Introductory lectures on hydro-acoustics

L

Office of Naval Research

Department of the Navy

Contract Nonr-22O(35)

INTRODUCTORY LECTURES ON HYDRO-ACOUSTICS

by

T, Yao-tsu Wu

Reproduction in whole or in part is permitted for

any purpose of the United States Government

Division of Engineering and Applied Science

California Institute of Te chnology

Pasadena, California

1. Introduction

It is an honour to me that this leading institution, Institut

für Schiffbau der Universität Hamburg, and my gracious hosts, Prof. G. Weinbium, Prof. O. Grim, Prof. K. Wieg-hardt, who have helped provide a stimulating research atmosphere in every respect to me during my academic

visit on leave from the California Institute of Technology with a Guggenheim Fellowship, have asked me to under-take the task of giving a few introductory talks on hydro-acoustics. Being assured that this is meant to be informal. expository in nature, with the primary objective to stimu-late further research interests in this growing field which is

playing an increasingly important role in naval hydro-dynamics, I accept it with pleasure, and regard it a good opportunity for myself to learn more about this subject

with those who are concerned with this common interest.

As the subject matter embraces a vast scope, choice of the material has to be made on a selective basis. In order

to

explain and elucidate the fundamentals of

hydro-acoustics,

a terminology used here to infer combined

acoustical motions and hydrodynamical flows, emphasis is laid on the generation and propagation of sound and noises. together with some useful mathematical methods of solu-tion. The acoustical motion is basically a small pertubation of density and pressure in a continuous medium with an

in-trinsically unsteady nature. The effects of small

inhomogenei-ties in a limited region, and a basic convective current, such as in the midst of turbulence, or in an unsteady jetstream, or in a vortex shedding wake, will be discussed. On the other hand, the problems of sound scattering and refraction, which generally call for an extensive mathematical analysis, and

those of sound absorption and dispersion in real fluids,

which require detailed physical and chemical

con-siderations, are left out on account of the time limitation. The materials presented here are arranged with the aid from some standard textbooks and reference books: e.g.,

Hydrodynamics by Lamb (1932), Fluid Mechanics by Landau

and Lifshitz (1959), Handbuch der Physik. Vol. XI, by

Flügge (1961). Methods of Mathematical Physics by

Cou-rant and Hilbert (1953), and other cited references on

special topics. I would regard it encouraging if I could

suc-ceed in stimulating, even slightly, some interest in future

research from this group gathered here.

2. The Basic EqtLations

In order to visualize the mechanism of sound generation

and the propagation of sound waves through a fluid, we

derive first the basic equations of acoustic motion in their general form. The Eulerian description will be adopted so

that the fluid motion is expressed in terms of a velocity

field q (x, t) as a function of the position vector Xof a fixed

point in an inertial frame and the time t. From the prin-ciple of conservation of mass, momentum, and energy applied to a compressible, viscous and heat-conducting

fluid, one may write

CllIFFSTECllIK

FORSCIIUNGSIIEFTE FUIt SCHIFFBAU UND SCHIFFSMASCHIENBAU

Heft 68. September 1966 (13. Band)

introductory Lectures on Hydro-Acoustics

T. Yao-tsu Wu

+

= Q (x, t) (1) 3t 3x 3(1q1)

a(pqq)

+ F1(x,t) (i = 1,2,3) St Sx 5X1 SXk (2) T ( - + q = K V2T + (l + H (x, t) (3)

where, as usual, denotes the fluid density, P the pressure,

T the temperature. S the specific entropy, r1. the

compo-nents of the viscous stress tensor, the dissipation

func-tion, K the coefficient of heat conducfunc-tion, and finally

Q, F = (F1, F.,, F.1), H represent the source terms which are respectively the time-rate of introduction of mass, momen-tum and heat energy into the fluid system per unit volume. The quantity F includes the momentum source due to mass introduction; it may include also the external body force, such as gravity. The heat source H may include, whenever

significant, the thermal radiation. In the above equations

and in what follows, the usual convention of summation by

dummy indices is adopted. For the Navier-Stokes

equa-tions, one has further the relations

= (divq) òjk + 2t ik'

= (div q) - 2 u rk2 (3a)

i.k.1 i.k

where 81k = 11f i k and Òjk = O if i ± k, u. and r are called

the first and second coefficient of viscosity, which may

de-pend on the thermodynamic state of the fluid element in

question. Finally, the set of equations governing the flow

is completed by including the thermodynamic equation

of state, which may be written in general as

f(P,Q,T)=O. (4)

Explicit expressions of (4) may be written for specific

media, for example, for an ideal gas,

PR9T.

(4a)

Equations (l)(4) are a set of four nonlinear differential

equations for the four unknowns q, P, . T.

It may be remarked here that the thermodynamic

pro-cesses actually involved in acoustic motions may in general

be so fast that the thermal equilibrium state can not be

maintained at all instants. Finite relaxation time for inter-nal degrees of freedom (in particular the vibratiointer-nal modes)

of the constituent molecules must be considered for the

actual motion. In the phenomenon of sound absorption and dispersion, the equation of state is often not precise enough

to account for the details and is necessary to be supple-mented by measurements of some physical quantities on

the phenomenological basis. In this regard we may mention

the notion of the bulk viscosity coefficient 111i defined as ik = i

/3q.

I + 2

\x1

- 93 - Schitfstechnik Bd. 13 - 1966 - Heft 68 3qk _(2a)

rl8 = _{(3îi + 2e) = i ± 2/tL,}

which is related to the coefficient of the trace of the viscous stress tensor by

trace

t = t =

(3 + 2 t) div q.

If î = 0, as Stokes assumed for an ideal gas, then i 2.t/3. However, experiments show that lB is in general

not zero (in fact it may be quite large compared with .

as for water); this may be attributed to the effect of the

relaxation phenomenon, an effect which causes the acoustic streaming, for example.

3. The Linear Approximation

When we consider the motion due to some infinitesimal, time-dependent fluctuations about the equilibrium state of the system, which is a case already embracing a wide scope of interest, we may make the following assumptions.

In the region free of the sound-making sources, there

exists an equilibrium state which is homogeneous and

static, characterized by q = 0, po, p.o, T0, S0; and any in-homogeneities of the physical quantities are confined to a finite region of space in which the acoustic disturbances are

originated.

We define the acoustic velocity u and other acoustic variables by

u = q,

_{p = PP,,, a = pp,,,}

_{= T - T,}

s = SS0,

(5)

and assume u/c0, pIP,,, alp,,, l/T,,, s/S0 all to be sufficiently small, where c,,2 is equal to (dP/dp)0.

We assume the processes to be irreversible (i. e. the viscous dissipation, and thermal conduction and radiation are all negligible) and also adiabatic (no external heat addi-tion) so that s = O by virtue of (3). Then it follows, by ex-panding the thermodynamic function p = p (a, s), that

p = c,2 a, c,, = (dP/dp),= . (6)

In particular, c,,2 = y P0/p,, for an ideal gas, 'y being the

ratio of the specific heats.

Using the above assumptions, (1) and (2) are linearized, up to the first order, to

3a 3u1 + Po 3x1 3u1 ap 0 _at

+

ax1

Upon elimination of u from (7) and (8) by cross differentia-tions, and by using (6), we obtain the wave equation for p as

i

_aP

+divF.

(9)

As Q and F are supposed to be known, the right hand side of (9) represents the source, or forcing function, for gene-rating the pressure disturbances. After p is solved from (9),

the other acoustic variables in the source-free region

fol-low simply from (6), (8) and (4), giving

a pic02,

u =

-J (VP)dt.

f aT\

T,,u

ap)8 p,,cp

where u is the coefficient of thermal expansion. The last expression comes from the well-known thermodynamic

relationship

f3T\

T

a(i\

ai)

c

aTk

which reduces to [(y 1)/y] T0/P,, for an ideal gas.

Within the linear approximation, we note from (8) that

curl u = O provided F is a conservative field, i. e. F = Vx.

Then a velocity potential q exists such that u = v and

at

''

=Q

(7)

(i=1,2,3).

