Copyright by the Acoustical Society of America. Tarkenton, G. M. (1990). Remarks on "On the coefficient of nonlinearity" in nonlinear acoustics.
Journal of the Acoustical Society of America, 88(4), 2024-2025. doi: 10.1121/1.400178
III. CONCLUSIONS
Acoust.Soc.Am. Suppl. 1 84, S218 (1988).
The generalized
smoothed
boundaryconditionto rough
surfacescatterappearscapableof explainingquiteadequately the observedscatterfrom the roughinterfacebetweenwater and an overlyingthin plate, for ka < 1, i.e., in the "lowfrequency"limit. The observations
usedin this paperwere
thosereportedby McClanahan
2andMcClanahan
andDiachok• fora modelconsisting
ofa thinluciteplatewithcorrugationsoverwater.Thisstronglysuggests
that thetechnique
shouldbe appliedin the contextof low-frequencyArctic
data--for whicha varietyof otherfactorsmayalsoenterinto
play.
IT. P. McClanahan
andO. I. Diachok,"Scattering
andreflection
ofacoustic wavesby striatedsurfacesdepictingthe Arctic Oceanice cover," J.
-•T.P.McLanahan,
"Scattering
andreflection
ofacoustic
waves
bystriated
surfaces
depictingtheArtic Oceanicecover,"Thesis,CatholicUniversity
of America,Washington,DC (1988).
3I.Tolstoy,
"Effectofroughness
onsubsonic
flexural
waves
in a thinplate
overlyinga compressible
fluid," J. Acoust.Soc.Am. 79, 666-672 (1986).
aM. A. Biot,"Generalized
boundary
conditions
for multiplescatterin
acousticreflection,"J. Acoust. Soc.Am. 44, 1616-1622 (1968).
5I.Tolstoy,
"Smoothed
boundary
conditions,
coherent
low-frequency
scatter, and boundary modes,"J. Acoust. Soc.Am. 75, 1-22 (1984).
6I. Tolstoy,"Longwavelength
acoustical
scatter
fromroughsurfaces,"
in
Frontiersof Acoustics,
Proceedings
of theInternationalSchoolof Physics
'EnricoFermi,'editedbyD. Sette(North Holland,Amsterdam,1986),pp.
209-254.
7M.J. Lighthill,Waves
in Fluids(Cambridge
U.P.,Cambridge,
England,
1978).
•V. Twersky,
"Onscattering
andreflection
ofsound
fromroughsurfaces,"
J. Acoust. Soc. Am. 29, 209-225 (1957).
Remarks on "On the coefficient of nonlinearity
acoustics" [J. Acoust. Soc. Am. 83, 74-77 (1988)]
in nonlinear
Grey M. Tarkenton
Department
ofPhysics,
VirginiaPolytechnic
InstituteandStateUniversity,
Blacksburg,
Virginia24061
( Received14August1989;acceptedfor publication25 June1990)
Throughan alternateanalysis,the resultsof Hamilton and Blackstock[J. Acoust.Soc.Am.
83, 74-77 (1988) ] are reviewedanda newinterpretationis presented.In thisview,the
nonlinearityof the momentumequationis examined,and its effecton the coefficientof
nonlinearity is revealed.
PACS
numbers: 43.25.Ba
In theirpaper,'HamiltonandBlackstock
suggest
that
the nonlinearterm in the conservation
of momentumequation plays no role in the coefficient of nonlinearity
13= 1 q- (B/2A ), and that this nonlinear term need not be
includedin the analysisarrivingat the wavespeed.The calculationspresentedhere indicatethat the term in question
doesplay a vital role in both the coefficientof nonlinearity
andthe propagationspeed.To illustratethispoint,an alternate derivationis providedbasedon the methodof characteristicsincludinga shadowfactor to follow the pertinent
nonlinear
term.
•p
8t
--q-p-•xxxq-V
8x =0,
8v
--q8t
av
cTVt a2(p) cTp=0,
8x
p
8x
(la)
(lb)
where
a(p)
-- dp
term. It follows that if the coefficient of nonlin-
-
dp+p(A+-v) dv_Oon
a2(p)
dx
dt
=A+,
(2)
sity,p is the pressure,v is the particlevelocity,and a is the
J. Acoust. Soc. Am. 88(4), Oct. 1990;
A -+ = [ (a + 1)/2]v __+
x/(a - 1)2U2q- 4a2/2.
Keepingonly first-ordertermsin v, the expressionfor A becomes
A+- (aq1)v+(aoq-B
a(•(p_po)q(H.O.T.)),
52po
wherethe soundspeedhasbeenexpandedaboutPoandB /
2A is as in Ref. 1. The nonlinearacousticapproximationis
now employedby linearizing the compatibilitycondition
and retaining the first nonlinear term in the soundspeed.
This processyields,after integratingthe compatibilitycondition,
istheisentropic
density-dependent
soundspeed,
p istheden-
2024
momentum
earityis independentof the momentumnonlinearity,then/3
will be independentofa. It will alsobe shownthat a mustbe
1 in order to obtainthe correctwavespeed.
