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4.1Stop-banddynamicGreen’sfunctionsandexponential

4.2Dynamicanisotropyandlocalisationneardefects...... 101

4.3Localisationnearcracks/inclusionsinalattice........ 115

5Cloakingandchannellingofelasticwavesinstructuredsolids 141

5.1Acloakisnotashield......................

5.2Cloakingasachannellingmethodforincidentwaves.....

5.3Boundaryconditionsontheinteriorcontourofacloak... 164

5.4Cloakinginelasticplates....................

5.5Singularperturbationanalysisofanapproximatecloak... 182

6Structuredinterfacesandchiralsystemsindynamicsof elasticsolids 199

6.1Structuredinterfaceasapolarisingfilter........... 199

6.2Vortex-typeresonatorsandchiralpolarisersofelasticwaves. 208

6.3Discretestructuredinterface:shielding,negativerefraction,and focusing..............................

Preface

Thestudyofwavepropagationinstructuredmediacanbetracedasfarbackas theseventeenthcenturywiththepublicationofNewton’s PhilosophiæNaturalisPrincipiaMathematica.Forinstancein Principia,Newtonexamined the,nowclassical,one-dimensionalmass-springlatticesystemandderivedan expressionforthespeedofpropagationofsoundwaves.Despitebeingstudiedforseveralcenturies,wavepropagationinmulti-scalestructuredmedia remainsanactive,exciting,andchallengingareaofresearch.

Indeed,recentdevelopmentstowardthecontrolofwavepropagationin multi-scalesolidshaveledtoconsiderableprogressinthedevelopmentand applicationofanalyticaltechniquesforthemodellingofwavepropagation inmulti-scalestructuredmedia.Theprincipalaimofthepresenttextisto provideaunifiedaccountofseveraloftheseanalyticaltechniquesasapplied toacollectionoffascinatingphysicalproblems.

Followingtheintroduction, Chapter2 summarisesseveralfundamental methods,concepts,andapproachesrequiredfortheanalysisofwavepropagationinmulti-scalesolids.Thischapterisdesignedtofurnishtheuninitiated readerwithasuccinctintroductiontotheprimarymethodsnecessaryforthe materialpresentedlaterinthetext.Itishopedthatgraduatestudentsand thoseresearchersunfamiliarwiththetopicscoveredinthepresentmonograph willfind Chapter2 particularlyuseful.

Themainbodyofthepresenttextrepresentsacoherentaccountofaselectionofinterestingproblemstackledoverthelast15years.Wewereprivileged tohaveworkedwithwonderfulcolleagueswithwhomwehavecollaborated overthepastyears,andweareverygratefultothem.Thebookisbasedon aseriesofresearchpaperscitedthroughoutthetext.

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Chapter1 Introduction

Thestudyofwavepropagationinmulti-scalestructuredmediahasproven arichareaofresearch,attractingtheattentionofresearchersfromabroad rangeofdisciplinesincludingmathematics,physics,andengineering.Froma mathematicalstandpoint,themodellingofdynamicphenomenainstructured mediaisbothinterestingandchallengingandhascaptivatedtheappliedmathematicscommunityformanydecades.Inthelastcentury,thefieldofwaves instructuredmediahasundergonearapidseriesofdevelopmentswithboth theoreticalandexperimentaladvancesactivelydrivingthefieldforward;see, forexample,[12, 43, 58, 98, 106, 119–121, 137, 143, 158, 174, 175].Morerecently, advancesinfabricationmethodsandmaterialstechnology,coupledwiththe desireandabilitytocreate“designermaterials”haveledtoasignificantexpansioninexperimentalresearchforwavesinmulti-scalestructuredmedia. Elasticstructuredmediapresentsauniquechallengeintheanalysisof wavepropagationinmulti-scalesolids.Inthecaseofelectromagnetism,all wavestravelatthesamespeed(thespeedoflight)inagivenmedium;the sameistrueofacousticwavesandwaterwaves.Incontrast,elasticbodies generallysupportthepropagationoftwodifferenttypesofwave:pressure wavesandshearwaves.Thesetwowavestravelatdifferentspeeds,evenin thissamemedium,andarecoupledatboundariesandinterfaces;thismakes theanalysisofwavepropagationinstructuredmedia,wheretherearemany interfacesandboundaries,singularlydifficult.Inturn,thisuniquecoupling leadstoparticularlyfascinatingdispersionpropertiesand,notonlycreates severalnewdirectionsinthemathematicalmodellingofthedynamicresponse ofmulti-scalestructuredsolids,butalsoallowsforthedevelopmentofmany interestingapplicationsinmechanicsandengineering.

Oneparticularlyinterestingareaofresearchrelatesto “high-contrast” structures.Inthecaseofmechanicalsystems, “high-contrast” referstomaterialsofstructureswithlargespatialvariationsindensityand/orstiffness. Suchstructuresareofparticularinterestbecausetheycansupportverylow frequencystandingwavesandlocalisedwaveforms.Amongtheplethoraof engineeringapplications,earthquakeprotectionsystemsare,understandably, ofparticularinterestandmulti-scalesystemsofresonatorscanbeusedto efficientlyfilter,reflect,polarise,modeconvert,andotherwisecontrolelastic wavesoveradesiredfrequencyrange(see,forexample,[44, 46]).

Inthepresentbook,weconsiderabroadrangeoftheoreticalapproaches tothechallengingproblemsofmodellingwavepropagationandlocalisationin

MathematicalModellingofWavesinMulti–ScaleStructuredMedia structuredelasticmedia.Inparticular,wewillfocusonmulti-scalematerials, withadynamicresponsethatmayappearunusualandoftencounterintuitive; usingtherigorousmethodsdevelopedhere,wewillshowthatthese,apparently strange,phenomenacanbeeasilyunderstoodandare,infact,entirelynatural. Examplesofsuchphenomenaincludedynamicanisotropy,focussingbyflat interfacesandcloaking.

1.1Bloch–Floquetwaves

Therearemanyclassicalmonographs,suchas[26],[85],[91],and[187], whichaddresswavepropagationinstructuredmedia;atopicthathasproven tobeanexceptionallyrichandattractiveareaofresearch.Motivated,notonly bycontemporarytheoreticalandtechnologicaladvances,butalsobywork goingbacktotheseventeenthcentury[135],therehasbeenaparticularly highlevelofscientificinterestintheanalysisofsystemsexhibitingphotonic andphononicbandgapsforBloch–Floquetwaves.Thescholarlyinterestin theanalysisofsuchphenomenaisunderlinedbytheextensivebibliography summarisedin[60].Inparticularfocussinganddiffraction,whicharewell knownforopticalsystems(see,forexample,[25]and[145]),canhavenew andinterestingaspectsinproblemsofvectorelasticity.Asystematicanalysis ofthescatteringofwavesinsolidswithperiodicarraysofdefectswaspresented in[35].Forproblemsofthemathematicaltheoryofelasticity,thepaper[166] extendedtheRayleighmethod[169]toexaminethescatteringofelasticwaves indoublyperiodicstructures.

Forthecaseoflatticedynamics, localisedprimitivewaveforms havebeen identified[8, 151]andstudiedindetail.Inparticular,itwasdemonstrated thattheselocalisedvibrationsareassociatedwithstationarypointsonthe dispersionsurfaces.Forboth,time-harmonicandtransientexcitations,the papers[92]and[93]haveanalysedthedynamicresponseofasquarelattice, withaparticularemphasisonthenatureofthecaustics,whichrequirecareful considerationwhenapplyingthemethodofsteepestdescent.

Forsystemsofelasticrods,whichincludeperiodicarraysoftrussesand frames,agenericalgorithmfortreatingthedispersivepropertiesofvector latticesandthedesignofstructurespossessingbandgapsindesiredrangesof frequencieswasdevelopedin[108].

