CORNELLUNIVERSITYLIBRARYBOUGHTWITHTHEINCOMEOFTHESAGEENDOWMENTrUND GIVEN IN 189I BYHENRY WILLIAMS SAGEMATHlSMATICSCornellUniversityLibraryQC321.C321921Introductiontothe mathematicaltheory3 1924 001 599 806Theoriginal oftiiis bookis intine Cornell University Library.Therearenoknowncopyrightrestrictions intheUnitedStatesontheuseofthetext.http://www.archive.org/details/cu31924001599806ONTOTHEMATHEMATICALTHE:!ONDUCTION OPHEAT IN SOLIDMACMILLAN AND CO., LimitedLONDON BOMBAYCALCUTTA MADRASMELBOURNETHE MACMILLAN COMPANYNEW YORK BOSTON CHICAGODALLAS SAN FRANCISCOTHEMACMILLAN CO. OF CANADA, Ltd.TORONTOINTRODUCTION TO THEMATHEMATICALTHEORYOFTHECONDUCTIONOFHEATIN SOLIDSBYH. S! CARSLAWScD.(Cambridge), D.Sc.(Glasgow), F.R.S.E.PROFESSOR OF MATHEMATICS IN THEUNIVERSITYOFSYDNEVFORMERLYFELLOWOF EMMANUELCOLLEGE, CAMBRIDGE, ANDLECTURER IN MATHEMATICSINTHEUNIVERSITYOFGLASGOWSECONDEDITION, COMPLETELYREVISEDMACMILLAN AND CO., LIMITEDST. MARTIN'S STREET,LONDON1921COPYRIGHTFirst Published, 1906.PRINTED IN GREAT BRITAINPEEFACEThisvolumecomplekssthe newedition of inybookonFourier'sSeries andIntegrals andthe Mathematical Theoryofthe Conductionof Heat. Theoriginalworkwasfirstpublishedin1906andhasnowfor some time been out of print. The first volume of the newedition appeared towards the middle of this year, and deals withthe Theory of Infinite Series and Integrals, with special referenceto Fourier's Series and Integrals. This formed a completelynewwork with the title Fourier's Series and Integrals. The secondvolume is devoted wholly to the Mathematical Theory of theConductionofHeatin Solids. Thispartof thebookhas alsobeencompletelyrewrittenandmuchenlarged. It nowincludesadiscus-sion of all the important boundary problems associated with theEquation of Conduction. The treatment of these questions,especiallyinthelaterchapters,shouldbeof usetothoseinterestedinthe applicationof modernanalysistothesolutionof thediffer-ential equationsofmathematicalphysics.InChapterI. theDifferentialEquationofConductionis obtainedand somegeneral theoremsastoitssolutionareestablished. ChapterII. deals withFourier's Eing. Thenexttwochapters aredevotedtoLinearFlow. Theprincipalchangesmadeinthesechaptersarecpnnected-withthemoreexacttreatmentoftheInfinite SeriesandIntegrals which enter into the solutions. Chapters V. andVI.,which deal with Two-Dimensional Problems andFlow of Heat ina Kectangular Parallelepiped, differ little from the correspondingchapters in the first edition.ChapterVII.dealswiththeCircularCylinder,ChapterVIII.withthe Sphere and Cone, Chapter IX. with Sources and Sinks, andChapterX.withGreen'sFunctions. Thesechapterscontainmuchadditional matter.VI PREFACEChapters XI. and XII. are quite new. The former is entitled"TheUseof ContourIntegrals inthe Solution of theEquation ofConduction." Bromwich's recent work has directed attentiontothe"operationalmethod"ofHeaviside. It is truethat all thequestionsexaminedinthischaptercouldbesolvedbythatmethod.Butto justify the operationalmethodwemustrelyuponcontourintegration, andthechief differencebetweenthemethoddevelopedbyme, asillustratedinthischapter, andtheoperationalmethodisthatIpreferineachcasetoturntothestandardpathintheplaneof the complex variable instead of using^kind of mathematicalshorthand.Inthe last chapterChapterXII.a sketch is given of the useofIntegralEquationsinthesolutionoftheEquationofConduction.Thissecondvolumecouldnothaveappearedsosoonafterthefirsthad I not been privileged to spend this year on leave of absencefrom the University of Sydney in myold College at Cambridge.ForthefacihtiessofuUygrantedto methereItakethisopportunityof expressingmyheartfeltthanks.EmmanuelCollege,Cambridge, October, 1921.CONTENTSTHEMATHEMATICALTHEORYOFTHE CONDUCTION OFHEATINSOLIDSCHAPTERITHE DIFFERENTIAL EQUATION OF THE MATHEMATICAL THEORYOFTHE.CONDUCTIONOFHEATSECTIONPAGE11. Introductory2. Conductivityj3. FlowofHeatacrossanIsothermalSurface-4. 4. FlowofHeatacrossanySurfaceg5. TheEquationofConduction76. TheTransformationofCoordinatesjq7. InitialandBoundaryConditions-128. TheSolutionisUnique149. SimplificationoftheProblemIgCHAPTERIIFOURIER'SRING10. Introductory2011. TheEquationofConductionintheRing2012. VariableTemperature-2213. SteadyTemperature2514. Neumann'sRingMethodfor Determiningthe ConductivityandEmissivity '26CONTENTSCHAPTERIIILINEARFLOWOPHEAT. INFINITEANDSEffl-INFINITESOLIDANDBODi'KGE15. Introductory2916-17. TheInfiniteSoUd2918. TheSemi-InfiniteSoUd - 3319. TheInfiniteorSemi-InfiniteRod - - 3820. ConductivityExperimentsuponBars. SteadyTemperature 39.21-22. Conductivity Experiments upon Bars. Variable Temperature-4123. Semi-Infinite Solid. Initial Temperature Zero. Surface Tempera-tureaFunctionoftheTime - 4624. Semi-Infinite Solid. Sxirface TemperatureaHarmonicFunctionoftheTime -4725. Semi-Infinite Solid. Initial Temperature Constant. RadiationatSurfaceintoMediumatZero...5026. Semi-Infinite Solid. Initial Temperature Zero. Radiation atSurface into Medium whose Temperature is a FunctionoftheTime - -5327. TerrestrialTemperature5328. TheAgeoftheEarth-57CHAPTERIVLINEARFLOWOFHEAT. SOLIDBOUNDEDBYTWOPARALLELPLANES. FINITEROD.29. Introductory- --6130-31. Finite Rod. Initial Temperaturefix).Ends at Zero. NoRadiation -6132. FiniteRod. EndsatFixedTemperatures. Radiation.SteadyTemperature - - -6633. FiniteRod. EndsatFixedTemperatures. InitialTemperaturef{x).NoRadiation.....g^34. Finite Rod. Ends at Temperaturesi{t)and(fiiit). InitialTemperature/(.r). NoRadiation - .gS35. Neumann's Bar Method for Determining the ConductivityandEmissivity-" -70CONTENTS ixSECTION KAGB36. Finite Kod. Radiation at Ends into Medium at Zero. InitialTemperaturef{x).NoRadiationat Surface 7437. Application of this Solution to the Determination of the Con-ductivityandEmissivity - - 7938. EquationofConductioninaWireheatedbyaConstantElectricCurrent 8139-40. DeterminationofBlpctrioalandThermalConductivities-83CHAPTEKVtwo-dimbnsionaIjproblems41. Introductory - - 8842-43. InfiniteRectangularSolid. SteadyTemperature - 8944-45. The Use of Conjugate Eunctions in Problems of SteadyTemperature.... 9146. SourcesandSinksinSteadyTemperature 10247. VariableTemperature..., 103CHAPTERVIFLOWOFHEATINARECTANGULARPARALLELEPIPED48. Introductory10549-50. SteadyTemperature-10551-52. VariableTemperature108CHAPTERVIIFLOWOFHEATINACIRCULARCYLINDER53. Introductory11364. InfiniteCylinder. SteadyTemperature 11455. InfiniteCylinder. VariableTemperature 11466. The Integrals [" rJ{ar)J{^r)dr and/rJ^{ar)dr 116JoJO57. ApplicationsoftheseIntegrals11758. Semi-InfiniteCylinder. SteadyTemperature 12159. Semi-InfiniteCylinder. VariableTemperature 12260. FiniteCylinder. InitialTemperaturef(r, 0,z). SurfaceatZero 124CONTENTSPAGE61. FiniteCylinder. Radiation -- 12562. GeneralProblemsontheCylinder- 12763. DeterminationoftheConductivityfromCylinders131CHAPTERVIIIFLOWOFHEATINASPHEREANDCONE64. Introductory-13565. Radiation at the Surface of a Sphere into Medium at Zero.InitialTemperature/(r).