## 1. Introduction

[2] From radiotherapy to nuclear physics research, particle accelerators constitute a powerful device under continuous development. Actual applications of accelerators and storage rings introduce high restrictive design constraints on beam intensity and emittance [*Salah and Dolique*, 1999; *Salah*, 2004; *Gai et al.*, 1997]. In order to achieve optimum performance, an accurate understanding of the involved physics is required. In this sense, the evaluation of the fields radiated by a charged particle moving linearly at constant velocity within a beampipe is particularly important, since it may influence the motion of trailing particles [*Panofsky and Wenzel*, 1956; *Wangler*, 2008; *Zotter and Kheifets*, 1998]. In a particle accelerator, the beampipe is excited by an electromagnetic field during the accelerating stage to store and accelerate beams of charged particles. The particles are usually packed into bunches and launched at the appropriate time to take advantage of the accelerating phases, in order to achieve relativistic velocities [*Bane et al.*, 1985]. After the accelerating circuit, the velocity of a beam should remain constant; but the electromagnetic radiation of moving charges may influence the motion of other particles and bunches. The electromagnetic field created by a charge is scattered on the metallic walls of the waveguide and acts back on trailing charges, thus inducing energy loss, beam instabilities and some secondary effects like the heating of sensitive components [*Figueroa et al.*, 1988]. The interaction with the structure can be described by impedances in the frequency domain, or equivalently by wakefields in the time domain [*Yokoya*, 1993a; *Stupakov et al.*, 2007]. These parameters have to be taken into account during the design of an accelerator, as they restrict the choice of materials and shape of components [*Burov and Danilov*, 1999].

[3] The wakefields critically depend on the geometry of the structure [*Danilov*, 2000]. The radiation of particles within waveguides has deserved the attention of many researchers in different fields of the electromagnetic theory [*Rosing and Gai*, 1990; *Ng*, 1990; *Xiao et al.*, 2001; *Hess et al.*, 2007]. The solution of wakefields for rectangular and circular waveguides is well known [*Wangler*, 2008; *Zotter and Kheifets*, 1998; *Gluckstern et al.*, 1993]; moreover, in *Gluckstern et al.* [1993] a formula is given for the specific case of charges traveling on the symmetry axes within an elliptical guide. This formula has been extensively followed to estimate the wakefields in beampipes similar to the elliptical geometry [*Rumolo et al.*, 2001; *Zimmermann*, 1997]. The absence of analytical expressions for predicting the radiation within many geometries demands the development of numerical techniques for the analysis of arbitrary waveguides [*Kim et al.*, 1990; *Jing et al.*, 2003; *Lutman et al.*, 2008; *Zagorodnov*, 2006]. Usually, particle-in-cell codes are used to compute wakefields. In this article, we present an alternative full-wave modal method for the analysis of the electromagnetic radiation of charges in motion within uniform waveguides with arbitrary cross-section. The method here presented is based on the dyadic three-dimensional electric and magnetic Green's functions formulation in the frequency domain. The radiated fields are obtained from the convolution of the Green's functions with the current distribution of the bunch. Then, the fields are expressed in the time domain by means of the Fourier's Transform technique, from which the wakefields are finally derived. The effect of the velocity on beams has been traditionally omitted in the accelerators publications, where the ultrarelativistic approach is assumed [*Gluckstern et al.*, 1993; *Yokoya*, 1993b; *Kim et al.*, 1990; *Bane et al.*, 1985; *Rosing and Gai*, 1990; *Iriso-Ariz et al.*, 2003; *Lutman et al.*, 2008; *Palumbo et al.*, 1984]. In this sense, the present work represents a contribution since different velocities for the particles can be used in the proposed formulation. The presented formulation is completely analytic except in the calculation of the modes of the waveguide, which must be numerically computed. This fact reduces significantly the associated error of the method to the accuracy of the numerical technique used for the modal analysis of the structure. Moreover, the proposed formulation can treat waveguides of arbitrary shapes, what makes it a suitable method when non-canonical geometries are considered. In this sense, this method gives a practical contribution, since many modern accelerating structures are based on non-canonical waveguides.

[4] The paper has been organized in three sections. In the next one, the theoretical formulation of the problem is detailed. This section is divided in three blocks, corresponding to the dyadic electric and magnetic Green's functions, the fields created by moving charged particles, and the resulting wakefields. Afterwards, some examples of charges moving lengthwise within arbitrary waveguides are tackled by the presented derivation. Finally, conclusions are outlined.