On linear and non-linear wave equations for the acoustics of high-speed potential flows

Journal of Sound and Vibration (1986) 110(1), 41-57

ON LINEAR AND NON-LINEAR WAVE EQUATIONS FOR THE ACOUSTICS OF HIGH-SPEED POTENTIAL FLOWSt L. M. B. C. CAMPOS lnstituto Superior Tdcnico and lnstituto de F[sica-Matem6tica, 1096 Lisboa Codex, Portugal (Received 5 September 1985) The w.ave equation is derived for the acoustic potential in the following cases: (a) linear/non-linear sound waves, respectively of small/large amplitude; (b) medium at re.st, or steady potential flow, the latter either of low Mach number, or high speed; (c) three-dimensional propagation in free space, or quasi-one-dimensional acoustics of ducts of varying cross-section. Thus there are 2 x 3 x 2 = 12 cases, for which 34 distinct forms of the wave equations are derived (Table 1);"of these, 16 forms of the wave equation in 10 cases~ appear explicitly in the references given. The wave equations are dedved from a variational principle for linear sound (Part 1), and checked by elimination among the equations of potential, unstefidy flow for non-linear acoustics (Part I1). It is shown that the non-linear wave equations (partly new), can be Written in a form similar to the linear acoustics ones (naostly known), by taking into account the self-convection of sound by sound; this implies replacing the linea/local and material derivatives, by self-convected and non-linear extensions of these (Table 2), which also affect the sound speed.

1. INTRODUCTION TO ACOUSTICS OF POTENTIAL FLOWS The wave equation for the acoustic potential, for the linear case o f sound of small amplitude, has appeared in the literature in many forms, all of Which can be considered as generalizations of the classical wave equation [1]; for example, three-dimensional propagation in potential flows is described, at low Mach number, by the convected wave equation [2, 3], and at high speed, by a generalized form in~colving scattering by the mean flow pressure gradient [4, 5]. For quasi-one-dimensional propagation, of the fundamental logitudinal acoustic mode, in a duct of varying cross-section, the classical wave equation i s replaced by the horn equation [6, 7], which has also been studied extensively (see, e.g., references [8, 9]); the addition of a mean flow leads to the wave equation for quasi-onedimensional nozzles, either of low [10] or high [11, 12] Mach number. This variety of linear wave equations [13, 14] raises the issue (Question I) of whether or not there is a single, unified method of cterivation, at least in the case of potential flow. The non-linear acoustics of potential flOWS, though more difficult, is no less relevant to practical applications than its linear counterpart, which has been studied more extensively for three-dimensional configurations; for example, one area of application is the aeroacoustics of engine inlets and nozzles [15] where linear theories have been used to model engine noise (e.g., [16, 17]), and a non-linear approach could be relevant to investigating large amplitude pressure pulses whicfi can cause engine surges or stall. The non. linear acoustic equation in a medium at rest is, in principle, similar to the exact equation for the potential in a comp/essible, unsteady flow, which is known in high-speed gas dynamics, viz., both in the three-dimensional case [18] and for quasi-one-dimensional 1"Work supported by C.A.U.T.L./I.N.I.C. 41 0022-460X/86/190041 + 17 S03.00/0

9 1986 Academic Press Inc. (London) Limited

42

L.M.B.C.

CAMPOS

ducts [ 18]. The case of non-linear sound superimposed upon a steady, compressible flow, has also been considered ([12], see the appendix), for a quasi-one-dimensional nozzle, an extension of the study of the non-linear horn [20,21]. This raises a second issue (Question II), namely, the derivation of the wave equation for the acoustic potential in the general case of non-linear, three-dimensional sound in a steady, compressible and irrotational mean flow. The aim here is to address the two issues raised in this introduction in the simplest and most direct way possible. In respect to Question I, the variational principle for horns [22] is generalized to three-dimensional propagation in an homentropic flow, by using a simpler method, viz., the introduction of a quadratic, acoustic Lagrangian (section 2.2); the same form of the Euler-Lagrange equation is applied in every case, yielding a unified method, in which all assumptions are built into a single Lagrangian, in general or approximate forms. The latter are given for linear sound in a fluid at rest (section 2.2), or low (section 2.3) or high Mach (section 2.4) number potential flow, in three-dimensional space and quasi.one-dimensional ducts, the latter appearing as a particular type of inhomogeneous "medium"; once the acoustic Lagrangian has been established, the variational method leads, without any further eliminations, directly to the wave equation, which may need, at most, only simplifications and/or re-arrangements. For non-linear sound (section 3), elimination between the equations of fluid mechanics leads to an equation for the unsteady potential, which, when decomposed into a steady mean flow and an acoustic perturbation of arbitrary amplitude, yields the general form of the wave equation (Question II). The latter applies to the non-linear acoustics of a fluid at rest (section 3.2) and low or high Mach number flow (section 3.3), in free space or in ducts (section 3.4); the linear terms serve as a check on the validity of the variational principle used for sound of small amplitude. It is shown that the wave equations for linear and non-linear acoustics can be written in a similar form, by modifying the local time and material derivatives, to account (section 3.5) for self-convection of sound by sound. 2. VARIATIONAL PRINCIPLE FOR LINEAR ACOUSTICS 2.1. INTRODUCTION Before concentrating on the acoustics of potential flows, it is perhaps appropriate to mention some of the differences from the rotational case. In a vortical flow sound waves may be rotational [23]: i.e., acoustic propagation is generally coupled with vorticity transport, viz., in a non-constant shear flow the "wave" equation is of third-order, involving both "acoustic" and "vortical" modes; the acoustic wave equation is always of second Order in a potential flow, since only two purely acoustic modes exist, viz., propagating in opposite directions. A variational principle can be formulated for waves in a vortical [24] or dissipative [25] fl0w, but it involves Clebsch potentials, which are integral, i.e., non-local, properties of the mean flow; in a potential flow, the variational principle for acoustic waves can be formulated in terms of local quantities only. Among the acoustic quantities, viz., potential, displacement, velocity and pressure, the first-named variable is chosen as characterizing the acoustic field; the Lagrangian for the potential is quadratic, i.e., of order one unit up o n the perturbation equations, which are linear, in the case of sound of small amplitude, which is considered first. 2.2. Q U A D R A T I C L A G R A N G I A N FOR I N H O M O G E N E O U S F L U I D The total flow variables are denoted by capital letters, namely, the potential ~/,,velocity V, pressure P, density F, and sound speed C; the total flow corresponds to the superposition of an acoustic perturbation (denoted by small letters 4~, ,3,p, p, c) upon a steady mean

POTENTIAL FLOW ACOUSTIC WAVE EQUATIONS

43

flow (denoted with subscript zero, e.g., 4o, Vo, Po, Po, Co): i.e.,

cl), V, P,F, C(s t)=d?o, vo, Po, Po, Co(~T)+ ~b, 5, p,p, c(.~, t).

