo Po

w here we h av e gro u p ed th e term s n o t co n tain in g d e riv a tiv e s of th e unknow n fun c­

tio n s pi, pi, Vi, Ti in K . Such term s can obviously c o n trib u te n o th in g w hen th e c h a r­

a c te ristic s m eth o d is applied. In E q . (3a) ( y — l ) a / R is a ra tio of specific h e a t to th erm a l c o n d u c tiv ity , an d x i involves p ro d u c ts of th e form (dtio/dy) (dvi/dx), (du0/d y ) ( d u i/ d y ) , • • • .

12 G. F. CARRIER A N D F. D . CARLSON

If we can show th e th ird an d fo u rth term s of th is e q u a tio n to be negligible, in ­ te g ra tio n of E q . (3a) leads to th e te rm s w ith coefficient e in th e expansion of E q . (30- Using a dim ensional tre a tm e n t analogous to P r a n d tl’s b o u n d a ry lay er an alysis, we define a ty p ical len g th I for th e d istu rb a n c e (say th e w av e len g th , if a co n tin u o u s w ave is considered, or th e b re a d th of a pulse, etc.) an d com p are first th e term s (y —l ) _1v 0' g r a d ( r i / r 0) an d (p / p0)(,du0/ d y ) ( d u1/ d y ). W e n o te t h a t (y —l ) -1v 0-grad (7 V 7 ’o ) ~ ( 7 '— I ) “ 1 ( 2 0 /7 0 )(| »01 /I) a n d

M dic0 du 1 ^ p. | V0 | | V: |

po dy dy floPo 5 I

so we h av e th e re q u irem en t, if th e la tte r term is to be o m itted , t h a t po5a0/M |vi | ii>T o / T i . Since T1/ T0 a n d |i y |/a o are of ap p reciab le m a g n itu d e (of o rd e r u n ity ) in th e sam e region, th is in eq u ality is essen tially p05ao//P£>l. H ere, 5 is th e b o u n d a ry lay er th ick n ess o r o th e r ty p ical dim ension of th e stre a m . T h u s we see t h a t ex cep t for very re stric te d regions th e ab o v e in e q u a lity will hold a n d th e te rm in. q u estio n m ay be o m itte d . T h e o th e r term s in x i a d m it a sim ilar tre a tm e n t.

W e now com pare th e term s d ( T \ / T 0) /d t an d a A T i / R T0po- N o te t h a t th ese are ex actly th e term s w hich w ould need to be com pared in th e still a ir case; t h a t is, no fu n c tio n s ch a rac te rizin g th e strea m a p p e a r excep t th e slowly v a ry in g Oq. U sing th e ty p ical len g th I an d th e fa c t t h a t th e p ro p a g atio n occurs a t essen tially v elo city do, we o b tain a A T i / R T 0p o ~ a T i / R T o l 2po<& d ( T \ / T ( i ) / d t ~ a o T / l T o as a necessary con ditio n.

T h a t is, th e n u m b e r a / a0lRpo<ZA ■ T his, of c o u rs e /is th e a c tu a l situ a tio n * an d hence b o th te rm s in questio n m ay be o m itte d . T h is n u m b e r is essentially th e reciprocal of a P ra n d tl n u m b er-R e y n o ld s n u m b e r p ro d u ct.

Acknowledgment. T h e a u th o rs wish to express th e ir a p p re c ia tio n to P ro f. H . W . E m m o n s for v alu ab le suggestions a n d for th e use of th e H a rv a rd high speed w ind tu n n el.

Bi b l i o g r a p h y

1. J. H a d a m a r d , Lemons sur la propagation des ondes, A. H erm ann, Paris 1903, p. 362.

2. H . B a t e m a n , Sound waves as extremals, J. A coust. Soc. Am er. 2, 468-4 7 5 (1930).

3. H . B a t e m a n , M onthly W eather R ev. 46, 4-11 (1918).

4. E . A. M i l n e , Sound waves in the atmosphere, Phil. M ag. Ser. 6, 42, 9 6 -1 1 4 (1921).

5. L o r d R a y l e i g h , Collected Papers, Cam bridge U n iv. Press, Cam bridge, 1920.

6. N . R o t t , Das Feld einer rascli bewegten Schallquelle, Leem ann, Zurich, 1945. 88 pp. (E . T . H ., In st.

A erodyn., N o . 9).

7. H . L a m b , Hydrodynamics, D over Publications, N ew Y ork, 1945, p. 576.

8. A. V a z s o n y i, On rotational gas flows, Quart. Appl. M ath. 3, 29-3 7 (1945).

9. J . C . E i s e n s t e i n , K . C . C l a r k , F . D . C a r l s o n , O .S .R .D . R eport N o. 5369, A u g., 1945.

