Sound radiation generated by mechanical or vibratory source arrangements and, in some instances of planar or piston types, encompasses all magnitudes of the wavelength or frequency as well as disposition of the observation site. The comparative increase in average total radiated power from infinitesimally small or point sources of periodic strength, on passing from rest to steady rectilinear motion, depends only on Mach number in the noncompact case; and a formally analogous rise in output is linked, to the presence of a steady parallel flow past a rigid plane wall in which a compact circular piston executes normal oscillations. The perspective of effects connected with such a background is flow is widened by regard for an elongated or strip piston, which prototype permits a straightforward and general analysis

Levine, H.

English

NASA Technical Reports Server

I 
I I 
JOINT INSTITUTE FOR AERONAUTICS AND ACOUSTICS 
NI\S/\ 
National Aeronautics and 
Space Administration 
Ames Research Center 
NASA-CR-166177 
19810015326 
JIAA TR - 31 
A NOTE ON 
Stanford University 
-.. ~- .. ~ - ... ....,. --:' 
_ j, ... - ---' • _ _ _ .i '"'- -...J 
SOUND RADIATION INTO A UNIFORMLY FLOWING FLUID 
; , 
'- -
-, , 
11, -4,..... ...... 1 
HAROLD LEVINE 
JANUARY 1980 
1111111111111 1111 11111 11111 11111 111111111 1111 
NF02470 
https://ntrs.nasa.gov/search.jsp?R=19810015326 2020-03-21T14:04:33+00:00Z
JIAA TR - 31 
A NOTE ON 
SOUND RADIATION INTO A UNIFORMLY FLOWING FLUID 
HAROLD LEVINE 
JANUARY 1980 
A NOTE ON 
SOUND RADIATION INTO A UNIFORMLY FLOWING FLUID 
by 
Harold LeV1ne 
Introduct1on and Summary 
There exists a considerable body of analys1s pertaining to the 
sound rad1ation generated by mechanical or vibratory source arrangements 
and, in some 1nstances of planar or piston types, the results encompass 
all magn1tudes of the wavelength or frequency as well as d1spos1tion of 
the observat10n s1te. Fewer deta11s are ava11able in the c1rcumstance 
of relat1ve mot10n between the source and 1tS surround1ngs and, partic-
ularly, with an aptness for both the compact and noncompact character1za-
t10ns of a given source model. The comparat1ve 1ncrease 1n average total 
radiated power from 1nf1n1tesimally small or point sources of peri-
od1c strength, on pass1ng from rest to steady rect111near motion, de-
pends only on the Mach number 1n the latter state; and a formally analo-
gous r1se in output 1S 11nked, by Ffowcs W1111ams and Lovely (1975), 
to the presence of a steady parallel flow past a r1g1d plane wall in 
which a compact c1rcular piston executes normal osc111ations. It is 
the intention here to w1den the perspective of effects connected with 
such a background flow by regard for an elongated or strip piston, 
wh1ch prototype permits a straightforward and general analysis. 
The t1me average net power output of a str1p p1ston that vibrates 
normally to 1tself w1th uniform amplitude, has a f1n1te w1dth 0 and 1S 
located 1n an otherwise r1g1d plane wall, adm1ts d1fferent expans10ns 
according as 
1 
ko « 1, M < 1, (i) 
or 
Mko « 1, 
or 
ko » 1, M < 1 
respectlvely, where k,M are the wave and Mach number, the last belng a 
measure of the ratio between flow and sound speeds. In the absence of 
flow, M vanishes and classlcal (or 11near acoustical) theory predicts 
that the total rate of energy radlatlon (WhlCh lS dlstr1buted nonunlformly 
over a semicircular range) amounts to 
p (1) 
11 - r ko (2) 
per unit amplitude and length of the plston; here Po speclfles the equl-
11brium density of the medlum, c is the sound speed and J
o
' J l are 
Bessel functions wlth the deslgnated orders. The development which 
sUltS a compact piston, 
p ko + 0 (3) 
lS a ready consequence of (1), whl1e that obtalned from (2), 
2 
represents the noncompact alternative. 