(1)

For potential flow one has ~'=V(/),

and

89

(2a, b)

where k - - - { y / ( y - 1 ) } P . / F , is a stagnation constant in the Bernoulli equation (2b), for an homentropic fluid, generally compressible. The perturbation equations, obtained by substituting expression (1) into equations (2a, b) and subtracting the mean state, are

5=V~b,

(1-1/y)[ 89189

(3a, b)

where in equation (3b) the density perturbation is p =p/c 2, with Co the adiabatic sound speed cg-~ (Opo/OPo)s. The equation (3a) for the acoustic velocity 13 as gradient of the perturbation potential ~b is of course linear, and equation (3b) can be linearized as

p=-poddp/dt,

dqb/dt=O~b/Ot+~o O~b/0~= ~ + ~o" V~b,

(4a, b)

showing that the acoustic pressure (4a) scales on the material derivative (4b) of the perturbation potential. The results that, for a potential flow of arbitrary Mach number, the acoustic velocity (3a) and pressure (4a) scale respectively on the gradient and material derivative of the perturbation potential imply that the kinetic and compression energies (per unit volume) scale on their square: Ev - 89

2 89 =

(#)2,

Ep =-p2/2poc~ 89 =

2.

(Sa, b)

Thus, for linear sound, the kinetic (Sa) and compression (5b) energies, specify, through their sum, the total energy E--'Ev + Ep., which will not be considered further here, and, through their difference, the acoustic Lagrangian L(~, V~; ~)--- E~-Ep = 89

ac~ = (b, Vc~, ~o" r

Co2(d~b/dt)2}{1 + 0(0~)}, =--Oc~/Ot,oc~/o~, 30" O~b/o~

(6a) (6b)

The acoustic Lagrangian for linear sound is a quadratic function of the time d and space V~ derivatives of the perturbation potential, and is given by equatio n (6a) with a relative error of order (6b): i.e., with omission of terms of third and higher orders, which measure the absolute error. The general acoustic Lagrangian (6a) for linear sound in a potential flow is similar to the well-known (see, e.g., reference [26]) Lagrangian for the classical wave equation (see equation (10a) below), with the sole difference that, in the second term, the local time derivative ~ is replaced by the material derivative dc~/dt (4b). All wave equations for linear sound (in section 2) will be derived from the quadratic Lagrangian, in the general form (6a), or its low Mach number or no-flow approximations; in equation (6a) three-dimensional propagation in free space is considered, and in the case of quasi-one-dimensional propagation in a duct of varying cross-section S(x), the equation is replaced by the Lagrangian per unit length, L(d, 6'; x ) - S(x)L(~b, 6'; x) = 89

'2- co2(ddp/dt)2]{1 + O(O~b)},

(7)

which is equal to the product of the cross-sectional area S(x) by the Lagrangian per unit volume (6a), the latter in one-dimensional form, with a prime denoting the derivative with respect to the longitudinal co-ordinate x: e.g., dp'-dd~/dx. The Lagrangians in free-space (6a) or in a duct (7) may depend explicitly on position s (or x), for the steady flow of an inhomogeneous fluid, of non-uniform density po(x) and sound speed co(~).

44

L.M.B.C.

CAMPOS

Generally, the variational principle applies to a Lagrangian that may also depend explicitly on the potential r (besides its derivatives r Vr and on time t (besides position :D, and requires [27-29] that the action, i.e., its integral over space-time, be stationary: i.e.,

6 I L(r162162

d3xdt=O'

6 f L*(r162

dxdt=O,

(8a, b)

respectively for the eases of free-space and a duet; the corresponding Euler-Lagrange equations [30, 31] are, respectively,

a(OL/O(5)/Ot+V. [OL]O(Vr

O(OL*/Or162

(9a, b)

which become, after the explicit form of the Lagrangian is substituted,:the wave equation in three-dimensional free space (9a) and quasi-one-dimensional ducts (9b). T h e operation of the variational principle can be checked, in the first instance, in the simplest case of linear sound in an inhomogeneous fluid at rest, for which the free-space (6a) and duct (7) Lagrangians simplify respectiyely to L = 89

r

2 - c0252i[1 +

O(Mo, 0r

L* =

89162162 '2 -- C 0 2 6 2 ) [ 1 +

O(Mo, 0r (10a, b)

where Mo=- Vo/Codenotes the mean flow Mach number. Substituting expression (10a) in the Euler-Lagrange equation (9a) yields, for three-dimensional sound,

(c2/po)V 9(poVr - q~= O(Mo, (0r

(lla)

which, in the case of an homogeneous fluid po-const., is the classical wave equation [1]; in the case of an inhomogeneous fluid, of non-uniform density po(x), the threedimensional lipear wave equation in a medium at rest has, in addition, to the classical form, a scattering term c~V2r r + Co2Vr 9 V log Po = O(Mo, (0r

(1 Ib)

A simple example of the relevance of the scattering term is provided by quasi-onedimensional, linear sound propagation in an homogeneous fluid, in a duct of varying cross-section S(x), i.e., the acoustic.horn [20]; besides the three- to one-dimensional substitution ~2r o r in the first term of equation (1 lb), the mass density tier unit volume Po is replaced bY the mass density per unit lengths poS, SOthat in the third term C7r V log po becomes r (poS)}' = r for constant Po. Thus one obtains the homogeneous horn wave equation [6, 7]

c2r "- ~ + c~(S'/ S)r t= O( Mo, (0r

(12a)

it can be verified that the wave equation (12a) coincides with the EulerzLagrange equation (9b) for the duct Lagrangian (10b). The homogeneous horn wave equation can be written in the equivalent form

c2S7'(Sr ' - r =-O( Mo, 0r

(12b)

which is more compact, and resembles the classical wave equation, by replacing the Laplacian ~ 2 r r by the non-uniform duct operator S-~(Sr '= r (S'/S)r 2.3. C O N V E C T I O N O F S O U N D IN A LOW M A C H N U M B E R F L O W In this section the effects of convection on linear, three-dimensional acousiic propagation are to be considered in the case Of low Math number flow, for which the density Po and sound speed Co are independent of the mean flow velocity Vo(OD;thus Po and Co are constant for an homogeneous fluid at low Mach number, and are functions of position,

POTENTIAL FLOW ACOUSTIC WAVE EQUATIONS

45

po(s and Co(s only in the presence of inhomogeneities of composition. From expression (6a) follows the low Mach number Lagrangian

L=89

2 - C2o(q~2+2(~o 9V4>)~)}{1 + O ( M 2, 0~b)},

(laa)

which, substituted in the Euler-Lagrange equation (9a), yields the wave equation

(c2/po)V 9 (poVd?)- ~ - ~ o " V(5- (C2o/po)V 9 (poCo2(bSo)= O ( M 2, (a~b)2),

(13b)

where the first two terms coincide with the case (lla) of a medium at rest, and the last two represent the effect of convection by the mean flow t~o. Using the condition of mass conservation V 9 (poVo)= 0 for the mean flow in the last term of (13b), one obtains the linear, three-dimensional wave equation for sound in a low Mach number, inhomogeneous potential flow,

(c2/po)V 9 (poV ~b)- ~ - 2 5 0 - V ~ + q[(Vo" V)log co2= O(M~, (0~b)2),

(14a)

where the "static" (first two) terms coincide with the case (lla) of a medium at rest, and the "convected" (last two) terms consist of an homogeneous transport of the potential by the mean flow velocity (third term), and an inhomogeneous convection effect if the sound speed is non-uniform (fourth term). An alternative form of equation (14a) is c2V24~- ~ - 2t~o- V~ + co204>9V log p0+ ~Vo" V log c 2 = O(MO2, (O~b)2),

(14b)

which separates the classical wave operator for an homogeneous fluid at rest (first two terms) from terms representing homogeneous convection at low Math number (third term), static effects of inhomogeneity (fourth term) as in equation (lib), and coupling of convection and inhomogeneity (last term). Introducing the double material derivative (4b) in the low Mach number approximation

d2qb/dt = ( 6 + 2TJo9Vt~){1 + O(M2)},

(15a)

allows the wave equation to be written in the more compact form C2V2 4 -- d2~b/dt2+ co2V4 9 V log Po+ 6t3o" V log Co 2=

O ( M 2, (04)2).