(8)

ScIiiffstechnik Bd. 13 - 1966 Heft 68 ₉₄

-Also, cp satisfies the wave equation in thesource-freeregio

Dp0.

(121

Of fundamental importance in acoustic theory is the mono-1 chromatic traveling wave,

p = Aei(k x,ot),

(13) it being understood that the real part is always taken for

the physical interpretation. This is a solution of the

homo-geneous equation of (9) provided

k2 = k12 + k' + k32= w2/C2 or

knk, kHk1w/c.

(14) Thus (13) represents a simple wave propagating in the direction of the wave number vector k with circular

fre-quency w, _{corresponding to period t and wavelength ?.}

given by

t 2t/w, ?. = 2't/k.

Obviously, _{p is normal to the planes of constant phase,}

k xut = const., and hence is parallel to k. From (10)

it therefore follows that u is parallel to k, thus charac-terizing the acoustic waves as longitudinal waves. In fact

u n or p = p,, c, u. (15)

The quantitiy p,, e,,, which is the ratio of the pressure p to the velocity u in small simple waves, is called the charac-teristic acoustic impedance of the medium. This result (15) has been used by Kármán (1930) to determine the impact pressure on a fiat plate coming in contact with a flat water

surface at velocity u.

By considering the time-rate of work done by the acoustic pressure p over an arbitrary closed surface in the medium,

it can be readily shown, by using (7), (8), (10), and the

diver-gence theorem, that in a source-free region

3w -

+divl=0

et (16) where i w = '/2p,,u° + _-p2, (17) poco p-I = pu = p,, C,,

u-n =

n. (18) Poco

Here w is the acoustic energy density, containing the

kinetic and potential components (which are equal in plane waves). The quantity (div I) may be regarded as the work

done by the acoustic pressure per unit volume per unit time. Thus, (16) expresses the energy balance in acoustic motion; it signifies I to be the sound energy flow vector. The last two expressions can be shown to be valid up to

the second order by starting from the exact equations. In concluding our discussion of the linear approximation, we make the following remarks.

Limitation of the adiabatic assumption (effect of heat

conduction). - In a fluid with thermometric conductivity x = K/pc, the effective distance penetrated by heat con-duction from hot to cold spots in a wave period t is

mea-sured by the so-called diffusion length Lj) = Vxt = J/2tx/w,

The effect of heat conduction cannot be neglected if í

becomes comparable or larger than the wavelength

X( 2tc/w), i.e. D or w 2'rc°/x. Therefore, the

adiabatic assumption becomes increasingly poor for the very high, ultrasonic frequency range. The process tends

towards the isothermal limit with increasing frequency.

Effects of nonlinearity and viscosity. - It is a

well-known study of Riemann that the nonlinearity of the exact

equations has an important effect of steepening the wave

front, causing the formation of shock waves when the wave

amplitude is sufficiently large. The viscous effect, on the other hand, tends to diffuse and flaten out a wave peak, thus counter-balances the steepening effect due to

non-linearity.

e) Effect of body forces. - The gravity force, for example, s an indirect effect on the variation of with height, thus

king the medium non-isotropic. In this case, F = ere g is the gravitational acceleration, hence div F =

approximately. By using this expression in (9), and

eking a solution in the form of a vertically traveling

mple wave, as given by (13), one finds that

(1) / ¡ g \T g

-

1

2 (2)

I 1

2 c,,

or w > g/2c2, the imaginary part k1 gives the attenuation

actor exp(k1z); such attenuations (or amplifications) of waves are in balance with the work done by the pressure and gravity. Furthermore, the real part kr gives rise to a

dispersion phenomenon since the phase velocity w/kr and the group velocity din/dkr both depend on frequency. Aside from these effects, it should be noted that a more significant effect arises from the spatial variation of c,,.

(d) Effect of basic inhomogeneity.When the medium

has an inhomogeneity in one or more of its basic properties,

such as a non-uniform distribution of temperature T,, or entropy S,, it can be shown that it has a similar effect as the gravity in causing the attenuation, dispersion,

scat-tering, and variation in C,,.

4. The General

Wave Equation for Acoustical Motions

Returning to the basic equations, by cross differentiation

of (1). (2), to eliminate the term 32(Qq1)/3x1 0t, we obtain

32c

72P--t_ =

3Q 32

-

+ divF

_{3x1 ,}

(Qq1qT11).

(19)

As suggested by the linear approximation, we introduce the space average of sound velocity c1,, and re-write the above equation for the purpose of describing the general

acoustical motions as

j2p72p_1

(20) 3Q H-clivF 32 (Q q1q - T11.) 32

/ P

3x1 3x1

- ---I

3t2 \

Co-The right hand terms will vanish for small motions in a

homogeneous, isentropic (without dissipation, thermal con-duction and radiation), source-free medium at rest. Under

other circumstances, when these terms are appreciable only in a confined region. they may be regarded as some sort of 'sound sources", and (20) then becomes the wave equation under a rather general condition. These source terms also provide some insight about the mechanism of

sound generation.

3Q

unsteady injection of fluid (such as air-flow siren, pulsating jet air-flow), gives rise

to a monopole radiation.

(liv F spatial variation of body force (such as gravity), gives rise to a dipole radiation. div div (q q) sound generation due to momentum

fluctuations, including the Reynolds

stress, in a gross manner. This term,

being a major noise source in turbulent

flow and turbulent jets, gives rise to a

quadruple radiation.

represents sound generation due to

fluc-tuations of viscous stress; this effect is

generally negligible.

may represent the effect of entropy

fluc-tuation. temperature fluctuation (as in

turbulent flows) or change in constituents.

k = kr + i k = ±

5. Boundary Conditions

The boundary conditions on a solid surface submerged in an acoustic field depend in general on the boundary material (such as its porosity, whether or not the pores

being interconnected, modulus of elasticity, etc.) and

some-time on the incident wave (its frequency and incidence

angle) as well. The property of the boundary material can be characterized by the ratio of the pressure to the normal velocity at the surface, which is called the normal acoustic

impedance Z11 of the surface, or

p/u11=Z,. (21)

Evidently, the value of Z2 of the fluid and that of the solid must be equal at the boundary surface since there must be continuity in both normal velocity and pressure across the

surface. For a simple wave, with wave number k, it is readily seen from the solution (13) and Eq. (15) that just

outside a solid surface having the outward normal n,

p Z1,

- ik,,c,,

If Z0 is a point property of the surface, independent of the

incident wave form (this being nearly the case for many porous surfaces through which no net flow takes place),

then (22) serves as a boundary condition for

single-frequency incident waves. This is usually referred to as the

classical boundary condition for it actually reduces the

problem to a classical boundary-value problem. Use of this

condition also curtails the consideration, which is

other-wise necessary, of the acoustic waves inside the solid.

A particularly simple ease is obtained when a surface

may be idealized to be perfectly rigid, so that, if held fixed, we have on its surface

U11 = O or Z,, = . (23)

There are, however, other types of boundary materials

whose surface impedance depends on the configuration of the incident wave. For this type of boundaries, it is neces-sary to consider the waves inside the solid simultaneously

in order to match both p and U,, at the surface.

In what immediately follows, we shall discuss some com-mon methods of solution for the initial-value problems and radiation problems.

6. Initial-Value Problem of One-Dimensional Waves

The problem may be stated as

e2

uO (<x<, t>0),

(24)

u(x,0)q(x),

u(x,0)=p(x) (--<x<ec).

(25)

Here the subscripts x, t denote partial differentiations, and e replaces e,2 for brevity. It is here implied that u is at least twice differentiable with respect to x and t. Equation (24) may also be written

(+

-

-)(-In terms of a new set of variables (X, i), defined as

Xx-f-ct,

11xct,

(26)

the above equation becomes

3u

3x3

= 0. (27)

Upon integration twice, one obtains the general solution as u = f (X) -I-- g (i) = f (x -f et) + g (x - et) (28)

where f and g are two arbitrary functions. In order to

verify this solution, it is necessary to assume f, g to be at

least twice continuously differentiable. This restriction can

be further relaxed. In fact, it is sufficient that f, g b only piecewise continuous (see Courant and Hubert, Vol. II).