-13666. DeterminationoftheConductivity andEmissivityfromSpheres-14067. Surfaceof theSphere keptatZero. InitialTemperature/(r,d, 4>)14168. Solid bounded by a Sphere and Cone. Initial Temperature/{',d>4>)-SizrfaceatZero- -14469. TheCone. InitialTemperaturef{r, 6, (j)).SurfaceatZero 147CHAPTERIXTHEUSEOPSOURCESANDSINKSINCASESOFVARIABLETEMPERATURE70. InstantaneousPointSource -14971. SphericalSurfaceSource - - 15172. InstantaneousLineSource 15273. InstantaneousPlaneSource15374. Doublets -15675-78. TheMethodofImages15879. Sommerfeld'sExtensionoftheMethodofImages - 166CHAPTERXTHEUSEOFGREEN'SFUNCTIONSINTHESOLUTIONOFTHEEQUATIONOFCONDUCTION80.Introductory - -.X6981. Linear Flow. Semi-Infinite Solid bounded by x=0 at 4>{t).InitialTemperature f(x)-.17282. TheSameSolid. Sourceatx' att=Q. RadiationintoMediumatZero - -- -. 173CONTENTSxi83. The Same Solid. Radiation into Mediumatcjb().Initial Tempera-ture/(a;)17684. Finite Solid. Source at x' at =0. Boundariesa;=0 and x^aatZeroI7785. TheSameSolid. Sourceat x' at=0. RadiationintoMediumatZero18086. Two-DimensionalProblems18287. Three-DimensionalProblems18388. InfiniteCylinder. Initial Temperaturef{r, d).SurfaceatZero-18489. Infinite Cylinder. Initial Temperature/(r,, 5). Radiation intoMediumatZero18790. TheWedgeofanyAngle18991. InfiniteCylinder. Surfacer=aand Planes 6=0,5=00at Zero.InitialTemperaturef{r, d)-19492. ExtensionsofthePreviousResults19693. Sphere. Initial TemperatureJ{r, d,).SurfaceatZero 198CHAPTERXITHEUSEOPCONTOURINTEGRALSINTHESOLUTIONOPTHEEQUATIONOPCONDUCTION94. Introductory -20195. Semi-InfiniteRod. Endatv^. InitialTemperatureZero 20296. TheSameRod. Endatacos oit. InitialTemperatureZero 20397. TheSameRod. Radiation at Endinto Medium atv. InitialTemperatureZero20598. The Same Rod. Radiation at End into Medium at a cos at.InitialTemperatureZero.- - -20699. Semi-InfiniteRodof twoDifferentMaterials. Endat. InitialTemperatureZero 206100-101. Pinite Rod. Ends at Zero and v. Initial TemperatureZero 210102. The Same Rod. Ends at Zero and Ct. Initial TemperatureZero- -....211103. The Same Rod. Ends at Zero and acoscui. Initial TemperatureZero- 212104. TheSameRod. OneEndatZero. RadiationattheotherintoMediumatv^. InitialTemperatureZero .212xii CONTENTSSKCTION PAGK105-106. FiniteRod of two Different Materials. The Ends atZeroandtig. InitialTemperatureZero--213107. Sphere. InitialTemperatureZero. Surfaceat'217108-109. Sphere of two Different Materials. Initial TemperatureZero. SurfaceatUo--2181 10. SomeProblemsontheCylinder222CHAPTERXIIINTEGRALEQUATIONSANDTHEEQUATIONOPCONDUCTION111. Introductory 225112-114. IntegralEquationsandLinearPlowofHeat-227115. Fourier'sRing 232116-117. Two-DimensionalProblems 234118. Three-DimensionalProblems 237Examples onthe Condttction ofHeat 238AppendixI. NoteonBessel'sFunction 248AppendixII. Bibliography 250ListopAuthorsQuoted 265GeneralIndex 267[Inthisvolumetheauthor'sbookFourier'sSeries and Integrals (2nd Ed.), 1921, will bereferredtoasF.8.]CHAPTER ITHEDIFFERENTIALEQUATIONOFTHEMATHEMATICAL^^THEORYOFTHECONDUCTIONOFHEAT1. Introductory.When different parts of a bodyare at different temperatures,heat flows from the hotter to the colder. Consider the metalrod ABCD,FIG. 1.andsuppose it is heatedat the endAfromsome external source.Forsometimethetemperatureoftherodgraduallyrises,thepartsnear Abeingheatedfirst,butnochangetakesplaceat CDtill BChashadits temperatureraised. Ultimately,if theend Ais heatedlongenough,itis found thatasteady stateof temperatureisreached,in which, while the temperature mayvaryfrom poiat to point,itremainsthesameateachpointasthetimechanges.This transference of heat from the hotter portions of a bodytothecolderiscalled ConductionofHeat. It mustbedistinguishedfrom Convection, onthe one hand, andEadiation, on the other.InConvectionthetransference ofheatis duetothemotionoftheheatedbody itself, as, for example,whenthe different parts of aliquidareatdifferenttemperatures,currentsareproduced by meansof which a uniform temperature is reached. In Eadiation thehotterbodyloses heatandthe colderbodygains it bymeansof aprocessoccurringinsomeinterveningmedium.2. Conductivity.The Mathematical Theory of the Conduction of Heatmay besaid to befoundedupon a hypothesis suggested by the followingexperiment:2 THEDIFFERENTIALEQUATIONOFTHEAmetal plate is given,boundedbytwo parallel planes of suchan extent that, so far as points well in the centre of the planesare concerned, these bounding surfaces maybe supposedinfinite.Thetwoplanes are kept at different temperatures, the differencenotbeingsogreatas to causeanysensiblechangeinthepropertiesof thesoUd. For example,the upper surfacemaybekept at thetemperature of melting ice by a supply of pounded ice packedupon it, andthelower at afixedtemperaturebyhaving a streamofwarmwatercontinuallyflowingover it. Whentheseconditionshaveenduredfor a sufficient timethetemperature of thedifferentpoints of the solid settles downtowards its steadyvalue,and atpointswellremovedfromtheendsthetemperaturewillremainthesamealongplanesparalleltothesurfacesoftheplate.Considerthepart of the solid boundedby animaginarycylinderof cross-sectionSwhoseaxis is normaltothesurface ofthe plate.This cylinder is supposed sofar inthe centre of theplate thatnoflow of heattakes place across its generating lines. Letthetem-peratureofthe lowersurface beoC.andofthe upperViC. {V(,>V]},andlet thethickness oftheplatebedcentimetres. Theresults ofexperiments upondifferent metals suggest that when the steadystate of temperature has been reached, the quantityQof heatwhich flows upthrpugh the plate in t seconds over the surface SisequaltoT;r, ^ o_.d'where X is a constant, caUed the Thermal Conductivity of thesubstance, dependinguponthe material of which it is made. Inother words,the flow of heat between these two surfaces is pro-portionalta thedifferenceoftemperatureofthesurfaces.Thisresult mustnotberegardedasproved by theseexperiments.They suggest thelaw ratherthanverifyit. Themoreexactverifica-tionistobefoundinthe agreementofexperimentwithcalculationsobtained fromthe mathematical theorybased on the assumptionofthetruthofthislaw.Strictly speaking, the conductivity K is not constant for thesame substance, but depends upon the temperature.However,whentherangeoftemperature is Umited,thischangein Kmaybeneglected, andin the ordinarymathematicaltheoryit is assumedthattheconductivity doesnotvary withthetemperature.AnearerMATHEMATICALTHEORYOPCONDUCTIONOFHEAT 3approximationto the actual state maybeobtaiaedbymaking Kaliaearfunctionofthetemperaturev,e.g.,K==K^{l+av),wherea is small.It isimportanttonoticethedimensionsof K,as it is frequentlynecessaryto changethe units of length, mass and time in termsofwhichit is stated.SmceZ=^K-'^i)St'itsdimensionswilldependuponthoseofQI{voVj}.Theunit of heat is taken as that quantitywhichwill raiseunitmass of water1C. Thedimensions ofQ/{vo~Vj)arethensimply[M], since the unit of heat varies jointly as the unit ofmassandthevalueof thedegree.It followsthat[K]^^^^'mmOnthec.G.s. systemtheunit ofheatis theCalory, thequantitywhichwill raise 1 grammeofwater1C*If it is desired to measureheatbythework necessary to pro-duce it, the dynamicalunit in this systemwouldbethe erg. Therelation between the calory and this unit is given to a sufficientapproximationbytheequation1 calory=4'2x10'ergs,andthe numerical value of K,whenheat is measured in calories,will be 