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At that time, it will become clearwhy we had to use Poisson’s equation in the derivation of Debye shielding.Chapter 4Waves in Plasmas4.1Representation of WavesAny periodic motion of a fluid can be decomposed by Fourier analysis into asuperposition of sinusoidal oscillations with different frequencies ω and wavelengths λ.
A simple wave is any one of these components. When the oscillationamplitude is small, the waveform is generally sinusoidal; and there is only onecomponent. This is the situation we shall consider.Any sinusoidally oscillating quantity—say, the density n—can be represented asfollows:n ¼ n exp ½iðk r ωtÞð4:1Þwhere, in Cartesian coordinates,k r ¼ kx x þ k y y þ kz zð4:2ÞHere n is a constant defining the amplitude of the wave, and k is called thepropagation constant. If the wave propagates in the x direction, k has only anx component, and Eq. (4.1) becomesn ¼ n eiðkxωtÞBy convention, the exponential notation means that the real part of the expressionis to be taken as the measurable quantity.
Let us choose n to be real; we shallsoon see that this corresponds to a choice of the origins of x and t. The real part ofn is thenThe original version of this chapter was revised. An erratum to this chapter can be found at https://doi.org/10.1007/978-3-319-22309-4_11© Springer International Publishing Switzerland 2016F.F. Chen, Introduction to Plasma Physics and Controlled Fusion,DOI 10.1007/978-3-319-22309-4_475764 Waves in PlasmasRe ðnÞ ¼ n cos ðkx ωtÞð4:3ÞA point of constant phase on the wave moves so that (d/dt)(kx ωt) ¼ 0, ordx ω¼ υφdtkð4:4ÞThis is called the phase velocity. If ω/k is positive, the wave moves to the right; thatis, x increases as t increases, so as to keep kx ωt constant.
If ω/k is negative, thewave moves to the left. We could equally well have takenn ¼ n eiðkxþωtÞin which case positive ω/k would have meant negative phase velocity. This is aconvention that is sometimes used, but we shall not adopt it. From Eq. (4.3), it isclear that reversing the sign of both ω and k makes no difference.Consider now another oscillating quantity in the wave, say the electric field E.Since we have already chosen the phase of n to be zero, we must allow E to have adifferent phase δ:E ¼ E cos ðkx ωt þ δÞor E ¼ E eiðkxωtþδÞð4:5Þwhere E is a real, constant vector.It is customary to incorporate the phase information into E by allowing E to becomplex. We can writeE ¼ E eiδ eiðkxωtÞ Ec eiðkxωtÞwhere Ec is a complex amplitude. The phase δ can be recovered from Ec, since Re Ec ¼ E cos δ and Im Ec ¼ E sin δ; so that Im Ec tan δ ¼Re Ecð4:6Þg1 ¼ g1 exp ½iðk r ωtÞð4:7ÞFrom now on, we shall assume that all amplitudes are complex and drop thesubscript c.
Any oscillating quantity g1 will be writtenso that g1 can stand for either the complex amplitude or the entire expressionEq. (4.7). There can be no confusion, because in linear wave theory the sameexponential factor will occur on both sides of any equation and can becancelled out.4.2 Group Velocity77Problem4.1 The oscillating density n1 and potential ϕ1 in a “drift wave” are related byn1 eϕ1 ω* þ ia¼n0 KT e ω þ iawhere it is only necessary to know that all the other symbols (except i) stand forpositive constants.(a) Find an expression for the phase δ of ϕ1 relative to n1. (For simplicity,assume that n1 is real.)(b) If ω < ω* , does ϕ1 lead or lag n1?4.2Group VelocityThe phase velocity of a wave in a plasma often exceeds the velocity of light c. Thisdoes not violate the theory of relativity, because an infinitely long wave train ofconstant amplitude cannot carry information.
The carrier of a radio wave, forinstance, carries no information until it is modulated. The modulation informationdoes not travel at the phase velocity but at the group velocity, which is always lessthan c. To illustrate this, we may consider a modulated wave formed by adding(“beating”) two waves of nearly equal frequencies. Let these waves beE1 ¼ E0 cos ½ðk þ ΔkÞx ðω þ ΔωÞtE2 ¼ E0 cos ½ðk ΔkÞx ðω ΔωÞtð4:8ÞE1 and E2 differ in frequency by 2Δω. Since each wave must have the phasevelocity ω/k appropriate to the medium in which they propagate, one must allowfor a difference 2Δk in propagation constant.
Using the abbreviationsa ¼ kx ωtb ¼ ðΔkÞx ðΔωÞtwe haveE1 þ E2 ¼ E0 cos ða þ bÞ þ E0 cos ða bÞ¼ E0 ð cos a cos b sin a sin b þ cos a cos b þ sin a sin bÞ¼ 2E0 cos a cos bE1 þ E2 ¼ 2E0 cos ½ðΔkÞx ðΔωÞt cos ðkx ωtÞð4:9Þ784 Waves in PlasmasFig. 4.1 Spatial variation of the electric field of two waves with a frequency differenceThis is a sinusoidally modulated wave (Fig.