(15b)

The first two terms correspond to the well-known [26, 32] convected wave equation, which applies to three-dimensional, linear sound in the non-uniform, low Mach number potential flow of an homogeneous fluid [2, 5]; the convected wave equation also applies to a uniform flow of arbitrary Mach number [33, 34], but in the case of an inhomogeneous fluid, the last two terms in equation (15b) should be considered; relative to the first two they scale as (V~b- V log po)/(V2~b) ~ ( A / L ) M 2, (613o 9V log co2)/~ ~ (A/L)MO2, (16a, b) where 9 log P o - V log co2- (poC2)-lVpo ~ (Vo/Co)2/L ~ M2/L, with L the length scale of the flow, and A the acoustic wavelength; thus the inhomogeneous terms only matter at low Mach number, MO2<<1, if the wavelength is much larger than the scale of the flow, A >>L. In the context of the preceding statement, inhomogeneities are being considered in a rather restricted sense, as changes of composition in a potential flow, in homentropic conditions, with gas pressure as the sole restoring force. The interaction of waves with inhomogeneities is much more complex in the presence of (i) turbulence or vorticity (see, e.g., references [3, 35]), (ii) viscous, thermal or resistive damping [36, 37], and (iii) other restoring forces, such as gravity and magnetic field [38, 39]. The last term in equation (15b) appears together with the second in C2od(co 2 dck/dt)/dt = d2$/dt 2 - (d~b/dt) - t3o" V log c 2 = d2cb/dt 2 - 6t3o. V log co2+O(MO2),

(17a)

46

L.M.B.C.

CAMPOS

so that the wave equation (15b) can be written in the compact form r

-- d ( c o 2 d~b/dt)/dt

+ (poC2)-IVpo 9 V c~ = O( M 2, (0~b)2).

(17b)

This equation was originally derived [4] for the stagnation enthalpy Q, but in an homentropic flow Q = - ~ so that the acoustic potential [5] satisfies the same equation. In the case of quasi-one-dimensional acoustic propagation, i.e., the fundamental longitudinal acoustic mode [13] in a low Mach number nozzle containing an homogeneous fluid, the Lagrangian (13a) is replaced (see equation (7)) by

L* = 89

dp'2 - Co2(d2+ 2d0o6')}{ 1 + O( M 2, 04~)},

(18)

where Po is the constant density per unit volume, and poS(x) is the density per unit length, which is not constant as a consequence of variations of cross-sectional area S(x) along the duct. The Euler-Lagrange equation (gb) for equation (18) leads to the nozzle wave equation [10]

c2 c~" - ~ + c~( S'/ S)c~'- 2voc~'= O( U 2, (Odp)2),

(19a)

which consists of the same static terms (first three terms) as the horn (12a), plus an homogeneous convection effect (last term). Comparison of the three-dimensional (14b) and quasi,one-dimensional (19a) cases, shows that the classical wave equation and the convection term, i.e., the homogeneous terms, are the same; concerning the inhomogeneous terms, the last in equation (14b) is absent from equation (19a) since the fluid is assumed to be homogeneous and the sound speed constant, and the third term Co2~b'(logPo)' becomes C~o(S'/S)qb ', since Po is replaced by poS with constant Po. Using material derivatives (15a) yields the wave equation (19a) in the most compact form,

c2S-'( S(a ')'-d2dp / dt 2= O( M 2, (0if)2),

(19b)

which combines the duct operator (first term) as for a horn in equation (12b), with the double material derivative (second term) as for homogeneous low Mach number convection in a free flow (second term) in equation (15b). 2.4. A C O U S T I C S O F H I G H - S P E E D H O M E N T R O P I C F L O W In a compressible, non-uniform mean flow, the density Po and sound speed depend on the velocity Vo(~), and thus are always functions of position Po(~), Co(~): i.e., dynamically non-uniform for an homogeneous fluid, and dynamic and statically non-uniform for an inhomogeneous fluid. For arbitrary Mach number, all terms should be retained in the quadratic acoustic Lagrangian (6a), viz., by equation (4b),

L=89

9 V~b)+ (~o 9V~b)2]}{1+ O(0q~)},

(20a)

where the first two terms apply to a medium at rest (10a), the third corresponds to low Mach number convection (13a), and the fourth to high-speed flow. The Euler-Lagrange equation (9a) for equation (20a) yields

(d/po)~. (vo~ 4,) -

6 - ,~o- v d -

(d/po)~" {(po/d)(d

+ 30. ~,)~o} = o((o,p)~), (20b)

which consists of the static terms (first two) as in equation (lla), and a mean flow effect (third term) and inhomogeneous term (the fourth) which differ from the low Mach number case (13b) in that the local ~ is replaced by the material derivative (4b), to introduce an O(M~) term. Using the conservation of mass for the mean flow V 9 (poVo)= 0, simplifies

47

P O T E N T I A L FLOW ACOUSTIC WAVE E Q U A T I O N S

(20b) to

(C2o/Po)V 9 (poVck)- ~-260" Vq~- 030" V)2~b+ (q~ + ~3o"V~b)t7o. V log Co2= O((0q5)2), (21a) where the first two terms apply to a medium at rest (lla), the third and fourth terms represent respectively low and high Mach number homogeneous convection, and the last term, due to inhomogeneous convection, also differs from those in equation (14a). An alternative form of equation (21a), the equation for linear, three-dimensional sound in a high-speed potential flow, is Co272q~- ~-2t3o 9 Vq~-(Z3o" V)2~b+ Co2Vq~"V log Po+ (q~ + ~5o"V4')Vo" V log Co 2= O((3q5)2), (21b) in which the homogeneous terms in a medium at rest (first two) and representing convection at low and high Math number (respectively third and fourth) appear separately from inhomogeneous terms, static and convected (respectively fifth and sixth). The exact form of the double material derivative (4b), d2qS/dt 2= ~+2~o" Vq5 +(~o 9 V)2~b,

(22a)

adds the last, O(M2o) term to the low Mach number approximation (15a), and allows equation (21b) to be written in the equivalent form

c2V2q5-d2c~/dt2+c2Vq5 9Vlog po+(dqb/dt)~o . V log c 2= O((04)2),

(22b)

where the homogeneous and inhomogeneous terms (respectively first two and last two) are all present in a high-speed flow. Using equation (17a) yields the more compact form of the wave equation for linear, three-dimensional sound in a high-speed potential flow: V2~b-d(co 2 dqb/dt)/dt+Vqb. V log po = O((Oq~)2);