Having appliedthe methodof characteristics
to the system ( 1), the resultingsolutionis
where
The salientpointsof the argumentcanbe madeusinga
one-dimensional
analysisthat generalizesdirectlyto the full
three-dimensional
case.To thisend,the equationsof motion
are written in the followingform:
•v
shadow factor used to follow the influence of the nonlinear
[ (P -- Po)/Po] _ v/ao= R _+,
( 3)
whereR + andR_ arethe Riemanninvariants.To represent
0001-4966/90/102024-02500.80;
© 1990 Acoust. Soc. Am.; Letters to the Editor
2024
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.173.126.47 On: Mon, 11 May 2015 17:49:57
a simple right-moving wave, set R_ to zero and then the
integrated expression(3) is equivalentto Hamilton and
Blackstock's
Eq. ( 11) for a right-movingprogressive
plane
wave.Havingdroppedthe higher-ordertermsin the propagationspeedand thenusingthecompatibilityconditionfor a
simpleright-movingwave,one seesthat
dx_ (a+ l) v+ ao+ B ao(p Po),
dt
dt
2
= ao ß
2APo
2
•
(•,
v,
whichshowsthat the wavespeedis dependenton a, indicating that the nonlinearmomentumterm doeshave an effect.
Further, the effectis presentin the coefficientprecedingv,
whichisthenonlinearitycoe•cient.The correctwavespeed
demandsthat a be 1; then
d••ao• 1•
The critical role of the momentumnonlinearityalsobecomes evident after a detailed review of Hamilton
and Black-
stock'soriginalderivation.It is easilydemonstratedthat the
originofthekineticenergy
intheLagrangian
termofEq.(8)
of Ref. 1 is just the momentum nonlinearity in the exact
equations.The right-handsideof Eq. (8) vanishesonly if a
precisecancellationbetweenthemomentumnonlinearity,as
represented
by the kineticenergy,andthe pressure
potential
energyt•ko• pl•,•
It mustbe emphasizedthat the issuehereis oneof interpretation and that the basicresultsdescribedin Ref. 1 must
be regardedassoundin all respects.In particular,the useof
the linearizedmomentumequation[Eq. (13) of Ref. 1] is
entirely appropriate.However, the point made here is that
the linearnatureof the momentumequationis dueto a cancellationof the nonlinearpressurework term by the convective nonlinearity rather than somelinearization scheme.
v,
asin Ref. 1.This analysisalsoshowsthat the constantterm 1
in the nonlinearity coe•cient arisesfrom nonlinearitiesin
both the massandmomentumequations,notjust in the mass
equation.
tM. F. HamiltonandD. T. Blackstock,
"On thecoefficient
of nonlinearity
/3 in nonlinear acoustics,"J. Acoust. Soc. Am. 83, 74-77 (1988).
On the linearity of the momentum equation for progressive plane
waves of finite amplitude
Mark F. Hamilton and David T. Blackstock
DepartmentofMechanicalEngineering,
The University
of Texasat Austin,Austin,Texas78712-1063,and
AppliedResearch
Laboratories,
The University
of Texasat Austin,Austin,Texas78713-8029
(Received14June1990;acceptedfor publication25 June1990)
This Letter is primarilyin response
to the Letter to the Editor by G. M. Tarkentonentitled
"Remarkson 'On the coefficientof nonlinearity/3in nonlinearacoustics'"[J. Acoust.Soc.
Am. 88, 2024-2025 (1990) ]. It is reaffirmedthat for progressive
planewaves,the momentum
equationcontributesnothingat secondorderto the expressions
for the coefficient
of
nonlinearityandfiniteamplitudepropagation
speed.
PACS numbers:43.25.Ba,43.25.Lj
We aregratefulto G. M. Tarkenton
• for pointingout
that our commentin Ref. 2 about the origin of the convection term in the expressionfor/3 might be misconstruedby
readers.Following Eq. (14) (in Ref. 2), we claim that "To
parison,we useTarkenton'snotation,includingthe shadow
factor a introducedto keep track of the convectionterm.
[ However, we note that to obtain the form of the momentum
mentumcontributesnothingto the coefficientof nonlinearity." The parametere characterizes
the dimensionless
mag-
equationgivenby Eq. ( 1b) of Ref. 1, Tarkenton useda thermodynamicrelationto eliminatethe pressurein favor of the
density, a modification that is discussedbelow.]
The momentumequationcontainstwononlinearterms
nitude of the acoustical field variables. Tarkenton asserts
ofordere•, notjusttheconvection
terminvestigated
byTar-
that "the nonlinearterm (emphasisadded) in the conservation of momentumequation...doesplay a vital role in both
the coefficientof nonlinearityand the propagationspeed."
The analysisin questionis that for progressive
planewaves.
The momentumequationis
kenton.The two terms are easilyidentifiedby writing Eq.
( 1), correctto ordere2, asfollows:
ordere2 (for progressive
planewaves),conservation
of mo-
p •-av
•- c•x =0,
(1)
where all quantitiesare definedin Ref. 1. For easeof com2025
J. Acoust. Soc. Am. 88(4), Oct. 1990;
Ov• 1 Op (P--Po)O(p
--po)_ave,
e7t Po e7x
po•
ao
•
e7x
e7x
(2)
wherePo,Po,and aoare the ambientvaluesof the pressure,
density,and soundspeed,respectively.Sincefor progressive
waves,correctto ordere2,thefirstnonlinear
termisequalto
v(3v/3x), Eq. ( 2 ) becomes
0001-4966/90/102025-02500.80;
@ 1990 Acoust. Soc. Am.- Letters to the Editor
2025
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.173.126.47 On: Mon, 11 May 2015 17:49:57