It appears, ln what follows, that the development containing 
powers of the Mach number and conformlng with (il), 
p \ 
ko 
J J (V)dV - Jl(kO) (0 0 
extends the prlor result (1) and lmplles that 
p 
MkO « 1 
ln the 11mit ko + 0; lf kO » 1, on the other hand, the expressl0n 
p ko + ex> 
, (5) 
(6) 
( 7) 
takes the place of (4). The prlnclpal effect of flow in (6) is merely 
to ralse the long wave power estlmate by a numerlcal factor 1 + 3M2 , 
whereas the comparison of (4) and (7) dlscloses that the flow brlngs 
about a slgniflcant change ln the relative order of magnltude of the 
second term ln the short wave estimate, namely from (ko)-3/2 to (kO)-l. 
Inasmuch as the second term of (7) lS, in the absolute sense, lndepen-
dent of the plston geometry (and, speclfically, of ltS breadth), a cor-
responding feature may be presumed for other shapes. A further 
estimate, 
p ! k 3(kO)2 4 Po c 
11M2 
+ "2 + 0(kO)4 , 
( 2)5/2 1 - M 
ko « 1 , o < M < 1 , (8) 
3 
wh1ch accompanies the stipulat10n (1), has a more extens1ve val1d1ty 1n 
respect to the Mach number than does (6). 
The Analys1s 
Let the und1splaced p1ston occupy a strip Ixl < a, - 00 < Y < 00, in 
the fixed plane (or wall) z = 0, and assume the eX1stence of a steady 
flow w1th un1form speed U Mc in the parallel (or x-) d1rect10n. 
G1ven the normal p1ston d1splacement (exclus1ve of a per10d1c t1me fac-
-1wt tor e ) , 
where 
z = n (x) 
I A (l;) = 
2TI 
00 
J 
_00 
for the 'top-hat' prof1le 
n (x) 
-1l;x 
n(x)e dx 
I , Ixl < a 
a , Ixl > a 
s1n l;a 
TIl; 
(9) 
(10) 
(11) 
then the corresponding acoust1cal excitat10n, der1vable from a veloc1ty 
. ¢ ) -iwt potent1al (x,z e , 1S 1mpl1cit 1n the convected wave equat10n 
V2¢ (-1k + M a:t ¢ k w/c 
or (12) 
(I - M2) a2¢ a2¢ 2lMk a¢ k2¢ a , z > a , --+ --+ -+ 
ax
2 
az
2 ax 
4 
together with the boundary or wall condit10n 
~I = 
az z=O 
-iwn + U an ax 
and an outgoing wave cond1tion at 1nf1n1ty. 
A solut1on of the I1near part1al d1fferential equat10n (12) has 
the form 
CXl 
cp (x, z) 
_ CXl 
where 
arg K(I;;) > 0 
(13) 
(14) 
(15) 
and compl~ance w~th the boundary cond~t~on (13) necess~tates a propor-
I;;U - w A(I;;) 
K (I;;) (16) 
between the we~ghting factors A (1;;), B (I;;) of the respective 1ntegrals (9), 
(14). The normal der~vative of cp, as computed from (14), (15) and (10), 
proves to be 
CXl CXl 
~I i f B (I;;) K (I;;) e1 1;;x dl;; - i f (w - I;;U)A(l;)e1 I;;x d l;; 
az z=O _ 00 - 00 
00 00 
= 
_ 1W f S1n I;;a e1 1;;x dl;; + 1U f 11;;x - s~n I;;a e dl;; 7r I;; 7r _ 00 
_00 
- 12W {sgn(a+ x) +sgn(a-x)}+U{cS(x+a) - cS(x-a)} (17) 
5 
where sgn V = ± 1, V ~ 0 and 0 s~gn~fies a Dirac delta funct~on; it is 
thus ev~dent that a~/az van~shes on the r~gld section of wall (Ixl > a), 
has a un1form magn1tude over the p1ston (Ixl < a) and man1fests the 
joint presence of a line slnk/source at the lead1ng/tra111ng edges of 
the p1ston (Ixl = =fa). 