This property of the solution of the wave equation (hyper-bolic type) is in a drastic contrast to the fact that the solu-tion of the Laplace equasolu-tion (elliptic type) must be an

ana-(22)

- 95 - Schiffsteehnik Ed. 13 - 1966 - Heft 68

div div T

32 _{/ p}

lytic function. Physically, f (x + et)

and g (x - et)

re-present respectively a wave traveling with velocity e in the negative, and positive, x-direction, without change in wave form. These wave forms can be determined uniquely by the set of two initial conditions (25). Application of these con-.

ditions to (28) gives u(x,o)

f(x)+g(x) =ç(x),

u1(x,o)=c[f'(x)g'(x)]=p(x).

Hence X 2 f (x) 1 2g(x) (x)± a

Using this result in (28), we obtain the unique solution

X rt

ir

u(x, t) = V2{q (x -F- et) + p

(xet)] +

2e jil'(X)d?. . (29)

x-et

The above solution may also be expressed as

u(x,t)

=

tMi{;x,t} +

_{[tMi {;x,t}]}

(30a)

where

X + et

M1 {;x,t}

=

2ctJ tO)dX.

(30b)

xet

When understood, the coordinates (x, t) can be omitted in

the functional argument of M1. The quantitiy M ()} has

the significance of a one-dimensional mean value, over the range from x - et to x + et, of the initial valuew (x),which

was given at time t before the present moment. This expres-sion thus provides a clear interpretation of the result, ana-logous to which a similar interpretation can be given for the

case of higher space dimensions (e.g., see Eq. 45 later).

Domain of Influence

and Domain of Dependence

The fact that when u (x, 0)= tp(x) is given for all x, the

solution u at (x1, t1) is independent of p(x) for x < x1ct1

or x > x1 + et1 is a significant feature of the wave

equa-tion. The lines X=X1 (or x+ct=x1+ct1) and 1=111

(x - et

-= x1 - et1) are called the mathematical

charac-teristics passing through the point (x1, t1). The region in

be-tween these two characteristics for t > t1 is called the

domain of influence of the point (x1 , t1); it is the only

region in which an initial disturbance at (x1, t1) may gene-rate any motion for t > t1 (see Fig. la). The region bounded

by these two characteristics and O < t < t1 is called the

domain of dependence; it is the bei of all the points whose domain of influence includes (x1, t1), (see Fig. lb). M1 {w} thus signifies a one-dimensional mean of the initial signal

it(x) over the domain of dependence of (x, t).

DQMAItd OF INFLUENCE OF (x,, I I ,tI O I -ct I 1r + rt I Fig. i a Fig. t b

A discontinuity in some physical variables will propagate along the characteristics, the variation of the jump in these variables across the characteristics is governed by an

equa-tion which can be derived from the differential equati

Furthermore, the initial value can be prescribed on a curl which does not coincide, in part or in whole, with a chara teristic line. Otherwise the solution cannot be determined.

7.

_{Radiation of One-dimensional Waves}

The radiation of one-dimensional waves generated by forcing disturbance may be expressed as follows

u--u11=Q(x,t) (<x<,t>o)

(3la

u(x,o)= u1(x,o) = 0. (31b)

There are several methods of solution in common use, such as by certain integral transforms, or by variation of

para-meters, or by using Green's functions. We choose to des-cribe here the impulse method on account of its merit in

exhibiting some underlying mechanism of wave radiation. Furthermore, this method can be applied to linear

differen-tial equations of higher order in t. whether it is a pardifferen-tial

or ordinary differential equation. This method is based on

the representation of an arbitrary function by an

inte-gration of successive impulses

Q(x,t) =jQ(x,t)ò(tr)dT (32)

where ò (t) denotes the Dirac delta function, and the upper limit is to be interpreted as t + 0.

Let us consider an associated problem for the variable

U (x,t;t) such that

- --U11 -

i Q(x,t)

b (tt)

(o<t<t+O,

_(33a)

e

U (x, to; r)

=

U1(x, to; t)

= O (33b)

in which r serves as a parameter. It is readily shown that if

¡J(x, t; t) is the solution of (33), then t+o

u (x, t) = j'U (x, t; r) dt (34)

O

is the solution of the original problem (31). Now by inte-grating (33a) from t =

tF to t

= t+ 2. using (33b) as the

initial condition, and by eventually letting r tend to O +,

the problem (33) is seen to be equivalent to the following

problem

U11 O

(t> r + o, x <

_{), (35a)}

U(x,t+o;r)

= 0. U(x,t+o;t)'= c2Q(x,r). (35b) But this is just an initial value problem with the origin of

time set at t =t. Its solution, by (30), is simply

U(x.t;r)(tt)M1 .tc2Q;x,tt}

a c e Q(X, r) dX. (36) 2 -i

tri

The solution of the original problem (31) is therefore, by (34).

t

xc(ta)

u (x, t) dt r

Q(X. r) dX. (37)

This result means that the field u at (x, t) due to a

con-tinuous source function Q(x, t) for t > O is equal to the

functional average of the source over the entire domain

of dependence pertaining to lx. t).

8. Spherical Waves

Take, in general, a space of n-dimensions with the posi-tion vector x

(x1 ...x), and consider the spherically

symmetric wave field u (r, t) which depends on the space only through the radial distance r, where r2=X12

+..

f X12.

Since

-3u r x1 3u

-ar ax1 -- -

r - 9r

32u f x. \2 S2u I i

x \ 3u

r' )

ar2

+ LI

-- r) ar

homogeneous wave equation

i 31i " 3U i 32U

3t2 = O

ornes

Urr + ul_!Ur

jUt1

= 0. (38)

s is the equation for the spherical waves in a n-dimen-'ial space, though only n= 1, 2, 3 correspond to physical

lity. Aside from the case of n

=1, which we have 3ady discussed, the three-dimensional spherical wave

a unique significance. When n -= 3, (38) can be rewritten

1:2

(ru) ---(ru)

= 0. (39)

is the quantity (ru) satisfies the same equation as the -dimensional wave (see Eq. 24); and therefore, by the

ument leading to (28), the general solution of (39) can written as

u (r,t)=

f(r + Ct) + ---g(rct),

(40) ere f and g are two arbitrary functions. The first term the right side represents a spherical wave converging the origin; and the second, a wave diverging from the gin, both parts being now weighted by the geometric

tortion factor 1/r, which is singular at r = 0. This result

useful in solving the general three-dimensional wave

blems, even when there is no spherical symmetry.

Vhen n 1, 3, one cannot obtain spherical waves having arbitrary form f and g. In those cases, the simple

stand-waves however are easy to attain. Take n

=2 for imple, one can readily show that the simple outgoing

I incoming cylindrical waves are

e-101 H0(1i

(wr)

e-° H,)2

ere H0' i (z) and H,, ' (z) are the Hankel functions of the

t and second kind.

The Initial Value Problems in Three-dimensions

rhis problem of the wave field u (x1, x.,, x1, t) may be

ted as

V2u-- '2_u,=0

(o<r<)

(4la)

u (x, o)= (x) , u1 (x, o) = 'ip (x) . (41b) present below Poisson's solution. By going over the alysis, it is hoped that some basic features of the wave

)pagation can be clearly seen. Following Poisson, we first

egrate (41a) throughout a sphere of radius R centered r= 0, giving 32

Çr2drudQ=V2udV

o dS

Ç3u

R2dQ= R2

--

IudQ j n

J 3R

RJ

s o Q

ere dS= R2 d is a surface element, d the correspond-solid angle, on the spherical surface of radius R,

en-oping the Volume V. This operation amounts to taking

average of the wave property about the origin. To

aluate the rate of change of this average property with reasing R, we differentiate the above expression with

pect to R,

-

R2

Iu(R,ft,l3)d1

R2

Iudl

3R

aR J

f e2 at i

u u

in which (r, l, 3) denote the usual spherical coordinates. As this result is true for arbitrary R, one may also write

r for R. By dividing both sides of the above equation by r,

one finds that

ji (r, t) u (r, t}, 1)

d

(dQ =sin i dii d13) (42)

satisfies the spherical wave equation (39). The function ji (r. t) is of course the average of u over a spherical sur-face of radius r. This result reveals a very interesting, and important, property of wave motion that about any point in the space, which is here taken as the origin of r,

the average value of u over a spherical surface propagates

always as a spherically symmetric wave even when the

motion has no local spherical symmetry. The general

solu-tion of¡j(r, t) is therefore

r ji (r, t)= f (r + et) -4- g (r - et).