1.2Structuredinterfacesandlocalisation

Theconceptofstructuredinterfacesoffinitethickness,whichjointwo continuousregionsofanelasticmedium,wasintroducedin[21];whereinthe authorsnotethattheinertialpropertiesoftheinterfacesignificantlyaffect thedynamicresponseofthestructureandleadtounusualfilteringproperties forelasticwaves.Extendingtheworkof[21],astructuredinterfacebetween twocontinuousdomainswasemployedin[29]todemonstratethefocussingof elasticwavesvianegativerefraction.Suchstructuredinterfacesinsolidmedia aresometimesreferredtoas“flatlenses”forelasticwaves.Similareffects havealsobeendemonstratedinacoustics,asshown,forexample,in[76].The effectsoffocussingandfilteringofelasticwaveshavebeenextendedtoentirely discretestructures:adiatomicinterfacelatticeembeddedinamonatomic ambientlatticeofthesamegeometrywasconsideredin[49];anditwasshown that,forcertainfrequencies,theinterfacelatticeactsasaflatelasticlens. Dynamichomogenisation,whichalsoincorporatesdirectionalpreferencesand effectsofnegativerefraction,wasaddressedintherecentpapers[54–56, 139].

Anisotropicinhomogeneousinterfacesareuseful,forexample,forthereductionofstressconcentrationinsolids. Figure1.1 givesasimpleexample forauni-axiallyloadedstaticsolidcontainingasetofcoatedinclusions.The materialofthering-coatingisorthotropicandtheYoung’smodulusinthe tangentialdirectionwithinthecoatingincreasestowardthecentre,similar to[58, 98].Asshownin Figures1.1c and 1.1d,thestressconcentrationhas beenreduceddramaticallycomparedtothecasesin Figures1.1a and 1.1b.

Thepresenceoflocalisedwaveformswaspreviouslyillustratedforscalar lattices[8,92,93,151].Theresultinganisotropy,diffractionpatternsandaberrationsareoftenexplainedusingthedispersionsurfacesandslownesscontours.Twogeometricallyidenticallattices,withthesameelasticstiffnessmay providedifferentdynamicresponsesduetodifferentdistributionsofinertia andembeddedresonatorsofdifferentkind.Instaticssuchlattices,whichwe denoteastypeIandtypeII,maybegeometricallyidentical.However,embeddinganinterfacecomposedofalatticeoftypeIintoanambientlatticeof typeIIwillresultinhighlyfascinatingdynamicfeatures.Althoughinstatics wewouldhaveauniformlatticewhichishomogenous,itsdynamicresponse mayexhibitthetotalreflectionofwavesbytheinterface,aswellasnegative refractionandfocussing.

Imperfectinterfacesmayinfluencetheoverallpropertiesofthehomogenisedcomposites.Inparticular,thereareinterestingexamplesconcerningthenotionofneutralinclusionsand,inastaticsetting,thesewerestudied in[14, 23, 100, 122].

Dynamiccompositestructurescontainingcoatedinclusionshavebeen analysedin[160,161],wherethenotionofneutralitywaslinkedtotheeffective refractiveindexinthelong-wavelimit;thecoatedinclusionswereconsidered

FIGURE1.1:Panels(a)and(b)showthestress σ11 and σ22 fieldplotsaround aninclusioninauni-axiallyloadedsolid;(c)and(d)panelsshowthestress σ11 and σ22 fieldplotsaroundaninclusionsurroundedbyanorthotropicinhomogeneouscoatinginthesameuni-axiallyloadedsolid.Thestressconcentration hasbeendramaticallyreduced.

(a)
(b)
(c)
(d)

asneutraliftheeffectiverefractiveindexofthecompositemediumwasthe sameasthatofthehomogenoussolidwithoutinclusions.Inthesepapers,the elasticcoatingaroundaninclusionwasconsideredasanisotropic,anditwas demonstratedthatthegeometricalandphysicalparametersofthecoatings couldbechosentomatchtheeffectiverefractiveindexofthedoublyperiodicarrayofcoatedinclusionswiththerefractiveindexoftheunperturbed homogeneousmedium.Acousticbandgapsforwavesinmediawithneutral inclusionswereconsideredin[77].

Instaticsonecanintroduceso-called imperfectinterfaces inordertodescribehigh-contrastcoatingswithelasticpropertiesthataresignificantlydifferentfromthoseoftheambientelasticmedium.Inthecontextofhomogenisationandboundsoneffectivemoduliandneutrality,staticmodelsofcompositescontainingcoatedinclusionshavebeenanalysedin[13, 15, 78, 79, 101]; includingbothstiffandsoft,aswellashighlyanisotropicimperfectinterfaces intwo-dimensionalelasticityandproblemsoftorsion.

Amodelof non-localstructuredinterfaces wasdevelopedin[21]forstatic anddynamicproblemsinelasticcompositemedia.Specialfeatures,highlightedinthatpaper,includestronganisotropyandafinite-widthinterface.In particular,fordynamicproblemsthestructuredinterfacemaypossesstrapped waveforms,whichenhancetransmission.Thepapers[16–18]studiednewstatic modelsofnon-localstructuredinterfaces,andaddressedevaluationoftheir effectivemechanicalpropertiesandtheirinfluenceontheelasticstressconcentrationarounddefects.Structuredinterfacespossessinertia,andhence, tractionsmaybecomediscontinuousacrosssuchinterfacesindynamicproblemsofwavepropagation.

Theinteractionbetweenperiodicdiscreteandcontinuoussystemswas studiedin[21, 29]whereintheexistanceof“slowwaves”wasnoted,andthe transmissionpropertiesofafinite-widthdynamicstructuredinterfacewas analysedleadingtounusualdynamiceffectssuchasnegativerefraction.

1.3Multi-physicsproblemsandphononiccrystal structures

Theinteractionofelectromagneticwaveswithphotoniccrystalstructures hasbeenthesubjectofrapiddevelopmentinthelastdecade.Inparticular, cloakingofelectromagneticwaveshasbeenanalysedbothexperimentallyand theoretically;someoftheearliestinvestigationsinthisareaofwavestudies werereportedin[58,174].Beingnaturalandwell-studiedforproblemsofelectromagnetismandacoustics,theconceptofwavecloakingwasalsoaddressed forotherclassesofphysicalproblems,whichincludelinearwaterwaves[63] andflexuralvibrationofelasticplates[64].Theequationsofvectorelasticity

FIGURE1.2:Structuredinterfacecoating:(a)theradialanisotropicstructure,(b)thesquareanisotropiccoating.

bringadditionalchallenges,comparedtotheMaxwellsystemortheequations describingflexuralwavesinelasticstructures;thetensorstructureofthegoverningequationsofmathematicalelasticityincorporatesthenotionofshear. Asaresult,pressureandshearwavespropagatewithdifferentspeedsand couplethroughtheboundaryconditions.Thismakesthemodellingofelastic metamaterialsystems,whichpossesscloakingproperties,difficultasoutlined, forexample,in[119, 143].

Thedynamicresponseofvectorelasticlatticesystems,interactingwith waves,hasbeenthesubjectofclassicalinvestigations,asdiscussed,forexamplein[91],aswellas[49, 177–179].Wenotesubstantialdifferencesbetween scalarproblemsinvolvingthevibrationofsystemsofharmonicspringsand vectorproblemsofelasticity,referringtoelasticrodsandbeams,whichconnectasystemoffinitesolidsorpointmasses.Thenotionofshearstressand shearstrainisabsentinscalarlatticesystemsandthereisnosuchanalogue inmodelsofelectromagnetism.Themisconceptionofpredictabilityofdynamicpropertiesofvectorelasticlatticesissometimesbasedonanintuitive extrapolationofresultsavailableforthescalarsystemsorforproblemsofelectromagnetism.Asillustratedin[49],resultsobservedformicro-polarelastic latticestructuresdonotalwaysfollowfromthesimplerscalarphysicalmodels orintuitiveassumptions.