The t1ffie average power output of the piston lS conveniently found 
through 1tS rate of local working on the adJacent med1um (with an equ1-
co 
p J 
-co 
ct> * (x,o) a<p (x,o) 
dZ dx (18) 
and, consequent to the use of the representation (14) for ~ and the con-
nect10n (16), th1S becomes 
p TIp W Re 
o 
00 
J 
-co 
co 
J 
_ 00
(19) 
the same result obta1ns 1f the transformat10n of (18) commences w1th a 
vers10n 
p 1 
- P W Im 2 0 
00 00 
J dx 
_00 
J 
- 00 
* -ll,;X { iw B (l,;)e - :2 [sgn(a + x) + sgn(a - x)] 
+ U[o(x + a) - o(x - a)]} dZ;; 
6 
that 1ncorporates the explicit determ1nation of d¢/dZ given in (17). On 
substitut1ng the expression (10) for A(~) and s1mpl1fY1ng (19), it ap-
pears that 
p = 
1 
l-M 
f 
1 
l-M 
1 
f 
-1 
2 (1 - M~) 
/1 _ ~2 
dT 
M < 1 
1n wh1ch the change of var1able T = (~ - M)/(l - M2) figures. 
To reduce the latter 1ntegral put ~ = cos e and 1ntroduce the ab-
brev1at10ns 
whence 
w1th 
F (a,M) 
n 
a 
'IT f cosne 
o 
ka , 1 
sw 
2 (as (cos e - M» 
(cos e - M) 2 
de, 
(20) 
(21) 
n 0,1,2 (22) 
An alternat1ve single 1ntegral representat10n for each of the funct10ns 
F (a,M), wh1ch 1S the more sU1table 1n seek1ng est1mates appropr1ate to 
n 
d1fferent hypotheses regard1ng the magn1tude of the argument var1ables, 
rests on the fact that the second order a - derivat1ves of the Fn are 
expl1c1tly known 1n terms of tr1gonometric and Bessel funct1ons, viz. 
7 
and 
2TI~2 cos(2a~M)J (2a~) 
o 
2TI~2 s1n(2a~M)J (2a~) 
1 
(23) 
(24) 
(25) 
The two-stage integrat10n of (23), wh1ch serves as a model for that of 
(23), (24) and 1S rendered def1n1te by the un1versal cond1t10ns 
dF n (0 ,M) 
Fn(O,M) = oa = 0, beg1ns w1th the express10n 
dF 
o 2TIQ2 
--aa = fJ 
a f COS (2BMT)J
o
(2BT) dT 
o 
and carr1es on via the sequential formulas 
a v 
f dv f cos(2BMT)J (2BT) dT 
o 0 0 
TIaB 
2a~ 2a~ 
f J
o 
(v)cos MV dv - % f 
o 0 
VJo(V)cos MV dV 
2aB l ~ J o (v) cos MV dV - ~ d~ 2aB } f J (v) S1n MV dV - 4a~2M S1n (2a~M) J (2a~) o 0 
o 
TIaB P (a, M) - f ! Q (a, M) + 2TIa~2M sin (2a~M) J 0 (2aB) (26) 
8 
where 
2aS 
p(a,M) = f 
o 
and 
2aS 
Q(a,M) f 
o 
J (V) cos MV dV 
o 
J (V)Sl.ll MV dv 
o 
00 
f 
2aS 
The analogues of (26) for Fl and F2 are 
and 
respectl.vely. 
