Wake-fields in the Main Superconducting L-Band Linacs of the ILC
Nonultrarelativistic resistive wall wake fields
Transcript of Nonultrarelativistic resistive wall wake fields
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
Nonultrarelativistic resistive wallwake fields
Diego QuatraroCERN
&Bologna University
April 7, 2008Many thanks to...G. Rumolo, E. Metral, B. Salvant, B. Zotter..
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
Outline
Motivations and concepts
The fields and the their calculation
Possible effects on bunch dynamics
Conclusions
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
The ideas...1/2
Ring Momentum β γ
PSB @ Injection 0.31 GeV/c ∼ 0.31 ∼ 1.05PSB @ Extraction 1.4 GeV/c ∼ 0.84 ∼ 1.79PS @ Extraction 14 GeV/c ∼ 0.9977 ∼ 14.954
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
The ideas...1/2
Ring Momentum β γ
PSB @ Injection 0.31 GeV/c ∼ 0.31 ∼ 1.05PSB @ Extraction 1.4 GeV/c ∼ 0.84 ∼ 1.79PS @ Extraction 14 GeV/c ∼ 0.9977 ∼ 14.954
Nonultrarelativistic regime
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
The ideas...1/2
Ring Momentum β γ
PSB @ Injection 0.31 GeV/c ∼ 0.31 ∼ 1.05PSB @ Extraction 1.4 GeV/c ∼ 0.84 ∼ 1.79PS @ Extraction 14 GeV/c ∼ 0.9977 ∼ 14.954
Nonultrarelativistic regime⇒ different approach should be considered
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
The ideas...2/2
i) Electromagnetic field travels with the speed of light...
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
The ideas...2/2
i) Electromagnetic field travels with the speed of light...
ii) Bunches could travel slower than light...PSB injection β ' 0.31.
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
The ideas...2/2
i) Electromagnetic field travels with the speed of light...
ii) Bunches could travel slower than light...PSB injection β ' 0.31.
iii) The tail can have an effect on the head dynamics.
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
The ideas...2/2
i) Electromagnetic field travels with the speed of light...
ii) Bunches could travel slower than light...PSB injection β ' 0.31.
iii) The tail can have an effect on the head dynamics.
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
The ideas...2/2
i) Electromagnetic field travels with the speed of light...
ii) Bunches could travel slower than light...PSB injection β ' 0.31.
iii) The tail can have an effect on the head dynamics.
PSfrag replacements
γ
p+
β ' 1
γ
γ
p+p+
β < 1
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
Outline
Motivations and concepts
The fields and the their calculation
Possible effects on bunch dynamics
Conclusions
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
Longitudinal wake field 1/7
Definition of the wake field
W ′0(z) =
12π
∫ ∞
−∞Z‖(ω)eiωz/βcdω (1)
non ultrarelativistic beam F. Zimmermann and K. Oide, PRLSTAB 7, 044201 (2004)
Z‖(ω) = iZ0ck2
r
2πω[K0 (kr r) +
I0(kr r)ω2λK1(bkr )K0(bλ) + kr c2
(
λ2 − k2)
K0(bkr )K1(bλ)
ω2λI1(bkr )K0(bλ)− kr c2 (λ2 − k2) I0(bkr )K1(bλ)
]
(2)kr = |ω|/γβc, k = ω/βc, λ2 = −iµ0σω + k2
r andb = beam pipe radius.
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
Longitudinal wake field 2/7
Subtracting the space charge term from Eq. (2) andletting
|λb| 1 → PSB case√
ωµ0σ ' 2 · 10−2[s1/2]√
ω
|kr b| 1 → PSB case ωb/βγc ' 5.3× 10−10[s]ω(3)
we have
Z‖(ω) = iZ0ω
cβ2γ2 I0(r |ω|/γβc)e−2b|ω|/γβc ·
·[
√
|ω|σµ0 (1− i · sgn (ω))√
|ω|σµ0 (1− i · sgn (ω)) +√
2 · sgn (ω) c/βγ (iµ0σ + ω/c2)
]
.
(4)range of validity ω such that |ωb/βγc| 1: shortdistances.
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
Longitudinal wake field 3/7
log− log plot of Z‖(ω) function
1e-07
1e-06
1e-05
0.0001
0.001
0.01
0.1
1000 1e+06 1e+09
Z⁄
⁄ (ω
)[Ω
]
ω [Hz]
exact Re(Z) β = 0.31
approximate Re(Z) β = 0.31
exact Im(Z) β = 0.31
approximate Im(Z) β = 0.31
1e-07
1e-06
1e-05
0.0001
0.001
0.01
0.1
1
1000 1e+06 1e+09 1e+12
Z⁄
⁄ (ω
)[Ω
]
ω [Hz]
exact Re(Z) β = 0.999
approximate Re(Z) β = 0.999
exact Im(Z) β = 0.999
approximate Im(Z) β = 0.999
ωsup as β → 1.