10. A . C o h e n , Differential equations, H eath , 1933, p. 263.

11. T il. v . I \ a r ma n, Compressibility effects in aerodynamics, J . A ero Sci. 8 , 3 3 7 -3 5 6 (1941).

* N u m e ric a l s u b s titu tio n in d ic a te s fo r a ir t h a t C§>10-6 in ch e s is a sufficient c o n d itio n .

13

T H IN CYLINDRICAL SHELLS SUBJECTED TO CONCENTRATED LOADS*

BY

SH A O W E N Y U A N f California Institute of Technology

A bstract. A single differential equation of the eigh th order in the radial disp lacem en t is given for the equilibrium of an elem en t of a cylindrical shell undergoing sm all disp lacem en ts due to a laterally d istributed external load. T h e radial deflection of thin cylindrical shells subjected to concentrated, equal and op p osite forces, actin g a t the ends of a vertical diam eter, is analyzed b y the Fourier m ethod. A pplica­

tions of th e solution o f th e problem o f the infinitely long cylind er to the problem s of a couple actin g on an infinitely long cylinder in th e direction of eith er the generatrix or the circum ference are also discussed.

1. In tro d u c tio n . T h e bend in g problem of an in fin itely long cy lin d er loaded w ith co n c en trate d , equal an d opposite forces, ac tin g a t th e ends of a v ertic al d iam eter, is discussed first. T h e eq u atio n s of equilibrium of an elem en t of a cylindrical shell u n ­ dergoing sm all disp lacem en ts d u e to a la te ra lly d istrib u te d ex tern al load are reduced to a single differential eq u a tio n of th e eighth o rd e r in th e ra d ia l d isp lacem en t. In th is eq u atio n th e vario u s te rm s are com pared as to th e ord er of m ag n itu d e and it is found th a t som e of th e term s are negligible.

T h e specified loading fu n ctio n is re p resen ted b y a F o u rie r integral in th e lo n g itu ­ dinal d irection, an d b y a F o u rier series in th e circu m ferential direction . T h e in teg ral re p resen ta tio n has th e a d v a n ta g e t h a t th e b o u n d a ry con ditio ns are a u to m a tic a lly ta k e n care of, and no su b seq u e n t d e te rm in a tio n of F o u rier coefficients is necessary.

T h e F o u rier coefficients an d th e u n d eterm in ed functio n in th e F o u rier integral in th is case are d eterm in ed sim ply from th e loading condition. T h e rad ial disp lacem en t is represented in a lijre m an n er w ith th e aid of an u n d eterm in ed fu nction w hich is o b ­ tain ed b y s u b stitu tin g b o th ra d ia l displacem en t an d loading expressions in th e differ­

en tial eq u atio n . T h e definite in teg ra ls involved in th e expression for radial deflection are ev a lu a ted by m eans of C a u c h y ’s theorem of residues.

T h e problem of th e inextensional d eform ation of cylindrical and spherical shells was tre a te d in d etail by Lord R ayleigh in his “T h e o ry of so u n d .” T h e assu m ptio n of th is ty p e of d efo rm ation underlies th e solution of m an y problem s of p ractical im ­ p o rtan c e, such as th e d e te rm in a tio n of stresses in th in cylindrical shells su bjected to tw o eq ual and opposite forces ac tin g a t th e ends of a d ia m e te r o r to in tern a l h y d ro ­ sta tic pressure. I t is found t h a t th e re su lts o b tain ed in th e case of inextensional de­

fo rm ations correspond only to a first ap p ro x im a tio n of th e solution in th is p ap e r, and th e stresses in th e p ro x im ity of th e po in ts of application of th e forces are n o t given w ith sufficient accuracy.

T h e expression for th e ra d ia l deflection of a th in cylinder of finite length is ob­

tain ed from th e corresponding solution for an infinitely long cylinder b y using th e m eth o d of im ages. I t is seen th a t th e difference of these tw o rad ial deflections can be given b y a correction fa cto r included in th e expression for a cylin der of finite leng th .

* R eceived A pril 20, 1945.

t N o w a t P olytech n ic In stitu te of B rooklyn.

14 SHAO W EN Y U A N [Vol. IV, N o. 1

T h e difference is believed to re su lt from restrain in g th e edges a t th e tw o ends of th e finite cylinder. T h e resu lts in d icate th a t th e rad ial deflection of an infinitely long cyl­

inder has a v ery long w ave length along th e g en e ratrix ; how ever, th e w av e length decreases as th e ra tio of ra d iu s o ver th ick n ess decreases. I t is believed t h a t th e long w ave length phenom enon is d u e to th e elastic reactio n along th e circum ference of th e shell w hich can be explained b y th e ra d ia l deflection along th e circum ference.