00 
f Jo(V)cos MV dV 
2aS 
J (V)Sl.ll MV dV 
o 
M < 1 
Hence, the three l.ntegrals Fo' Fl , F2 are chara~terized, apart 
(27) 
(28) 
(29) 
(30) 
from expll.cl.t terms, through the s1ffipler pair, namely P(a,M) and Q(a,M); 
and the representatl.on for the power whl.ch l.nvolves the latter turns out 
to be 
3 
+ aS2 M(2 - M2)Sl.n(2aSM)J
o
(2aS) + as; M2 cos(2aSM)J1 (2aS) 
+ ~ S~ M2(1 - COS(2aSM)Jo(2aS»)f 
9 
( 31) 
Mult1ple differentiation of p(a,M) and Q(a,M) with respect to M, at the 
specif1c value M = 0, perm1ts the formation of developments conta1ning 
powers of the Mach number, which commence 1n the fash10n 
P (a,M) 
and 
EmploY1ng these est1mates and approx1mat1ng the other terms of (31) 1n 
accordance w1th the hypothes1s that a8M « 1, or s1mply Mka « 1, the 
development (5) c1ted in the 1ntroduct10n emerges after the d1mens1on-
less magnitUde 
2a 2ka ko 
is redefined in terms of the strip breadth o. 
The same development for the average rate of energy rad1at10n can 
be secured, as a matter of techn1cal 1nterest, by the Jud1c1ous extrac-
t10n of fin1te parts for d1vergent 1ntegrals; thus, recal11ng the 
Legendre polynom1al generat1ng funct10n 
1 
(1 - 2zt + t 2 r"2 = 
and wr1t1ng 
00 
L 
n=O 
10 
p (z)tn 
n 
It I < 1 
- 2zt + 
(32) 
w1th the 1dent1f1cat1ons 
t = MI'; ( 33) 
1t 1S formally 1nferred from (19) that 
p Re 
(34) 
k 2 
2 (k) 2 J Sln sa 
=-p w c 
7T 0 0 1';2 
( 35) 
Slnce the restriction It I < 1 which assures convergence of the expansion 
(32) does not hold un1formly in the context of the pert1nent formulas 
(33), (34) all save a pa1r of terms 1n (35) possess non1ntegrable singu-
larit1es (at s k). The proper 1ntegrals are 
F(ka) 
and 
k 
G(ka) J 
o 
dl'; 
• 27" Sln sa d s 
after noting an alternat1ve expreSS10n 
k 
J 
k 2 
- J sln
2
1';a 
o o S 
11 
d ( k ) dl'; 
dk / 2 2 
k -I'; 
for the first improper integral, its finite part may be identified 
k 
. 2Z;; -~+ f Sl.n a dZ;; _ ~ {F (ka) } = _ 'ITa dk r,;2 /k2 _ dk k 2k J l (2ka) 0 7;2 
Likewl.se, the prescriptl.on 
k 
. 2r,; 1 k 2 Sl.n a (3 - 1) f 
r,;2 k2 _ r,;2 
dr,; 
0 
Ik2 _ r,;2 
= 
= 
k 
f 
0 
d2 
dk2 
2 
TIa 
2 1 (1 -Sl.n r,;a Z;;2 /k2 _ 7;2 
k 
sin
2
1;. dl; l k 2 f 
r,;2/ k2 _ Z;;2 0 
k 2 
4 k 2 _ r,;2 + 3 
k4 ) 
(k2 _ r,;2)2 dZ;; 
k 
. 2r,; 
dl; J d k f Sl.n a dk 
r,;2/k2 _ r,;2 0 
as 
supplies a value for the other l.mproper l.ntegral whl.ch appears l.n (35). 
When the various integral determl.natl.ons are brought together with (35), 
the resultant power estl.mate, 
2f 3 1 - J o (2ka)] 
+ M LkaJo(2ka) + 2 J 1 (2ka) + 2ka + 0 
Mka « 1, 
agrees wl.th (5). 