(23)

it coincides with the expression (17b) for inhomogeneous low Mach number flow, since the material derivatives include all O(M~) terms, which are neglected in equation (17b) and included in equation (23), so that the difference is not apparent. The general, linear wave equation [4, 5] includes, besides the potential or Laplacian term, the double material derivative as in the convected wave operator, with the inverse square of the sound speed in between if it is not uniform, and a term representing scattering by the inhomogeneous mean flow. In the case of quasi-one-dimensional, linear sound in a nozzle of varying cross-section containing a high-speed mean flow, the Lagrangian (20a) is replaced by (see equation (7)) L* = 89

'2 - co2• z) Z co2(2q~Vo~b,+~,2)}{ 1 + O(0~b)},

(24a)

comprising the static terms (first two) for a horn (10b), plus low (18) and high Mach number convection (respectively third and fourth terms). Substitution in the EulerLagrange equation (9b), yields, after some manipulation involving the conservation of mean mass flux (poSvo) '= O, the high-speed nozzle equation

c~b"- )~ + c2( S'/ S)ck'- 2VoC~' - v~qb"- 2voV~qb'+ 2Moc~( (b + rock') = O((34,)2),

(24b)

consisting of static terms (first three) as for a horn (12a), low-speed convection (fourth term) as for the low Mach number nozzle (19a), with high-speed effects appearing both in homogeneous convection (fifth term) and inhomogeneous terms (last two terms). Alternative forms of equation (24b) appearing in the literature include (Co2 - 0o2)05"- 6 -2Voq~'+ c~(S'/S-2MoM~)ck'+2Moc~(b = O((O~b)2),

(25a)

48

L.M.B.C.

CAMP O S

(equivalent to equations (8)-(10) in reference [12]), and also Co2{~b"+S'/S)d?'} - d2qS/dt 2 - VoV~qb'+2Moc~ dqb/dt = O((Oqb)2),

(25b)

(equivalent to equation (16) in reference [11], and reproduced in reference [15], where the undefined quantity h~ = p~/po = -vov~ should be identified with the homentropie form of the enthalpy). The most compact form of the general, linear nozzle wave equation (24b), analogous to equation (23), is

S-~( Sd?') ' - d(co 2 ddp/ dt)/ dt + (poc2)-tp'od?' = O((0~b)2),

(26)

combining the duct wave operator (first term) replacing the Laplacian as in equation (12b), the double material derivative with intermediate sound speed (second term) as for high-speed convection (17a), and the scattering by the mean flow pressure or density gradient (third term)as in equation (23). 3. EQUATIONS. OF MOTION FOR NON-LINEAR ACOUSTICS 3.1. INTRODUCTION Non-linear acoustics coincides with the gas dynamics of compressible, unsteady potential flows, for which variational principles are known [18, 40-42]. However, itis simpler to eliminate between the exact, non-linear equations of continuity and momentum for compressible, irrotational flow, to obtain a non-linear wave equation for the unsteady potential, which describes acoustic waves of finite amplitude in a medium at rest. Elimination between the equations of fluid mechanics, involving use of scalar and vector potentials [43], has been used to derive the equations of non-linear, dissipative acoustics to second-order [44]; in what follows here exact non-dissipative waves are considered, described by the scalar potential alone, with all non-linear terms retained, which turn out to be of order up to, and including the third. Also, the total fluid variables are decomposed into a steady, non-uniform mean flow, and an unsteady, non-uniform perturbation, of arbitrary amplitude. The linear terms coincide with the wave equations for sound of small amplitude, thus providing an independent check on the correctness of the variational principle used before (in section 2); the non-linear terms can all be collected into non-linear and self-convected derivatives, in such a way that the exact wave equations become formally similar to linear forms. This analogy applies to all cases, of sound in a fluid at rest (section 3.2), or in a low- or high-speed flow (section 3.3), in three-dimensions in free space, or in quasi-one-dimensional ducts (section 3.4). 3.2. E X A C T WAVE E Q U A T I O N FOR L A R G E A M P L I T U D E The equations of continuity and momentum (for an inviscid fluid) are

[-'+Ce. V F + F ( V . 17)=0,

V+(Ce. V) Ce+F-~VP =0,

(27a, b)

where as before in equation (1), V, P, F denote respectively th.e total velocity, pressure and density, and a dot denotes a partial time derivative: e.g., ce-aCe~at. For a potential flow V^ V = 0 and Ce=V~, the momentum equation (27b) reduces to the Bernoulli equation (5a): i.e., ~b+-~(V@)2+H = H , ,

H=-fr-'dP=f

(C2/F) dF,

(28a, b)

where H denotes the enthalpy for a homentropic flow as defined in equation (28b), and C denotes the adiabatic sound speed C 2--- (OP/OF)s; H , in equation (28a) is a constant, the stagnation value of the enthalpy, for a fluid at rest. From equations (28a, b) one can

POTENTIAL FLOW ACOUSTIC WAVE EQUATIONS

49

calculate the spatial and time derivatives of the enthalpy:

(C2/F){/~', VF} = {/2/, VH} = - ~ - V ~ .

~qb, - ~ t ~ - ( ~ .

~)f7 qo;

(29a, b)

these relations may be used to eliminate the mass density F from the equation of continuity (27a) multiplied by (C2/F), i.e.,

C2

lr+ (c21r)

r 9 9 + c2 2q = 0,

(30a)

yielding the exact, non-linear equation for the unsteady potential in a compressible flow: C2~2qb- ~ 6 - 2 V r

V~-Vr

{(~ qb. ~)~ qb} = 0.

(30b)

From the expression (28b) for the enthalpy, H = { 3 , / ( e - 1 ) } P / F = C2/(3,-1),

C2= C,2-(y'l){~b+-~(V~b)2}, (31a, b)

it follows that, for a perfect gas, the exact, non-linear sound speed C differs from the constant, stagnation value C., by a term depending on the potential. The non-linear and self-convected local time derivatives are defined, respectively, by 6dP/St-- qb+ (V~b)2,

8cP/st = qb +89 qb)2;

(32a, b)

they both coincide with the local time derivative qb _=Or in the linear case, to O((V 4)2), but they differ in the additional, non-linear term: i.e., (a) the non-linear local derivative (32a) adds the term (~TcP)2= V r V r corresponding to the passive convection by sound V ~ . V, of sound qb, as if it were not a self-interaction; (b) the self-convected local derivative (32b) adds instead the term _~(~)2, which introduces a factor 1/2 relative to "passive" convection, since "self-convection" of sound by sound is determined by the kinetic energy (per unit mass) of the disturbance. The following two examples illustrate the use of these concepts: C 2= C 2 - ( y - 1)6c19/8t,

8(rcPlrt)lSt = ~ + 2 V r

~tb+~.

[(~r

~)~]. (33a, b)

These show the following: (a) the correction (33a) of the exact sound speed C relative to the stagnation value C., is determined (3ib) by the self-c0nvected derivative (32b), rather than the "passive" convection form; (b) is the non-linear local derivative (32a) is applied to the self-convected form (32b), one obtains in equation (33b) the last three (i.e., the non-Laplacian) terms in the exact equatiofi for the potential (30b), which includes, besides the linear term ~, two non-linear ones. The pair of equations (30b) and (31b) has been used to study three-dimensional, compressible potential flow, in the context of steady [45] and unsteady [46] gas dynamics. Since for non-linear sound in a medium at rest the total potential coincides with the acoustic potential, q~ = r the same equations, i.e. Co 2r -

- [ ( y - 1)12](Vr162

-(r-

= O(Mo),

(34)

describe three-dimensional sound of large amplitude in a medium at rest, where the (linear) sound speed is co; in equation (34) the linear terms (first two) correspond to the classical wave equation, for sound of small amplitude in an homogeneous medium (1 lb), with po constant, and there are, in addition, for sound of large amplitude, a number of non-linear terms (last four). The linear (first two) and second-order non-linear (third and fifth) terms in equation (34) coincide with the non.dissipative case of the usual wave equation for non-linear acoustics (formula 12 in reference [44]). All non-linear terms, of

50

L.M.B.C.