4.1). The envelope of the wave, givenby cos [(Δk)x (Δω)t], is what carries information; it travels at velocity Δω/Δk.Taking the limit Δω ! 0, we define the group velocity to bevg ¼ dω=dkð4:10ÞIt is this quantity that cannot exceed c.4.3Plasma OscillationsIf the electrons in a plasma are displaced from a uniform background of ions,electric fields will be built up in such a direction as to restore the neutrality of theplasma by pulling the electrons back to their original positions.
Because of theirinertia, the electrons will overshoot and oscillate around their equilibrium positionswith a characteristic frequency known as the plasma frequency. This oscillation isso fast that the massive ions do not have time to respond to the oscillating field andmay be considered as fixed. In Fig. 4.2, the open rectangles represent typicalelements of the ion fluid, and the darkened rectangles the alternately displacedelements of the electron fluid. The resulting charge bunching causes a spatiallyperiodic E field, which tends to restore the electrons to their neutral positions.We shall derive an expression for the plasma frequency ωp in the simplest case,making the following assumptions: (1) There is no magnetic field; (2) there are nothermal motions (KT ¼ 0); (3) the ions are fixed in space in a uniform distribution;(4) the plasma is infinite in extent; and (5) the electron motions occur only in thex direction.
As a consequence of the last assumption, we have∇¼^x ∂=∂x E ¼ E^x∇E¼0E ¼ ∇ϕð4:11ÞThere is, therefore, no fluctuating magnetic field; this is an electrostatic oscillation.The electron equations of motion and continuity aremne∂veþ ðve ∇ Þve ¼ ene E∂tð4:12Þ4.3 Plasma Oscillations79Fig. 4.2 Mechanism of plasma oscillations∂neþ ∇ ð ne ve Þ ¼ 0∂tð4:13ÞThe only Maxwell equation we shall need is the one that does not involve B:Poisson’s equation. This case is an exception to the general rule of Sect. 3.6 thatPoisson’s equation cannot be used to find E.
This is a high-frequency oscillation;electron inertia is important, and the deviation from neutrality is the main effect inthis particular case. Consequently, we writeε0 ∇ E ¼ ε0 ∂E=∂x ¼ eðni ne Þð4:14ÞEquations (4.12)–(4.14) can easily be solved by the procedure of linearization.By this we mean that the amplitude of oscillation is small, and terms containinghigher powers of amplitude factors can be neglected. We first separate the dependent variables into two parts: an “equilibrium” part indicated by a subscript 0, and a“perturbation” part indicated by a subscript 1:ne ¼ n0 þ n 1ve ¼ v0 þ v1E ¼ E0 þ E1ð4:15ÞThe equilibrium quantities express the state of the plasma in the absence of theoscillation. Since we have assumed a uniform neutral plasma at rest before theelectrons are displaced, we have∇n0 ¼ v0 ¼ E0 ¼ 0∂n0 ∂v0 ∂E0¼¼¼0∂t∂t∂tð4:16Þ804 Waves in PlasmasEquation (4.12) now becomesð4:17ÞThe term (v1 · ∇)v1 is seen to be quadratic in an amplitude quantity, and we shalllinearize by neglecting it.
The linear theory is valid as long as |v1| is small enoughthat such quadratic terms are indeed negligible. Similarly, Eq. (4.13) becomesð4:18ÞIn Poisson’s equation (4.14), we note that ni0 ¼ ne0 in equilibrium and that ni1 ¼ 0by the assumption of fixed ions, so we haveε0 ∇ E1 ¼ en1ð4:19ÞThe oscillating quantities are assumed to behave sinusoidally:v1 ¼ v1 eiðkxωtÞ ^xn1 ¼ n1 eiðkxωtÞE ¼ E1 eiðkxωtÞð4:20Þ^xThe time derivative ∂/∂t can therefore be replaced by iω, and the gradient ∇ byik^x : Equations (4.17)–(4.19) now becomeimωv1 ¼ eE1ð4:21Þiωn1 ¼ n0 ikv1ð4:22Þikε0 E1 ¼ en1ð4:23ÞEliminating n1 and E1, we have for Eq. (4.21)imωv1 ¼ ee n0 ikv1n0 e2v1¼ iikε0 iωε0 ωIf v1 does not vanish, we must haveω2 ¼ n0 e2 =mε0ð4:24Þ4.3 Plasma Oscillations81The plasma frequency is therefore 2 1=2n0 eωp ¼ε0 mrad= secð4:25ÞNumerically, one can use the approximate formulapffiffiffi ω p =2π ¼ f p 9 n n in m3ð4:26ÞThis frequency, depending only on the plasma density, is one of the fundamentalparameters of a plasma.
Because of the smallness of m, the plasma frequency isusually very high. For instance, in a plasma of density n ¼ 1018 m3, we have1=2¼ 9 109 sec 1 ¼ 9 GHzf p 9 1018Radiation at fp normally lies in the microwave range. We can compare this withanother electron frequency: ωc. A useful numerical formula isf ce ’ 28 GHz = Teslað4:27ÞThus if B 0.32 T and n 1018 m3, the cyclotron frequency is approximatelyequal to the plasma frequency for electrons.Equation (4.25) tells us that if a plasma oscillation is to occur at all, it must have afrequency depending only on n.
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