Fundamentals of Plasma Physics

14.3 Echoes 415

̃v 1 (ω,t)consists of a particular solution satisfying the inhomogeneous part of the equation
(i.e., balances the driving term on the right hand side) and a homogeneous solution (i.e., a
solution of the homogeneous equation (left hand side of Eq.(14.87))). The coefficient of the
homogeneous solution is chosen to satisfy the boundary condition att=0. The particular
solution is assumed to vary aseikx−iωtand so is the solution of the equation

(−iω+ikv 0 ) ̃v 1 =−

eE ̄

m

e−iωt. (14.89)

The homogenous solution is the solution of

∂ ̃v 1

∂t

+ikv 0 ̃v 1 =0 (14.90)

and has the formv ̃ 1 h=λexp(−ikv 0 t)whereλis a constant to be determined. Adding the
particular and homogeneous solutions together gives the general solution

v ̃ 1 =−

eE ̄

m

ie−iωt

(ω−kv 0 )

+λe−ikv^0 t (14.91)

whereλis chosen to satisfy the initial condition. The initial conditionv 1 = 0att= 0
determinesλand gives

̃v 1 (ω,t)=−

ieE ̄

m

(

e−iωt−e−ikv^0 t

)

(ω−kv 0 )

(14.92)

as the solution which satisfies both Eq.(14.87) and the initial condition. The term involving
e−ikvtis called the ballistic term. This term contains information about the initial condi-
tions, is a solution of the homogeneous equation, is missed by Fourier treatments, is incor-
porated by Laplace transform treatments, and keepsv 1 from diverging whenω−kv→ 0.
If we wished to revert to the time domain, then the contributions of all the harmonics
would have to be summed, giving

v 1 (t)=−

ieE ̄

2 πm

∫

dω

(

e−iωt−e−ikv^0 t

)

(ω−kv 0 )

. (14.93)

14.3.1Ballistic terms and Laplace transforms

The discussion above used an approach related to Fourier transforms, but added additional
structure to account for the initial condition thatv 1 =0att=0.This suggests that Laplace
transforms ought to be used, since Laplace transforms automatically take into account ini-
tial conditions. Let us therefore Laplace transform Eq.(14.87) to see ifindeed the particular
and ballistic terms are appropriately characterized. The Laplace transform of Eq.(14.85)
gives

p ̃v 1 +ikv 0 v ̃ 1 = −

eE ̄

m

∫∞

0

dte−iωt−pt

= −

eE ̄

m

1

iω+p

(14.94)