CAMPOS

second- and third-order, are included if the local time derivative (~ is replaced by self-convected (32b) and non-linear (32a) derivatives as in equations (33a, b): i.e., - 6(64/St)/6t = O(Mo),

[Co2 - ( y - 1)8-'~-~]r

(35)

shows that the "exact" classical wave equation (35), for sound of large amplitude, can be written as the usual classical wave equation, for sound of small amplitude, with the following substitutions, to account for non-linear effects in the two terms: (i) the Laplacian r is multiplied by the sound speed Co squared, with a correction involving the selfconvected local derivative; (ii) the double time derivative ~ =024/0t 2 is replaced, first by a self-convected 6/6t and then by a non-linear local derivative 6/6t. 3.3. C O U P L I N G OF A C O U S T I C N O N - L I N E A R I T Y A N D S TEA D Y F LO W For the sound of arbitrary amplitude, with acoustic potential 407, t), propagating on a steady flow of potential 4o(f), the total flow potential 9 = 4 + 40 satisfies eqb = t5o+r

r162

~5o+r

r

= eq(,,

qb = (~,

(36)

where )5o=V4o is the mean flow velocity. Substituting in the expression (31b) for the exact sound speed C, one obtains

C2=C2o-(T-1)D4/Dt,

c~=-C~,-(T-1)v~/2,

(37a, b)

where co is the linear sound speed, corrected relative to the stagnation value C . , by the mean flow velocity. In equation (37a) the self-convected material derivative D/Dt, has been used, which, together with the non-linear material derivative D/Dt, defined respectively by

D 4 / D t - qb+~o. r D4/Dt-- qb+~o"r +89

r162 2, 2= &k/~t+~o" r = d 4 / d t + 8 9 1 6 2 2,

(38a) (38b)

replace the corresponding local forms (32a, b) when a mean flow t3o# 0 is present. The non-linear (38a) and self-convected (38b) material derivatives both reduce to the material derivative (4b) for the mean flow, in the case of linear sound. Adding the non-linear correction yields the exact material derivative, with convection velocity t7o+r for the non-linear materialderivative (38a), for which "passive" convection of sound by sound r r is assumed, as for convection of sound by flow Z3o"r viz., the latter is the cross-term in the total kinetic energy I(t3o+ r 4)2; for the self-convected material derivative, there is a distinction between passive convection of sound by flow Vo" r and selfconvection 89162r of sound by sound, since the latter corresponds to the kinetic energy of the disturbance 891622 alone. Substituting (36) and (37a, b) into (30b) and (31b), one obtains, for three-dimensional, non-linear sound in a low Mach number flow, the wave equation c~r

-

$- 2~o" r

2r

-- [(3~-- 1)12](r162

Cq;~- r

[(r r162

-- (T -- 1)Clio 9r 1 6 2

-V.4" [(Bo" r 1 6 2 = OCM~),

-- ( r

(~/-1)6r r

"r (39)

where the linear part (first three terms) consists of the same static (first two) and convected (third term) as the convected wave equation for an homogeneous medium (14b), with go, co constant; the additional, non-linear terms in equation (39), can also be split into a "static" part (fourth to sixth terms) applying to a medium at rest, as in equation (34), and a "convected" part, introducing new terms (the last three), to account for the

51

P O T E N T I A L FLOW A C O U S T I C WAVE E Q U A T I O N S

interaction of acousitc non-linearity with low Mach number mean flow. The non-linear terms in equation (39) occur either as a correction to the sound speed (37a) or are contained in the expression D(~)/Dt

= ~ + ~o" V~ + (Vo" ~)2~ + 2 V ~ . V~ + ( V ~ . V)(~o" V~)

+v~" {E(~o+V~,) 9rive} --d~,/dt~+2v4, 9V~+(V~" v)(~o" %~)+v~," {[(~o+~,)" V]V,~}, (40a, b) for the successive application of the self-convected (38b) and non-linear (38a) material derivatives; the linear part, as expected, coincides with the double material derivative (22a), and the non-linear terms are double and triple products of first- and second-order derivatives. Substituting expressions (40a) and (37a) into equation (39), one obtains the compact form of the exact acoustic wave equation for three-dimensional sound of large amplitude, in a steady, homogeneous, low Mach number potential flow, [Co2 - (5"- 1) D~b/Dt]V2~b - D(D--~'/Dt)/Dt = O(M2),

(41)

similar to the ordinary convected wave equation, applying to sound of small amplitude, with the non-linear effects included in the two terms by (i) multiplying the Laplacian V2~b by the sound speed Co squared corrected by (38b) the self-convected material derivative, (ii) replacing the double material derivative d2/dt 2 by a self-convected and a non-linear (38a) material derivative (applied in this order, which should not be changed, since they do not generally commute). In the case of non-linear sound in a steady, compressible mean flow, after substitution of equation (36) into equations (36b) and (3 lb), it is necessary to simplify the coefficients involving mean flow quantities, such as mass density Po and sound speed co, which are non-uniform at high Mach number. The general, explicit form of the exact threedimensional wave equation for the potential, c2V2q5-~-2t7o - Vq~-(t~o- V)2~b+ co2V~b"V log po+ (6 + Vo" V~b)t~o" V log Co2-2Vff 9 Vq~ - V 4 ~" [(V4~" V)V,~]-(5"- 1 ) ~ V 2 ~ - [ ( Y - 1)/2](V4~)2V2~-(5"- 1)(rio" V4~)V24~ -Vq~" [(t~o- V)V~b]-(Vq~. V)Cv0" V~b)+ [(5"-1)/2](Vq~)2~5o " ~ log po=0,

(42)

contains the same linear terms (first six) as the equation (21b) deduced from the variational principle (8a) and (9a) in section 2, so that one has an independent check on its validity; the non-linear terms in equation (42) can be grouped, as their linear counterparts, in three sets: namely, those corresponding to (i) a medium at rest, viz., first, second and fifth terms, linear as in the classical wave equation ( l l b ) and seventh to tenth terms, non-linear as in the "exact" form (34); (ii) low Mach number convection, viz., third a n d sixth terms, linear as in the convected wave equation (14b), and eleventh to thirteenth terms non-linear as the exact form (39); (iii) high-speed flow effects, viz., fourth term, linear as in the high-speed wave operator (21b), and fourteenth (i.e., the last) term, non-linear, which only appears as a coupling of finite amplitude and high Mach number effects. Use of equations (37a) and (40a) yields the compact form of the exact high-speed wave equation: [co~ - ( 5 ' - 1) ~ ] ~ 2 ~ _

D ( D c k / D t ) / D t + c2V~b " V log po+ (D~/Dt)t~o" V log c2-- 0,

(43) which applies to non-linear, three-dimensional sound in a steady high-speed potential flow, and resembles the linear form (22b), with thg following substitutions in the four

52

L.M.B.C.

