Atmospheric Dynamics 2
Atmospheric Dynamics 2 ESCI 343
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ESCI 343 Atmospheric Dynamics II Lesson 1 QG Heighttendency Equation Reference An Introduction to Dynamic Meteorology 3ml edition JR Holton Synopticdynamic Meteorology in Midlatitudes Vol 1 HR Bluestein Reading Holton Chapter 6 Section 63 THE QUASIGEOSTROPHIC THERMODYNAMIC ENERGY EQUATION The quasi geostrophic form of the thermodynamic energy equation in pressure coordinates is 7 Quasigeostrophic thermodynamic energy equation where ois the static stability parameter defined as am 3p The static stability parameter is positive for a stable atmosphere and negative for an unstable atmosphere Note that for horizontal advection only the geostrophic wind is used 02 From the hydrostatic equation in pressure coordinates we have M R dT a 3p p Solving for T gives T La Rd aP Using this expression for temperature in the Q G thermodynamic energy equation gives R 3Vg Vpa aw ii at 3p op p Rearranging terms we get 1822 0V 82 0w Rd 3p at g p 3p pep And finally if we define the geopotential tendency as I E a then the Q G thermodynamic energy equation becomes Equation 1 Keep in mind that equation I is really nothing7 more than the the 39 energy equation THE QG VORTICITY EQUATION REVISITED The Q G vorticity equation in pressure coordinates is a a 43 V Vsi ffo a The geostrophic vorticity in terms of the geopotential is 1 g 7V2 CD 8 f0 17 Substituting this into the Q G vorticity equation gives another form of the Q G vorticity equation 1 a Vzlz fng VpiVi gtff02 7 quot Equation 2 f0 3p Equations 1 and 2 are an alternate form of the Q G system They are two equations with two dependent variables x and a If we know the what the geopotential field is then these equations form a complete system which can be solved for either 1 or a THE GEOPOTENTIAL TENDENCY EQUATION The first equation we will derive is the geopotential tendency equation found by eliminating a between Eqns 1 and 2 The idea behind this is simple but the individual mathematical steps become complicated None the less by differentiating Eqn 1 with respect to pressure then multiplying it by fez0 and then adding this to Eqn 2 a very ugly equation is found note that for ease of notation the subscript p is no longer written on the del operator but is implied vhfii z f 7 ov iv2qgtf Jig 47 ov g f 2R 3 i Jig 039 ap2 0 3 f0 039 3p 3 3p o cp 3p p 039 3p The static stability parameter normally increases with height however analysis of the Q G tendency equation is slightly easier if we assume that 039is constant so that the last term disappears In this case the equation becomes 2 2 2 2 V2 f7 ai2 f0l7g 0V iV2ltIgtf fi3 47 0V 83 f0 R i i 039 3p f0 039 3p 3 3p o cp 3p p QG geopotential tendency equatwn Though ugly this equation has a sort of inner beauty We first try to see this beauty by analyzing the terms of the equation in a qualitative fashion We do this by imaging that the horizontal structure of disturbances in the atmosphere can be approximated by sinusoidal functions such as 2506 y M X01 texpikx ly If we ignore the pressure derivatives on the left hand side LHS of the tendency equation then the LHS becomes V2 106 y p70 V2 Xptexpikx 1M k2 12Xptexpikx 1 k2 12Zx y m or more simply szcx 1 What this means is that for a sinusoidal disturbance having a zero mean value the horizontal Laplacian of a field is proportional to the negative of the field So we can qualitatively think of the LHS of the equation as being nothing more than ix so that if the LHS is negative it means that the geopotential tendency is positive It may help to write the equation in the following manner i2 Jilij g 31 z olt f0v vf0vlt1gtf Ni v v ap my ap If we can find the signs of the terms on the right hand side RHS of the equation we will be able to tell whether heights are going to rise or fall a The vorticity advection term Though the terms on the RHS look intimidating they really aren t The first term on the RHS is nothing more than absolute vorticity advection Absolute vorticity advection 41 0 V g f 429 0 VEfiVZ f 0 So if vorticity advection is positive this means that the geopotential tendency I is negative falling heights b The differential thermal advection term Remember that we earlier found that p 3 1 R 3p This means that the second term on the RHS is proportional to the vertical derivative of temperature advection If temperature advection decreases with height increases