20 Doppler shift and Doppler radars

Transcription

1 20 Doppler shift and Doppler radars Doppler radars make a use of the Doppler shift phenomenon to detet the motion of EM wave refletors of interest e.g., a polie Doppler radar aims to identify the speed of a vehile in relative motion. In this leture we will desribe the general priniple of how a Doppler radar works and also learn about the Doppler shift phenomenon in non-relativisti and relativisti limits. Doppler radar: Consider a stationary dipole loated at the origin exited by a osinusoidal input urrent i(t) =I o os(ωt) e jωt + e jωt e jωt + where refers to the omplex onjugate of the term preeding it. The dipole will radiate a spherial wave field E(r,t) os(ωt kr) e j(ωt kr) + where ω k = assuming propagation in vauum or air. 1

2 Consider now a ar speeding away with veloity v from the dipole along the x-axis having an instantaneous loation E i = E o os(ωt kx) x(t) =x o + vt E i = E o os(ω t + k x) x at time t. Thefieldattheloationofthearattimet will then be x o + vt os(ωt k(x o + vt)) = os((ω kv)t kx o ) e j((ω kv)t kx o) +. An indued surfae urrent os(ω t kx o ) on the ar s body osillating at a frequeny ω = ω kv will then radiate like a olletion of dipoles, produing a refleted field os(ω t kx o k(x o +vt)) = os((ω 2kv)t 2kx o ) e j((ω 2kv)t 2kx o) + deteted bak at the loation of the original dipole in this waveform we have inluded an additional phase delay of k(x o + vt) to aount for the return trip of the refleted wave. Clearly, the refleted field osillates with the frequeny ω = ω kv = ω 2kv in the referene frame of the stationary dipole. If the dipole is arranged to detet the refleted wave field (using a T/R swith a radar jargon implying that the antenna is 2

3 swithed to onnet to the input port of a reeiving devie shortly after the transmission of a burst of EM wave), then the veloity of the ar, v, anbeobtainedfrom two-way Dopplershifted frequeny ω.that showpolieradarswork. Note that positive v (motion away from the radar antenna) auses ω <ω and is referred to as redshift, whereas negative v (motion toward the radar antenna) auses ω >ωand is referred to as blueshift. ω = ω kv ω = ω 2kv Doppler shift in relativisti and non-relativisti limits: The one-way and two-way Doppler shift formulae ω = ω kv and ω = ω 2kv obtained above, where v is the relative 1 radial reession veloity of the radiator and the observer, are valid only when v. The reason for this is, our analysis above, leading to these formulae, negleted an important detail that aording to Maxwell s equations we need to have not only ω k =, but also ω k =, 1 It does not matter whether the radiator or the observer is moving sine motion is always relative. 3

4 whereas we have, in effet, used an inonsistent relation ω k intermediate stage. = at an This inonsisteny produes a negligible error if v (the usual ase pertinent for polie radar appliations) but the errors are unaeptably large if v approahes (like in Fermilab). We will refer to the approximate Doppler shift formulae given above as non-relativisti Doppler formulae theyaretobeusedifand only if v/ 1, i.e.,inthenon-relativistilimit. Relativisti Doppler formulae that an be used unonditionally (and most importantly for v/ approahing unity) are ω 1 v 1 = ω 1+ v and ω = ω v 1+ v = ω 1 v 1+ v. Before deriving these relativisti formulae (orret for all v), let us note that they redue to the non-relativisti formula if v/ 1. In that ase we have, for instane, ω = ω 1 v 1+ v = ω (1 v )1/2 (1 + v )1/2 = ω(1 v )1/2 (1 + v ) 1/2 ω(1 v 2 )(1 v 2 ) ω(1 v )=ω kv. 4

5 Derivation of the relativisti formula: To derive the relativisti Doppler shift formulae we will not need ompliated relativisti transformation formulae disussed in PHYS 325 (also summaried in ECE 329 notes). It is suffiient that we make a areful use of Maxwell s equations in developing an aurate model of a field refleted from a refletor in motion as shown next: Consider a plane TEM wave in free-spae, (a) Stationary refletor (in lab frame) E i (x, t) =ẑe o os(ωt kx), inident on a onduting surfae at x =0plane from the left suh that E i = E o os(ωt kx) E i = E o os(ωt + kx) x k = ω. The wave will be refleted to produe E r (x, t) = ẑe o os(ωt + kx) (b) Moving refletor E i = E o os(ωt kx) E i = E o os(ω t + k x) so that the total tangential field at x =0plane vt ẑ (E i (0,t)+E r (0,t)) = E o os(ωt 0) E o os(ωt +0)=0. Now, what would E r (x, t) be if the onduting refletor were not stationary on the x =0plane, but rather moving with a steady veloity v to the right, having a trajetory x = vt as depited in the margin? 5