Thepapers[99, 129, 160–162, 166, 190]haveaddressedtheproblemsofdispersionofBloch–Floquetwavesandphononicbandgapsforelasticwavesin doublyperiodicsolidscontainingarraysofvoidsorinclusions.Transmission problemsforwavesinteractingwitharraysofelasticgratingsofinclusionswere studiedin[163, 164],includingacomparativeanalysisofthefilteringproper-

(a)
(b)

FIGURE1.3:Wave-cloakingofasquarescatterer:(a)adistortedwavefront withinthecloakinglayer;(b)theraydiagramshowingthedistortedmetrics inthecloakingregion.

tiesofelasticwavesindoublyperiodicmediaandthetransmissionproperties forthecorrespondingsinglyperiodicgratingsconsistingofcircularinclusions.

Anelegantlydesignedelasticcoating,withanisotropicinertialproperties, hasbeenproposedin[123]tomodelanelastic“invisibilitycloak”,whichtakes anincidentwavearoundascatterer.Thepaper[28]showsanalternativedesignofanelasticcloakingcoatingwhichengagesamicro-polarinhomogenous compositestructure.Structuredinterfaces,andtheirdiscreteapproximations, canbeeffectivelyusedtoconstructdynamicinvisibilitycloaks,withtwoexamplesshownin Figure1.2,whichillustratearadiallysymmetricstructure (part(a))andasquarestructuredinterface(part(b)).Theradiallysymmetric wavecloaksarediscussed,forexample,in[43, 58, 98],andthesquarecloak hasbeenmodelledin[48].

Thepapers[30, 39, 40]haveaddressedpropagationofelasticwaves,their dispersionproperties,andtheirinteractionwithdefectsinatwo-dimensional structuredmediumendowedwithmicro-rotationsassociatedwithspinning gyroscopesembeddedintothelatticesystem.Theanalysisofgyroscopicmotionofanindividualspinnerwasincorporatedintothesystemofconservation oflinearandangularmomentawithinthelatticesystems.Consequently,it hasledtoaspecialdesignofchiralmedia,whicharecharacterisedby“handedness”intheirmicrostructures.Suchmediapossessspecialpropertiesrelated toshielding,polarisationandfilteringofelasticwaves.

(a)
(b)

1.4Designermulti-scalematerials

Theterm “metamaterials” isoftenusedformulti-scalematerials,which areengineeredtopossessspeciallydesignedpropertiesthatarenotfoundin nature.Examplesofinterestingdesignsincludestructureswhichexhibitnegativerefraction[176],filtering,polarisationandfocussingofwaves[62, 74, 118]. Inparticular,metamaterialwavecloakingstructureshavebeendesignedand implementedinmanyphysicalapplications;theexamplein Figure1.3 shows adistortedwavewithinasquareinhomogeneousandanisotropiccloakinginterfaceandtheraydiagramillustrating“diversion”ofanincidentwave[48].

Anotherterm,“hyperbolicmetamaterials”,isusedforaspecialclassof structuredmulti-scalemedia,whichexhibitastrongdynamicanisotropyaccompaniedbystar-shapedwaveforms.Suchmaterialsweredesignedand studiedinproblemsofoptics[165],aswellasinacousticsandlinearelasticity[8,50,92,94,131,151,170].Inparticular,intheframeworkoftime-harmonic fields,thewavesinhyperbolicmetamaterialsaremodulatedbyfunctions, whichsatisfyhyperbolicpartialdifferentialequations.

Thepaper[52]introducedtheconceptof“parabolicmetamaterials”as multi-scalestructureswhichsupportuni-directionallylocalisedwaveforms. ThisworkalsodemonstratedinterestingconnectionswiththeDiracpointson thedispersionsurfacesforBloch–Floquetwavesinaspecialfrequencyrange. WealsonotethatDiracconesonthedispersionsurfacesoccurastheresultof degeneraciesinthedispersionequationscharacterisingBloch–Floquetwaves inphotonic[117],phononic[42],andplatoniccrystals[112].

Thedynamichomogenisationschemedevelopedanddemonstratedin [5,6,47,55,56,103]isalsousefulinthedescriptionofhyperbolicandparabolic metamaterials.Classicaltwo-scaleasymptotichomogenisationapproachesfor microstructuredmedia(see,forexample,[12, 154])usuallyinvolvethestudy ofaclassofmodelproblemsonanelementarycellandcanbeusedeffectively inthelong-wavequasi-staticregime.Incontrast,theapproachofdynamichomogenisationaddressesperturbationsawayfromknownresonancesandmay beusedtostudydynamicanisotropyandlocalisationofwavesindispersive media.

1.5Dynamicanisotropyanddefectsinlatticesystems

Classicalapplicationsinthetheoryofdefectsincrystalsanddislocations followfromthefundamentalwork[105],whereexplicitclosed-formsolutions werederivedforaheterogeneouslatticesystemwhentwodistantparticlesof differentmassesareinterchanged.Theenvelopefunction-basedperturbation

approachwasdevelopedin[59, 102]foranalysisofwaveguidesinphotonic crystalstructures.Inthelattercase,anarrayofcylindricalinclusionsembeddedintoanambientmediumrepresentsawaveguide,wherethefrequencies oftheguidedmodesareclosetothebandedgeoftheunperturbeddoubly periodicsystem.

Periodiclatticesmayexhibitverydifferentbehaviourindynamicscomparedtotheirstaticresponse.Inparticular,dynamicanisotropyanddifferent classesoflocalisationhavebeenidentifiedinperiodiclattices,asoutlined in[8, 50, 92, 94, 131, 151, 170].Insomefrequencyregimes,correspondingto neighbourhoodsofresonances,hyperbolicmetamaterialscanbeviewedas multi-scalestructures,whereBloch–Floquetwavesexhibitbehaviourassociatedwithlocallyhyperbolicdispersionsurfacesintheneighbourhoodofcertainresonances.Strongspatiallocalisationmayoccurwherewavepropagation ispermittedonlyalongdirectionsassociatedwiththeprincipalcurvaturesof thehyperbolicdispersionsurface.ImportantconnectionsbetweenthedispersionpropertiesofBloch–Floquetwavesinaperiodiclatticeandthesolutions offorcedproblemsarediscussedin[50].

ThedynamiclatticeGreen’sfunctiondescribesthevibrationofalattice withasingle-pointdefect,orappliedpointforce.Green’sfunctionshavebeen studiedin[107]fortwo-dimensionalsquarelatticesinpassbandregimes.The paper[131]addressedcontinuousanddiscretemodelsforexponentiallylocalisedwaveforms,withvariousforcingordefecttypes.Localisedwaveforms havebeenidentifiedforthecaseswhentheforcingfrequencyand/orthenaturalfrequencyofthedefectareplacedinthebandgap.Suchdefectmodes canbelinkedtothestop-bandGreen’sfunctions.

Usinganasymptoticapproach,thepapers[20, 68]consideredtheeffectof apre-stressonthepropagationofflexuralwavesthroughanelasticbeamon aWinklerfoundation.Specialattentionwasdevotedtodynamicallylocalised waveformsandtocontrolthepositionofbandgapsviapre-stress.Itwasfound thatatensilepre-stresscanincreasethefrequencyatwhichaparticularband gapoccurs.

Themonograph[178]presentsadetaileddiscussionofapplicationsfordynamiclatticeproblemsinvolvingcracksmodelledassemi-infinitefaults,advancingintwo-dimensionalperiodicelasticlattices.Forastructuredinterface andacrackpropagatingwithanaverageconstantspeedthroughasquare lattice,localisedmodeswereanalysedin[124].Thearticle[138]presented thestudyofasemi-infinitedynamiccrackinanon-uniformelasticlattice. Thecrackstabilitywasanalysedandaconnectionhasbeenestablishedbetweentheunstablecrackgrowthandtheenergyofwavesemanatingfromthe crack-tipinthesteady-stateregime.