with pressure then i 47 0V 33 gt0 3p 3 3p and sox will be positive so heights will be rising Two important points to note 1 Remember that we are using pressure coordinates so if something is increasing with height it is decreasing with pressure and therefore aap lt 0 2 It is the vertical derivative of the advection that matters Strong cold advection over weak cold advection has the same effect as weak warm advection over strong warm advection because in both cases the derivative has the same value c The differential diabatic heating term The differential heating term third term on RHS behaves similarly to the differential thermal advection term It the heating decreases with height or cooling increases with height then heights will rise Another useful way of writing the essence of the QrG tendency equation is Mat olt absolute vorticity advection 8l8pthermal advection 88pheating or awe x iabsolute vorticity advection iaazthermal advection Q uheating The previous analysis leads us to a very important conclusion In quasi geostrophic theory there are only three ways for heights to fall either through positive vorticity advection through warm advection that increases with height or through diabatic heating that increases with height A PHYSICAL INTERPRETATION OF THE TENDENCY EQUATION The effects of the terms on the RHS of the mndency equation can be explained physically as well as mathematically a Vonieity advection The physical explanation of the vorticity advection term is rather obviou we now that positive vorticity is associated with low heights so if vortici yi 39 quot quot t quot quot the hpi ht will fallr 39 r absolute vorticity of the disturbance but only serves to propagate itr L me 17 Differential theml advection The effects of differential thermal advection can be thought of as follows Imagine a scenario where there is net warm advection in the lower levels below 500 mb and net cold advection in the upper levels above 500 mb Sinc the thickness between two pressure surfaces is proportional to temperature the lowrlevel warm advection will lead to increased thickness of the 1000 r 500 mb layer while the upper level cold advection will lead to decreased thickness of the 500 r 200 mb layer The net result is height rises at 500 mb see diagram below 200 ml 200 ml cold mlvecrmu 500 mb A 500 mb advection 1000 mb wiluuo mb The same result will occur with weak warm advection aloft and stronger warm advection in the lowrlevels see diagram below 7 200 ml 200 m0 ear mnn mlmrinu 500 m0 500 ml mm lesrrinn 1000 ml wimno m0 If the advection is the same strength both aloft and below then there is no change in height at 500 mb see diagram below 100 ml 77 200 m0 x mm mlmnnu nun advertinn 1000 m0 wimnn m0 Typically thermal advection is very small in the upper troposphere above 500 mb compared to that in the lowerrlevels so it is really the low level advection at determines the 500 mb geopotential tendencyr Co advection in the lower levels will decrease the thickness of the 1000 7500 mb layer and lower the heights at 500 mb as would be expected since cold advection decreasing with height is the same as w nn advection increasing with height Warm advection in the lower levels will increase the thickness of the 1000 7500 mb layer and result in height rises at 500 mb 4 Di erential diabatic heating The physical interpretation for the differential diabatic heating term is similar to that for differential advection If there is more heating above a level than below it the heights at that level will fall Phrased another way we can say that above the level of maximum heating Jp the heights will rise and below the level of maximum heating heights will fall Le CHATELIER S PRINCIPLE Le Chamlier39s Principle named for Henry Louis Le Chatelier stams that a thermodynamic system will resist changes in temperature and other thermodynamic properties and if forced such that the temperature changes will react with process that try to restore the original temperature Though Le Chamlier39s Principle isn39t as rigorous and general as often thou t o be we can see Le Chamlier39s principle at workin the differential thermal advection and diabatic heating terms of the QrG tendency equation For example cold advection or diabatic cooling over warm advection or diabch heating forces height rises at 500 mb as well as height falls at 200 and 1000 mb as per the diagram below see1de Hear J Chem Edna 3A 375 1957 gt9 100 mb 200 mb mm