PA214 Waves and FieldsFourier MethodsBlue bookNew chapter 12Fourier sine seriesApplication to the wave equationFourier cosine seriesFourier full range seriesComplex form of Fourier seriesIntrouction to Fourier transforms an the convolution theoremFourier Methods!r "erv#n $o# %&'(www2)le)ac)uk*epartments*ph#sics*people*acaemic+sta,*mr'PA214 Waves and FieldsFourier MethodsLecture noteswww2)le)ac)uk*epartments*ph#sics*people*acaemic+sta,*mr'214 course texts Blue book- new chapter 12 available on BlackboarNotes on Blackboard Notes on s#mmetr# an on trigonometric ientities Computing exercises .xam tipsmock papersBooks "athematical "ethos in the /h#sical &ciences %"ar# 0) Boas( 0ibrar#1ResourcesPA214 Waves and FieldsFourier Methods2he wave equation for a string 3xe atanhas harmonic solutions Introduction&uperposition tells us that sums of such terms must also be solutions-PA214 Waves and FieldsFourier Methods&et coe4cients from initial conitions- e)g) string release from rest with thenPA214 Waves and FieldsFourier Methods5hat happens if the initial shape of the string is something more complex6In general can be an# function 2he implication is that we can represent any functionas a sum of sines7 an*or cosines or complex exponentials)his is Fourier!s theore"PA214 Waves and FieldsFourier Methods5e will 3n that a function in the rangecan be represente b# the Fourier sine serieswhereFourier sine series #hal$%ran&e'PA214 Waves and FieldsFourier Methods8ow oes this work 6Nee a stanar integral %new chapter 12 9 A)2(7Fourier sine seriesPA214 Waves and FieldsFourier Methods()uare *ave+ PA214 Waves and FieldsFourier Methods()uare *ave+ PA214 Waves and FieldsFourier Methods()uare *ave+ PA214 Waves and FieldsFourier Methods()uare *ave+ PA214 Waves and FieldsFourier Methods()uare *ave+ PA214 Waves and FieldsFourier Methods()uare *ave+ PA214 Waves and FieldsFourier MethodsPeriodic extension o$ Fourier sine seriesan that5e know that sine waves have odd s#mmetr#-PA214 Waves and FieldsFourier Methods5ithin can expan any function as a sum of sine waves-8ow oes this expansion behave outsie of the range6PA214 Waves and FieldsFourier Methodssquare wavesawtooth waveexp wave %o(PA214 Waves and FieldsFourier Methods&tring 3xe atan he *ave e)uationInitial conitions anPA214 Waves and FieldsFourier Methodse)g) an- then %see workshop 1- exercises 1 : 2(PA214 Waves and FieldsFourier Methodse)g) an- then %see workshop 1- exercises 1 : 2(PA214 Waves and FieldsFourier Methodse)g) an %see new chapter 12- exercise 12);(PA214 Waves and FieldsFourier Methodse)g) an %see new chapter 12- exercise 12);(PA214 Waves and FieldsFourier MethodsCan go through the same proceure with the solutions to other /!.se)g) 0aplace equation %see workshop 1 exercise