4'2xl0' times its value when this dynamical unit isemployed.!''Another unit sometimes used is the British Thermal Unit (b.t.u.), i.e. thequantitj' requiredtoraise1poundof water atits maximumdensity(39 P.) by1F.1 B.T.U. =252'0 oal.fExperimentsshowthat the amount of heat required to raise 1 gramme ofwater1arenotquitethesameat different temperatures,andinanexactdefini-tionofthecalorythetemperatureofthewaterwouldneedtobespecified. Itisusualtotakeforthisspecifiedtemperature15C,andthecalorywillthenbethequantityof heat requiredto raise 1 grammeofwaterfrom15C. to16C. Forthis15calorywehavetheequation1 calory =4-184x10^ergs.SeeKayeandLaby, TablesofPhysicaland ChemicalConstants{4thEd.),p. 5.4 THEDIFFERENTIALEQUATIONOFTHEInthefundamentalexperimentfromwhichourdefinitionof theconductivity is derived, the solid is supposed to be homogeneousandof such a materialthat,whenapoint within it is heated,thehqatspreadsoutequallywell in all directions. Suchasolid is saidto *be isotropic, as opposedto crystalline andnon-isotropic soUds,inwhichcertain directions aremorefavourable for theconductionof heatthanothers. Thereare also heterogeneous solids,inwhichtheconditionsofconductionvary frompointtopointas well as indirection at each point. In this book we shall examine onlytheTheoryofConductioninHomogeneousIsotropicSolids.3. TheFlowofHeatacrossanIsothermalSurface.ConsideranisotropicsoUdwithadistribution of temperature atthetime t givenby'v=f{x,y,z, t).Wemaysuppose a surface described in the soUd, such that ateverypoint upon it the temperature at this instant is the same,say F. Such a surface is called the Isothermal Surface for thetemperature F, and it may be looked upon as separating theparts of thebodywhich are hotter than7from the parts whichare cooler than F. Wemayimagine the isothermals drawn forthis instantfor difierent degrees andfractions of a degree. Thesesurfaces maybe formed in anyway, but no two isothermals cancuteachother,sincenopartofthe body canhavetwotemperaturesat the sametime. The solid is thus pictured as divided upintothin shells byits isothermals. Heat is flowing from one shell toanother, this flowof heatbeing along the normalsto the surfaces,as no transference of heat takes place along thesurfaces of equaltemperature.GeneraUsingtheresultof 2we- takeasour fundamentalhypothesisforthe Mathematical TheoryoftheConductionofHeat that the raleat whichheat crossesfromthe inside to the outsideofan isothermalsurfaceperunitareaperunittimeisequaltodnwherevis thetemperatureofthesurface, KtheThermalConductivityofthe substance, and =- denotesdifferentiation along theoutward'drawnnormalto the surface.MATHEMATICALTHEORYOFCONDUCTIONOPHEAT 5Asaparticularcase, whenthe isothermalsareplanes perpendiculartotheaxis of x,therate offlowofheatperunitareaperunittime's ~^-^ inthedirectionofthepositiveaxisof x. Ifisdecreasingas Xincreases,thisratewiU bepositive. Ifv increasesasxincreases,the rate will be negative,meamngthat the flow'of heat is in thedirectionofthenegativeaxisofx.4. TheFlowo!HeatacrossanySurface.Wehave stated in the preceding article that we assumethat4herate of flow ofheatacrossanisothermalat apoint Pisonperunitareaperunittime, or, inthelanguageofdifferentials,on .dS being an element of the isothermal surrounding the point P.Weproceedtoobtainananalogousexpressionfortherateatwhichheat flows across anysurface, notnecessarilyisothermal, per unitareaperunittimeatanypointP.We shall denote this rate of flow by/.The value of/willdepend upon the position of the point, the direction of thenormalto the surfaceat that point, andthe time. Weshallnowshow that,if the values of/are givenforthree mutually perpendicularplanes meeting at a point, its value for any other plane throughthepoint maybewrittendown.Consider the elementarytetrahedron PABC, whose three facesPBC,PCA,PABare parallel to the coordinate planes, while theperpendiculartotheface ABCfromthepoint Phas the direction-cosines (A, yu, v),andis oflengthp.(Fig.2.)Let the area ofABCbeA; then the areasofPBC,PCAandPABarerespectivelyXA,/xAandi^A.If wedenote the rates of flow for the elementary areas PBC,PCA,PABandABCbyf^,fy,/,and/,the rate at which heatis gainedbythetetrahedronis ultimatelygivenby6 THEDIFFERENTIALEQUATIONOFTHEHowever, if c andpare the specific heat and density, this ratofgainofheatis equalto1 . dvProceeding to the limit whenjj^O, this expression becomeszeroand/a.,/^,/jand/become therates of flow atthepoint Pacros2Fig. 2.planes parallel to the coordinate planes and a plane through JFthe perpendicular to which is in the direction (A, /x, v). Thuwehavex/.+m/.+./.=/Now, according to our fundamental hypothesis, the rate cflow of heat across an isothermal surface per unit area per unitime is equal to the product of the conductivityand the rate cdiminution of the temperature in the direction of thenormal tthesurface. Let Pbeapointupontheisothermal, andthenomasatPtheaxis of z, the axes of xandybeing in thetangentplaDthrough P- Then/a;and/arebothzero,sincenoflowtakesplacalongthesurface.Therefore/="/ dvaz-V^'"MATHEMATICALTHEORYOFCONDUCTIONOFHEATdh3where^denotesdifferentiation in the direction(\, fx,v),sincedv^ dv dv,dv , dv dvThusthe rateofflow ofheat at apoint across anysurfacefromtheinsideto theoutsideperunita/reaperunittimeis-K-,dnwhere^denotesdifferentiation along the outward-drawn nmmal tothesurface at thepoint.5. TheEquationof Conduction.Consider the case of a homogeneous isotropic sohd heated inany wayand"then allowed to cool. The temperature v at thepointP{x,y,z) will beacontinuous functionof x,y,zandt, andthefirst differentialcoefficientsofvwillalsobecontinuous.ConsideranelementofvolumeofthesohdatthepointP, namely,the rectangular parallelepipedwith this point as centre, its edgesbeing parallel tothe coordinate axes, andof lengths 2dx, 2dyand2dz.LetABCDand A'B'G'D' be the faces to whichtheaxis of xis perpendicular. Then the rate at which heat is flowing intothe parallelepiped over the faceABCD(xdx)will ultimately begi^^^^yUydz{f,-'^dxwheref^is the rate of flow atPacrossthecorresponding plane.Similarly the rate at which heat is flowing out across the faceA'B'G'D' is given byidydz{f,+^^dx).Thustherateofgainofheatfromthesetwofacesis equal' to8dxdydz^.Similarlyfromtheothersweobtain-Sdxdydz^and -Sdxdydz^.Butthiselementofvolumeis gainingheatattherateof8dx dy dzcp~r-8 THEDIFFERENTIALEQUATIONOPTHEThereforewehaveButwehaveseenthatand K is independentof x,yandz.Thereforeourequationbecomeswhere /c=.cpTheconstant k was calledbyKelvinthe DifEusivity of the sub-stance,andbyClerk-MaxwellitsThermometricConductivity.Thedimensionsofthediffusivity kareobtainedat oncefromthoseoftheconductivity A' (of.p. 3). Since c above istheratioofthequantityofheatrequiredtoraise unit mass of the substance1C. to the quantity requiredto raise unit mass of water 1C., it is of zero dimensions in mass, lengthandtime. Alsothedimensionsofthedensitypare[Jf]/[i'].Thuswehave rriiIt follows that if the units of length and time are the foot and yearinstead of the centimetre and second, the value of k for these units willhave to be multiplied by (30'48)'''/3-1557x10^ to reduce it to the.c.G.s.system. (Cf.p. 58.)Bysome of the