allvecrinn adiabatic arming 500 mb 500 mb mnn 9 gt nllVPt nll nl oba cTcnn ng 1000 ml 71000 ml Hr mm 39 39 39 and rm W 39 t it mm 1 t WM h man A L L L h hcri x T UniCity L l c rh39 mm nrk m m h man A m r a h hcri w 1 ummjon and adiabatic cooling in the lower levels and subsidence and adiabch warming in the upper levels both of which served to oppose the temperature change due to advectionr LeChatelier s Principle doesn t mean that the effects of the differential heating advection will be completely cancelled by the adiabch heatinchooling from the quotquothntdne39 quot39L imposed by the thermal forcing and will respond with a secondary circulation EFFECTS OF STATIC STABILITY The static stability of the troposphere on the synoptic scale is rarely negative Since static stability appears in the denominator of the heating mrms an increase in static stability will cause the height rises or falls from these mum to be of a less magnitude than in a less stable atmosphere This is intuitively satisfying The vertical change of static stability is a little more complex The static stability term which we ve pie iou I 39 39 2 I x Lo miquot l 7 8p Siquot 39 39 igh this L vertical motion will lead to a lowering of the geopotential heights while upward motion will lead to raising of geopotential heights To understand this physically recall the from the 39 1 at u u a 39 a motion 39 ia at X on The impact of the vertical motion is enhanced as the static stability increasesr Therefore if quot quot erewillbem39e 39 quot quot hr i larger than at lower altitudes where o39is smallerr Thus there will be heating increasing with height which we have already seen leads to height fallsr For upward motion there will be more cooling aloft than below which will lead to height risesr If static stability decreases with height then the effect of vertical motion is opposite from that just described since there will be larger heating or cooling below rather than aloft AN ADVANCED TREATMENT OF THE TENDENCY EQUATION Since we ve previously assumed that disturbances in the atmosphere have a sinusoidal horizontal structure we will also assume that the forcing terms on the RHS of the tendency equation also have a sinusoidal structure So we assume 106 y p t Xptexpkx ly vorticity advection term Fv p eXp kx l y thermal advection term eXp kx l y P diabatic heating term eXp kx l y P and put these into the tendency equation This gives an ordinary differential equation 2 d X K2X 12 Fv didi div2 f0 dp dp K2 2 k2 12039 f o2 we ve assumed the static stability parameter a is constant with height The solutions to this equation are hyperbolic sines and cosines with a characteristic vertical length scale of 121K So the longer the wavelength of a disturbance lower values of K indicate longer wavelengths the deeper the e ects of its forcing terms are felt in the atmosphere QUASIGEOSTROPHIC POTENTIAL VORTICITY The tendency equation ignoring the diabatic heating term J can be written as D 73 ivchfi 0 Dt f0 8p 039 8p The quantity in brackets is called the quasigeostrophic potential vorticity quV2 fi f0 3p 0 3p and is conserved following a uid parcel EXERCISES 1 Derive the Q G tendency equation showing all steps 2 Is the vertical extent of the forcing terms in the Q G tendency equation larger or smaller in the tropics as compared to the middle latitudes 3 Holton 67 ESCI 343 Atmospheric Dynamics II Lesson 8 Sound Waves References An Introduction to Dynamic Meteorology 3rd edition JR Holton Waves in Fluids J Lighthill Reading Holton Section 731 SOUND WAVES We will limit our analysis to sound waves traveling only along the x axis but keep in mind that we could easily extend this to waves traveling in an arbitrary direction We start with the linearized equations of motion for the case of zero mean ow and for which the reference density is constant with height These are LIl3 at 0 a 1 2231 3 pax We also have an equation of state that relates any three of the thermodynamic variables We will use 0 p and Gas our thermodynamic variables so our equation of state can be written as pp00 The equation of state can be written in differential form as 3p 3p d 7 d 7 do 2 P M feel 0 Sound waves are adiabatic so that 0is constant Therefore we can write 1591 LP dt apgdt 0f szipLP39 a apgat If we substitute this into the continuity equation then linearized set of equations for one dimensional sound waves are then Bu39l3i 3 ax alzsip 31 at 80 08x which are two equations