To determine the arbitrary functions f, g, we let the origin of r O be the field point P)) in question, which will be

de-signated by the position vector x,. and let r tend to zero.

Then ¡j (r, t) U (X)), t)

as r - 0,

assuming such a limit exists. We further assume that r ¡j, r fl, r ji all tend to

zero as r- 0. From these limiting values one easily finds

that (i) g (

X) = f (X), consequently

r ji(r, t)= f (r + et) f (et - r) (43)

and (ii)

u (x, t)=hm ji (r, t) = 2f' (et) . (44)

From (43) it follows that

¡a

i

3\

i

- +

j (rji) 2f (r + et).

\8r

e

3t/

By letting t - 0, and applying the initial conditions (41b).

one finds 2 f'(r) r Ç ,(r,31,

) d +

a ¶ (r,*, 13) d12 4tc) 3r 4x Q O

Substituting this result in (44), by changing r into et, one

obtains the final solution

U (X)), t) = t M:1 {'ip; X)), t} + - [t M.1 {w; X0,

tfl

(45a)

where

M:1 {i; X)), t} = [5'ip(X)dQ] (45b)

xx,1

et

This is the Poisson solution of the initial value problem,

which is expressed in a similar form as in the

one-dimen-sional case (cf. Eq. 30). Physically this means that in a three-dimensional space which includes P,1, the distur-bances due to the initial values cç and i' which just reach

P,) (X),) at t are those situated initially on a spherical surface

with center atX,, and radius et. Here the domain of

depen-dence is

the expanding sphere

(r < et), and M1 {ap}

signifies the average of j' taken over the whole spherical

surface r =et. (see Fig. 2 for the geometry.)

Fig. 2

10. Radiation of Three-dimensional Waves

Radiation of waves in an unbounded space may be stated

as

ElI2u = \72U

u11 =Q(x,t) (OZ x < oc)

(46a)

u (x, 0)= ut (x, 0) 0 (46b)

Like earlier in the one-dimensional case, we adopt here again the impulse method. The associated problem of

radiation of an impulse at t = t

is a function U (x, t;t)

satisfying

EJ2U = - Q (x,t)

ô (t -

t) (t > 0,

t >

0) (47a)

U(x, rO; t) = Ut

(x,TO;T) = O, (47b)

which is equivalent to the following initial value problem

fl2U= O

_{(t> t + 0)}

Ut(x,T+ O; t) = c2Q(x,T), U (x, t + 0; t) = O.

The solution of this problem is, by (45),

IJ(x,t;T) = c2(tT)M. (Q;x,t_t}

c (tt) I

_{(X+ nc}

_(tT);

_{t) d (48)}

J

o

where n denotes the unit outward normal to the spherical

surface centered at x. Therefore the solution of the original

problem (46a, b) follows as

u(x,t)

₌

_IU(x,t;r)dT

where in the last step use is made of the change of

variables: R = c(tt). By writing n = (u, )l, ?), and X0=

(x, y0, z))) where

x)=x+aR, y0=y+3R, z0z+yR

so that R2dRdQ = dx), dy0 dz(, = dv0, (49) can also be writ-ten et / R 1'

Qixo,t-1 C

u(x,t)=

_4t)

-R R et (R =

The physical significance of this result is very clear. Due to an elementary source in volume dV located at point P), or X the signal at another point P. or x, is

du Q

(x0.t_ R

dv,.

4.'tR _\

ej

A part of this contribution is j/4jtR, which is inversely

pro-portional to the distance R between P and P,, like the

gravitational potential of a single source. But the signal

strength at P at time t is due to the source strength Q at x

emitted at an earlier instant T = tR/c such that the

dis-turbance, while propagating with finite velocity c, reaches

P at t. For this reason, the integrand of (50) is called the

retarded potential.

The total signal u (x, t) comes from the integral of the retarded potential throughout the

entire domain of dependence, which is R Z et. If the motion

started from t = - c, and the initial data have already

died out, the region of integration in (50) is infinite.

Simple Harmonic Motion

A particular case of fundamental importance is

the simple harmonic motion with circular frequency w given by

Q (x, t) = Q, (x) e-iot, u (x, t) ip (x) eirnt,

(t > - ''). (51)

For linear acoustics, one needs to consider only this case

since the more complicated solution can be constructed by

spectral analysis. Substituting (51) in (50), we obtain the

amplitude function

_.14Re1JtQ (x) dV,, lR

/

Ix-x,, k

_{= -j,}

(iJ\ (52) vo

the range of integration including all the space V0 where

Q0 0.

nR,t_R )d

dv,.

(50)

where 72 is the Laplacian in the x coordinates

ô (x-x0) = ô (xx,). ô (yy0)

ô (zz0). From

and (55) it follows that

G(X; X))) _{7xö2 i' (x,) - li! (x0) G (x; x,,) =}

- Q,, (x,) G (x; x),) + il' (x,) ô (x -x0).

(49) Integrating the equation with respect to X,, throughout

and using the Green's identity, one readily obtains

11. Boundary-value Problems; Green's Functi

Consider again the fundamental case of monochrom

waves, with the time factor e-it (as shown in (51)). boundary-value problem of the amplitude function can be generally stated as follows. Let the boundary given domain D be denoted by B; it may be a mate boundary, an interface, or merely a geometrical surfi

such as the point of infinity. Then

(V°+ k°) ip = Q),(x) for xin D

where k = to/c. As explained earlier, there are two ty

of boundary conditions for ip:

classical type aj' + bt',, specified on B non-classical type a' +bìj', not specified on B. The Green's function G (x; X,,), which is associated u the same time factor, satisfies the equation

(V\2 + k2) G (x; x,,) = - ô (x - X,,) r

('f

G (x; x,) Q,, (x,,)d2 x, + I [G (x; x,) I _E 3G (x; x ) '1 I ' (x) x in D - tp (x0) - - ' dS,, = 3n,, j i O x outside D

in which the symbol 3/3n,, denotes the x,,-derivative alt

the outward normal n,, to the surface element dS,, on

There is no definite rule for prescribing the boundary e' ditions of G; usually they are so chosen as to simplify solution. If, for example, the boundary condition is of classical type, (54a), then by choosing aG + bG0 = O on the surface integral over B in (56) will vanish. And if source term Q,, has only inhomogeneous terms (complet specified functions of x0, not involving '), then the probl

is reduced to a quadrature after G is determined. If, ht

ever, some other boundary condition is chosen for G,

surface integral in (56) will remain, with the integrI

]inear in ip or its derivatives, and hence (56) becomes integral equation for '. A more complicated integral eqi

tion may also arise if the original boundary conditior

non-classical so that it necessitates evaluation of the w field in the complementary domain of D and requires condition of continuity in pressure and normal velocity

the boundary B. It may also happen that the source te

Q,, involves the unknown function il, as illustrated by ( To determine the Green's function, it often is conveni

to split it into two parts, one representing a singular o going wave and the other a complementary regular so

tion, that is,

G (X; x,,) = G,, (x; x,) + g (x; x,) (5

(V2 + k°) G,, (x; x,,) ô (x - x,,)

(0< lxx,,< 00)

(72 + k2) g (x; x,,) = O. (5

The boundary conditions of g follows from those of G (57a) after G,, is determined. Clearly, G,, depends on x,

only through R = x - x, in terms of which (5Th) may

written

( dR2 + kO) (RG,,) = O

d2 \

(R >0).

To ensure G0 to be an outgoing wave, it is necessary to force the so-called Sommerfeld radiation condition:

hm

(3G,).kGRO

R\3R

'J

Schiffsteehnik Bd. 13 - 1966 - Heft 68 ₉₈

-4 't RdR

Q(x±

he solution of (58) with the singular behavior at R = O as

pecified by (57b) and with the radiation property (59) is

G,, e1kT. (60)

4 rR

For the particular case of an unlimited space, g 0. and

the surface integral in (56) vanishes in the limit as R-'cx,

thus reducing (56) once again to (52).

