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<b>General theory of Wake fields</b>

<p>What are they? When charged particle is moving through linac or any other structure in accelerator it generates fields in the structures. The generated fields will affect any trailing particle following the leading one. 

<p>We can integrate over the structure EM fields generated by the bunch of relativistic particles to get total effect of fields on particles from the same or the next bunches. 

<p>Total effects are usually expressed in loss and kick factors. 
<p>Los factor is the integrated energy loss by the bunch over passing the structure. 
<p>Kick factor is usually average transverse angle gain by the bunch passing the structure.

<p>Fields of the leading particle are calculated assuming there is no transverse movement of this particle during passing the structure due to the Wake fields.

<p><b>Superposition</b>

<p>Superposition principle allows us to calculate one particle fields and then integrate over all the other particles in the bunch:

<p>We can split bunch particles into transverse slices. 

<p>Wake fields of the slice can be received by convolution over transverse distribution of the particles in the slice.

<p>Wake fields of the bunch can be received by convolution over longitudinal distribution of the particles in the bunch.

<p><b>Uniform structures</b>
<p>For uniform structures we don't need to integrate over longitudinal direction of the structure as the fields don't depend on this so we need only get the filed itself. Wake fields are usually expressed here in the fields over the unit of the structure.

<p>The task is then just to find eigenfunctions and eigenvalues of the problem. The next step is to get delta function distribution over these functions and get non uniform equation solution. Received during this process Green-functions are, in fact, Wake-fields we need and the problem is solved.

<p>Circular and rectangular cases

<p>Eigenfunctions for Circular case are Bessel functions
<p>Eigenfunctions for Rectangular case - Sine and Cosine functions

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