CAMPUS

terms, to account for finite amplitude effects: (i) correction of the factor of the Laplacian ~2r from the linear Co 2 to the disturbed (37a) square of sound speed as in the static case (35), with D/Dt instead of 8/3t; (ii) substitution of the double material derivative by the self-convected (38b) and non-linear (38a) material derivative, as at low Mach number (41); (iii) the inhomogeneous terms (third and fourth) are unchanged, except for the replacement in the latter of the material derivative by a self-convected form. 3.4. Q U A S I - O N E - D I M E N S I O N A L PROPAGATION I N HORNS A N D NOZZLES

The preceding three-dimensional, non-linear wave equations apply in free space, and also in the presence of obstacles or walls, with appropriate boundary conditions. The latter can be dispensed with, i.e., rigid wall effects included in the wave equation, in the case [10, 13, 201 at the fundamental longitudinal acoustic m o d e in a duct of varying cross-section S(x). The momentum equation (27b) is valid in one-dimensional form, but in the continuity equation (27a) the density per unit volume F should be replaced by the density per unit length IS, so that it takes either of the equivalent forms

o = acrs)/at + a(rsv)/ax,

0 = ["IF § (F'/F + S'/S) v + V'.

(44a, b)

Substituting the one-dimensional form of equations (29a, b) into equation (44b) one obtains the equation for the unsteady potential,

C2~"-~-~'(~'-~'2r

=O,

C2=C~-(y-1)(~+ 89 (45a, b)

where the exact sound speed C is related to the constant, stagnation value C. by equation (45b). The equations (45a, b) have been used (e.g. [19]) to study unsteady, compressible potential flow in ducts, and they also apply to non-linear quasi-one-dimensional sound in a horn, i.e., a duct of varying cross-section without mean flow: c~[r

(S'/S)r

- 6 " 2 r 1 6 2 r162

(y _ 1)[r

(S'/S)r

+89162 = O(Mo). (46)

This may be designated the exact horn equations, since it coincides, in the linear terms (first two), with the horn wave operator for sound of small amplitude (12a), and adds, for sound of large amplitude, non-linear terms (last three) similar to the free-space equation (34), in one dimension, with the Laplacian r replaced by the duct operator r162 The non-linear terms are included in equations (33a, b), through the self-convected (32b) and non-linear (32a) local derivatives, which yield the compact form of the exact horn equation, [Co2 - (y - 1) 8-'r

{S-t (Sr

'} - 8(6q~/6t)/St = O(Mo),

(47)

which can be obtained from the classical wave equation by correcting the sound speed for non-linear effects, replacing the Laplacian by the duct operator and replacing two partial time derivatives by a self-convected and a non-linear local derivative. If one considers sound propagation in an unsteady flow, there is no basis for distinguishing between "acoustic perturbation" and "mean flow", in particular if the former is of amplitude comparable to the latter; in this case, the wave equation should be applied to the total potential, in the same form as for a medium at rest, viz., equations (34) and (35) for three-dimensional sound in free space, and equations (46) and (47) for quasi-onedimensional sound in a duct. If the "mean flow" is steady, even in the compressible case of high Math number, the acoustic perturbation can be distinguished as the unsteady effect, regardless of whether it is of small or large amplitude. Thus in the case of steady flow, the decomposition (36) can be applied to equations (45a, b), and one obtains, at

P O T E N T I A L FLOW A C O U S T I C WAVE E Q U A T I O N S

53

low Mach number, the exact nozzle wave equation Co2[r

(S'/S)r

- 3 v0~b'r

- r -2Vo" q~'- 2r ( 2 - 7) Vor162

r162

(S'/S)r

( y _ 1)(q~+ 89162162 (S'/S)r

= O(M~),

(48)

where the linear terms (first three) correspond to a static effect (first two) as for a horn (12a) and a convection term (third) as for a nozzle (19a), and the non-linear terms (last five), also include a static contribution (fourth to sixth term) as for a horn (46), and two new terms (last) due to coupling of acoustic non-linearity, low Mach number convection and changes in cross-sectional area. Using the self-convected (38b) and non-linear (38a) material derivatives, one obtains the compact form of the exact nozzle wave equation, [c 2 - (~/- 1) Dr162

'] - D ( ~ ) / D t

= O(M2),

(49)

which is similar tO the case of a horn (47), with local 8/8t replaced by material D/Dt derivatives; this implies that the effects of non-linearity of sound are similar for horns and nozzles, and the usual procedure, in the linear case, of accounting for mean flow effects, by replacing local by material derivatives, also extends to sound of large amplitude, by using non-linear and self-convected derivatives. The most general duct wave equation, corresponding to equations (42) and (43) in free space, applies to quasi-one-dimensional acoustic waves of finite amplitude, in a steady, high-speed potential flow, viz.,

c2r 1 6 2 + c~(S'/ S) r - ( T - 1)[ r

2Vo6' - v2r

(S'IS)r

2VoV~r + 2Moc'o(~ + roe') - 2 r

+89162 - 3 0or162 (2 - 30 0or162

r162

(S'IS)(b']

+ Moc~r '2 = 0

(50)

which has appeared in the literature ([12], see the appendix,, where the signs of the coefficients f3 to f9 shohld be reversed); the terms in equation (50) may be interpreted by comparison with earlier expressionsas (first three) linear acoustics of horns (12a), linear convection at (fourth term) low Mach number (19a) and (fifth to "seventh term) high-speed (24b), non-linear (eighth to tenth term) sound in horns (46), non-linear, 10w Mach number (eleventh and twelfth terms) convection (48),-with the last (thirteenth) term being new, and non-vanishing only for non:linear sound in high-speed flow. The compact form of equation (50), [Co2 - ( ' y - 1) ~ ]

S-I(Sqb') ' - D(Dr

VoV~r

0,

(51)

can be compared with the linear case (25b). The exact, high-speed nozzle wave equation (51), demonstrates all the transformations that may be needed in the acoustics of potential flows: (i) correction of the sound speed Co for the mean flow (37b), to account for non-linear effects (37a) in the exact sound speed C; (ii) replacement of the Laplacian ~2r for three-dimensional waves in free space, by the duct operator S-l(Sr ', for quasi-one:dimensional nozzles; (iii)substitution Of two partial time derivatirces, by one self-convected (38b) and one non-linear (38a) material derivative; (iv) inclusion of scattedngby the mean flow, in the case of high-speed convection, i.e., homentropic flow with Mach number not small. 3.5. D I S C U S S I O N OF T H E S E L F - C O N V E C T E D A N D N O N - L I N E A R DERIVATIVES The wave equations for the acoustics of potential flows have been derived in the following cases: (i) linear and non-linear sound, respectively of small/large amplitude; (ii) the medium is a fluid at rest or a low or a high Mach number steady flow; (iii)

54

L. M. B. C. CAMPOS TABLE 1

Wave equation for acoustic potential Three-dimensional free space

Model

Quasi-one-dimensional duct

Case *Unsteady, homentropic flow *Linear sound: Fluid at rest Low Mach number convection High-speed flow *Non-linear acoustics Fluid at rest Low Mach number convection High-speed flow

(30b, 31b)

(45a, b)

(lla; 1lb) (13b, 14a; 14b; 15b; 17b) (20b; 21a; 21b; 22b; 23)