Conversely, these spectral components can be converted to the general case by using the notion of the retarded time

R

t = t

_c

Restoring the time factor in (60), the Greens function for

an unbounded space is

ei (kRwt) e-1° G (R, r) =G(R, t

-4-iR 4irR

= G (R, t) eikll

Similarly, restoring the time factor in (52), we have

u (x, t) = ei (kliwt) Q,, (x,) dV0 =

4R

eikI Q (x0, t) dV,,, V,) and we infer Q (x, t) eikil Q,, (x) e1 (kRot) Q,, (x) e /

R\

= Q(x,r) = Ql x,t

_ej

I. (62)

Since the retarded time is independent of the frequency,

the above result must hold valid by virtue of Fourier ana-lysis. Substitution of (62) in (61) gives the result identical

to (50).

Using the spherical coordinates (r, l, w)

for x and

(r,,, r,,, q,,) for x,, the function G, has the following useful

expansion in terms of the spherical harmonics Ye,,, and

spherical Bessel function f, hg,

eikR G,, (x; x,,) =

4 R

(63) I k Ym (*, w) YEm (no, w,,)h' (kr) j (kr,), (r,, < r) L, rn where

Yñ (, q) = ei

Pei,, (0), j p (x) - - J( 1/2 (x), 2x

h(x) =

Hp ií.,(1)(x). 2x

For prescribed domain D and material property of

boun-dary B, determination of g therefore constitutes the whole

matter in this method of solution. When the region D is finite, particularly for some simple shapes, G can also be

expanded in a series of normalized eigen-functions, 111,,, which are the solution of (\72 + ?) I',, O (n = 0, 1, 2,. .

satisfying the required conditions for G. Then

G (x; x,,) = fl',, (x) W,, (x,) / (k2 .,,2) ₍₆₄₎ Green's function G satisfies the reciprocity relationship

G (X; x0) = G (X,,; x).

The general unsteady motion is prescribed by

/'

I i

3\

k c2 3t2)

= Q (x, t)

(xinD,t>0)

(65)

together with appropriate initial values and boundary con-ditions. In this case the Green's function satisfies

/ 1

32\

Vx

-

_{- at )}G (x, t; x0, t,,) = - ì (x - x,) ô (t - t,,)

(66)

From these two equations one can show that for x within D and t> 0, u (x, t) dt,, G (x, t; x,,, t,,) Q (x, t,,) d3 X,, + r C Ou 3G + dt,, I

---- Gu

dS,, (67)

j

On0 an,, O li i

I

Ou 0G + ., .,

G(x,t;x,,,t,,)u(x,,,t,,)

d2x,,. C- 2t,, t0 t,, o In the particular case of an unbounded region together

with homogeneous initial values (zero), we have

u (x, t) = dt0 .f G (x, t; x,,, t,,) Q (x,, t) d8 x, (68a)

G (x,t;x,,,t,,)

_{= 4R}

('p t + t,)

(68b)

where ô (x) is the Dirac delta function, which has the fol-lowing properties:

Ç f (x + y) ô (x) dx = f (y) (if a >0 and f (y) exists), (69a)

f (x) ô {g (x) dx = f (x1) / 1g' (x1)1 (69b)

(provided g' (x1) 0),

x1 being the real zeros of g (x) in (a, b). By using these

properties, it can be readily verified that the integration

with respect to t,, reduces (68) to (50). Conversely, (50) also

implies (68) by virtue of (69).

12. The Source Terms; Wave Zone

We consider the case when the source system has a

characteristic frequency , located in a region of a typical linear dimension f (thus occupying a volume of order /').

In general a wave zone is established at some large

distan-ces from the source system. It is noted that this problem has two characteristic lengths: one is f, the linear extent of the source system, and the other is the wave length

= 2 /k = 2 tcIo).

(a) W a y e Z on e - As y m p t o t ie s o lu

t ion

for the far f ield

We choose, for convenience, the origin r = O to be at a point inside the source system, and consider a field point x at a large distance such that

r» f, and r» ?, but with no restriction on eI)..

(70)

Then for a point x,, located within the source system, the

vector R = xx,, can have its length R = x-x,,

approxi-Fig. 3

- 99 - Schiffstechnik Bd. il - 1966 - Heft 68

mated (see Fig. 3) under condition (70) as

R

r -

x, (71)

where n = x / xl = x/r, a unit vector along x. With this

approximation (52) becomes C _e11 j' (x) = J Q,, (x,,)--- dV,,4 tR vo eikr (' e-iknX,,Q,, (x,) dx,, [1 + 0(1/Ir)]. (72) 47rr j vo

We note that the above integral is a function of n (or

rather a function of the spherical coordinates i) and í) but

R

C

is independent of r. This result shows that in the far field, a wave zone is established, with the amplitude and phase

given by the integral in (72), regardless of the value f/X.

(b) Radiation due to poles

of different orders

Suppose, in addition to (70),

the wavelength of the

radiated waves is large compared with the body size f,

that is, r»?.» L, or

kJ'«l«kr.

(73)

Such is the case for the electrodynamic radiation from an antena, or the acoustic radiation of a turbulent flow. Then

we can further expand the exponential in the integral of

(72) into a power series in (kf),

f

x0 \ (kf)2 / x1 \

e-lkflx2 = 1_ikLn.-j--.)_._2-

n i) + .

. (74) Whence

eikriC

p(x)=

_4tr

I Q0(x)dV,ikn

Q0(x0)x dV, ,j

j

vo vo

The first term is independent of the direction of x. and is

called monopole radiation. The directional variation of radiation of the higher-order poles arises from the devia-tion of Q (x) from the spherical symmetry. For instance,

the first moment of Q, (x) gives the dipole strength, a

vec-tor quantity; and the second moment of Q, in dyadic

form, gives the quadruple strength. When kf « 1, the

successive poles are increasingly weaker and the series converges rapidly, thus leaving the first non-vanishing

term to be predominating one at low frequencies.

Further-more, the condition of kf << I implies that the

retarda-tion is almost uniform over the source system, giving near-ly coherent radiation. When kf > 1, it is necessary to use the more exact expansion of G0 in terms of the spherical

harmonics.

Let us now return to (20) and examine the different

mechanisms of the source terms, (the boundary effect may

be neglected here).

(i) Oscillatory fluid injection (or laminar jet): 3Q/3t.

Obviously the leading term due to this sound source

is the monopole radiation.

(ii) Oscillatory body force: Q, (x) - div F (x). The corresponding pressure field is

p (x) = j' G (x; x) 7o F(x0) dV,, =

= JF' 70G(x; x) dV j'G(x; x)F(x) dS

II

upon integration by parts. The surface integral may b put to zero if B is chosen sufficiently large so that either F = O

on B or the surface integral is negligible. By using the

approximation (71) in grad G,

7G = 7(4

e1k1)

= rrR\R

R uk

I

-

R2/

1\

I

=

ikn/

I

i - - I elk

n .X0)

42tr\

ikrJ

Further expansion of exp ( ik n x,) for kf «1 « k

yields eikr Ç p(x)

--ikn

F(x,)dV0 42tr

j

vo + k2n

.[JF(x)

x dV n + O(kL)} (76)

The leading term is therefore a dipole radiation; its field

strength varies with a factor cos i, where ( is the angle between n and the direction of the volume integral of F.

(iii) Momentum fluctuation: Q,(x) = 32 (q qk)/ 3x 3x1 This is the most significant part of sound generation by

turbulent flow, or by turbulent jet. Assuming again that

this term vanishes outside of V,, we can integrate the

volume integral in (72) twice by parts, giving

eikr 1 32 p (x) -g-- j qj q 3xj 0xk eikn X,dV,, + -vn k2

eikrn [j (P)dVj. n +

. (77)

The leading term is seen to be a quadrupole radiation, whilst the monopole and dipole have disappeared due to

the double differentiation. (iv) Bulk and surface scattering.

The source terms 72 (P e02 o) and (div div r) are

rela-(75) tively unimportant in sound generation. the first being

around its zero average and the latter depending on fluc-tuations of the viscous stress, which is usually small.