Acomprehensivemathematicaltheorydevelopinganasymptoticanalysis offieldsinmulti-structuresispresentedinthemonograph[90],whereboundarylayersnearjunctionsareshowntohavespecialimportanceinbothstatic problemsaswellasinproblemsoftime-harmonicvibrations.

FIGURE1.4:Panel(a)showsaphotographofatypicalweldwhichexhibits thecharacteristiccolumnargrainalignments.Panel(b)isaplotofthegrain orientationsgeneratedbyascanningelectronmicroscopefromanactualweld. ImagescourtesyofAmecFosterWheelerplc.

1.6Modelsandphysicalapplicationsinmaterials science

Studiesofthepropagationofelasticwavesininhomogeneousand anisotropicmaterialsareoftenmotivatedbytherequirementforeffectiveultrasonicnon-destructiveexaminationofstronglyanisotropicsteelweldsand othercomponents.Inparticular,aninhomogeneitymayarisewhenthecastingorweldisformedandthemetalliccrystalsalignsuchthattheirsymmetry axesareparalleltothemaximalthermalgradient. Figure1.4 showsatypical weldandgrainmap,illustratingtheinhomogeneity.

Insomeapplications,ultrasonicprobesandsensorsareusedtodetect defectsinweldedstructures,andthisinturnmayrequiremeasurementsof thescatteredfields.Anumericalray-tracingmodelwasdevelopedin[146] toaccountfortheinhomogeneousandanisotropicnatureoftheweld.The paper[140]presentedaninvestigationintotheeffectofbeamdistortionin anisotropic,buthomogeneousmaterials.Furtherstudies[2, 186]suggested thattheweldmaybemodelledasamaterialwithconstantdensityandcrystallinemoduli,butwiththeorientationsofthecrystalssmoothlyvaryingwith position.Inaddition,thepaper[144]developedaraytheoryforelasticwaves propagatingthroughtheinhomogeneousandanisotropicmedium,withthe emphasisonthespecialcaseofSH-waves.

ApplicationsoftheBloch–Floquetwavetheorytothestudyofthedynamic localisationinweldshasbeenillustratedin[51],whichhassurpassedthetraditionalray-tracingapproachesandrevealednewfeatureslinkedtodispersion andlocalisationofelasticwavesinaweldwithagranularstructure.Inpar-

FIGURE1.5:Anillustrationoftheeffectofgrainstructureonthepathof ultrasonicwavesthroughgranularwelds.Hereweplotthemagnitudeofthe dilatation.

ticular,theauthorsof[51]developedananalyticalandcomputationalmodel capableofusingdatatakenfromactualwelds(suchasthegrainstructure shownin Figure1.4)toanalysethepropagationofultrasonicwavesthrough granularwelds. Figure1.5 showsanillustrativecomputationhighlightingthe localisationandcomplexwavebehaviourexhibitedinsuchwelds.

1.7Structureofthebook

Thistextisbasedonrecenttheoreticaladvancesleadingtomodellingof dynamicanisotropy,localisationanddesignofmulti-scaleheterogeneousmaterials,whichpossessunusualdynamicpropertiessuchasnegativerefraction, filtering,andfocussingofwavesbyflatinterfacesaswellasthecreatingof invisibilitycloaks.Somebackgroundmaterialandatheoreticalintroduction arealsoincludedinthebook.Someofthekeytopicspresentedhereinclude Green’sfunctionsandlocalisedwaveforms,dynamicresponseofdynamicelasticstructureswiththinligaments,dynamicanisotropyandlocalisationinlatticesystems.

Furthermore,thestate-of-the-artispresentedformodelsofhyperbolicand parabolicmodesanduni-directionalorstar-shapedwaveforms,disintegrating elasticsolidsandeffectivejunctionconditionsformulti-structuresindynamics.Multi-resonatorsystems,whichalsoincludetuneablethermo-elasticdynamicsystems,arestudiedusingasymptoticapproximations,combinedwith numericalsimulations.

Asymptoticanalysisisalsousedheretostudycontinuousanddiscrete media,dynamicinclusionsandcracksinlatticesversusBloch–Floquetwaves,

12 MathematicalModellingofWavesinMulti–ScaleStructuredMedia platoniccrystalsandlocaliseddefectmodes,chiralsystemsandmodelsof vibratinggyro-lattices,approximatecloakinginlatticestructures,aswellas scatteringreductioninstructuredplates.

Modellingfoundationsandmethodsofanalysisofwavesinmulti-scale mediaarediscussedin Chapter2 Chapter3 includesmodelsofwavesin so-called“disintegratingsolids”,whicharemulti-scalestructureswiththin ligamentsanddisintegratingjunctions.Dynamicanisotropy,wavelocalisation anddefectwaveformsinlatticestructuresarepresentedin Chapter4. Chapter 5 dealswiththetopicofcloakingviatheintroductionofspeciallydesigned multi-scaleheterogeneouscoatings.Finally,in Chapter6,thereaderwillfind modelsofstructuredinterfacesandspecialchiralmediadevelopedformultiscaleelasticsystems.

Thebookisaimedatawideaudience,includingappliedmathematicians, physicistsandengineers.Itmaybeespeciallyusefultoresearchstudentswho wouldliketostudydispersionandlocalisationofwavesinstructuredmedia.

Enjoyreading!

Chapter2

Foundations,methodsofanalysisof wavesandanalyticalapproachesto modellingofmulti-scalesolids

Thispreliminarychapterintroducessomefundamentalconceptsandmethodsrequiredfortheanalysisofwavepropagationinmulti-scalesolids.We beginin Section2.1 byintroducingtheconceptofdispersion,initiallyfor theclassicalexampleoflinearwaterwaves,thenfollowedbyone-dimensional mass-springchains.TheelementsofBloch–Floquettheoryforinfiniteperiodic mediaaresummarisedin Section2.2,followedbyadiscussionofasymptotic approximationsofhigh-contrastcontinuabydiscretelatticesin Section2.3. Wethenproceedtostudyaone-dimensionaltransmissionproblemwheretwo dissimilarlatticesarejoinedbya “structuredinterface” in Section2.4;we alsointroducethenotionofthetransmissionmatrix.AnimportantconnectionisestablishedbetweenformallydistinctBloch–Floquetproblemandthe transmissionproblem.

In Sections2.5 and 2.6 wemoveontoproblemsfordefectsinotherwise periodicmediaandanalysetheassociatedlocalisedwaves.Finally,in Section2.7,webrieflysummarisetheapproachofCrasteretal.[47, 55]forthe asymptotichomogenisationofperiodicmediaatfinitefrequencies,whichhas provenextremelyusefulintheanalysisofdynamicproblemsinmulti-scale media.

Thepresentchapterisintendedtoprovidetheuninitiatedreaderwitha conciseintroductiontotheprincipalmethodsrequiredforthematerialpresentedinlaterchapters.Theinterestedreaderisalsoreferredtothereferences providedforfurtherinformation.Incontrast,thereaderfamiliarwiththisarea ofresearchmayconsiderproceedingdirectlyto Chapter3

2.1Wavedispersion

Thisintroductorysectiondiscussessomeclassicalexamplesofdispersive waves,i.e.waveswhosespeedsaredifferentfordifferentfrequencies.Atthe beginningwerefertolinearwaterwavetheory,whichiswellknowninmodels

MathematicalModellingofWavesinMulti–ScaleStructuredMedia offluidflowasdiscussed,forexample,in[24].Anotherexample,includedin Section2.2,isaone-dimensionallatticesystem,knownfromclassicaltexts, suchas[85]and[26].Thedispersionrelations,thatistheequationsrelatingthe permissiblevaluesoffrequencyandwavenumber,arewritteninexplicitform andformationofastopband,i.e.rangesoffrequencywithwhichnowaves canpropagate,isdiscussedforheterogeneoussystems.Anasymptoticmodel isgivenforahigh-contraststratifiedstructurein Section2.2.Following[128] wediscussthelatticeapproximationofsuchasystem,whichiscapableof reproducingthedynamicresponseoftheheterogeneouscontinuumsystemin thelowfrequencyrange.