early writers whodid not employ the o.o.s. system,thethermal unit was taken as the amount of heat which would raise unitvolumeofwater1C.Let of these units be required to raise unit volume of the substance1C.Thentheequationofconductiontakestheformwhere Kis theconductivityintermsofthenewunit.It is clear that Kjc in this system is equal to the diffusivity Kjpcdiscussed above.Ontheotherhand,whenthe thermal unitistheamountofheatrequiredtoraiseunit volume of water 1C., the numerical value ofthe. conductivitywillnotagreewiththatobtainedwhen the unit is the amountrequired toraiseunitmassofwater1C,unlessthelinearunitis thecentimetre.MATHEMATICALTHEORYOPCONDUCTIONOPHEAT 9If the solid is isotropic,butnothomogeneous, the equation forVbecomesInthe case of SteadyTemperature, whenthe temperature doesnot vary with the time, the equation becomes that of Potential.Alsoif atthepointP{x,y,z)thereexistsasourceof heatsupplyingin the time dt the quantity Adt of heat per unit volume, theequationbecomes^'^=a^(^aJ+a^(^a^)+a^(^a-J+^-Such a condition is realised whenconduction takes place alongawirealongwhichanelectric current is flowing,since this currentisgeneratingheatinaccordancewithJoule'sLaw.Theseresults may alsobeobtainedbytheapplicationofGreen'sTheorem,*that when^,ij,^,as well as their first differential ooefSoients, are con-tinuousfunctionsofx,yandz, insideaclosedsurface,j\%+mr,+nC) dS=///(!+!+1)dx dy &,(I,m,n) being the direction-cosines of the outward-drawnnormal,and theintegrationsbeingtakenoverthesurface,andthroughoutitsvolume.SupposeanysuchsurfacedrawnlyingwhollyinsidethegivensoM.Therateatwhichheatflowsoutacrosstheelement dSofthesurfaceism+mf,+nf,)dS.Thereforethetotalrateofgainofheatwithinthesurfaceis-jjilL+rnfy+nf,)dS.Butthisrateofgainofheat mayalsobeexpressedbyfff{op^^dxdydz,theintegrationbeingtakenthroughtheregionboundedbythissurface.ThusJjjcp'^^dxdydz +fj{lfj,.+ /;+/.) dS =0.ThereforebyGreen'sTheoremandthis holds whatever closed surface we consider, provided it lies whollywithinthesolidandnosourceofheatexistswithinit.*Cf. Lamb,Hydrodynamics, 42.10 THEDIFFERENTIALEQUATIONOFTHEApplythis result to theelementsurroundingthepoint P(x,y,z), andweobtaintheequation"P-dt^-dx^-dy^-dz"'asbefore.6. TheTransformationofCoordinates.These equations maybe easily transformed into other systemsof orthogonal coordinates, the most useful being the SphericalPolar System, in which the position of the point is determinedbyits distancerfromthe origin, its latitude 0,andits azimuth0,andthe CyUndricalSystem, in whichits position is determinedbythe polar coordinates r, Q of its projection onthe plane of x,y,andthecoordinatez.These are special cases of the general system of orthogonal co-ordinates, in which the position of a point is given bythe iater-sectionofthethreeorthogonalsurfaces,^=const., >;=const., f=const.Weproceed to showhowthis transformationmaymost easily beaffected.Considerthe element of volumeboundedbythe surfacesid^,ridr,,feZf,andletA'B'C'D'and ABCDbethefacesi+d^.Let ds^=X''di^+/ui.''d,,^+v^d^^be the equation giving the length of the elementary arc joiningthepoints(^, ri,f)and{i+d^,rj+drj,C+d^).Thenthearea of thesection of the^surfacethroughP{^, r/,f)cutoffbythesurfacesijd>i, ^d^isgivenby4/xi/dt]d^,andtherateatwhich heatflowsacrossthis sectionperunittimeisiflVdr]d^fi,f{beingtherate offlowofheatat Pacrossthesurface^.Therefore the rate at which heat flows intothe element acrosstheface ABCDisultimately4[/^"/^-^(fxvfi)dijdr,d^,andtherateatwhichheatflowsoutacrossthefaceA'B'C'D'is^{f^vh+YMh)di)dr,d^.MATHEMATICALTHEORYOFCONDUCTIONOFHEAT 11HencethetotalrateofgainofheatfromthesetwofacesisTheotherfacesgiverespectively-8 1(.A/,)didr, dl-8 |(XmA)didr,d^.Insertingthevaluesof/^,/,and/f,namely,f__Kdv f__Kdv f__Kdv^^~Xdi'J-'~M9'?'^^~'"H'andequatingtheexpressionwethusobtaintoSXfiv d^ d>] d^cp ^,wehavewhichreducesto. dvwhen Kisconstant,andasusualwehavewrittenk=cpSpherical PolarCoordinates.Inthissystem a; =rsin 61 cos^,y=rsindsin(p,z=rcos6,andds^=dr^+rHe^+r'sm^9d^.Thereforetheequationforvbecomesdv K[d(. dv\. 1 d(. 3d\ 1dH^TrT^ldrV dr) +ii5^30 V'^doJ "^sin^6d-0at allpointsofthesohd.If the initial distribution is discontinuousat points or surfaces,these discontinuities must disappear after ever so short a time,andin this case our solutionmustconverge tothe valuegivenbythe initial temperature at all points where this distribution iscontinuous.III. BoundaryorSurfaceConditions.(A) The SurfaceofSeparationoftwo Mediaof Different Con-ductivitiesKiandK^.MATHEMATICALTHEORYOFCONDUCTIONOFHEAT 13Let^1 andv^ denotethe temperatures in thetwomedia. Thenit is assumedthatatthesurface ofseparation thetemperaturesarethesame.Supposeanelementof areadS taken uponthesurfaceofsepara-tion, andthat an element of volume is constructed bymeasuringoff lengths e alongthenormalsoverthis areaintobothmedia,thequantity e being an infinitesimal of a lower orderthan the lineardimensions ofdS.FIG. 3.Thenthe rate atwhichheatis gainedbythis elementofvolumefromtheflowoverthesurfacewillultimatelybethe differentiations being takenalongthenormals fromthe com-monsurface into each medium, the contribution from the endsbeingnegligible.EquatingthistotheexpressionCi,Cg being the specific heats andpi,p^the densities of the twomedia,andproceedingtotheHmitwhenevanishes,wehaveand'yi=2.astheconditionsatthesurfaceofseparationofthetwosubstances.(B) Whenradiation takes place at the surface of the soUd iatoa gas at the temperature v^^, it is assumed, and the assumption14THEDIFFERENTIALEQUATIONOFTHEis suggestedbyexperiment,thatthe loss of heatperunitareaperunit time is proportional to the difference of the temperaturesof the surface and the gas. In other words this loss of heat isH{vVq), whereHis a, constant associated with the state of thesolid and its surface. This quantity is called the Bmissivity orExterior Conductivity, and it is foimd to vary considerably withthetemperatureandthestateofthesurface,sothatinexperimentson conduction it is best, as far as possible, always to reduce theloss ofheatbyradiationatthesurfacetothemagnitudeofasmallcorrectionbytreatingthesurfacewithasuitablematerial.Theconditions at the surface follow in the samewayas above, .andwehaveg^^+h{v-Vo)=0,whereH/K=h,andthe differentiation is takenalongtheoutward-drawnnormal.(C) There are other possible surface conditions. Theboundarymay bekeptataconstanttemperature, orat atemperaturewhichvaries with the position of the point and with the time; or theboundary may be rendered impervious to heat. The analyticalexpressionsforthesecasesareobvious.In the mathematical treatment of the question these surface and initialconditionsarenotregardedasconditionswhich vmustsatisfy onttiesurfaceitself or at the instant t=0. Theyare taken as limiting conditions, and itisrequiredintheonecasethatoursolutionshallconvergetothegivensurfaceorinitial value, andin theothercasethatthe differential coefficients inthelimit as weapproach the surface shall satisfy the corresponding conditions.8. The Solution of the Problem is Unique.We shall now show that the general problem of conduction,basedupontheequationofconductionandtheseinitialandsurfaceconditions, admitsofonlyonesolution.If possible, let there betwo independent solutions v^, v^ of theequations~,=kVHinthesolid,v=f(x,y,z) fort=0inthesolid,v=(p{x,y,z, t) atthesurface.Let V=ViV2.MATHEMATICALTHEORYOFCONDUCTIONOPHEAT 15ThenVsatisfiesdV-^=kV^Vinthesolid,F=0fort=0inthesolid,F=0atthesurface.WeshallprovethatVmustbezeroeverywhereinthesolid.Considerthevolumeintegraltheintegrationbeingtakenthroughthesolid.