in two unknowns To find the dispersion relation for sound waves we assume sinusoidal solutions of 14 Adam p B Kiowa and substitute them into the two prior equations to find that these are nondispersive waves travelling at a phase speed of c i 8p 0 Since we know that these are sound waves we have shown that for a general uid the speed of sound is given by 052 E 3 8p 0 This shows us that the speed of sound is fundamental property of a uid It also shows us that sound waves are non dispersive1 THE CONTINUITY EQUATION WRITTEN WITH THE SPEED OF SOUND Using equations 3 and 2 we can write Dp LE E cf Dt and using this the fully compressible continuity equation can be written as This is a commonly used way of writing the continuity equation THE SPEED OF SOUND IN AN IDEAL GAS In an ideal gas the equation of state has the form p pR T In terms of potential temperature this can be written as 1 Actually we ve only shown that linear sound waves are nondispersive We haven t and won t discuss any of the effects of nonlinear sound waves R p pR 0amp quot P Taking the partial derivative of this with respect to density at constant potential temperature and making use of the fact that for an ideal gas cp 2 CV R after some patience you can find that for an ideal gas the speed of sound is given by where SOUND WAVES WITH A NONZERO MEAN FLOW So far we ve ignored the mean ow If there is a basic state mean ow then the analysis is slightly more complex Our linearized equations of motion become Lu39 Lugag at 3x 53x 1 339 7339 731439 71 p p 2 iui 05 at 3x 3x Remember that our goal is to find the dispersion relation for the waves We do so by assuming a sinusoidal form for both dependent variables of the form u Aeichrw p Beukxwx We then substitute these directly into the two equations to find the dispersion relation for sound waves in a mean ow a kIZ i cs 0 12 i cs Note that the effect of the mean ow is additive This is a property of linear waves For linear waves the phase speed with mean ow is just the phase speed without the mean ow plus the mean ow itself VERTICALLY PROPOGATING SOUND WAVES For sound waves propagating non horizontally the linearized governing equations are Bu39 l 31139 Bit 3x 3v39l3p39 at 3 3y 3w 1 Bp39p39 at 5 E 19p39 B au39 av39 Bw 72 w 0 7 7 7 cs at dz 3x 3y dz For sufficiently large wave number sufficiently small wavelengths it turns out that we can ignore the effects of buoyancy and vertical gradient of the reference pressure In practical terms this means as long as the waves are sufficiently short such that the wavelength is small compared to the scale over which pressure and density change with height or 1 85 l7 31 details can be found in Lighthill Section 42 So as long as we limit ourselves to sound waves in the normal range of human hearing and are not concerned with infra sonic waves long wavelength sound waves our equations are Kgtgt 3u39 lap39 Bit E 3x 3v39l3p39 af ay Bw l3p39 at az iap39 au39 av39 Bw 4777 p7777 cs at 3x 3y dz Substituting sinusoidal forms for 14 v w and 0 of 14 Aeikxlywz7w v Beikxlywzrw W Ceikxyw7w p Deikxlywz7u yields the following dispersion relation 02 05 k2lzm2ch2 EXERCISES 1 The linearized governing equations for one dimensional sound waves with zero mean oware LI19 at 53x 18p39 781439 of at pax Substitute the assumed solutions 1439 Ae lk quot Beluga into these equations to derive the dispersion relation for sound waves 2 The linearized governing equations for three dimensional sound waves with zero 3 4 mean ow are Bu lap 3v39 lap39 335 3w39 lap39 33 Leggy of at 3x 3y dz Derive the dispersion relation 602 052k2 l2 m2ch2 for these waves Show that the speed of sound in an ideal gas is 05 inR T 3 Find the scale height and speed of sound for an isothermal atmosphere with a temperature of 255K b For this atmosphere find out how large the wavelength of an acoustic wave would need to be before we started concerning ourselves with the effects of gravity and buoyancy on these waves ESCI 343 Atmospheric Dynamics II Lesson 9 Internal Gravity Waves References An Introduction to Dynamic Meteorology 3rd edition JR Holton AtmosphereOcean Dynamics AE Gill Waves in Fluids J Lighthill Reading Holton 741 THE BRUNTVAISALA FREQUENCY Before progressing with an analysis of internal waves we should review the important concept of the Brunt Vaisala frequency The linearized vertical momentum equation is DW39 1 3p 0 Dr 5 3z 0 8 Assuming that the atmosphere is in hydrostatic balance