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E. Metral et al.
beam. The computation of these wake fields is quite involved and two fundamental
approximations are introduced:
1. The rigid-beam approximation: The beam traverses a piece of equipment rigidly,
i.e. the wake field perturbation does not affect the motion of the beam during the
traversal of the impedance. The distance z of the test particle behind some source
particle does not change.
2. The impulse approximation: As the test particle moves at the fixed velocity v =
βc through a piece of equipment, the important quantity is the impulse (and not
the force) given by
Δp (x, y, z) =
+∞
−∞
dtF (x, y, s = z + βct, t) =
+∞
−∞
dte (E + v × B) ,
(4.8)
where vectors are designated by boldtype letters. Starting from the four Maxwell
equations for a particle in the beam, it can be shown that for a constant β (which
does not need to be 1) [9]
∇ × Δp (x, y, z) = 0,
(4.9)
which is known as Panofsky-Wenzel theorem. This relation is very general, as no
boundary conditions have been imposed. Only the two fundamental approximations
have been made. Another important relation can be obtained when β = 1, taking the
divergence of the impulse, which is
∇ ⊥ · Δp ⊥ = 0.
(4.10)
Considering the case of a cylindrically symmetric chamber (using the cylindrical
coordinates r, θ , z), yields the following three equations from Panofsky-Wenzel
theorem
1
r
∂Δp z
∂θ
=
∂Δp θ
∂z
,
∂Δp r
∂z
=
∂Δp z
∂r
,
∂ (rΔp θ )
∂r
=
∂Δp r
∂θ
.
(4.11)
The fourth relation when β = 1 writes
∂ (rΔp r )
∂r
= −
∂Δp θ
∂θ
.
(4.12)
Consider now as a source charge density a macro-particle of charge Q = N b e
moving along the pipe (in the s-direction) with an offset r = a in the ϑ = 0 direction
E. Metral et al.
beam. The computation of these wake fields is quite involved and two fundamental
approximations are introduced:
1. The rigid-beam approximation: The beam traverses a piece of equipment rigidly,
i.e. the wake field perturbation does not affect the motion of the beam during the
traversal of the impedance. The distance z of the test particle behind some source
particle does not change.
2. The impulse approximation: As the test particle moves at the fixed velocity v =
βc through a piece of equipment, the important quantity is the impulse (and not
the force) given by
Δp (x, y, z) =
+∞
−∞
dtF (x, y, s = z + βct, t) =
+∞
−∞
dte (E + v × B) ,
(4.8)
where vectors are designated by boldtype letters. Starting from the four Maxwell
equations for a particle in the beam, it can be shown that for a constant β (which
does not need to be 1) [9]
∇ × Δp (x, y, z) = 0,
(4.9)
which is known as Panofsky-Wenzel theorem. This relation is very general, as no
boundary conditions have been imposed. Only the two fundamental approximations
have been made. Another important relation can be obtained when β = 1, taking the
divergence of the impulse, which is
∇ ⊥ · Δp ⊥ = 0.
(4.10)
Considering the case of a cylindrically symmetric chamber (using the cylindrical
coordinates r, θ , z), yields the following three equations from Panofsky-Wenzel
theorem
1
r
∂Δp z
∂θ
=
∂Δp θ
∂z
,
∂Δp r
∂z
=
∂Δp z
∂r
,
∂ (rΔp θ )
∂r
=
∂Δp r
∂θ
.
(4.11)
The fourth relation when β = 1 writes
∂ (rΔp r )
∂r
= −
∂Δp θ
∂θ
.
(4.12)
Consider now as a source charge density a macro-particle of charge Q = N b e
moving along the pipe (in the s-direction) with an offset r = a in the ϑ = 0 direction