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<H2><A NAME="SECTION00022000000000000000">
Superposition</A>
</H2>
<P>
Superposition principle allows us to calculate one particle fields
and then integrate over all the other particles in the bunch so:
<P>
We can split bunch of relativistic particles into transverse slices.
Here <IMG
WIDTH="18" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img4.gif"
ALT="$s_i$"> � slice position in the bunch.
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<!-- MATH
\begin{displaymath}
\lambda _i (s_i )
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\lambda _i (s_i )
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Wake fields of the slice can be received by integration over
transverse distribution of the particles in the slice.
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Wake fields of the bunch can be received by convolution over
longitudinal distribution of the particles in the bunch.
<P>
Slices cannot mix so we can express all the field components in a
frequency domain in the following way:
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<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
e^{i(k(z - vt))}
\end{displaymath}
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e^{i(k(z - vt))}
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<P>
Useful but not obligatory simplification - ultrarelativictic limit:
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\begin{displaymath}
v \approx c;\gamma > > 1
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v \approx c;\gamma > > 1
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<P>
Additional simplification without specific geometry assumed �
relation between longitudinal and transverse wakes stated as
Panofsky-Wenzel theorem
<P>
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\begin{displaymath}
\frac{{\partial w_t }}{{\partial s}} = \nabla _\rho w_l
\end{displaymath}
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SRC="img8.gif"
ALT="\begin{displaymath}
\frac{{\partial w_t }}{{\partial s}} = \nabla _\rho w_l
\end{displaymath}">
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<ADDRESS>
german_kourevlev
2006-01-19
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