2.1.1Elementaryconsiderationsforlinearwaterwaves

Oneofthestraightforwardexamplesofdispersionisbasedonthelinear theoryofwavespropagatingalongthesurfaceofanincompressibleinviscid fluidofuniformdensity.Forconvenience,wecallsuchafluid“water”.The continuityequationandtheequationofmotionforthevelocity u andpressure p havetheform[3]

where F isthebodyforcedensity, ρ isthemassdensity,and t istime.In particular,ifthebodyforcerepresentsgravitywehave F = ge(3),and

where g isapositiveconstant,normalisedgravitationalacceleration,theupwardvertical x3-axisisorthogonaltotheunperturbedwatersurface,and e(3) istheunitvectoralongthisaxis.

Assumingthatthefluidflowisirrotationalandusingthenotation φ for thevelocitypotentialwehave u = ∇φ. Henceaccordingto(2.1)thefunction φ isharmonic

(2.4) withinthefluid.Thenon-lineartermin(2.2)becomes

Thus,(2.2)maybewrittenas

whichleadstoBernoulli’sequation

Freetopsurface

Water

p = p0 = const S2

Fixedbottomsurface

FIGURE2.1:Surfacewaterwavesinaflowoffinitedepth.

where f (t) isafunctionof t only.

S1

Theboundaryconditionsaresetatthebottomanduppersurfaces S1 and S2,respectively,asshownin Figure2.1.Itisassumedthatthebottomsurface S1 ofthechannelisfixedandhencethenormalcomponentofthevelocityis zero,thatis,

Itisalsoassumedthattheuppersurface S2 ischaracterisedbytheequation x3 = ζ(x1,x2,t), andthereisnovariationinpressureon S2: p = p0 = conston S2.

Hence,accordingto(2.6)onthefreesurfacewehave

Thisisaccompaniedbytheidentity

Inthelinearapproximation,wesimplifytheproblembyaddressingthecase when |u|,thesurfacefluctuations,andthederivatives ∂ζ/∂x1 and ∂ζ/∂x2 are allsmall.

Inparticular,iftheunperturbedsurfaceis x3 = h = const,thenforthe freesurfacewecanwrite

(x,t)= φ(x1,x2,ζ(x1,x2,t),t)= = φ(x1,x2,h,t)+(ζ h)

Totheleadingorderapproximation,therelations(2.8)and(2.9)areseton theunperturbedsurface x3 = h.Thesecond-orderterm 1 2 |∇φ|2 isneglected, andEquation(2.8)becomes

.

Sincetheterm p0 ρ f (t)+ gh isafunctionof t only,itdoesnotgive anycontributionto u,andmaybe“absorbed”into ∂φ ∂t .Thus,thevelocity potential φ canbechoseninsuchawaythat

Byneglectingthesecond-orderterms (∂φ/∂x1)(∂ζ/∂x1), (∂φ/∂x2)(∂ζ/∂x2) in(2.9),involvingthepartialderivativesof ζ and φ,wededucethattothe leadingorder

andthereforeEquations(2.10),(2.11)yield

Thus,inthe“ocean”withtheflatbottomsurface x3 =0 andtheunperturbeduppersurface x3 = h,thelinearisedformulationforthevelocity potential φ iscomprisedofLaplace’sequation(2.4)withinthefluidlayer 0 <x3 <h,theboundarycondition n ·∇φ =0 atthefixedbottomsurface x3 =0,andthe“freesurface”boundarycondition(2.12).Notethatthisformulationinvolvesthesecond-ordertimederivativeofthevelocitypotential ontheupperfreesurface.Inturn,afterevaluationof φ,the“waveprofile”on thefreeuppersurfaceisdefinedby(2.10).

2.1.2Dispersionequation

Consideratime-harmonicwaterwaveofradianfrequency ω,withthe straightfrontperpendiculartothe x1 axis.Weseeksolutionsindependentof x2 withthevelocitypotential

Itisconvenienttoworkwiththecomplexpotential Ψ oftheform Ψ= W (x3)e i(ωt kx1) , andassumethat φ =ReΨ.Consideringaharmonicsolution(i.e. ∇2Ψ=0) weobtain W k2W =0,

whichyields W = C1 cosh(kx3)+ C2 sinh(kx3).

Theboundaryconditionatthebottomsurface(x3 =0)leadsto W (0)=0, andhence C2 =0 and

Inturn,thefreesurfacecondition

whichsimplifiestotheform

Equation(2.16)relatesthewavenumber k andtheradianfrequency ω, andthusthe[phase]wavespeed c = ω k isingeneralfrequencydependent. Thisshowsthatthewatersurfacewavesare dispersive.Therelation(2.16)is calledthe dispersionequation

2.1.2.1Asymptotics:deepandshallowwaterwaves Wewouldliketonotetwoimportantparticularcases,whichare

(a)Deepwater,whenthenon-dimensionalquantity kh islarge(kh 1), andhence tanh(kh) 1.Thustoleadingorderthedispersionequation (2.16)yields ω 2 = gk.

Inthiscase,thewavespeedincreaseswhenthefrequencydecreases, i.e. c = gω 1 , whichisfullyconsistentwiththephysicalobservation thatlowfrequencywavesintheoceanpropagatefasterthanthehigh frequencyripples.

(b)Shallowwater,when kh 1,andhence tanh(kh) ∼ kh.Inthiscase,to leadingorderthedispersionequation(2.16)takestheform

ω 2 = gk2h, whichimplies c = √gh. Intheframeworkofthisapproximation,the wavespeedinshallowwaterisfrequency-independentandhencethe shallowwaterwavescanbetreatedasnon-dispersive.

FIGURE2.2:Periodicchainofspringsandmasses.

Itisnotedthatthedispersionofsurfacewaterwavesintheaboveillustrativeexampleislinkedtotheboundarycondition(2.12),whichinvolvesthe second-ordertimederivativeofthevelocitypotential.

Thephenomenonofwavedispersioncanalsobeobservedinsimpleperiodicstructures.Conventionally,dispersiondiagramsareusedtodescribethe changeinfrequencywiththewavenumber,whichwillbeillustratedinthe nextsection.

2.2Bloch–Floquetwaves

ABloch–Floquetwave(sometimesreferredtoasaBlochwaveoraFloquet wave)inaperiodicsystemisanobjectdiscussedinclassicaltextbooks,such as[85]and[26].Hereweshowelementaryexamplesreferringtothetimeharmonicmotionofaone-dimensionalperiodiclattice,consistingofpoint masses(rigidparticles)connectedbymasslesssprings.Thesketchofsucha systemisshownin Figure2.2.

Westartbyconsideringamass-springchainwhereallparticleshavethe samemass, m,andareconnectedtotheirnearestneighboursbyspringsof stiffness µ andlength b

Let un bethelongitudinaldisplacementfromequilibriumofthe nth node withinthechain,withtheinteratomicspacing b.Thentheequationsofmotion taketheform

with n beinginteger.(2.17) Ifthemotionistimeharmonicthen un = Un exp( iωt), andtherefore

Inthecaseoftravellingwaves,wehave

andtheequationofmotionbecomes

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CHAPTER VII.