How-ever, if a primary traveling wave exists, then these terms

may be regarded as corresponding to a local change in the

refractive index, causing diffraction and scattering of the

primary wave.

4

13. Radiation of a Moving Source

(Lienard-Wiechart Potential)

Consider the radiation of a sound source, of a negligible physical dimension, moving with an arbitrary velocity V (t) for t > 0. The analysis will be carried out here for the case

< e, the case of V > e can be evaluated in a similar

manner*). Referring to Fig. 4, we denote the location of the point source by x = X (t). or dropping the subscript,

x = X (t) dX / dt = V (t). (78)

The velocity potential q. satisfies the equation

= Q(x,t) = Q, b(xX(t))

(t >0) (79)

where the function Q for the moving point source has been

described by a delta function with a constant coefficient Q,. Substituting this source function in the general solu-tion (68), and integrating first with respect to the spatial

variables, one obtains

so that R is the distance between the field point x and the

source, (R (x. t) will also be written as R (t) for brevity).

Now we evaluate the above integral by using the property

(69) of the delta function. Write

g(r)

rt +

R(x.r).

(81)

e

*) The mathematical problem of a moving electric charge in a dielectric is the same as presented here, its electric potential is

named after Lienard-Wiechart. The case of V1 > e, where c

is the speed of light in the dielectric medium, gives rise to the phenomenon of Cherenkov radiation.

(x, t) = Q0

ô --

(t

1 C C R(x,T))

dr,

(80a)

4t

J o - R(x,r) where

R(x,t) = xX(t)

3 (80b) R (x, t) = RI = [

(X

-i=.1 X (t))2]'', Schiffstechnik d. 13 1966 - Heft 68 100 -monopole dipole kl

n.

fQ,, (x,) x0x0dv0] n + O (k[)8 quadrupole

Noting that

dg

i /R\

i RV

g'(-v)=---=l+

_dt

_{C \a/}

(--1=1--

_{C R}

which is always positive for IVI< e, g (t) can therefore

have only one real zero, which is the real root of

R(x,-r) = c(tu).

(82)

Let this root be denoted by -r = t0 (x, t). Differentiating

(82) with respect to t and taking the spatial gradient of(82),

we obtain respectively

a\

ai

VR at

ckl_ at)a

at)

R

at'

3R aR R 3R

- c7-r

+ --

7-r = -

+

Vt.

Whence

(

R

=

R

i C (Vt)t _{/ 1}

VR

dR-- - R-V

_C

Application of formula (69b) to (80a) finally yields

Q0

(x,t)

= 5(t)

S(t) = R(t) -

V(T)R(t),

in which t = -t (x, t) can be determined from (82) for given x, t and X (t). As compared with the incompressible case,

= QI4tR(t), the above result exhibits the effect

of

motion of the source on the retardation of the signal trans-mission. The field strength changes not only with distance,

but also with the direction measured from the path of the

source motion.

The following geometrical significance can be attached to the above result. Referring to Fig. 4, P is the field point

Fig. 4

with position x,Qtis the present position X (t) of the source

at time t, Q0 is the position X (-r) of the point source at the retarded time -t, so that Q0P R(t), which is the position

of the source at time t relative to P, satisfying the

condi-tion R(-r) = c(tt). We introduce Q

to be the virtual

present position defined as the position which the source

would occupy at present instant t had it continued to move

from Q0 with a uniform velocity equal to V (-r). Then

Q0Qv

= V(t)(tt)

R(r)

Draw

QS

perpendicular to QP, intercepting Qn1' at S,

then

R(t) V(r)

PS = R (t) - Q0Qv _R(-t)

= R (t) -

_c R (t) = s (-r).

By comparing the acoustic solution (84) with the

incom-pressible limit, Q0 / 431 R (t), the effect of compressibility

alone (with the source at rest, V = O) is to replace R (t) by

the retarded distance R (t), and the additional effect of

motion of the source is to replace R (t) by s (r).

In case the moving source is also pulsating with a

characteristic frequency u, it may be described as

Q (x, t) = Q0 e-lot ô (xX (t)) . (85) Then by following the same analysis as above, one finds

the potential to be

w (x, t) =

4()

e-iWt, (86)

in which

t

= r (x, t) is given by (82). Since the time factor in the above solution is periodic in the retarded time r, the

apparent frequency actually encountered at x in terms of

the physical time scale t is clearly

¡

V(r)R(-r) i

O)' ' ., - (,) I

i - -

- -- - I , (87)

t i

cR()

j

which is of course the well-known Doppler effect.

If we further combine sources and sinks to create a

doublet.

The higher order singularities can be constructed in a

similar manner. Carrying out the differentiation in (89), and

V. R\ R e,,

es )

cs° making use of (83), we obtain

+

° J

wdoublet = -

+

14. Slightly Inhomogeneous Media;

Geometrical Acoustics

f V2

2s\Re, Vel

+

_{- e2}

_{R )}

I

+

_{C S2} j' e-1 (90)

in which e,, is the unit vector in the positive x,, direction,

and the arguments of V, R and s are r. The first term in the bracket is of order R' for R large; it therefore repre-sents the radiation field. The second and the last term in the bracket are of order R2 at large distances; they

con-stitute the induced field.

In many problems of sound propagation, in the

atmo-sphere or in the ocean, it is often the case that the medium is neither homogeneous nor quiescent. The inhomogeneities cause sound scattering and refraction, and the flow motion

of the medium makes it anisotropic. The problem of in-homogeneities will be discussed in this section, and the

effect of moving medium is to be treated in the next.

If the speed of sound e and the amplitude of the wave

change only slightly, due to a small inhomogeneity of the medium, in a distance of a wavelength X, the appropriate wave equation, in the source-free region, is

J2p72p

i (91)

It is necessary to consider the following two cases sepa-rately. The first is characterized by a slow and smooth

variation of e such that

X7°c «

7cl. The second is

appropriate when the variation of e is random and fine-grained such that X7°c and

7e

are of the same order in magnitude, as in a turbulent flow. In the latter case, the

approximation by an expansion of the solution into a

series of k (1 being the grain size, k the wave number, and k,( «1), as discussed in § 12, can be used.

Conse-quently we shall discuss only the first case here.

Under the aforementioned conditions for the first Case, we may introduce the idea of rays, which are lines every-where tangent to the direction of propagation, and ignore

the wave nature. This approximation is called the

geome-trical acoustics, which corresponds to the limit of small

wave-length, X -'.- O, or large wave number, k -- v. In this case we can keep a clear separation between the phase of a sound wave and its amplitude. We write

p (x, t) = A (x, t) ei'(x, t) (92)

- 101 - Schiffstechriik Bd. 13 1966 Heft 68

Q (x, t) e-i(ot _Qo

the corresponding potential is

i = a ô X (t)) i , (88) 3x,, (817) CPdoublet (x, t) s(t)e"'0

where W is a real function defining the surface of equal

phase, hence called the phase function or eikonal function,

and A is the amplitude function. Both W and A are

assumed to be slowly varying functions of x and t. Over small regions of space (several wavelengths in extent) and short interval of time, the waves must be

approxi-mately plane. Hence, upon expansion, up to the first order in x and t, we have

W = W + x grad W + t 3W/3t.

Comparing with a local plane wave '1' = k X - ut + u

(with constant k and w) we define the wave number vector

k and frequency w as

k = grad W = 3W/3x, w = - 3W/3t, (93)

both of which are supposed to be slowly varying functions of x and t. The local speed of sound c is the speed of pro-pagation of the equal phase W = constant, hence, by denot-ing the distance along k by s,

/ds\

3w j3W

3W ¡

e

dt

-

I. = -

_/gradW

which also reduces to c = u/k as for the plane wave. This is the basic equation (eikonal equation) of geometrical

acoustics for W,

3W 3W2 3W2

() +() +( 3z)

in which c is a function of x for an inhomogeneous medium. The well know analogy between the geometrical acoustics

and the classical mechanics comes from the fact that the action function S of a material particle also satisfies the

same equation as (94), which is there called the Hamilton-Jacobi equation. The momentum vector p and the

Hamil-tonian H (the energy) of the particle can be derived from

S by

p=3S/3xmgradS,

H=-3S/3t

which, upon comparison with (93), indicates that W is

ana-logous to S. p to k, and w to H. Furthermore, since the Hamilton-Jacobi equation is equivalent to Hamilton's canonical equations

3H . 3H

p=--_

x=

we can write by this analogy for the geometrical acoustics the ray equations

3w . 30)

k=

,

x=..,--.