=k{{{vWW dxdydz.ButbyGreen'sTheorem,\\v ~^dS={{{vVWdx dydztheintegralsbeingtakenoverthesurfaceandthroughthevolumeofthesolid.ThereforeSince Yis zero over the surface, the first integral vanishes,andweobtain-'M()"-f)"+>*-Therefore16 THEDIFFERENTIALEQUATIONOFTHEAsimilar discussion shows that there can be onlyonesolutionfor the problem with the otherBoundaryConditions and for thecase of Steady Temperature. To prove that the equations musthave a solution is another matter. Their physical interpretationrequires that this be true: the mathematical demonstration ofsuchExistenceTheoremsbelongstoPureAnalysis.9. Simplification of the General Problem of Conduction.Whenthe surface conditionsdonotvarywiththetime,wemayreduce the general problem to depend upon two simpler cases,oneofthesebeingacaseofSteadyTemperature.Forexample,whenwehavetosatisfy .3v=k:V^,throughthesoUd,'"=fi^>y>2)initially,and =c/)(a;,y,z) atthesurface,wemayput v=u-\-w,wheremis afunctionofx,y,z only,andsatisfies\I^u=Qthroughthesolid,and u=(j>{x,y,z) atthesurface;and wis afunctionofx,y,zandt, suchthat-=k'W^wthroughthesolid,wf{x,y,z)uinitially,and w=0atthesurface.The first is a case of Steady Temperature, and the second is acaseofVariableTemperaturewithzerosurfacetemperature.The case of Radiation into a mediumwhose temperature doesnotvarywiththetime maybetreatedinthesameway.When the surface temperature varies with the time, or whenradiation takes place at the surface into a mediumwhose tem-perature varies with the time, three different methods may beemployed. The first is due to Duhamel, who showed that thesetwocasescouldbe reducedtothoseof constantsurfacetemperatureorradiationintoamedium at constanttemperature.The secondmethodcorresponds tothe use of Green's Function in theTheoryMATHEMATICALTHEORYOFCONDUCTIONOFHEAT 17ofPotential.(Cf. Ch. X.) Thetliird involves the use of contourintegrals.(Cf.Ch.XL)At this stage we shall refer only to Duhamel's method, whichdependsuponthefollowingtheorem:*I.If v=F{x,y,z,\,t) representsthetemferatureat {x,y,z) atthetime t ma solid in which the initial temperature is zero, while itssurface temperature is {t)], isgivenby=!'4,{X)j^F{x,y,z, t-\)d\.Now the general problem with varying surface temperaturerequiresthesolutionoftheequationsdv""^=kVHthroughthesolid,=/(;,y,z) initially,v=(j){x,y,z, t) atthesurface.Putv=u-{-w,where=/cV%throughthesolid,M=0initially,u=(p(x,y,z, t) atthesurface;MATHEMATICALTHEORYOFCONDUCTIONOFHEAT 19and-^=(cV%throughthesolid,w=f{x,y,z) initially,w=0atthesurface.The equations for uwehave just discussed. Those for warein their simplest form. HenceDuhamel's Theorem simplifies thisproblem and reducesit to the caseof surfacetemperatureindependentofthetime.CHAPTER IIFOURIER'S RING10. Introductory.Fromreasons ofsymmetryin the initial distribution oftempera-tureortheformofthesoUd, itwilloftenhappenthattheequationsforthetemperaturewhichwehaveobtainedinthepreviouschaptermaybe simplified, and that one, and sometimes two, of the co-ordinates disappearfromthese equations. Forexample, if wearedealingwithasphereinwhichtheinitialtemperatureis afunctiononly of the distance r fromthe centre, andthe surface conditionsare the same all over the sphere, the temperature will dependonlyuponrandt. Similarly, ifthesoUdisboundedbytwoparallelplanes, x=0andx=a,and if the initial temperature is a functionof X only, andthe surfaces arekeptat constanttemperatures, theisothermals will remain planes parallel to the bounding planes,and the teniperature will depend only upon x and t. Further,in the case of an infinite cylinder whose generating hues areparallel to the axis of z, when the initial distribution is thesame at aUpoints onlines parallel to this axis, andtheboundaryconditions are of the same nature, the temperature will dependonlyuponx,yandt, andwiUbethesameatpointsinthecylinderwhichlieonhuesparalleltotheaxis.11. TheEquationof Conduction in Fourier's Ring.One of the simplest and most suggestive problems in the Con-duction of Heat, when the temperature depends only upon onecoordinate and the time, is Fourier's Problemof theRing. Thisproblem is also of special interest, as it was the first to whichFourierappliedhis mathematicaltheory, andforwhichtheresultsof his mathematicaliavestigation werecomparedwiththe facts ofexperiment.**Fourier, TMorieanalytiquedelachuleur, Ch. II. andIV.20FOURIER'SRING 2For simplicity we shall suppose the ring to be formed by ttrevolutionof its normal cross-section, a circle of small radiu!about an axis perpendicular to the plane of the ring, though thinvestigation will also apply to any curved bar of small crosssection,theaxisofwhichformsaclosedcurvewithnoloops.The cross-section is supposed so small that the temperatuimay be regarded as the same at all points of the section. Thinitial distribution is given, and the problem is to determine thtemperature at any point in the ring when it has been alloweto coolbyradiationand conduction, orbyconductionalone, whethesurfaceisimpervioustoheat.Wechoosethelengthxfromafixedpointonthe circle passinthroughthecentresofthenormalsectionsasthecoordinatedefininthe position of a pointonthis circle. Weexaminethemovemenof heat in an element of volume contained between the sectiorahanda'6'atdistancesxandx-\-dxfromtheorigin, theareaofthcross-sectionbeingo) andtheperimeterp.Therate at whichheat flows into this element over the face ais equaltodvandtherateatwhichitflowsoutovera'6'isHenceultimately the rate of gain of heat due to the two endsgivenby 92^K^^codx.,Therate at whichheat is being lostbyradiation at the surfacoftheelementisH{vVf,)pdx,.whereHis the emissivity : and the total rate of gain of heatthereforeultimately(K^o,-j)H{v-Voy)dx.But, if c is the specific heat andpthe density of the substancethisrateofgainofheatis alsoultimatelyequaltodvcp^.w dx.,, rdv KdH Hp. .Thereforea7=a",^(^-'"o)-dt cpdx^ cpw22 FOURIER'SRINGWritine;^=X and ~=k,cpuj cpwehave g-=/c ^^-\{v-Va).Wlieii the surface is renderedimpervious to heat His zero, andtheequationbecomesg^ g^^Also the case in which there is radiation maybe reduced to thisformbysubstituting^^^-\-ue->'Ktheexternaltemperatureobeingconstant.Itwinbenoticedthat when K isnotconstant,asimilardiscussioDleadstotheequationdv 1 d(dv\ Hp, ,at Cpax\ 0x1 cpoo12. TheVariable Temperature of the Ring.Considerthe distribution of temperaturein suchahomogeneousisotropic ringofunitradius, when thereisnoradiationatthesurface,andtheinitialtemperatureis anarbitrarycontinuousfimction/(a;),satisfyingDirichlet'sConditions(cf. F.S.,93),*while /(-7r)=/(7r).In thisproblemweshallsupposethisarbitraryfunctioncontinuous.In theothercases ofLinearFlowofHeatthedifficulties introducedbydiscontinuitiesintheinitialtemperaturewillbeexamined,fTheequationsforthetemperaturearethefollowing :,^.dv d^v(2)(3)(?)