ARMY RATIONS: WHAT THEY WERE.—HOW THEY WERE DISTRIBUTED.—HOW THEY WERE COOKED.

“Here’s a pretty mess!”

T M.

“God bless the pudding, God bless the meat, God bless us all; Sit down and eat.”

A H S’ B, 1796.

“Fall in for your rations, Company A!” My theme is Army Rations. And while what I have to say on this subject may be applicable to all

of the armies of the Union in large measure, yet, as they did not fare just alike, I will say, once for all, that my descriptions of army life pertain, when not otherwise specified, especially to that life as it was lived in the Army of the Potomac.

In beginning, I wish to say that a false impression has obtained more or less currency both with regard to the quantity and quality of the food furnished the soldiers. I have been asked a great many times whether I always got enough to eat in the army, and have surprised inquirers by answering in the affirmative. Now, some old soldier may say who sees my reply, “Well, you were lucky. I didn’t.” But I should at once ask him to tell me for how long a time his regiment was ever without food of some kind. Of course, I am not now referring to our prisoners of war, who starved by the thousands. And I should be very much surprised if he should say more than twenty-four or thirty hours, at the outside. I would grant that he himself might, perhaps, have been so situated as to be deprived of food a longer time, possibly when he was on an exposed picket post, or serving as rear-guard to the army, or doing something which separated him temporarily from his company; but his case would be the exception and not the rule. Sometimes, when active operations were in progress, the army was compelled to wait a few hours for its trains to come up, but no general hardship to the men ever ensued on this account. Such a contingency was usually known some time in advance, and the men would husband their last issue of rations, or, perhaps, if the country admitted, would make additions to their bill of fare in the shape of poultry or pork;—usually it was the latter, for the Southerners do not pen up their swine as do the Northerners, but let them go wandering about, getting their living much of the time as best they can. This led some one to say jocosely, with no disrespect intended to the people however, “that every other person one meets on a Southern street is a hog.” They certainly were quite abundant, and are to-day, in some form, the chief meat food of that section. But on the point of scarcity of rations I believe my statement will be generally agreed to by old soldiers.

THE “COOPER SHOP,” PHILADELPHIA.

Now, as to the quality the case is not quite so clear, but still the picture has been often overdrawn. There were, it is true, large quantities of stale beef or salt horse—as the men were wont to call it —served out, and also rusty, unwholesome pork; and I presume the word “hardtack” suggests to the uninitiated a piece of petrified bread honeycombed with bugs and maggots, so much has this article of army diet been reviled by soldier and civilian. Indeed, it is a rare occurrence for a soldier to allude to it, even at this late day, without some reference to its hardness, the date of its manufacture, or its propensity for travel. But in spite of these unwholesome rations, whose existence no one calls in question, of which I have seen—I must not say eaten—large quantities, I think the government did well, under the circumstances, to furnish the soldiers with so good a quality of food as they averaged to receive. Unwholesome rations were not the rule, they were the exception, and it was not the fault of the government that these were furnished, but very often the intent of the rascally, thieving contractors who supplied them, for which

they received the price of good rations; or, perhaps, of the inspectors, who were in league with the contractors, and who therefore did not always do their duty. No language can be too strong to express the contempt every patriotic man, woman, and child must feel for such small-souled creatures, many of whom are to-day rolling in the riches acquired in this way and other ways equally disreputable and dishonorable.

I will now give a complete list of the rations served out to the rank and file, as I remember them. They were salt pork, fresh beef, salt beef, rarely ham or bacon, hard bread, soft bread, potatoes, an occasional onion, flour, beans, split pease, rice, dried apples, dried peaches, desiccated vegetables, coffee, tea, sugar, molasses, vinegar, candles, soap, pepper, and salt.

It is scarcely necessary to state that these were not all served out at one time. There was but one kind of meat served at once, and this, to use a Hibernianism, was usually pork. When it was hard bread, it wasn’t soft bread or flour, and when it was pease or beans it wasn’t rice.

THE UNION VOLUNTEER SALOON, PHILADELPHIA.

Here is just what a single ration comprised, that is, what a soldier was entitled to have in one day. He should have had twelve ounces of pork or bacon, or one pound four ounces of salt or fresh beef; one pound six ounces of soft bread or flour, or one pound of hard bread, or one pound four ounces of corn meal. With every hundred such rations there should have been distributed one peck of beans or pease; ten pounds of rice or hominy; ten pounds of green coffee, or eight pounds of roasted and ground, or one pound eight ounces of tea; fifteen pounds of sugar; one pound four ounces of candles; four pounds of soap; two quarts of salt; four quarts of vinegar; four ounces of pepper; a half bushel of potatoes when practicable, and one quart of molasses. Desiccated potatoes or desiccated compressed vegetables might be substituted for the beans, pease, rice, hominy, or fresh potatoes. Vegetables, the dried fruits, pickles, and pickled cabbage were occasionally issued to prevent scurvy, but in small quantities.

But the ration thus indicated was a camp ration. Here is the marching ration: one pound of hard bread; three-fourths of a pound of salt pork, or one and one-fourth pounds of fresh meat; sugar, coffee, and salt. The beans, rice, soap, candles, etc., were not issued to the soldier when on the march, as he could not carry them; but, singularly enough, as it seems to me, unless the troops went into camp before the end of the month, where a regular depot of supplies might be established from which the other parts of the rations could be issued, they were forfeited, and reverted to the government—an injustice to the rank and file, who, through no fault of their own, were thus cut off from a part of their allowance at the time when they were giving most liberally of their strength and perhaps of their very heart’s blood. It was possible for company commanders and for no one else to receive the equivalent of these missing parts of the ration in cash from the brigade commissary, with the expectation that when thus received it would be distributed among the rank and file to whom it belonged. Many officers did not care to trouble themselves with it, but many others did, and—forgot to pay it out afterwards. I have yet to learn of the first company whose members ever received any revenue from such a source,

although the name of Company Fund is a familiar one to every veteran.

The commissioned officers fared better in camp than the enlisted men. Instead of drawing rations after the manner of the latter, they had a certain cash allowance, according to rank, with which to purchase supplies from the Brigade Commissary, an official whose province was to keep stores on sale for their convenience. The monthly allowance of officers in infantry, including servants, was as follows: Colonel, six rations worth $56, and two servants; LieutenantColonel, five rations worth $45, and two servants; Major, four rations worth $36, and two servants; Captain, four rations worth $36, and one servant; First and Second Lieutenants, jointly, the same as Captains. In addition to the above, the field officers had an allowance of horses and forage proportioned to their rank.

A BRIGADE COMMISSARY AT BRANDY STATION, VA

I will speak of the rations more in detail, beginning with the hard bread, or, to use the name by which it was known in the Army of the Potomac, Hardtack. What was hardtack? It was a plain flour-andwater biscuit. Two which I have in my possession as mementos measure three and one-eighth by two and seven-eighths inches, and are nearly half an inch thick. Although these biscuits were furnished to organizations by weight, they were dealt out to the men by number, nine constituting a ration in some regiments, and ten in

others; but there were usually enough for those who wanted more, as some men would not draw them. While hardtack was nutritious, yet a hungry man could eat his ten in a short time and still be hungry. When they were poor and fit objects for the soldiers’ wrath, it was due to one of three conditions: First, they may have been so hard that they could not be bitten; it then required a very strong blow of the fist to break them. The cause of this hardness it would be difficult for one not an expert to determine. This variety certainly well deserved their name. They could not be soaked soft, but after a time took on the elasticity of gutta-percha.

A HARD-TACK FULL SIZE.

The second condition was when they were mouldy or wet, as sometimes happened, and should not have been given to the soldiers. I think this condition was often due to their having been boxed up too soon after baking. It certainly was frequently due to exposure to the weather. It was no uncommon sight to see thousands of boxes of hard bread piled up at some railway station or

other place used as a base of supplies, where they were only imperfectly sheltered from the weather, and too often not sheltered at all. The failure of inspectors to do their full duty was one reason that so many of this sort reached the rank and file of the service.

