The transverse bunch spectrum and the transverse decoherence/recoherence following an initial bunch offset are important phenomena in synchrotrons and storage rings, and are widely used for beam and lattice measurements. Incoherent shifts of the particles betatron frequency and of the synchrotron frequency modify the transverse spectrum and the bunch decoherence. In this study we analyze the effects of transverse space charge and of the rf nonlinearity on the decoherence signals. The transverse bunch decoherence and the resulting coherent spectra are measured in the SIS18 synchrotron at GSI Darmstadt for different bunch parameters. The frequencies of the bunch head-tail modes provide a direct measure for the self-field space charge tune shift. Particle tracking simulations together with an analytical model are used to describe the modifications in the decoherence signals and in the coherent spectra due to space charge and the rf bucket nonlinearity. Transverse coherent oscillations of bunches induced by a fast kicker magnet are routinely used in synchrotrons or storage rings to measure, for example, the tune or other ring parameters, see e.g. (1). The transverse offset of a bunch, averaged over the bunch length, can be recorded every single turn. The spectrum is then concentrated around the base-band Q f0 f 0 , where Q f0 is the fractional part of the betatron tune Q0 and f0 is the revolution frequency. This diagnostics is usually used for time- resolved and very accurate measurements of the tune Qf0. Transverse bunch decoherence is a process of a turn-to- turn reduction of the total bunch offset signal after an initial bunch displacement. In a linear focusing lattice the bunch decoherence is a manifestation of the lattice chro- maticitywhere the synchrotron dynamics also plays an important role, causing the signal recoherence exactly after the synchrotron period. Other damping mechanisms, as due to lattice nonlinearities, additionally damp the trans- verse oscillations. Transverse decoherence is often used as a machine diagnostics tool. Undesired transverse bunch oscillations can also appear after the bunch-to-bucket transfer between synchrotrons. In order to use transverse decoherence as a diagnostics tool for intense bunches of arbitrary length and also to control undesired oscilla- tions of such bunches, it is important to understand the decoherence in the presence of transverse space charge and nonlinear synchrotron oscillations. We demonstrate that the decoherence signal can be explained in terms of the transverse head-tail bunch mode spectrum. For finite chromaticity also the k> 0 head-tail modes contribute to the bunch coherent spectrum. The shift of the head-tail mode frequencies due to space charge and wall currents can be well explained in terms of the analytical expressions for an airbag bunch distribution (2,3). The head-tail mode frequencies are also modified by changes in the individual particle synchrotron frequency. In long bunches, one has to account for the spread of the synchrotron frequencies. Both transverse space charge and nonlinear synchrotron oscillations are important to under- stand the decoherence signals and transverse spectra. We demonstrate that, once the spectrum and decoherence mod- ifications are understood, they can be used to extract useful information about the bunches. In this paper we describe measurements of transverse bunch spectra and decoherence signals obtained in the heavy-ion synchrotron SIS18 at GSI Darmstadt. The ob- served modification of the head-tail spectrum and of the decoherence signal caused by transverse space charge and nonlinear synchrotron oscillations are explained in terms of our theoretical approach. This approach is based on an expansion of an analytical theory for head-tail modes in combination with particle tracking simulations. In Sec. II we use theoretical and numerical approaches to analyze the effects of space charge and nonlinear syn- chrotron motion on the transverse spectra and on the bunch decoherence signal. We show that a simple model for the head-tail mode frequencies with fitting parameters can be Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distri- bution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. PHYSICAL REVIEW SPECIAL TOPICS - ACCELERATORS AND BEAMS 15, 114201 (2012)