=(?)\dx/x= \dxJxthethirdconditionsimplyexpressingthefactthatthetemperatureand the flow of heat must be continuous at the point given bya;=7r,ortheoppositeendofthediameterthroughtheorigin.LettheFourier'sSeriesfor/(a;)beo+(%''OSa;+6isinx)-\-{a^cosIx-^-h^sin2a3) +...,*In this volume the author's hook Fourier's Series and Integrals (2nd Ed.),1921,willhereferredtoasF.8.tCf. e.g.17. 30, 31.FOURIER'SRING23sothatand1 C"0'n=-fix')COSnx'dx',1fhn=-/(a;')sin nx' dx'.Considerthefunctionvdefinedbytheinfinite seriesao+(aicosx-\-\sina;)e-'"+(a2cos2a;+62sin2a;)e-''2'+...00^2(*flCos?ia;+6sinwa;)e-'"'-^'-rt=0It is clear that each term of this series satisfies the difierentialequation(1)and the conditions(3),and that if we were deaUngwiththesumofafinite numberofterms,thesumwouldalsosatisfytheseconditions.In the case of an infinite series wehaveseen*thatwemustproceedwithmorecaution.Sincef{x)is bounded,! there is apositivenumber Msuch that|/(a;)|0)But,usingtheCosine Seriesfor 5?^^, namely,cosh/xxcosh/xa;_2|tanh/^ttcosh^TT TT"1;^cosMTT]oursolutionforvfoUowsatonce,andisgivenby=^tanh7re-'^'TT1 , ^COSWttos+Zj2 I g coswxe-L2/x2Vm+*icn'^fAfter a considerable time has passed, the convergency of thisseries becomes veryrapid owingto thepresenceofthefactore-''''""''.Neglectingthetermsafterthefirsttwo,wehavetheequations,Vq+i>^=tanhmtte"'^',fjLTT-Vo+V^=fiVfX-tanhyU7re-wefind that v=^ff{x+2V{Kt)i)e-^'di.Inthe limitwhent^>-0,f{x-\-2\/(Kt)^)=f{x),if this function iscontinuous; and it is assumed that the Uniiting value of thisintegralisgivenbywhichisequaltof(x).Therefore the temperature in the Infinite Solid at time t, duetotheinitialtemperaturev=f(x),'' ^^^'^^^"=2vfe^)E/(^') ^''"^dx'.ThecorrespondingresultsfortwoandthreedimensionsareCO 00*^*^^=(2^(TKf))jj]f^'^'V''^')^*"'dx'dy'dz'.00 CO COSince e-"'''cos26a;(ia;=^e'^,*Jo2a*Cf. J./S.,p. 195,Ex. 13, andGibson,Treatiseonthe Calculus(2nd.Ed.),p.469.INFINITEANDSEMI-INFINITESOLIDANDEOD 31andthereforee-''''cosa {x' x) da=ir7r7K*^J\Kt)we maytransformtlieexpressionfori) into1poo poodx'\f{x')cosa(a;' x)e-""''(?a,TtJ-00Joa form which would be suggested by Fourier's Integralfor/(.t),namely,.]^(:Tl^"dx^'c/)=f{t)at x=0,^=0when i!=0.These equations have already been discussed in23,and wehaveseenthat2 r y x^ \,,Hence,asin25,27. TerrestrialTemperature.Observationsofthetemperatureatpointsnearthesurfaceoftheearth have been carried out at a large number of meteorologicalstations in difierent parts of the world for many years. Theseresults have estabHshed the existence of two distinct phenomenaof terrestrial temperature.The first is that the variations of thesurfacetemperature fromthe heat by day to the coldbynight do not affect the tempera-tures of points at a depth of more than 3-4 feet, while theyearly changes from the cold of winter to the heat of summermaybeobservedupto a depth of 60-70 feet. Below that depththe temperature remains practically constant from day to dayand is not subject to alterations due to the changes at thesurface. In other words, the heat waves due to the changes of54THELINEARFLOWOFHEATthe temperature at the surface die away before they penetratetoadepthofmorethan60-70feet, andtheheatwhichis thustrans-ferredtotheearthoscillatesintheuppercrust,andwhileit proceedsinwards at certain seasons of the year, at others it ascends andradiatesintospaceatthesurface.However, after we pass the limit at which the temperature isaffected bythese surface changes, andreach the depths at whichit remains constant from day to day and year to year, there isa marked increase in the temperature as we descend. The tem-peraturesobservedatagreat numberofpointsatconsiderabledepthandat manydifferentstationsleave no doubt uponthis phenomenon.It hasbeenobservednearthe equatoras well as in thetemperatezone, andalthoughthe rate of increase varies with different placesand is much greater in the neighbourhood of active volcanoes orthermal springs, it mayroughlybetaken as about1F. for every50feet of descent at depthsupto aboutone mile.* This rise oftemperature was ascribed, both by Fourier and Laplace, to thehigh initial temperature of the earth, this supply of heat beinggradually diffused outwards, and still to a great extent preservedat the centre of the earth. Suchan assumption does not requirethat the rate of increase of temperature should be uniform aswe continue to descend, and other physical phenomena showthat the interior of the earth cannot be a mass of moltenrock.The periodic changes in the temperature near the surface havebeen used by writers from Fourier and Poisson onwards in thedeterminationoftheconductivityoftheEarth,thesedeterminationsbecoming of increasing value in recent years owing to the growthinthenumberof stationsatwhichthermometricobservationshavebeenmade.Since these dailyand annual variations of surfacetemperaturearenoticeableonlyatpointscomparativelyneartheEarth'ssurface,the problemmaybe simplified byneglecting the curvature of theEarth and supposing the surface to be the plane x=0, which issubjected to a periodic change of temperature. Thisproblem has*Someauthoritiesnowregardtheaverageasmorenearly1F. for every60feetofdescent, oreven 1F. forevery70feet. Cf. Sollas, TheAgeof theEarthandotherGeologicalStudies, 1905.INFINITEANDSEMI-INFINITESOLIDANDROD 55beendiscussed in24,*andthetemperatureat thedepthxbelowthesurfacewasfoundtobeA^+A^e-^iT>cos (c-^()*-ei)+^2e~^^(^)''cos(^2toi- JQcc-eaj-f...,whenthesurfacetemperatureisV=^0+^1 cos(mi ei) -\-A^ COS(2&)iea)+. . . .ItistruethatthetemperatureatthesurfaceoftheEarthdependsnotonly upon thetime,but upon thepositionof theplaceofobserva-tion, andthattheconstantsA^^,Ai,... willbefunctions oftheposi-tion of these points ; but, if a comparativelysmall portion of thesurfaceisconsidered,thetemperatureoverthis maystill besupposeddependentonlyonthetime,andthegeneralprinciplesunderwhichtheperiodical surfacechanges aretransmittedintotheinteriorandtheredieawaywillbefullyillustratedbythe solutioninthisform.Thus the theory shows that each partial wave is propagatedwith unaltered period inwards; that the amplitudes. of thewavesof shorter period diminish more rapidly than those of greaterperiod;andthattheyalsohaveamorerapidalterationofphase :while,ontheotherhand,thevelocityoftheirpropagationis sinallerin the ratio of the square roots of the periodic times. It followsthat the periodical variation takes a simpler form as we descend,where the partial waves of smaller period become more rapidlynegligible, so that after a certain depth the principal wave withthe largest period and greatest amplitude will alone be found:while at a still greater depththis will also- havebecomenegligibleand the temperature will have become constant. The depth atwhichtheamplitudeoftheyearlyvariationis e.g.0-1 willbeabout19 times greater than that at which the corresponding amplitudeforthedailyvariationwilloccur,since'V(5)~'''VCr,gives the ratio of the depths and T'=365T. This result, it willbe seen, agrees with the temperature observations which haveshownthatwhilethedailyvariationis notnoticeableafteradepth*Seealso Bousainesq, loc. cit, T. I.,pp.210-228;andpapersin Bui. sci. math.,Paris(Ser.2), 39, 1915.56 THELINEARFLOWOFHEATof 3 or 4 feet, the annual variation maybe traced to a depth of60or70feet.These features of the problem were all noticed by Fourier andPoisson, towhomthis discussion is due. Thesimplest applicationofthesolutiontothedeterminationoftheConductivityoftheEarthatdifferentplacesonits surface willbefoundinKelvin'spaperonTheReduction of Observations ofUndergroundTemperature."