3x

In a homogeneous isotropic medium, w = ck with c constant,

so that k

0, X ck/k, and hence the rays are straight

lines advancing with constant c.

For unsteady motions in an inhomogeneous medium, the first equation of (95) gives the variation of k with time. The

second equation is also a very important formula, x is

actually the so-called "group velocity", which is the speed of propagation of the wave energy in a wave packet. The

dependence of w on k generally depends on the property of the medium. Together these two equations determine

the time-rate of change of w. Writing u = w (x, k, t), x and

k being the canonical variables for w by analogy to the

classical mechanics, we have

du 3w 3w 30)

dt=8t+

x+

k=

(96)

upon making use of (95). Therefore, in a steady state (i. e.

the properties of the medium is independent of t), 3w/3t = 0, then du/dt = 0, i. e. the frequency u remains

con-stant along a ray. This result is of course analogous to the law of change of Hamiltonian, dH/dt = 3H/3t, in the

clas-sical mechanics.

In steady propagation of sound in an inhomogeneous

non-dispersive (c is independent of k) medium at rest,

= ck, where c is a given function of x only and w is

con-stant along a ray. Then the equations (95) become

i (3W'\2 I e2 k 3t

1=0.

(94) (95) Schiffstechnik Bd. 13 - 1966 - Heft 68 102

-x=cn,

(97)

where n = k/k, a unit vector along the ray path in the

direction of propagation. The first equation of (97) gives the change of k along the ray. The magnitude of k varies

along a ray simply according to k = co/c, with o constant.

Hence

C,)

k=--,- c=-

_e-

e-in which the second equation of (97) has been used. To

determine the change of n along a ray, we substitute

k = kn in (97) and use (98), giving

n=-7c+n(n\7c)nx(nxVc).

(99)

The change in n is therefore always coplanar with the

vec-tors 7c and n, and is always perpendicular to n, turning always from \7c. Hence a ray will remain in a plane if, and only of, 7e is everywhere parallel to a fixed plane. Introducing the element of length along the ray ds = c dt,

and noting that dn/ds = N/R, where N is a unit vector

along the principal normal and R is the radius of

curva-ture of the ray, we can rewrite (99) as

N

R

nx(nxVlogc).

(100)

This equations determines the form of the rays, once the

initial n is given.

If equation (94) is solved, so that 111 is a known function of x and t. the distribution of sound intensity in space can be determined from the energy equation (16), which must hold in all space except at sources of sound. In steady

con-ditions, we have div! = 0, I = cEn, E = u2 = p2/c2.

Consequently, writing n = k/k = 7W! 7WF,we obtain

div (cE 7W /IV'11) 0. (101)

For the average intensity distribution, we may neglect the

variation of the phase. Hence by substituting E = A2 / c2

in (101), we have

div (A27W) = 0 (102)

which determines A in the space.

An alternative derivation of these formulae can be

obtained by writing

p = A eik0 (, t) -kot

where k is a constant reference wave number, k, = co/c,, with c being the value of e when the inhomogeneities dis-appear. By substituting this expression into (91), expanding

for large k0, neglecting the variations of u, and assuming

the motion to be steady, the real part of the resulting

equation gives (94), and the imaginary part, (102).

15. Propagation in a Moving Medium

When the medium has a basic flow motion with velocity

V (x), we must replace the first equation of (5) by q = V (x) + u (x, t), where u is the flow velocity due to the

acoustic motion. If V and e are slowly varying functions of

x compared with u, we can follow the same procedure as before, and the appropriate wave equation for p, in the

source-free region, now becomes

1/3

e2 3t

+v.7)P.

At an observation point x the medium may be reduced

to at rest, in another frame of reference K', by a local Gali-lean transformation. We take a fixed system K of

co-ordi-nates x, and a system K' of coordico-ordi-nates x' moving with

velocity V relative to K, so that

x' = xV(x)t,

t' = t

(104a) and hence 3 3

3'

3x' =

(104b)

V2p=

7c=k'7c

(98) (103)

ovided the variation of V is small over distances of the

der of the wavelength. In the system K' the fluid

is

cally at rest,

and by

this Galilean transformation.

viously (91) is transformed to (103). A monochromatic ave in the K' system has the usual form

p = A e (k.X' -kct') = A e1 lic.X -(kc + k V)t] , (105)

he last expression being the wave form in fixed system K. The coefficient of t in the exponent is clearly the frequency w of the wave as measured in the system K,

c'ikc+kV

(106)

in which the term k V gives the effect of the convective

current V on the encounter frequency w. The group velo-city of propagation is

3w 3 k

3k 3k (kc) + V k e + V; (107)

this is the vector sum of the group velocity relative to the moving medium and the flow velocity V by which the wave energy is simply carried along. The last part of (107) holds true only in nondispersive media.

Under the condition that the wavelength is everywhere

small compared to the distances over which V and c change

appreciable, the wave field in the system K can still

assume the form (92), and (93) again holds valid approxi-mately. Since M' satisfies (94) in the system K', it is

trans-formed by (104) to

(grad M')2 =

1 (3W

C 1 at + V grad M'

)2

(108) which is the eikonal equation in the system K. The cano-nical equations (95) can also be used in the system K for they do not contain partial differention with respect to t.

The formula (96) can be seen to remain valid in the system K under the conditions mentioned above, so that w is again constant along a ray in steady motions.

To obtain the equations of propagation of the rays, we

substitute (106) in (95), regarding x. k as independent cano-nical variables, we have

k = - k VC - (k V) V - k X (V X V),

k (109)

xeeU=c kV.

When V is everywhere much less than c. and c is

con-stant, it can be shown that

N

- = -

dn ₌ 1

- -- n X (curl V) (110)

R ds c

where n is a unit vector in the direction of U and R is the radius of curvature of the ray. If the changes in e and V are both small, their effects on the bending of the rays.

Eq. (100) and (110), can be superimposed.

16. Turbulent Flow Noise

In a turbulent flow the medium is broken up into a

large number of small eddies, each of which may be

re-garded as a single cell of random fluctuations in

momen-tum and pressure. The measurements of flow quantities taken simultaneously at different points in an eddy are well correlated, but those at points in different eddies are

almost uncorrelated. The random motion of the eddies re-lative to one another produces fluctuations of velocity and pressure in the main body of the fluid. The local pressure

fluctuations can be regarded as an acoustic near field. At far enough distances from the body of the turbulent fluid a radiation field is established, with the entire system of random pressure fluctuations acting as the sound source.

Our aim here is to find the expression for the acoustic far

field.

The principal sound source in a turbulent flow is

3 T

Q0(x, t) _{= 3xj} , T qj q . (111)

Therefore, by (77), the acoustic pressure in the far field is

p (x, t) =

-

n n

e'' Tlk (x,,, t) dV,,

Ve

02 1

= n1 t [Ti] dV,, (112)

4trc-

_J

where w is the characteristic frequency of fluctuation of

the pressure in the eddy volume V,., R = xx,», r =

ni = xi / r, and the square bracket denotes the function

evaluated at

the retarded time,

i. e. [Tt (x1,, t)] =

(x,, t - R/c). This result shows that the turbulent noise

is a quadruple radiation.

In the simplest form of the theory (see Lighthill (1952), (1954)), the difference in retardation time within each eddy may be neglected, and the value of [T11J can be approxi-mated by its value at the center. Thus

w2 Ve

p (x, t) = -

_4trc2

n nIC [TIk]. (113)

The contribution to the pressure at x and t by a given eddy is a stochastic function of t, and those by different eddies

are uncorrelated. This statistical property of a

homo-geneous, isotropic turbulence can be expressed in terms of the time correlation function

W(t) = (p(x,t)p(x.tr))

(114)

where the bracket indicates the time average. The mean

acoustic energy intensity at a given point is then M' (0) / e and the total acoustic power transported across a surface

is the surface integral of this (see Eqs (16)(18) for an

analogy to the deterministic case). The energy spectrum,

E (w), is related to '-I' by E (w) = W (r) e'"t dT, I =

-2t9c j

i I M'(o)

=

JE

(w) du. (115) Taking the operation of the time-correlation (114) on the

solution (113), one obtains the energy intensity at distance

r due to a single eddy as

w .i:' v,.'