The third condition was when from storage they had become infested with maggots and weevils. These weevils were, in my experience, more abundant than the maggots. They were a little, slim, brown bug an eighth of an inch in length, and were great bores on a small scale, having the ability to completely riddle the hardtack. I believe they never interfered with the hardest variety.

When the bread was mouldy or moist, it was thrown away and made good at the next drawing, so that the men were not the losers; but in the case of its being infested with the weevils, they had to stand it as a rule; for the biscuits had to be pretty thoroughly alive, and well covered with the webs which these creatures left, to insure condemnation. An exception occurs to me. Two cargoes of hard bread came to City Point, and on being examined by an inspector were found to be infested with weevils. This fact was brought to Grant’s attention, who would not allow it landed, greatly to the discomfiture of the contractor, who had been attempting to bulldoze the inspector to pass it.

The quartermasters did not always take as active an interest in righting such matters as they should have done; and when the men growled at them, of course they were virtuously indignant and prompt to shift the responsibility to the next higher power, and so it passed on until the real culprit could not be found.

But hardtack was not so bad an article of food, even when traversed by insects, as may be supposed. Eaten in the dark, no one could tell the difference between it and hardtack that was untenanted. It was no uncommon occurrence for a man to find the surface of his pot of coffee swimming with weevils, after breaking up hardtack in it, which had come out of the fragments only to drown; but they were easily skimmed off, and left no distinctive flavor behind. If a soldier cared to do so, he could expel the weevils by heating the bread at the fire. The maggots did not budge in that way.

The most of the hard bread was made in Baltimore, and put up in boxes of sixty pounds gross, fifty pounds net; and it is said that some of the storehouses in which it was kept would swarm with weevils in an incredibly short time after the first box was infested with them, so rapidly did these pests multiply.

A BOX OF HARDTACK.

Having gone so far, I know the reader will be interested to learn of the styles in which this particular article was served up by the soldiers. I say styles because I think there must have been at least a score of ways adopted to make this simple flour tile more edible. Of course, many of them were eaten just as they were received— hardtack plain; then I have already spoken of their being crumbed in coffee, giving the “hardtack and coffee.” Probably more were eaten in this way than in any other, for they thus frequently furnished the soldier his breakfast and supper. But there were other and more appetizing ways of preparing them. Many of the soldiers, partly through a slight taste for the business but more from force of circumstances, became in their way and opinion experts in the art of cooking the greatest variety of dishes with the smallest amount of capital.

FRYING HARDTACK

Some of these crumbed them in soups for want of other thickening. For this purpose they served very well. Some crumbed them in cold water, then fried the crumbs in the juice and fat of meat. A dish akin to this one, which was said to “make the hair curl,” and certainly was indigestible enough to satisfy the cravings of the most ambitious dyspeptic, was prepared by soaking hardtack in cold water, then frying them brown in pork fat, salting to taste. Another name for this dish was “skillygalee.” Some liked them toasted, either to crumb in coffee, or, if a sutler was at hand whom they could patronize, to butter. The toasting generally took place from the end of a split stick, and if perchance they dropped out of it into the campfire, and were not recovered quickly enough to prevent them from getting pretty well charred, they were not thrown away on that account, being then thought good for weak bowels.

Then they worked into milk-toast made of condensed milk at seventy-five cents a can; but only a recruit with a big bounty, or an old vet the child of wealthy parents, or a re-enlisted man did much in that way. A few who succeeded by hook or by crook in saving up a portion of their sugar ration spread it upon hardtack. The hodge-

podge of lobscouse also contained this edible among its divers other ingredients; and so in various ways the ingenuity of the men was taxed to make this plainest and commonest yet most serviceable of army food to do duty in every conceivable combination. There is an old song, entitled “Hard Times,” which some one in the army parodied. I do not remember the verses, but the men used to sing the following chorus:—

’Tis the song of the soldier, weary, hungry, and faint, Hardtack, hardtack, come again no more; Many days have I chewed you and uttered no complaint, O Greenbacks, come again once more!

It is possible at least that this song, sung by the soldiers of the Army of the Potomac, was an outgrowth of the following circumstance and song. I am quite sure, however, that the verses were different.

For some weeks before the battle of Wilson’s Creek, Mo., where the lamented Lyon fell, the First Iowa Regiment had been supplied with a very poor quality of hard bread (they were not then (1861) called hardtack). During this period of hardship to the regiment, so the story goes, one of its members was inspired to produce the following touching lamentation:—

Let us close our game of poker, Take our tin cups in our hand, While we gather round the cook’s tent door, Where dry mummies of hard crackers Are given to each man; O hard crackers, come again no more!

C:—’Tis the song and sigh of the hungry, “Hard crackers, hard crackers, come again no more! Many days have you lingered upon our stomachs sore, O hard crackers, come again no more!”

There’s a hungry, thirsty soldier

Who wears his life away, With torn clothes, whose better days are o’er; He is sighing now for whiskey, And, with throat as dry as hay, Sings, “Hard crackers, come again no more!”—C.

’Tis the song that is uttered In camp by night and day, ’Tis the wail that is mingled with each snore, ’Tis the sighing of the soul For spring chickens far away, “O hard crackers, come again no more!”—C.

When General Lyon heard the men singing these stanzas in their tents, he is said to have been moved by them to the extent of ordering the cook to serve up corn-meal mush, for a change, when the song received the following alteration:—

But to groans and to murmurs There has come a sudden hush, Our frail forms are fainting at the door; We are starving now on horse-feed That the cooks call mush, O hard crackers, come again once more!

C:—It is the dying wail of the starving, Hard crackers, hard crackers, come again once more; You were old and very wormy, but we pass your failings o’er.

O hard crackers, come again once more!

The name hardtack seems not to have been in general use among the men in the Western armies.

But I now pass to consider the other bread ration—the loaf or soft bread. Early in the war the ration of flour was served out to the men uncooked; but as the eighteen ounces allowed by the government more than met the needs of the troops, who at that time obtained

much of their living from outside sources (to be spoken of hereafter), it was allowed, as they innocently supposed, to be sold for the benefit of the Company Fund, already referred to. Some organizations drew, on requisition, ovens, semi-cylindrical in form, which were properly set in stone, and in these regimental cooks or bakers baked bread for the regiment. But all of this was in the tentative period of the war. As rapidly as the needs of the troops pressed home to the government, they were met with such despatch and efficiency as circumstances would permit. For a time, in 1861, the vaults under the broad terrace on the western front of the Capitol were converted into bakeries, where sixteen thousand loaves of bread were baked daily. The chimneys from the ovens pierced the terrace where now the freestone pavement joins the grassy slope, and for months smoke poured out of these in dense black volumes. The greater part of the loaves supplied to the Army of the Potomac up to the summer of 1864 were baked in Washington, Alexandria, and at Fort Monroe, Virginia. The ovens at the latter place had a capacity of thirty thousand loaves a day. But even with all these sources worked to their uttermost, brigade commissaries were obliged to set up ovens near their respective depots, to eke out enough bread to fill orders. These were erected on the sheltered side of a hill or woods, then enclosed in a stockade, and the whole covered with old canvas.

AN ARMY OVEN.

When the army reached the vicinity of Petersburg, the supply of fresh loaves became a matter of greater difficulty and delay, which Grant immediately obviated by ordering ovens built at City Point. A large number of citizen bakers were employed to run them night and day, and as a result one hundred and twenty-three thousand fresh loaves were furnished the army daily from this single source; and so closely did the delivery of these follow upon the manipulations of the bakers that the soldiers quite frequently received them while yet warm from the oven. Soft bread was always a very welcome change from hard bread; yet, on the other hand, I think the soldiers tired sooner of the former than of the latter. Men who had followed the sea preferred the hard bread. Jeffersonville, in Southern Indiana, was the headquarters from which bread was largely supplied to the Western armies.