*Thesubstance of theEarth is takentobeahomogeneousmass ofsuchrockaswehaveonthesurfaceattheplaceofobservation,andthevaluesobtainedfortheconductivity areto averyconsiderableextentaffectedbythenature of the soil orrockinwhichthether-mometersare imbedded. Thedataarethe temperatureobservationsatplacesonthesameverticalandatdifferentdepths,theseobserva-tions extending over a considerablenumberof years. InKelvin'smemoir, Forbes' Edinburgh Observations for a periodof 18 yearswereemployed.These observations allow the mean temperature curve for ayear to be drawn, and its harmonic components to be obtained.Inthis waywefind for the depths atx^ andX2, the temperaturesViandV2intheform,v,=A^'+AC cos(^ t-e{)+A^' cos(^ -e,") +... .Butaccordingtothesolutionof24,wehavev=A,+A,e-^i^)^'ooB(^t~^[^) x-e)+A2e-M>oos{';t-^(^)x-.:)Thereforeweshouldhave4'A"A:=A^e--l^>\A"=A,e-^(!^)^*Edinburgh, Trans. R. Soc.INFINITEANDSEMI-INFINITESOLIDANDROD 57Thuslog^'-log^/_e/-e'_ l/riTAThe results of thecalculations of the meantemperature curvesat differentdepths give values for Aq, Aq", ... , which varyonlyto a very slight extent. Theseagreewith the theoretical resultthat themeantemperature dueto the surface changesshouldnotvaryaswedescend.Thefirst harmonicterm, or the annual variation, is the largest,andobservationsbaseduponitwillthereforebemosttrustworthy.Kelvinfound that therewas almost complete agreement betweenthevalues oflog ^/-log ^/'and.!l!z:^',the two expressions which should each be equal to ^/{^r|K), theunitoftimebeingtheyear.Fromthese resultsthevalue of k, or Kjcp,wasobtainedforthematerialattheplaceof observation.Calculationswerealsomadeofthesecondharmonicamplitudeandepoch.Inthiscasethedifferenttemperaturecurvesforthedifferent years gavefunda-mental differences in the coefficients for the semi-annual period. Thesediscrepancies and others in the case of the higher harmonics arenot to bewonderedat,astheactualstate ofaffairs is nottheidealonewhichhasbeenpostulatedwith regard to the periodical variation of the temperature andthematerialoftheEarth.28. TheAgeof the Earth.We haveseenin 27thatafterthelimitsat which the temperatureis affected by the surface changes are passed, a marked increaseisobservedasthedepth increases, and thatthis temperaturegradienthasbeentaken, inordinarycircumstances, asabout1F.forevery50 feet of descent up to a depth of about one mile. Thatthisgradientmightbeusedto obtaina roughestimateofthetimethathas elapsed since the Earthi^egan to coolfrom its molten state,wasremarkedbyFourierhimseK.*Intheproblem,assimplifiedby himformathematicaltreatment.*"Extraitd'unMemoire sur le refroidisaement seoulaire du globe terrestre,''BvM. des sciencespar la Societe philoma&ique de Paris, 1820. Also (Euvres deFourier, T.II. (cf. p. 284).v=-f-\58 THELINEARFLOWOFHEATthe curvature of the Earth is neglected andthe conductivity(k)supposed constant. The surface is taken as the plane x=0,andradiation takes place into a medium at temperature zero ; thetemperature when cooling begantaken as the time t=0isconstant and equal to v^,. He obtained the result given in25thatforlargevalues of t thetemperaturegradientnearthesurfaceis approximatelyVQl'\/{TrKt).Kelvin*took the simpler problem of the semi-infinite sohdboundedbyx=0, the boundarybeingkeptatzerotemperature,theinitialtemperaturebeingconstantandequalto Vq. Wehave seenin18thatthetemperatureat thedepthx atthetimet is givenbyfJ;/2^/(()'3v V Hence^-= ,,"^.c'm.ax ^/(TrKt)WhenXissmallandt is large, thisbecomesapproximatelyS/ilTKt)'asinFourier'sproblem.With the value of k used in Kelvin's paper(cf. loc. cit.,15)namely400theunits oflengthandtime beingthefootandyear,fwehave9^ 1 __^dx-divt'""Taking=7000F. as a suitable temperature for melting rock,and 0)'obeinganypositivenumber. (Cf.12.)Thus the function v, defined by the series(4),is a continuousfunctionofx, andacontinuousfunctionof t, intheseintervals.*It is easy to show that the series obtained by term by termdifferentiation of(4)with respect to x and t are also uniformlyconvergent in these intervals ofx and t respectively. Thus theyareequaltothedifferential coefficients ofthefunctionv.HenceSOLIDBOUNDEDBYTWOPARALLELPLANES 63Sincethe seriesis imiformlyconvergentwithrespect toxin theintervalO^x^l,when i>0, it represents a continuous functionofa; iQthisinterval.Thus, Lti;=thevalueofthesumoftheserieswhena;=0=0,and Lt'y=thevalueofthesumoftheserieswhenx=l=0.Hencethe Boundary. Conditionsaresatisfied.With regard to the Initial Conditions, we may again use theextension ofAbel'sTheoremcontainedinF.S., 73I.Wehaveassumed thatf{x)is bounded and satisfies Dirichlet'sConditions in(0,I).ThereforetheSineSeriesforf(x),.ttx,.2x3;,ffljSm-=-+a2sm-T\-...,converges,anditssumisf(x)at everypoint between aadIwheref(x)is continuous, and |{/(a;+0)+/(a;0)}at all other points.*(Cf. F.S.,98.)ItfollowsfromtheextensionofAbel'sTheoremreferredtoabovethatwhenvis definedby(i),wehaveLt v=Lt2j"'nsm--^- xe' 1-2'(->-0 I->-0 1"=f{x)atapoiatofcontinuity=i-{/(a'+0)+/(a'-0)}atall otherpoints.Thus we have shown that if the initial temperature satisfiesDirichlet'sConditions,and is continuous from a;=to x=l,whiley(0) =f{l)=0,the functiondefined by(4)tsatisfies all theconditionsoftheproblem.If theinitialtemperaturehas discontinuities, thefunctiondefinedby(4)at these points tendsto\{f{x-{-0)-\-f{x0)}as 0. If t*ltf(x)isboundedandsatisfies Dirichlet's Conditions, it follows from F.8.93thatitcanonlyhaveordinarydiscontinuities.fThiscanbewrittenasv=yj/()jIsm-^^c sm-pxe*i-i \dx,since the seriesundertheintegral uniformlyconvergent. (F.S., 70.)64 THELINEARFLOWOFHEATis taken small enough, v will bridge the gap fromf{x0)to/(x+O), and the temperature curve will pass close to the point^{f(x+0)+f{x~0)}.It mustberememberedthatthe physical problem, as wehavestated it for discontinuity, either at the ends of the rod or in theroditself,isanidealone. Innaturetherecannotbeadiscontinuityin the temperature in the rod initially. In the physical problemwemustassumethat a suddenchangeof temperature takes placeat the instant from which our observations are measured, in theimmediateneighbourhoodofthepointofdiscontinuityortheends,if they are points of discontinuity. The gap in the temperatureis thussmoothedover. Thesolution of themathematicalproblemwehaveobtainedsatisfiestheseconditions,anditmaybetakenasrepresentingthephysicalprobleminthismodifiedaspect.31. Somefurther remarksmaybe madeas to the Ltv, andthewayinf--0inthatinterval. Inotherwords,giventhearbitrarypositive numbere, thereexistsapositivenumberx suchthat\v(x,t)-f{x)\ 4>,{t)=v{l,t),andthusthecoefficientofsin-=-a; in. theexpansionof^r-^,whichis equaltoisgivenby_!!LE!a,.+?