IC

1631- r-oc' (116)

where is a typical mean square value of the quadruple

tensor T'j. Any more detailed information about re-quires knowledge of the correlation function of the

velo-city. To obtain the total contribution of all the eddies

having the same frequency w we follow a general principle

of wave theory that the amplitudes resulted from

well-correlated sources combine linearly, but that, with uncor-related sources, energy intensities combine linearly.

Therefore the energy intensity generated at r by a unit volume of turbulence is 1,./V, and based on a unit mass

of fluid,

L w 72Ve

1=

=

,.,

., - . (117)

0V,

l6rrroc"

The total acoustic power output by a unit mass of

turbu-lence is

w4'tr2I =

(O .E2 V,

4t2e5

(118)

It should be noted that this approximation requires the

eddy size £ (V,. '' f ) to be small compared with the

wave-length of the sound it generates, so that variations of

re-tarded time within an eddy can be neglected. This assump-tion, that ul / e be small, is satisfied at low Mach numbers (based on the mean flow velocity), for, as observed, high

frequencies go with small eddy sizes, and hence wf is a

characteristic velocity, approximately a root-mean-square

velocity fluctuation, which is further related to the flow

velocity.

This estimation of acoustic intensity and power enables

one to derive useful information by some physical

argu-ments and dimensional considerations. For turbulent flows it is convenient to choose a typical root-mean-square

velo-city fluctuation, denoted by U, as the reference velovelo-city,

with the corresponding r. m. s. Mach number M = U/c. Then

since w - U. and the mean square fluctuations,

2, of

terms like

= q qj are in proportion to

Q1U4, hence (118) becomes

w=K1Is,

eU3Ii',

(119)

where K is a dimensionless constant, and the quantity s,

being proportional to (V2QU)

_(4t2)/

_{is a typical power}

transfer per unit mass of the fluid. The acoustic efficiency of conversion of fluid flow energy into sound. defined by

= WIE, is therefore proportional to the fifth power of the r. m. s. Mach number. In the special case of homo-geneous, isotropic turbulence, the constant K has been

determined by Proudman (1952) to be approximately 38, i. e.

w=38Mt.

(120)

The total acoustic power output, W w £, is clearly

pro-portional to M8.

17. Turbulent Jet Noise

In the previous case the mean velocity of the turbulent flow was taken to be very small relative to e, and hence

its effect could be neglected. Under this condition the radiation will be spherically symmetric, with total power

output proportional to the eighth power of the r.m.s. of the

velocity fluctuations. In a high speed turbulent jet flow, the convective velocity may be comparable to, or even

greater than c. The effect of this relative motion is to give rise to directionality of the radiated waves, as discussed in Sect. 13, more energy being radiated downstream than

up-stream.

The aerodynamic characteristics of subsonic turbulent jets at high Reynolds numbers have been observed by

Laurence (1956), Lassiter (1957), Townsend (1956). The

fol-lowing description follows closely the paper of Lighthill

(1963). The shear layer, or mixing region between the jet

and the atmosphere, becomes fully turbulent within half

a nozzle diameter (d) of the orifice. Over a distance of 4 or

5 d downstream (see Fig. 5), this turbulent flow region

0.1*

--0.04*

h

-I_{- MIXING REGION} dW 8G AOJLJSTMEI4T DECOY ._REGION ¡ REGION

Beyond x = 8d, the velocities fall off like x', but the co relation radius F varies only slightly up to x = 20 d. E

perimental evidences show that about half the total soun

emission comes from the mixing region O < x <4d, an only a very small fraction coming from the decay regio in x> 20d. These salient features of turbulent jets enabl us to evaluate some important properties of the jet soun

emission.

(I) Low-speed Jets

In low-speed jets (with U,) «e so that the convective motion of the eddies may be neglected), one can readily infer how the acoustic power output per unit volume of

the jet, which is w given by (118) multiplied by the density

, varies with position. In the mixing region, we have

found that the eddy size L grows like x and wF is charac-teristic velocity proportional to the jet velocity U0, hence

-' U0/x, Ve x3.

The fluctuations of the quadruple strength Tjj( are

how-ever in proportion to Q0 U)5, nearly independent of x.

Con-sequently, from (118), Qo w is expected to vary as p,,U08/c5x

per unit volume of fluid in the mixing region. But since the volume per unit length increases with x, the sound emitted per unit length of jet should remain constant in this region, the total up to x = 4d (the whole mixing region) being

ob-viously

Wnijxing region) = (const)QoU08 d2

i

e5

Since this amounts to about half the total acoustic power

output of the jet, the latter may be written

¡

o,U,,'\

f Q0c\

W = K0 M,,5 I,srd2 = K0 t d0

2 ₎M,,° (121)

where std2 is the jet cross-sectional area at exit, M,., = U0/c

is the mean exit Mach number, e the sound speed in the

region outside the jet, and K, is a dimensionless constant. Here (std2Q0U,'/2) represents the total hydro-mechanical

power of the jet and KM,,5 therefore

indicates the

"acoustic efficiency". This result that W is proportional to M18 has been observed experimentally by Mueller and

Mat-schat (1958). The constant K has been found to be of the

order of 10g.

It is of interest to compare this case with the noise

pro-duced by a homogeneous, isotropic turbulence of the

pre-vious section. Since the root-mean-square velocity

fluc-tuation, U, in jets is about 0.1 U (to a maximum of 0.14 U11),

K,, M,5 K0 (10) M5 10M5, where M = U/c; this acoustic

efficiency is of the same order as in the previous case, which

is 38M5.

(II) Subsonic Jet Noise

In this operating range, the mean jet exit Mach number

M0 = U/c may vary up to unity, and the effect of

convec-tion of the eddies, with velocity U,, becomes important.

As observed by Davies et al (1963), U varies from 0.2 U() to 0.7 U, (or 0.2 M,, < M0 < 0.7 M5). This relative motion between the eddies and a fixed observation point has two

effects on the measured sound intensity. One produces a

frequency shift (the Doppler effect, see Sect. 13, Eq. (87)) in the radiation field; and the other gives an effective length of eddies in place of their true length. In the moving frame

K' fixed relative to the eddies, let the characteristic

fre-quency be co' and the eddy volume be Ve'. Then the acoustic pressure p of (113) and the acoustic power output w of (118) must hold with w replaced by w', and V0 by V,,'. Now the relationship between w and80' is given by (87). As for the

effective size of an eddy in the fixed frame K, it is noted that only the correlation length L in the direction of the

relative motion is changed (see Fig. 6) according to

V0' L' 1

-=

-.-- (122)

Ve f

1Me coSt)

where M,. = U0/c, and * is the angle between the direction of emission and the jet direction. One therefore obtains

û)2V0 /

i

p (x, t) =

-4strc2 n nI [Till)

_{(lM0 cos)3}

(123) Schiffstechnik Bd. 13 - 1966 - Heft 60 ₁₀₄ -O 44 Sd Fig. 5

Stationary cold jet, M0 O,3, Re 6 10* (Laurence 1956)

grows linearly with the distance x from the nozzle, its

width being approximately x /4, so that this region reaches the jet axis at x = 4d. At any section of the mixing region, the root-mean-square velocity fluctuation reaches a

maxi-mum of about 0.14 U,, where U0 is the jet velocity. The

eddies,

being stretched out by the mean shear,

are

elongated; the longitudinal correlation radius of an eddy

is about 0.1 x, whereas its lateral correlation radius is only about 0.04 x. In this region the eddies are convected down-stream at velocity U ranging from 0.2 U to 0.7 U,, across the mixing region, an average Ur being U12.

From x 4d to x 8d, a region of adjustment exists, in