(12a; 12b) (19a; 19b) (24b; 25a; 25b; 26)

(34; 35) (39; 41) (42; 43)

(46; 47) (48; 49) (50; 51)

three-dimensional propagation in free space or quasi-one-dimensional fundamental mode in a duct, of varying cross-section. In Table 1, for each of these 2 x 3 x 2 = 12 combinations o f cases, at least two alternative forms of the wave equation, one explicit and one compact (viz., a total o f 34 formulas), are shown. The explicit wave equations naturally get more complicated as one proceeds down the table from linear to non-linear acoustics, and media at rest to low Mach number and high-speed convection; the more striking conclusion is that the most complicated wave equations (non-linear sound in high-speed flow) can be "inferred" from the simplest (classical wave equation) by a" sequence of progressive transformations as listed in Table 2. The partial time derivative, in a medium at rest, is TABLE 2

Definitions and symbols for time t derivatives Medium Designation Linear Non-linear Self-convected

Fluid at rest Local derivative O/Ot, (7b) 6/6t, (32a) 6/6t, (32b)

Potential flow Material derivative d/dt, (4b) D/Dr, (38a) D/Dt, (38b)

t Space derivatives: thiee-dimensional, free space--V2~b; quasi-one-dimensional

duct__S-I( S~b') '.

replaced by the material derivative, in the presence o f convection, as is well-known for linear sound; for non-linear acoustics it is feasible to introduce self-convected and non-linear derivatives (32a, b) and (38a, b) to account for the effect of sound on sound. Note that all time derivatives are particular cases of the non-linear and self-convected material derivatives (38a, b); concerning the spatial derivatives the Laplacian ~2 and duct operator S-I(Sgo') ' correspond to the transformations g r a d ~ - S d/dx and div~--~S -~ d/dx. Taken together, the Tables 1 and 2 provide a quick reference, to two to six alternative forms of the wave equation, for any of twelve cases of acoustics of potential flows. Since all the preceding results were deduced under (i) the assumption o f potential flow, (ii) the neglect o f dissipation effects and (iii) omission of all restoring forces other than pressure gradients, it may be worthwhile to indicate some consequences of these restrictions. The (i) assumption of potential flow, as pointed out before, allows the use of a

POTENTIAL FLOW ACOUSTIC WAVE EQUATIONS

55

single, local, potential, instead of three, non-local Clebsch potentials leading to a thirdorder wave equation, coupling sound and vorticity [47]; non-linearity can produce vorticity, e.g., across shock waves, invalidating the assumption of potential flow. Shock waves [48] are associated with entropy increase, as is (it) viscous, thermal or resistive dissipation; damping, even linear, is important in limiting the steepness of non-linear waveforms [49]. For scales much shorter than the dissipation scales, non-linear effects can be considered in isolation, but dissipation, if present, always causes an ultimate asymptotic decay of the disturbance, with a complex intermediate phase [50], in which coupling of non-linearity and damping is important. The non-linear acoustics of a dissipative fluid can be described in terms of scalar and vector potentials, which satisfy equations of second- or fourth-order, depending [44] on thermal boundary conditions. The simplest cases, of linear damping and non-linearity to second order, lead to Burger's equation [51, 52], which is a transformation of the classical heat, rather than wave equation. If, besides pressure (iii) other restoring forces (e.g., gravity, Coriolis or magnetic forces) are present, the wave field can no longer be represented by a single scalar potential, and a vector variable, such as the displacement, should be used; the vector wave equation is still of second order, but its elimination may lead [53] to scalar equations of higher order, and, in any case, additional terms appear. The preceding discussion suggests that the reduction of non-linear and convective wave equations to simple forms, which was ettected here, will require careful analysis to overcome any of the three restrictions (i) to (iii) indicated above. Concerning applications of the results derived, it seems likely that the non-linear wave equations to third-order will defy analytical solution, and can only be attacked by numerical methods, e.g., finite difterences, or finite elements, or boundary elements. The linear wave equations have been solved analytically in some cases [8-10, 54], and the numerical methods indicated above also apply. The availability of the variational principle, equivalent to the wave equation, and including convection effects, opens up at least two possibilities. On the theoretical side, the variational principle can be used to derive energy conservation equations [55, 56]; the latter are known in the "ray approximation" [57-61] but remain controversial for sound of arbitrary (not necessarily short) wavelength. For the purpose of applications, the variational principle can be used, together with "trial" functions, to obtain approximate solutions of acoustic problems. This type of approach is in widespread use in many fields, e.g., structural mechanics, and relies on the choice of a trial function satisfying the boundary (and initial) conditions, and including a number of arbitrary parameters (or functions); the latter are determined by substituting the trial function in the Lagrangian, and minimizing the action integral. This leads to the "best choice" of parameters for the assumed form of trial function; the exact solution would yield the lowest possible value for the action integral, and the restriction to a narrower class of trail functions raises this minimum. Both of these developments could be the subject of future work. REFERENCES 1. LORD RAYLEIGH1877 Theory of Sound (two volumes). New York: Dover Publications, second edition, 1945 re-issue. 2. K. TAYLOR 1978 Proceedings of the Royal Society A363, 271-281. A transformation of the acoustic equation with implications for wind-tunnel and low-speed flight tests. 3. L. M. B. C. CAMPOS 1978 Journal of Fluid Mechanics 89, 723-749. On the spectral broadening of sound by turbulent shear layers. Part 1: Scattering by interfaces and diffraction in turbulence. 4. M. S. HOWE 1975 Journal of Fluid Mechanics 71, 625-673. Contributions to the theory of aerodynamic sound, including excess jet noise and the theory of the flute.

