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Unformatted text preview: 15. Hydromagnetic waves Reading: Shu, Vol.II, Ch.22 15.1 Electrostatic waves Let us start with the simplest possible case of a vanishing external magnetic field, and a homogeneous ( ρ =const.) cold ( T = 0) fluid at rest ( V = 0). We are interested in longitudinal waves, so our ansatz is ~ E 1 = E ~e k exp( ı ~ k ~x- ı ω t ) ⇒ ~ ∇ × ~ E 1 = 0 = ∂ ~ B ∂t ⇒ ~ B = 0 ∀ t (15 . 1) and we are dealing with electrostatic waves. Let us further consider frequencies so high that only the electrons can follow. Then the problem is essentially one-dimensional. The relevant equations are ~ ∇ · ~ E = 4 π ρ e = 4 π e ( Z n i- n e ) (15 . 2) 4 π ~ j e =- ∂ ~ E ∂t ’ - 4 π e n e ~ V e (15 . 3) ∂n e ∂t =- ~ ∇ · ( n e ~ V e ) (15 . 4) m e n e ∂ ~ V e ∂t + ( ~ V e ·∇ ) ~ V e =- e n e ~ E (15 . 5) Suppose all variables are composed of the equilibrium value plus a small, wavelike perturbation according to Eq.15.1, so all derivatives turn into multiplications with ı k or- ı ω . Then retaining only terms linear in the perturbation we obtain ı k E 1 =- 4 π e n e, 1 ∧ ı ω E 1 =- 4 π e v 1 n e, ω n e, 1 = n e, k v 1 ∧ n e, ı ω v 1 = e n e, m e E 1 (15 . 6) This set of equations is solved by the dispersion relation ω 2 = 4 π e 2 n e, m e = ω 2 p,e (15 . 7) where we recover the electron plasma frequency ω p,e that we had used earlier. The longitudinal waves that follow this dispersion relation are called electron plasma waves or Langmuir waves. The phase velocity and group velocity of Langmuir waves are v Φ = ω k = ω p,e k v gr = ∂ω ∂k = 0 (14 . 8) so the wave energy does not propagate in the cold plasma limit. 1 15.2 Electromagnetic waves Now we are considering transverse waves, for which the magnetic components cannot be ne- glected. For the moment we will still assume that an external magnetic field does not exist....
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This note was uploaded on 02/29/2012 for the course PHYS 227 taught by Professor Rabe during the Fall '08 term at Rutgers.
- Fall '08