SOFT BREAD.

Commissary Department, Headquarters Army of the Potomac, Captain J R Coxe

I began my description of the rations with the bread as being the most important one to the soldier. Some old veterans may be disposed to question the judgment which gives it this rank, and claim that coffee, of which I shall speak next, should take first place in important: in reply to which I will simply say that he is wrong,

because coffee, being a stimulant, serves only a temporary purpose, while the bread has nearly or quite all the elements of nutrition necessary to build up the wasted tissues of the body, thus conferring a permanent benefit. Whatever words of condemnation or criticism may have been bestowed on other government rations, there was but one opinion of the coffee which was served out, and that was of unqualified approval.

APPORTIONING COFFEE AND SUGAR.

The rations may have been small, the commissary or quartermaster may have given us a short allowance, but what we got was good. And what a perfect Godsend it seemed to us at times! How often, after being completely jaded by a night march,—and this is an experience common to thousands,—have I had a wash, if there was water to be had, made and drunk my pint or so of coffee, and felt as fresh and invigorated as if just arisen from a night’s sound sleep! At such times it could seem to have had no substitute.

It would have interested a civilian to observe the manner in which this ration was served out when the army was in active service. It was usually brought to camp in an oat-sack, a regimental quartermaster receiving and apportioning his among the ten companies, and the quartermaster-sergeant of a battery apportioning his to the four or six detachments. Then the orderly-sergeant of a

company or the sergeant of a detachment must devote himself to dividing it. One method of accomplishing this purpose was to spread a rubber blanket on the ground,—more than one if the company was large,—and upon it were put as many piles of the coffee as there were men to receive rations; and the care taken to make the piles of the same size to the eye, to keep the men from growling, would remind one of a country physician making his powders, taking a little from one pile and adding to another. The sugar which always accompanied the coffee was spooned out at the same time on another blanket. When both were ready, they were given out, each man taking a pile, or, in some companies, to prevent any charge of unfairness or injustice, the sergeant would turn his back on the rations, and take out his roll of the company. Then, by request, some one else would point to a pile and ask, “Who shall have this?” and the sergeant, without turning, would call a name from his list of the company or detachment, and the person thus called would appropriate the pile specified. This process would be continued until the last pile was disposed of. There were other plans for distributing the rations; but I have described this one because of its being quite common.

The manner in which each man disposed of his coffee and sugar ration after receiving it is worth noting. Every soldier of a month’s experience in campaigning was provided with some sort of bag into which he spooned his coffee; but the kind of bag he used indicated pretty accurately, in a general way, the length of time he had been in the service. For example, a raw recruit just arrived would take it up in a paper, and stow it away in that well known receptacle for all eatables, the soldier’s haversack, only to find it a part of a general mixture of hardtack, salt pork, pepper, salt, knife, fork, spoon, sugar, and coffee by the time the next halt was made. A recruit of longer standing, who had been through this experience and had begun to feel his wisdom-teeth coming, would take his up in a bag made of a scrap of rubber blanket or a poncho; but after a few days carrying the rubber would peel off or the paint of the poncho would rub off from contact with the greasy pork or boiled meat ration which was its travelling companion, and make a black, dirty mess, besides leaving the coffee-bag unfit for further use. Now and then some young

soldier, a little starchier than his fellows, would bring out an oil-silk bag lined with cloth, which his mother had made and sent him; but even oil-silk couldn’t stand everything, certainly not the peculiar inside furnishings of the average soldier’s haversack, so it too was not long in yielding. But your plain, straightforward old veteran, who had shed all his poetry and romance, if he had ever possessed any, who had roughed it up and down “Old Virginny,” man and boy, for many months, and who had tried all plans under all circumstances, took out an oblong plain cloth bag, which looked as immaculate as the every-day shirt of a coal-heaver, and into it scooped without ceremony both his sugar and coffee, and stirred them thoroughly together.

There was method in this plan. He had learned from a hard experience that his sugar was a better investment thus disposed of than in any other way; for on several occasions he had eaten it with his hardtack a little at a time, had got it wet and melted in a rain, or, what happened fully as often, had sweetened his coffee to his taste when the sugar was kept separate, and in consequence had several messes of coffee to drink without sweetening, which was not to his taste. There was now and then a man who could keep the two separate, sometimes in different ends of the same bag, and serve them up proportionally. The reader already knows that milk was a luxury in the army. It was a new experience for all soldiers to drink coffee without milk. But they soon learned to make a virtue of a necessity, and I doubt whether one man in ten, before the war closed, would have used the lactic fluid in his coffee from choice. Condensed milk of two brands, the Lewis and Borden, was to be had at the sutler’s when sutlers were handy, and occasionally milk was brought in from the udders of stray cows, the men milking them into their canteens; but this was early in the war. Later, war-swept Virginia afforded very few of these brutes, for they were regarded by the armies as more valuable for beef than for milking purposes, and only those survived that were kept apart from lines of march. In many instances they were the chief reliance of Southern families, whose able-bodied men were in the Rebel army, serving both as a source of nourishment and as beasts of burden.

THE MILK RATION.

When the army was in settled camp, company cooks generally prepared the rations. These cooks were men selected from the company, who had a taste or an ambition for the business. If there were none such, turns were taken at it; but this did not often happen, as the office excused men from all other duty.

When company cooks prepared the food, the soldiers, at the bugle signal, formed single file at the cook-house door, in winter, or the cook’s open fire, in summer, where, with a long-handled dipper, he filled each man’s tin with coffee from the mess kettles, and dispensed to him such other food as was to be given out at that meal.

THE COMPANY COOK.

For various reasons, some of which I have previously hinted at, the coffee made by these cooks was of a very inferior quality and unpleasant to taste at times. It was not to be compared in excellence with what the men made for themselves. I think that when the soldiers were first thrown upon their own resources to prepare their food, they almost invariably cooked their coffee in the tin dipper with which all were provided, holding from a pint to a quart, perhaps. But it was an unfortunate dish for the purpose, forever tipping over and spilling the coffee into the fire, either because the coals burned away beneath, or because the Jonah upset it. Then if the fire was new and blazing, it sometimes needed a hand that could stand heat like a steam safe to get it when it was wanted, with the chance in favor of more than half of the coffee boiling out before it was rescued, all of which was conducive to ill-temper, so that such utensils would soon disappear, and a recruit would afterwards be seen with his pint or quart preserve can, its improvised wire bail held on the end of a stick, boiling his coffee at the camp-fire, happy in the security of his ration from Jonahs and other casualties. His can soon became as black as the blackest, inside and out. This was the typical coffeeboiler of the private soldier, and had the advantage of being easily replaced when lost, as canned goods were in very general use by

commissioned officers and hospitals. Besides this, each man was generally supplied with a small tin cup as a drinking-cup for his coffee and water.

GOING INTO CAMP.

The coffee ration was most heartily appreciated by the soldier. When tired and foot-sore, he would drop out of the marching column, build his little camp-fire, cook his mess of coffee, take a nap behind the nearest shelter, and, when he woke, hurry on to overtake his company. Such men were sometimes called stragglers; but it could, obviously, have no offensive meaning when applied to them. Tea was served so rarely that it does not merit any particular description. In the latter part of the war, it was rarely seen outside of hospitals.

One of the most interesting scenes presented in army life took place at night when the army was on the point of bivouacking. As soon as this fact became known along the column, each man would seize a rail from the nearest fence, and with this additional arm on the shoulder would enter the proposed camping-ground. In no more time than it takes to tell the story, the little camp-fires, rapidly increasing to hundreds in number, would shoot up along the hills and

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