D eflection curves of cylindrical shells w ith v ario u s leng ths are calcu lated an d th e resu lts show t h a t th e m axim um ra d ia l deflection occurs a t len g th over ra d iu s ra tio / / a ~ 20. T h e ra d ia l deflection of an infinitely long cylin der w ith th e ra d iu s o v er th ic k ­ ness ra tio a / h —100, becom es zero a t a b o u t x / a —15 and th e n reverses its sign. T h e edges of th e corresponding cylinder w ith finite len g th a re so re strain ed t h a t th e nega-

Fig. 1. F orces and m om en ts on elem en t of wall.

tiv e deflection portion of th e infinite cylinder is b ro u g h t to zero a t th e edges of th e cylinder w ith finite length. H ence, th e m axim um deflection of a cy lin d er w ith Z / a ~ 20 is g re a te r th a n th a t of th e corresponding infinitely long cylinder.

T h e problem s of a couple ac tin g on an infinitely long cy lin der in th e d irection of eith e r th e g en e ratrix or th e circum ference are also analyzed b y using th e co rresp o n d ­ ing solution for th e ra d ia l deflection u n d er a c o n c e n tra te d load. T h e ac tio n of th e couple is eq u iv ale n t to t h a t of tw o equal an d o pp osite forces a c tin g a t an infinitely sm all d istan ce a p a rt.

2. F u n d a m e n ta l eq u a tio n s. T h e fu n d a m e n ta l e q u a tio n s of a cy lind rical shell u n ­ d er th e specified loading are o b tain ed from considering th e equilibrium of an elem ent c u t o u t by tw o d iam etrica l sections an d tw o cross sections p erp en d icu lar to th e axis of th e cylindrical shell as show n in Fig. 1.

In th is discussion th e usual assu m p tio n s are m ad e; nam ely , t h a t th e m ateria l is isotropic an d follows H o o k e’s law, th e undeform ed tu b e is cylindrical, th e w all th ic k ­ ness is uniform an d sm all com pared to th e radius, th e deflections are sm all com pared to th is th ickness so t h a t second ord er stra in s can be neglected, an d t h a t s tra ig h t lines in th e cy lin d er w all and p erp en d icu lar to th e m id dle surface rem ain s tra ig h t a fte r disto rtio n .

T h e n o ta tio n used for re s u lta n t forces an d m o m en ts per u n it leng th of w all section are in d icated in Fig. 1. A fte r sim plification, th e following eq u atio n s of eq uilibrium are o btained:*

d N x dN<,t d M x4, d M *

u 1--- = 0, a ---f- aQ$ — 0,

dx d<f> d x dcf>

d N* 3 Z X* d M x d M* ,

——

<

«

o,

a<2*

o. (i)

d<f> ox ox d<f>

dQx 3Q*

a _^L + + N4 + qa = 0 , _ N ^ a = o,

dx d<f>

in w hich q is th e n orm al pressure on th e elem ent.

If Qx an d <2* are elim inated from E qs. (1) an d th e relatio ns

N x * ' N $ x I H£ x <t> == * X

a re used, th e six eq u atio n s in (1) can be reduced to th e following th ree : d N x d N x*

a — ---- 1--- — = °.

dx d<f>

d N * dNx* d M x<, 1 d M *

b a i = 0, (2)

d(f> dx dx a d<f>

2 3 W X* a w , 1 32M* Z*

--- - -+ --- + 1 + _ f + q = o.

a d<f>dx d x2 a2 d<f>2 a

T h e relation betw een th e re s u lta n t forces an d m o m en ts an d th e stra in s of th e m iddle surface will be tak en th e sam e as in th e case of a flat p late:

E h E h y E h

N x = (ex + re*), A * = (e* + vex), Arx* = IV** = ———— - >

1 — v2 1 — v 2 2(1 + v)

M x = - D ( X x + rZ*), Af* = - £ (Z * + v X x), Mx* = - iIf*, = D{ 1 - » ) ! ,* , w here D = E h2/ 12(1 — r 2) is th e flexural rig id ity of th e shell an d h is th e th ick ness.

R esolving th e disp lacem en t a t an a rb itra ry p o in t in th e m iddle surface d u rin g d e ­ form atio n in to th re e com ponents— u along th e g en e rato r, v along th e ta n g e n t to th e circu lar section, a n d w along th e norm al to th e surface d raw n inw ards— one finds t h a t th e extensional stra in s and changes of c u rv a tu re in th e m id dle surface are

du 1 dv w dv du

e* = — > e* = ) y** = --- 1--->

d x a dcf) a d x ad<f>

d 2w 1 d2w 1 dv 1 d2w 1 dv

X x — J Z * = --- 1--- ; Z i * = --- 1--- ’

d x2 a2 d<f>2 a2 d(f> a dxd<f> a dx

H ence, E qs. (2) can be p u t in t th e form of th re e eq u atio n s w ith th re e unknow ns u, v, w:

d2u 1 + y d2v v dw 1 — v d2u

d x2 2 dsdx a d x 2 ds2

1946] C YLIN DR IC AL SHELLS SU BJEC TED TO C O N CENTR ATED LOADS 15

* See R ef. 5, p. 440.