Turning attention next to a noncompact pl.ston, typl.fl.ed by the l.n-
equall.ty ex. ka » 1, and havl.ng regard for the ultl.mate stage of the 
successive equall.tl.es 
12 
00 00 
J Jo(V)COS MV dV 
Y 
cos MV 
V 
00 
- Jl(y)COS My +! VJl(V){CO~2MV + M Sl.nVMV}dV 
00 J
l 
(V) 00 
- J l (y)cosMY+ f -V- cosMVdv+M J d(-JO(V») Sl.n MV Y y 
00 Jl(V) 2 
- J 1 (y) cos My + f -V- cos MV dV + MJo (y) Sl.n My + M f y y 
00 
1t follows that 
Jo(V)COS MV dv 
00 1 1 00 J l (V) ~ ( J 0 (v) cos MV dV = --2 -Jl (y) cos My + MJ 0 (y) Sln My + yf V cos MV dV Y l-M 
whence the function P(a,M) def1ned 1n (30) lS express1ble as 
Correspondingly, 
00 { 00 J 1 (V) f J J 0 (V) sin MV dV = ~ 1-J l (y) Sln My - MJo (y) cos My + f -V- Sln MV dV Y l-M ( y 
and 
~ 00 J 1 (V) ~ Q(a,M) = S J (2aS)sin(2aSM) +MJ (2aS)cos(2aSM) - f sinMVdv 1 0 2aS V 
13 
(35) 
(36) 
(37) 
(38) 
from which 1t lS deduced (on employing the result dS/dM = 2S2M) that 
2 2 2 
+ 2aS MJ
o
(2aS)sln(2aSM) + 2aS J l (2aS)cos(2aSM) - MS 
co J l (v) I -- slnMVdV 
2aS V 
(39) 
Subst1tut1ng the expreSS10ns (36), (38) and (39) for p(a,M),Q(a,M) and 
dQ/dM 1nto the formula (31) const1tutes the last step of our power analysls, 
and Ylelds the generally valld representatlons 
and 
+ 
1 Q ~ 2 
a - IJ M 2 
co 
1 S~ M I 
2 2aS 
Sln MV dV 
ko 
a = 2 = ka, S 
~ 2aS J l (v) 1 ~ 
as J --V- COSMVdV+"2 MS J l (2aS)sln(2aSM) 
o 
+ 1:.. S~ M 
2 
2aS 
Jl(V) J Sln MV dV 
V 
o 
1 
where the former lends ltself to estimatlon lf as » 1 and the latter lf 
(40) 
(41) 
as «1. In partlcular, when a + 0 and M does not assume values close to 
unity (or S close to inf1nlty) the representat10n (41) 1mpl1es 
Pip kc 3 
o 
14 
M-+O 
as stated prev10usly. For small values of M, furthermore, the develop-
ments 
2aS J l (\I) f -\1-- cos M'V d'V = 
o 
and 
20. Jl(\I) 1 2J 20. f 4 f d'V +- M - f 'VJ (\I)d'V + 2Jl (20.) + 0 M o \I 20 1 
20. 2aS J
l 
(\I) J S1n M\I d\l \I M J J l (\I) d\l + 0{M3) 
o o 
support, 1n conJunction w1th (41), the propriety of the result (5). 
The terms of (40) are arranged 1n a sequence that bef1ts the1r 
relat1ve importance 1n the l1mit a -+ 00, M t OJ namely, the f1rst and 
second are 0(0.) and 0(1), respect1vely, wh1le the third and fourth 
are o(a-~) and the last 1S 0(0.-'*"). Of part1cular note, as prev10usly 
stated, 1S that the flow man1fests 1t presence, to the 1ead1ng order 
in the average power output at short wavelengths, by a contr1but10n 
wh1ch does not 1nvolve the piston scale. Estimates for the terms of 
(40) wh1ch conta1n Bessel functions are read1ly ga1ned through the 
ut111zat10n of large argument asymptot1c forms pert1nent to these 
funct10ns. 
Reference 
Ffowcs Wil11ams, J. E. and Lovely, D. J. 1975 J. Flu1d Mech. 71, 689. 
15 
End of Document 


A note on sound radiation into a uniformly flowing fluid

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server