^(0,(O-(-l)"2,{t),whiletheinitialtemperatureof the rod is zero, and radiation takes place into amediumatzerotemperature,isgiven bywhere K=K/cp and v=Hplcpa},InNeumann'sProblem^i{t)=viwhen 2rT5^+5-5-003ofoic orjaxanddx^~d^\dx)'^didr,dxdx'^dr,^\dx) '^di dx^^ dr, dx^'Similarlydy^^di^\dy)^d$dr,dydy'^dr,Ady) '^ d^dy^'^d^dy^'Adding these two results, and using(1), (2), (3), (4),and(5),weseethatdx^'^dy^Thus,if wecanobtain"a solutionoftheequation9p+9?~'TWO-DIMENSIONALPROBLEMS 93satisfying certainboundaryconditionsatthecurvesthis solution in the ^r/ plane maybe transferred to the xyplane,theboundaries beingthe curves in the ccy planewhichcorrespondbythetransformationi+iri=f{x+iy)to the curvesf=fi,etc., while the temperatures at these boun-daries correspond to the temperatures at the boundaries in thef;?plane.,Suppose that we have the case of the rectangleinthe^17planeandthato=fi{r,)atf=^1, (hin'g^^.94TWO-DIMENSIONALPROBLEMSSubstitutingforf,>; fromtherelationwehavethetemperatureintheregionboundedbythecurveswhichcorrespond tof=fi,etc.,these curves being keptatthe temperaturescorrespondingto/i(ij), etc.45. Applications of this Method.*I. TheSectorof aCircle.ConsiderthetransformationInthiscase ^=-0,;y=-log(-andthesectorofradiusaandangleacorrespondstotheregioninthe^t] plane.Thustheequationsde-^dr,^'''(0;=constant represents the system ofcircles passing through A, B, these two sets of curves, as in allcases ofconjugatefunctions, beingorthogonal. Withthis notajiionthexyplaneis givenby 7rwhereoj, ag, arethepositiveroots ofJo'(aa)=0.Then,sincer[Jo{ar,r)Yd/r=-^{Jn{arfl)Y,2frf{r)Jo{ar)d^wehave^=^S-"'' "'(j(,)).^oM-III. Radiation at surfacer=ainto amediumat zero temperaturInitialtemperaturev=f{r).Inthiscasewetake/(r) =^iJo(ai-)+^2'^o(a2-) +->whereai, oa, are thepositiveroots ofaJg(aa)+hJ^iaa)=0.f"a^Then,sincer{Jo{ar))^dr=^-i^{h^+aJ)(Jo{aa))^Jo -n2aj\ rf(r)Jo{anr)drwehave^=^2 S^-'--y;^2)(j(^,)).^o(a.r).IV. Surface r=aat zerotemperature. Initialtemperaturev=f{r,iInthiscasetheequationofconductionbecomesdv fd^v Idv. 1 dH''dt '^Kdr^'^rdr'^r^dd^J'andtheexpression e-"''/{ar)(^cosn0+5sinm6*)satisfies this equation, n being taken integral as the temperatuis periodicin withperiod 27r.NowtaketheFourier's Series forf{r, 6),namely,COf(r, 6)=2('*' ^ ^^+Ksin'^^).1f"where =-If(r, 6)cos nddO, (n^l)&,,=-Tf(r,6)smnede,"JITandao=^Wfir,6)de.FLOWOFHEATINACIRCULARCYLINDER 119Thesecoefficients arefunctions of r. Expandtheminthe seriesofBessel'sfunctionsofthewthorder, e.g.whereai, a^, ... , as, , arethepositiverootsofJ(aa)=0.Thenwehave^-'==7ra''(.7j(,))4r_/^''^^'='*^'^"('-)'-^'-^^'Thusweobtainoursolutionintheform^'=2S(^.^^ nd+Bn^ssinn6)J(agr)e~'"'t.V. Radiation at surface r=ainto a medium at zero temperature.Initialtemperaturev=f{r, d).InthiscasewetaketheFourier'sSeriesfor/(r,6),namely,00/(".S)=S('^^*^**^+^ '*^)'11=0asinIV.The coefficients are functions of r. Expandthem in the seriesofBessel'sfunctionsoftheith order, e.g. ,00whereai,a^, arethepositiveroots oftheequationaJ'(aa)+AJ(aa)=0.Thenwehave2^os=--^r-^~rmTT-nAvA''ff{r,Q)Jo{asr)rdrde,A,=7^^Jtf"r/(S)cosnejJasr)TdTde,7ra^[as'+h^~){Jn(asa))^''^-"'^'^r rf{r,6) sin neJ{asr)rdrdO,JoJ-"^a'^(as^+h^-^^Wn(asa))'-120 FLOWOFHEATINACIRCULARCYLINDERand^=2'^{^n,s'^osnd+Bn^sSiD.nd)Jn{asr)e- .linsUV.U .1/.,I.i^n guiij,,!,>,;., yuj,yrVI. Surfacer=aatzero. Initialtemperaturev=f{r,6, z).Inthiscasewehavedt~''\dr^'^rdr'^r^de^'^dz^)'and e-''('''+'''>*J(ur)*'P^w9 ?^azsin sinisaparticularintegral.Nowexpandf{r,6, z) intheFourier'sSeries2(acos nd-\-bnsin nd).?l=0The coefficients aand6are functions of r and z, denotedbyF{r,z) andff(r, z).Expandthese functions in the series of Bessel's functions givenbythepositive roots ofJ(f^a)=0,andlet F(,2)=2^"(^)"^n (^')Finally,takethe Fourier's Integralsfor when z=-Z.ThismaybesolvedbytakingwhereMj satisfiesV^Mj=0,hasthegivenvalueatr =a,andis zero at2 =Z,withsimilarconditionsfortheothers.InthiswayweobtainMi=2E/\sin-5j-(z+;)(^cosTOg+-B,sin?tg),whereAm,nand5,aredetermined bythe expansionofXi(^>2)asabove.Also M, =2I ^"^^o t^^''.(i"*")(^^'. " cs '^^+S''. '^)'"jn n=osmn//x(wherethesummationin/*isoverthepositiverootsofand.4^,,B^,aredeterminedbytheexpansionofXal*".6)-Similarly%=2I?i5j4^^J(/*-)(4M,',smFLOWOFHEATINACIRCULARCYLINDER 127andtheX'sarethepositiverootsofhcos XZ Xsin Xl=0.Thefunction\{f{r,6, z)+f(r, 6,-z)}is thenexpandedina serieswhosetermsare ofthis type, andwisgivenbytheequation*00K tL n=063. MoreGeneralProblemsontheCylinder.Themethods of the preceding sections maybe used in dealingwiththehollowcyHnder,orthesolidinwhichtheboundingsurfaceis formedbythe cylinder (or hollow cyhnder), twoplanesthroughtheaxis,andoneortwoplanesperpendiculartotheaxis.It will be sufficient to give here the solutions of the followingproblemsofthiskind:I.InfiniteHollowCylinder. Surfacesr=aandrbkeptat zero.Initial Temperaturef{r).Inthiscasewehavedv /d^v,ldv\,,.J=\W^+-rFr)^^^Ifwe putv=e~''-''^u, whereudependsonr only, theequationforuisdhi Idu . .orBessel'sequationoforderzero.Astherangeofrdoesnotextendtotheorigin,Bessel'sfunctionsof the secondkind are not excluded. Insteadof introducingthefunction r(cf. AppendixI.,2),it isbetter to take thefunctionF, where Zrw=J+*r (cf. Appendix I.,4),since H^^^z)vanishesatinfinityintheupperpartof thez-plane.Let U,(ar)=J,{ar)H^Hah)-J,{ah)H-'^4- I'^r Ml IT=^2fl/r/ ^u2/(' ^)sill-^OJmAar)r drdd,thesummationinabeingoverthepositiverootsofJ^aa)=0.0The solution for the wedge given by 9=0and6=0ocan bededucedfromtheabovebyletting a^oo . Cf.69, 90.III. Finite Cylinder. The ends z=l, the planes 6=0,6=6o,andthesurfacer=akeptatzero. Initialtemperature/{r, 6,z).Inthiscasewe. havedv_/3^,1 dv 1 d^v d'^v\dt~''\dF^'^r JrW' W^MJ'andaparticularintegralofthisequationis*' ''JnAXr) sm-s-dSin -^{z+l).daAlsotheconditionsatthesurfacearesatisfiedifm,narepositiveintegers,andAisarootofJ^(Xa)=0.Expand/(r,6, z)intheSineSeries2^asm-3-e,1 C'Oabeingafunctionofrandz,say^(r,z).FLOWOFHEATINACIRCULARCYLINDER 131ThenexpandFn{r,z) intheSineSerieswhosetermsaremultiplesof2j(z+0- The coefficients of this series will be functions of r,sayJ'.H.nCr).Finally expandJi,(r)in the series of Bessel's functions givenbythepositiveroots ofj^(xa) =o.hInthiswayweareledtothesolutionofourproblemintheformwhere^x,m,=a?/) ,t /^ winOoC^ wnrjmVinrf{r,9,z)sin-^(z+l)Bm.^-dJn,r{\'r)rdrdBdz.63. Determination of the Conductivity from Cylinders.Theresults ofthelast sectionscanbereducedtoasimplerformwhenthe initial temperatureis constant. Weproceedtoexaminethreecaseswhichlendthemselvestoexperiihentalinvestigation.I. Initial temperaturev=Vo.Radiation at r=a, z=l into amediitm,atzero.*Inthiscasewetaketheexpressioncos'\zJo(iuir)e-''^>'^+i^'^^,whichis aparticularintegraloftheequation=kWotThissatisfiesthe surface conditions, ifXis apositiverootoftheequation^hcosIX XsinZX=0andfxisa.positiverootofiJiJo'{jua)+}iJo(iUia)=0.Now,assumingasbeforethepossibihtyoftheexpansions.1=^1 cos Xi2:+^2 cos X2Z+...,l=BiJoW)+BMf,^r)+...,wemayobtainthevaluesofthesecoefficientsbyintegration.* Weber,H.P.,Ann.Physik,Leipzig (N.F.),10, p. Ill, 1880.132 FLOWOFHEATINACIRCULARCYLINDERTodetermineAi,A^, ... wefiistshowthatI cosXzcosA2(i!z=0 (m^n),and1cos'' A2 dz=^ ,..", , .Wehave I cos \^zcos \z dz=|-J{cos{X^+\)z+'cos(\^~\)z)dz2{Xm+^) 2(XmX)cosXZcosXZ,> 7 % J. ^ 7.-(XmtanXj-XtanX0(^m^^n)=0,since XtanX?=A.Also I cos2Xz