and that a rising parcel s pressure always adjusts to that of the environment then p 0 and we have DW 7 g 1 Dr 0 As a parcel of air rises from some original level z to a new level z Az p will change We can calculate this change as follows 1 First we assume that the density of the parcel remains unchanged as it rises so that 0 at the new level z is given by 7 7 7 7 d7 d7 p zAzgt pltzgt pz Azgt PZ 0Z AZ 372 2 But as the parcel rises it will also expand and its density will change as the pressure changes according to Ap dip Ap izAp i gAz dp 0 cs cs Adding this effect onto 2 yields p z Az 1 LfJAz 3 dz cs and so 1 becomes D7W 141872 Di 5 1 cf which can also be written as 2 DDt2AzN2Az0 4 where N2 E dli s 0 If N 2 is positive solutions to 4 are oscillatory and of the form Azt Acos Nt B sin Nt N is the fundamental frequency of the oscillation and is referred to as the BruntVdisa la frequency or buoyancy frequency For an ideal gas the Brunt Vaisala frequency can be written in terms of potential temperature as N2 g 6 0 dz Equation 6 is valid for an ideal gas only whereas equation 5 is true for any uid For an ideal gas equations 5 and 6 are equivalent see exercises DISPERSION RELATION FOR PURE INTERNAL WAVES For the present discussion we will ignore changes in density due to local compression or expansion which is a valid assumption as long as the waves are short compared to the scale at which the density changes with height large values of wave number We will therefore use the linearized anelastic continuity equation so that the governing equations are 314 1 31239 at 5 3x 31 1 31239 at 7 9y aw39 1 31239 039 573u39 31 aw39 w 7 which when written in ux form are 7 a pu 1 1a 3 v ai 7b a 7 a pw i pg 7x a 7 a 7 a 7 7puipvipw0 7d 3x 3y dz We ve also seen from equations 3 and 5 that p BNZAz 7e 8 Equation set 7a e are the governing equations for internal gravity waves They can be reduced in number as follows 1 take 33 of 7d and combining it with aax of 7a aay of 7b and 332 of 7c to get 4 v2 p 821 2 take aza of 7e to get 32 7 a N2 a 7 P N2l a g at 37 and combine with 7c to get 2 2 2 a I N2p NiaL 8b at g az Equations 8a and 8b are two equations in two unknowns Substituting the sinusoidal solutions p Aeikxlywzrw Beikxlyrrziwt into equation set 8 yields the following dispersion relation for internal gravity waves of 2 2 K N 602i klNiH 9 We2 12 m2 K where K H k2 12 is the horizontal wave number and K We2 12 m2 is the total wave number ANALYSIS OF INTERNAL WAVE DISPERSION The dispersion relation for internal waves 9 shows that for purely horizontal waves K KH the frequency is equal to the Brunt Vaisala frequency For non horizontally traveling waves the frequency is less than the Brunt Vaisala frequency Therefore the Brunt Vaisala frequency is an upper limiting frequency for internal waves In other words for internal waves 02 S N 2 The phase speed for internal waves is given by CQi7KHN K K2 The phase velocity is given by c a A K N A A A CFKi KH3 kllmk 10 The group velocity is a A A A 2 A A K2 A cga ia ja ki quot1N3 lawn Jig 11 8k Bl am KHK m Inspection of 10 and 11 shows a curious fact that for internal waves if there is a downward component to the phase velocity then there is an upward component to the group velocity and vice versa In fact by taking the dot product of 10 and 11 we find that Q cog 0 mh other in the vem39mlpme Th 1 mumted m the dtagmm below EXERC SES 1 Show that for an 1deal g3 2 relauon for mtemal gmvtty wavex Ki N2 72 K 3 a Show that the group veloctty for mtemalwave I 2 NkflrKi 1 m t MK I What I the mammde of the group veloctty for purely Vemcally propagatmg wave7 0 wave7 4 Uxe equattom 10 and 11 to how that for mtemalwavex r 5 For an ideal incompressible gas the linearized governing equations in the xz plane can be written as 31439 l 31239 at 5 3x 91 4 9 at 5 dz 0 81439 3w39 7 7 3x dz 85 0 7Az 3z a Substitute d into b and then take 33 of the resulting equation to get azw39 l 321239 N2 a 0 ataz a b C d e b Substitute sinusoidal solutions into a c and e to find the dispersion relation and phase speed c What kind of waves are these ESCI 343 Atmospheric Dynamics II Lesson 4 Introduction to Waves Reference An Introduction to Dynamic Meteorology 3 d edition JR Holton Waves in Fluids J Lighthill AtmosphereOcean Dynamics AE Gill Reading Holton Section 72 GENERAL The governing equations support many wavelike motions waves are broadly defined as oscillations of the dependent variables Some of