56

L. M. B. C. CAMPOS

5. L. M. B. C. CAMPOS 1978 Proceedings of the Royal Society A351, 65-91. On the emission of sound by an ionized inhomogeneity. 6. LORD RAYLEIGH 1916 Philosophical Magazine 31, 89-96 (Papers 6, 376-383). On the propagat i o n of sound in narrow tubes of variable section. 7. A . G. WEBSTER 1919 Proceedings National Academy of Sciences 5, 275-282. Acoustical impedance and the theory of hoi'ns and the phonograph. 8. E. EISNER 1966 Journal of the Acoustical Society of America 41, 1126-1146. Complete solutions of "Webster" horn equation. 9. L. M. B. C. CAMPOS 1984 Journal of Sound and Vibration 95, 177-201. Some general properties of the exact acoustic fields in horns and baffles. 10. L. M. B. C. CAMPOS 1984 Zeitschriftfur Flugwissenschaften und Weltraumforschung 8, 97-109. On the propagation of sound in nozzles of variable cross-section containing low Mach number mean flows. 11. P. HUERRE and K. KARAMCHETI 1973 Department of Transportation Symposium on University Research on Transportation Noise 2, 397-413. Propagation of sound through a fluid moving in a duct of varying area. 12. E. LUMSDAINE and S. RAGAB 1977 Journal of Sound and Vibration 53, 47-51. Effect of flow on quasi-one-dimensional acoustic wave propagation in a variable duct of finite length. 13. L. M. B. C. CAMPOS 1985 Progress in the Aerospace Sciences 22, 1-27. On the fundamental acoustic mode in variable-area low Mach number nozzles. 14. L. M. B. C. CAMPOS 1986 Reviews of Modern Physics. 58, 117-182. Waves in Gases. Part I: Acoustics of jets, turbulence and ducts. 15. A. H. NAYFEH, J. E. KAISER and D. TELIONIS 1975 American Institute of Aeronautics and Astronautics Journal 13, 130-153. Acoustics of aircraft engine-duct systems. 16. R. M. MUNT 1977 Journal of Fluid Mechanics 83, 609-640. The interaction of sound with a subsonic jet issuing from a semi-infinite jet pipe. 17. L. M. B. C. CAMPOS 1978 Journal of Fluid Mechanics 89, 751-787. On the spectral broadening of sound by turbulent shear layers. Part II: Comparison with experimental and aircraft noise. 18. H. S. TSIEN 1958 in Fundamentals of Gas Dynamics (editor H. Emmons) (eight volumes) 1, 3-63. Princeton University Press. The equations of gas dynamics. 19. A. H. SHAPIRO 1958 The Dynamics and 77~ermodynamics of Compressible Fluid Flow (two volumes). New York: Ronald Press. 20. N. W. MCLACHLAN 1934 Loudspeakers: Theory, Performance, Testing and Design. Oxford University Press. 21. S. GOLDSTEIN and N. W. MCLACHLAN 1935 Journal of the Acoustical Society of America 6, 275-278. Sound waves of finite amplitude in an exponential horn. 22. E. S. WEIBEL 1955 Journal of the Acoustical Society of America 27, 726-727. On Webster's horn equation. 23. W. MOHRING 1971 Journal of Sound and Vibration 18, 101-109. Energy flux in duct flow. 24. R. SEELIGER and G. WHITHAM 1968 Proceedings of the Royal Society A305, 1-15. Variational principles in continuum mechanics. 25. S. MOaBS 1982 Proceedings of the Royal Society A381, 457-468. Variational principles for perfect and dissipative fluid flows. 26. P. MORSE and K. INGARD 1968 Theoretical Acoustics. New York: McGraw-Hill. 27. O. BOLZA" 1904 Lectures on the Calculus of Variations. Chicago University Press, reprinted: New York, Dover Publications, 1961. 28. C. CARATHEODORY 1935 Calculus of Variations. Basel: Birkhauser; reprinted, New York: Chelsea Publications, 1972. 29. L. A. PARS 1960 Calculus of Variations. London: Heinemen. 30. A. R. FORSY-rH 1926 Calculus of Variations. Cambridge University Press, reprinted: New York, Dover, 1960. 31. L. ESGOLTS 1970 Differential Equations and Calculus of Variations. Moscow: Mir Publishers. 32. M . J . LIGHTHILL 1978 Waves in FluMs. Cambridge University Press. 33. A~ D. PIi~RCE 1981 Acoustics. New York: McGraw:Hill. 34. L. M. B. C. CAMPOS 1984 Revue d'Acoustique 67, 217-233. S u r la propagation du son dans les 6coulements non-uniformes et non-stationaires. 35. M. LIGHTHILL 1953 Proceedings of the Cambridge Philosophical Society 44, 531-551. On the energy scattered from the interaction of turbulence with sound and shock waves. 36. P. LYONS and M. YANOWITH 1974 Journal of Fluid Mechanics 66, 273-288. Vertical oscillations in a viscous and thermally conducting atmosphere.

POTENTIAL FLOW ACOUSTIC WAVE EQUATIONS

57

37. L. M. B. C. CAMPUS 1983 Journal de M3canique Th3orique et Appliqu3e 2, 861-891. On viscous and resistive dissipation of hydrodynamic and hydromagnetic waves in atmospheres. 38. L. M. B. C. CAMPUS 1983 Wave Motion 5, 1-14. On three-dimensional acoustic gravity waves in model non-isothermal atmospheres. 39. L. M. B. C. CAMPUS 1983 Journal of Physics A16, 417-427. On magnetoacoustic-gravity waves propagating or standing vertically in an atmosphere. 40. G. H. BRYAN 1918 Aeronautical Journal 22, 255-257. Compressible fluids. 41. H. BATEMAN 1929 Proceedings of the Royal Society AI25, 598-618. Notes on a differential equation which occurs in the two-dimensional motion of a compressible fluid and the associated variational problems. 42. H. BATEMAN, F. MURNAGHAN and H. L. DRYDEN 1956 Hydrodynamics. New York: Dover Publications. 43. D. BLACKSTOCK 1964 Journal of the Acoustical Society of America 36, 534-542. Thermoviscous attenuation of plane, periodic, finite-amplitude sound waves. 44. D. G. CRIGHTON 1979 AnnuaiReview of Fluid Mechanics 11, 11-33. Model equations for non-linear acoustics. 45. L. D. LANDAU and E. F. LIFSHITZ 1953 Fluid Mechanics. Moscow: Mir Publishers, translation, Oxford: Pergamon Press. 46. R. YON MISES 1958 Mathematical Theory of Compressible Fluid Flow. New York: Academic Press. 47. W. MOHRING, E. A. MULLER and F. OBERMEIR 1984 Reviewsof Modern Physics 55, 707-724. Problems in flow acoustics. 48. R. COURANT and K. FRIEDRICHTS 1948 Supersonic Flow and Shock Waves. New York: Interscience. 49. M. J. LtGHTHILL 1951 in Surveys inMechanics (editor G. K. Batchelor) 250-351. Cambridge University Press. Viscosity effects on sound waves of finite amplitude. 50. G~ B. WHITHAM 1974 Linear and Non-linear Waves. New York: Wiley. 51. J. M. I~URGERS1948 Advances in Applied Mathematics 10, 171-179. A mathematical model illustrating the theory of turbulence. 52. J. M. BURGERS 1974 The Non-linear Diffusion Equation. D0rdrecht: D. Reidel. 53. L. M. B. C. CAMpOS 1985 Geophysical and Astrophysical Fluid Dynamics 86, 217-272. On vertical hydromagnetic waves in a compressible atmosphere under an oblique magnetic field. 54. H. S. TSIEN 1952 Journal of the American Rocket Society 22, 139-143. The transfer functions of rocket nozzles. 55. D. LYNDEN-BELL and J. KA'rz 1981 Proceedingsof the RoyaISociety A378, 179-205. Isocirculational flows and their Lagrang!an and energy principles. 56. J. KATZ and D. LYNDE'N-BELL 1981 Proceedings of the Royal Society A381, 263-274. A Lagrangian for Eulerian fluid mechanics. 57. D. I. BLOKHINTSEV 1946 Journal of the Acoustical Society of America 18, 322-332. The propagation of sound in an inh0mogeneous and moving medium. 58. C. J. R. GARRET 1967 Proceedings of the Royal Society A299, 26-27. The adiabatic invariant for wave propagation in a non-uniform moving medium. 59. L. M. B. C. CAMPOS 1982 Portugaliae Mathematica 41, 14-32. On anisotropic, dispersive and dissipative wave equations. 60. L. M. B. (2. CAMPOS'1983 Portugaliae Physica 14, 121-143. Modern trends in research on waves in fluids. Part I: Generation and scatte.ring by turbulent and inhomogeneous flows. 61. L. M. B. C. CAMPOS 1983 Portugaliae Physica 14, 145-173. M o d e m trends in research on waves in fluids. Part II: Propagation and dissipation in compressible and ionized atmosphere~.