1. Status with Particle Studio simulations

2. Conclusion of previous meeting
A factor 4.4 (probably ) is observed between the amplitude of simulated wakes and theoretical wakes.
This amplitude factor aside, we have separated the dipolar and quadrupolar terms in the rectangular shape, and they agree with the theory.
The simulated wakes obtained for several rectangular shape form factors also agree with the theoretical curve.

3. Open questions Factor 4.4 between theory and simulations
- most likely a difference of convention
Issues with cylindrical shape

5. Longitudinal Wake
CST particle studio (Weiland and Wanzenberg, Wake fields and impedances, DESY M-91-06)
Vaccaro, Palumbo and Zobov (CAS 1994)
with
and
 Between the longitudinal wake convention of Vaccaro (theoretical wake) and Weiland (simulated wake), there is a minus sign difference.

6. Theoretical formula from Vaccaro (from Classical Thick wall regime)
F= geometry form factor
L=device length
b = radius (circular case)
= conductivity
Let’s choose a form factor = 0.5 (h=3b), so that F// ~ 1

7. Obtaining the transverse wake from the longitudinal wake with the Panofsky Wenzel theorem
So, with the beam at transverse location y=y1 we calculate W//(s) at 2 different y locations and we obtain

8. Comparison simulated transverse wake and calculated transverse wake from simulated longitudinal wakey x

9. Indirect calculation method from T
Indirect calculation method from T. Weiland (handbook of accelerator physics Ed. Chao and Tigner p.202)
Source beam at transverse location 1 b
Test path at transverse location 2
longitudinal
transverse

10. Indirect calculation method from T
Indirect calculation method from T. Weiland (handbook of accelerator physics Ed. Chao and Tigner p.202)
Source beam at y10
test path at y2=0 b
Longitudinal (all modes)
Transverse (all modes)

11. Switching to rectangular cross section
The simulations with the cylindrical shape are not stable, so we have to use the rectangular shape to benchmark the theory.
The theory is only valid for the cylindrical, but some form factors are known.
In our case, h=3b, i.e. q=0.5
Longitudinal
Transverse:
Cylinder (wquad=0):
rectangular
With

12. Relationships for the rectangular cross section
Longitudinal (mode 0)
Transverse (mode 1)

13. Relationships for the rectangular cross section
Longitudinal (mode 0)
Transverse (mode 1)
Longitudinal mode 1
Panofsky Wenzel:
and

14. Panofsky Wenzel using only the theoryx

17. Case 1
Case 2 y y x x
Case 3 y
Case 4 y x x