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