the waves supported by the equations are External surface gravity waves Internal gravity waves Inertia gravity waves Acoustic waves including Lamb waves Rossby waves Kelvin waves Kelvin Helmholtz waves Some of these waves are important for the dynamics of synoptic scale systems while others are merely noise In order to understand dynamic meteorology we must understand the waves that can occur in the atmosphere BASIC DEFINITIONS amplitude 7 half of the difference in height between a crest and a trough wavelength 1 7 the distance between crests or troughs wave number K 7 271791 the number of radians in a unit distance in the direction of wave propagation sometimes the wave number is just defined as 11 in which case it is the number of wavelengths per unit distance 0 A higher wave number means a shorter wavelength 0 Units are radians ml or sometimes written as just ml 0 We can also define wave numbers along each of the axes I k is the wave number in the x direction k 2777911 I l is the wave number in the y direction l 2 271791 I m is the wave number in the z direction m 2777912 0 The wave number vector is given by K E ki l ml don t confuse k and I and points in the direction of propagation of the wave I angular frequency 0 7 27rtimes the number of crests passing a point in a unit of time 0 Units are radians s sometimes just written as s l I phase speed c 7the speed of an individual crest or trough o For a wave traveling solely in the x direction c talk 1 For a wave traveling solely in the y direction e all For a wave traveling solely in the z direction e calm For a wave traveling in an arbitrary direction 0 wK where K is the total wave number given by K2 k2 12 m2 For a wave traveling in an arbitrary direction there is a phase speed along each axis given by ex wk ey all and 0z 2 calm Note that these are not the components of a vector c2 2 e cj c 000 O 5 exieyjezl The phase velocity vector is actually given by group velocity cg 7 the velocity at which the wave energy moves Its components are given by If the group velocity is the same as the phase speed of the individual waves making up the packet then the waves are nondispersive If the group velocity is different than the phase speed on the waves making up the packet then the waves are dispersive dispersion relation 7 an equation that gives the angular frequency of the wave as a function of wave number wF iml Each wave type has a unique dispersion relation One of our main goals when studying waves is to determine the dispersion relation THE EQUATION FOR A WAVE The equation for a wave traveling in the positive x direction is uxt Asinkx w tBcoskx w t An alternate way of writing this is uxt Asinkx etBcoskx et For a wave traveling in the negative X direction the equation is uxi Asinkxw tBc0skxw t EULER S FORMULA Euler s formula states that e cos0 isin0 2 From Euler s formula we have the following two identities i0 fie e e cos07 sin 0 i 67 2 Using Euler s formula a wave traveling in the positive x direction can be written as mm Ae l m a wave traveling in the negative x direction can be written as mm Aeilm where the amplitude A may itself be a complex number A a iai and gives information about the phase of the wave We will frequently use this complex notation for waves because it makes differentiation more straightforward because you don t have to remember whether or not to change the sign as you do when differentiating sine and cosine functions The complex amplitude A gives information about the phase of the wave In this form we have the following phase relations between two waves u and v given by u Ae lk v Be lkH u 0lt v in phase u 0lt iv 90 out of phase u 0lt v 180 out of phase u 0lt iv 270 out of phase SPECTRAL ANALYSIS It is rare to find a wave of a single wavelength in the atmosphere Instead there are many waves of different wavelengths superimposed on one another However we can use the concept of spectral analysis to isolate and study individual waves recognizing that we can later sum them up if need be So keep in mind that real atmospheric disturbances are a collection of many individual waves of differing wavelengths Fourier Series Applies to Continuous Periodic Functions Most continuous periodic functions period 2 L can be represented by an infinite sum of sine and cosine functions as fx a0 2a cos 272m 2b 8 711 n1 3 where the Fourier coefficients are given by 1Lz 7 d a0 Lil20 x LZ a 3 j fxcos 27 de nu L Lz b 3 j fxsin27mxdx L LZ L The Fourier coefficients give the amplitudes of the various sine and cosine waves needed to replicate the original function 0 The coefficient no is just the average of the function 0 The coefficients on are the coefficients of the cosine waves the even part of the function The coefficients 71 are the coefficients of the sine waves the odd part of the function For a completely even function the bn s would all be zero while for a completely odd function the on s would be zero Fourier series can also be represented using complex notation and in this notation fx 2 an exp 2 where the coefficients ocn are complex numbers with the real part representing the amplitudes of the cosine waves and the imaginary part representing the amplitudes of the sine waves 1 1 2 i27rnx a 4a 412 7 j fxexp7dx 2 L LZ L Each of the Fourier coefficients ocn are associated with a sinusoidal wave of a certain wavelength If the original function contained one pure wave then there would only be two Fourier coefficients m and 171 The more sinusoids more wave numbers needed to represent the function the more Fourier coefficients are necessary In general 0 Smoother functions require fewer waves to recreate and have fewer higher frequency components Sharper functions require more waves to recreate and have more higher frequency components Broad functions require fewer waves to recreate and have fewer higher frequency components Narrow functions require more waves to recreate and have more higher frequency components Fourier Transforms Applies to Continuous Aperioa39ic Functions Fourier analysis can be extended to functions that are continuous but not periodic aperiodic functions This is done by representing the function as an infinite integral 4 fx 2i jFltkgtexpiltxdk 1 72 7m where the Fourier coefficients are represented by Fk which is a complex number given by Fltkgt jfxgtexp ikxdx 2 Equations 1 and 1 are called the Fourier transform pairs Equation 1 is the representation of the function in physical space Equation 2 is the representation of the function in frequency or wave number space As with Fourier series the real part of the Fourier coefficient ReFk represents the cosine or even part of the function while the imaginary part lmFk represents the sine or odd part of the function FOURIER SPECTRA OF SOME EXAMPLE FUNCTIONS As mentioned previously sharp narrow functions have more and higher frequency waves in their Fourier spectra then do smooth broad functions The figures below shows some example functions and their associated Fourier spectra The first four figures show box functions of various width while the second four pictures show Gaussian curves of various width Things to note 0 In general the narrower the function the broader the spectrum and vice versa The power series of a Gaussian curve is also a Gaussian curve An impulse function has an infinitely broad power spectrum while an infinitely broad function has a single spike for its power spectrum xx Fx rm ftx rw fx foo x Nnrnnli ed Pn erSpeenum 06 06 04 00 02 04 06 k Normalized Power Specllum 1 h 704 702 00 02 04 06 k Normalized Power Speeuum 20 J 702 00 O 2 04 k Normalized Power Speeuum 70 4 4 1 0 397 04 06 Nnrm39lli ed Pn er Spectmm a 0 4 432 00 O 2 04 06 k Nnrm39lh39 ed Pn er Speouum 1 M A 70 4 J 2 0 0 O 2 O 4 06 EXERCISES 1 Show the following to be true cos kx at Rele lkx mj coskx at Re e ljmm sinkx at Re ieikx sinkx at Reieikx 2 Show the following to be true Biezlkmx ikeiomu x 72264th k26ieru 83 ezlkmx iw email 1 gawk a2 Emma 3 A wave is represented in complex notation as uxt Ae lk where A 2 3i Show that this is equivalent to representing the wave as uxt 2coskx cot3sinkx at 4 Find the phase difference between the following two waves uxt Ae lk vxt Be lk for the following values of A and B a A23i B 32i b A23i B 2 3i c A23i B3 2i d A23i B46i e A23i B9 6i 5 a Let a wave be represented by ux 6 Show that u and dudx are 270 out of phase b Let a wave be represented by ux coskx Show that u and dudx are 270 out of phase which shows the consistency of representing sinusoids using complex notation 6 A wave traveling in two dimensions is represented as ux yt Aellkmy Show that V214 k2 lzu demonstrating the Laplacian of a sinusoidal function is proportional to the negative of the original function 7 What is the physical meaning of a complex frequency In other words if cohas an imaginary part what does this imply Hint Put a a icoi into M eikxiwt and see what you get