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
Longitudinal wake field 4/7FFT of impedance Eq. (2) (Exact) andEq. (4)(Approximate)
1⋅103
1⋅104
1⋅105
1⋅106
1⋅107
1⋅108
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
ln
W0’(z)
[Ω
s-1
]
z[m]
β = 0.1
Exact Wake
Approximate Wake
1⋅103
1⋅104
1⋅105
1⋅106
1⋅107
1⋅108
1⋅109
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
ln
W0’(z)
[Ω
s-1
]z[m]
β = 0.2
Exact Wake
Approximate Wake
1⋅105
1⋅106
1⋅107
1⋅108
1⋅109
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
ln
W0’(z)
[Ω
s-1
]
z[m]
β = 0.3
Exact Wake
Approximate Wake
1⋅104
1⋅105
1⋅106
1⋅107
1⋅108
1⋅109
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
ln
W0’(z)
[Ω
s-1
]
z[m]
β = 0.4
Exact Wake
Approximate Wake
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
Longitudinal wake field 5/7
Field in front of the bunch, FFT of exact formula...
1e+06
1e+09
1e+12
0 0.05 0.1 0.15 0.2 0.25
ln
W0’(z)
[Ω
s-1
]
z[m]
Front field
PS Extraction ⇒ β = 0.998PSB Extraction ⇒ β = 0.83
PSB Injection ⇒ β = 0.31
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
Longitudinal wake field 5/7
Field in front of the bunch, FFT of exact formula...
1e+06
1e+09
1e+12
0 0.05 0.1 0.15 0.2 0.25
ln
W0’(z)
[Ω
s-1
]
z[m]
Front field
PS Extraction ⇒ β = 0.998PSB Extraction ⇒ β = 0.83
PSB Injection ⇒ β = 0.31
..for the Head and the Tail of the bunch...
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
Longitudinal wake field 6/7
10000
1e+06
1e+08
1e+10
1e+12
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
ln
W0’(z)
[Ω
s-1
]
z[m]
LongitudinalField
longitudinal CHAOT. field at β = 0.31F. field at β = 0.31
T. field at β = 0.998F. field at β = 0.998 1e+09
1e+12
-0.005 0.000 0.005
Agreement with well-known Chao’s ultrarelativistic formula
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
Longitudinal wake field 7/7
For the ultrarelativistic case...
1⋅108
1⋅1010
1⋅1012
1⋅1014
-0.1 -0.08 -0.06 -0.04 -0.02 0
ln W
0’(z
) [Ω
s-1
]
z[m]
b = 5 cm
β ≈ 1
Chao’s formula
1⋅1010
1⋅1012
1⋅1014
-0.008 -0.004 0
we have a more complete description of the field close tothe bunch.
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
Outline
Motivations and concepts
The fields and the their calculation
Possible effects on bunch dynamics
Conclusions
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
Strong head-tail instability 1/3TRANSVERSE plane simple model
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
Strong head-tail instability 1/3TRANSVERSE plane simple model → two particles beam
A. Chao, Physics of Collective Instabilities in High Energy Accelerators
i) two particle beam→ charge Ne/2
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
Strong head-tail instability 1/3TRANSVERSE plane simple model → two particles beam
A. Chao, Physics of Collective Instabilities in High Energy Accelerators
i) two particle beam→ charge Ne/2ii) s/c ∈ [0; Ts/2] 1 leads 2 and vice versa for
s/c ∈ [Ts/2; Ts]
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
Strong head-tail instability 1/3TRANSVERSE plane simple model → two particles beam
A. Chao, Physics of Collective Instabilities in High Energy Accelerators
i) two particle beam→ charge Ne/2ii) s/c ∈ [0; Ts/2] 1 leads 2 and vice versa for
s/c ∈ [Ts/2; Ts]iii) constant wake functions
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
Strong head-tail instability 1/3TRANSVERSE plane simple model → two particles beam
A. Chao, Physics of Collective Instabilities in High Energy Accelerators
i) two particle beam→ charge Ne/2ii) s/c ∈ [0; Ts/2] 1 leads 2 and vice versa for
s/c ∈ [Ts/2; Ts]iii) constant wake functions
a) WT (z) = −W0 → action from the leading to thetrailing
b) WH (z) = W1 → action from the trailing to the leading
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
Strong head-tail instability 1/3TRANSVERSE plane simple model → two particles beam
A. Chao, Physics of Collective Instabilities in High Energy Accelerators
i) two particle beam→ charge Ne/2ii) s/c ∈ [0; Ts/2] 1 leads 2 and vice versa for
s/c ∈ [Ts/2; Ts]iii) constant wake functions
a) WT (z) = −W0 → action from the leading to thetrailing
b) WH (z) = W1 → action from the trailing to the leadingiv) betatron oscillation with ωβ frequency
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
Strong head-tail instability 1/3TRANSVERSE plane simple model → two particles beam
A. Chao, Physics of Collective Instabilities in High Energy Accelerators
i) two particle beam→ charge Ne/2ii) s/c ∈ [0; Ts/2] 1 leads 2 and vice versa for
s/c ∈ [Ts/2; Ts]iii) constant wake functions
a) WT (z) = −W0 → action from the leading to thetrailing
b) WH (z) = W1 → action from the trailing to the leadingiv) betatron oscillation with ωβ frequency
~x =(
x1, x ′1, x2, x ′2)
⇒ ~x = ~Φ(~x)Equation of motion for half a period TS/2 = π/ωS , then1 → 2
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
Strong head-tail instability 1/3TRANSVERSE plane simple model → two particles beam
A. Chao, Physics of Collective Instabilities in High Energy Accelerators
i) two particle beam→ charge Ne/2ii) s/c ∈ [0; Ts/2] 1 leads 2 and vice versa for
s/c ∈ [Ts/2; Ts]iii) constant wake functions
a) WT (z) = −W0 → action from the leading to thetrailing
b) WH (z) = W1 → action from the trailing to the leadingiv) betatron oscillation with ωβ frequency
~x =(
x1, x ′1, x2, x ′2)
⇒ ~x = ~Φ(~x)Equation of motion for half a period TS/2 = π/ωS , then1 → 2
~Φ =
x ′1− (ωβ/c)2 x1 + α1x2
x ′2− (ωβ/c)2 x2 + α2x1
, αj =Nr0Wj
2γCj = 1, 2.
(5)
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
Strong head-tail instability 2/3
Basin of stability obtained from 4-th order Runge-Kuttaintegration of Eq. (5).Γ = πNr0W1c2
4γCωsωβ
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
Strong head-tail instability 2/3
Basin of stability obtained from 4-th order Runge-Kuttaintegration of Eq. (5).Γ = πNr0W1c2
4γCωsωβUltrarelativistic beam ⇒
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
Strong head-tail instability 2/3
Basin of stability obtained from 4-th order Runge-Kuttaintegration of Eq. (5).Γ = πNr0W1c2
4γCωsωβUltrarelativistic beam ⇒
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
Γ
ωβ / ωs
α1 = 0.0
Dotted region ⇒ unstable system.
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
Strong head-tail instability 3/3
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
Γ
ωβ / ωs
α1 = 0.1α2
Border of stability for u. r. case
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6Γ
ωβ / ωs
α1 = 0.5α2
Border of stability for u. r. case
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
Γ
ωβ / ωs
α1 = -0.1α2
Border of stability for u. r. case
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
Γ
ωβ / ωs
α1 = -0.5α2
Border of stability for u. r. case
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Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
Outline
Motivations and concepts
The fields and the their calculation
Possible effects on bunch dynamics
Conclusions
![Page 34: Nonultrarelativistic resistive wall wake fields](https://reader031.fdocuments.in/reader031/viewer/2022012008/61d99be648aeaf27cb7de2cb/html5/thumbnails/34.jpg)
Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
I Extent of the wake field concept fornonultrarelativistic case
![Page 35: Nonultrarelativistic resistive wall wake fields](https://reader031.fdocuments.in/reader031/viewer/2022012008/61d99be648aeaf27cb7de2cb/html5/thumbnails/35.jpg)
Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
I Extent of the wake field concept fornonultrarelativistic case
I Numerical evidence of front field fornonultrarelativistic bunches
![Page 36: Nonultrarelativistic resistive wall wake fields](https://reader031.fdocuments.in/reader031/viewer/2022012008/61d99be648aeaf27cb7de2cb/html5/thumbnails/36.jpg)
Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
I Extent of the wake field concept fornonultrarelativistic case
I Numerical evidence of front field fornonultrarelativistic bunches
I For the longitudinal plane the resistive wall front fieldseems to be relevant
![Page 37: Nonultrarelativistic resistive wall wake fields](https://reader031.fdocuments.in/reader031/viewer/2022012008/61d99be648aeaf27cb7de2cb/html5/thumbnails/37.jpg)
Nonultrarelativisticresistive wallwake fields
Diego QuatraroCERN
&Bologna University
Motivations andconcepts
The fields and thetheir calculation
Possible effects onbunch dynamics
Conclusions
I Extent of the wake field concept fornonultrarelativistic case
I Numerical evidence of front field fornonultrarelativistic bunches
I For the longitudinal plane the resistive wall front fieldseems to be relevant
I Work in the transverse plane still ongoingHowever preliminary simulations based on twoparticle model seem to show a different behaviourconcerning the beam stability