6 The answer of the question raised above is quite simple: We would have E r (x, t) =ẑf(t + x ), where f(t) is to be determined, so that ẑ (E i (vt, t)+e r (vt, t)) = E o os(ωt kvt)+f(t + vt )=0, beause 1. E r (x, t) =ẑf(t + x ) is a viable (and the only viable) ẑ-polaried wave solution of Maxwell s equations propagating in the x diretion in free spae, and 2. the seond equation above is the relevant boundary ondition to be fulfilled on the surfae of the moving refletor at every instant in time. E i = E o os(ωt kx) E i = E o os(ω t + k x) vt The boundary ondition equation above implies that f(t(1+ v )) = E o os(ωt kvt) = E o os(ωt(1 v )) = E o os(ω 1 v 1+ v t(1+ v )). Thus, and f(t) = E o os(ω 1 v 1+ v t), E r (x, t) =ẑf(t+ x )= ẑe o os(ω 1 v 1+ v (t+ x )) = ẑe o os(ω t+k x), 6

7 with ω = ω 1 v 1+ v and k = ω = ω 1 v 1+ v = k 1 v 1+ v. E i = E o os(ωt kx) E i = E o os(ω t + k x) With vt ω = ω 1 v 1+ v and k = k 1 v 1+ v the refleted wave E r (x, t) =ẑf(t + x )= ẑe o os(ω t + k x) wave is learly a o-sinusoid just like the inident wave but with Doppler shifted frequeny and wavenumbers ω and k,respetively, aused by the motion of the refletor surfae (as disussed below). The result an also be used with negative v orresponding to a refletor moving to the left. The Doppler shift formulae given above are relativistially orret that is, they are valid for all possible values of v eventhoughwedid not invoke any relativisti argument above. This is true beause relativity derives from the Maxwell s equations and the aompanying boundary onditions, and so any rigorous dedution derived from Maxwell s equations will be by default relativistially valid. 7

8 Fousing next on the Doppler shifted frequeny formula ω = ω 1 v 1 v 1+ v 1 v = ω 1+ v 1+ v, we an re-express ω as ω = ω 1 v 1+ v with ω ω 1 v 1+ v. E i = E o os(ωt kx) E i = E o os(ω t + k x) We now reognie the Doppler shifted frequeny vt ω = ω 1 v 1+ v of the refleted wave as a Doppler shifted version of the wave frequeny ω 1 v = ω 1+ v seen in the refletor frame, whihisinturnadopplershifted version of the frequeny ω of the in the inident wave field E i (x, t) defined in the so-alled 2 lab frame. This onludes our derivation of the relativisti Doppler shift formulae stated earlier on. 2 By definition the frame where the unprimed frequeny ω is observed is the lab frame; itanalsobe alled the unprimed frame. 8

9 One-way Doppler shift: When a TEM wave is observed to have a frequeny ω in the lab frame (and wavenumber k = ω/ sine we are onerned with free-spae propagation at this point), the same TEM wave will appear to have a frequeny ω in a seond referene frame whih is in motion within the lab frame. The one-way Doppler shifted frequeny ω will be related to the labframe frequeny ω as ω 1 v = ω 1+ v if the moving observer has a veloity v in the lab frame defined to be positive in the diretion of wave propagation (away from the wave soure). For non-relativisti speeds suh that v 1 we have ω ω(1 v )=ω kv as already seen. This simplified Doppler formula is easy to understand sine E i (x, t) =ẑe o os(ωt kx) E i = E o os(ωt kx) E i = E o os(ω t + k x) vt (see margin) implies that the inident field at the loation x = vt of the refletor must vary with time t as E i (vt, t) =ẑe o os(ωt kvt) =ẑe o os((ω kv)t) =ẑe o os(ω t) 9

10 where as obtained above 3. ω = ω kv Relativisti Doppler shift equations given above are appliable for freespae propagation only the reason is, a frequeny independent propagation veloity was assumed in the derivation of ω. The equations take modified forms 4 for propagation in material media. However, nonrelativisti Doppler equations as the time-rate-of-hange of wave phase are found to be valid in material media where ω/s generally ω dependent. 3 1 v An astute student may ask at this point: how ome E i (vt, t) os((ω kv)t) and not os(ω t) 1+ v if a rigorous appliation of eletromagneti solutions shouldproduerelativistiallyaurateresults(as laimed earlier on)? This is the sort of question Albert Einstein asked to himself in his free time at work in a Swiss patent offie and figured out that the rigorous onlusion ought to be (ω kv)t = ω 1 v 1+ v t t = 1+ v 1 v (1 v )t = 1 v2 t, 2 where t is the time kept by a lok attahed to the refleting surfae. The fat that loks in relative motion keep time at different rates see the relativisti transformation formula between t (measured on the refletor) and t (measured in the lab where the refletor is moving with veloity v) giveninafootnote of Leture 12 in ECE 329 notes was one of the surprising results of the work Einstein published in 1905 under the title On the Eletrodynamis of Moving Bodies, popularly known as the relativity paper. 4 The modified form ω = ω 1 v n, 1 v2 2 where n v p is the refrative index of the medium in terms of propagation speed v p = ω,hardlyomes k up in pratie beause relativisti veloities are rarely enountered within material media. 10