1 Introduction

The majority of spiral galaxies, instead of showing two long and symmetric arms originating in the nucleus and extending all over the disc, present a more complex structure, sometimes with multiple arms, which can depart as satellites from a main arm. Johnson B-band images trace the distribution of young stellar population of spiral galaxies, and consequently, the arms, reasonably well. This is why we have chosen this band. Arms are fairly well described not only by logarithmic spiral but also by a number of other functions, such as Archimedean, Cotes, hyperbolic, gravitational and parabolic spirals (Danver 1942). The inherent problem in this kind of fit is that uncertainties set by arm distortions are too large to determine which is the best one (Kennicutt 1981). Another fundamental problem is the loss of bi-dimensional information, i.e. the radial and azimuthal intensity related to a particular fit. In order to solve these problems, Kalnajs (1975) introduced the Fourier transform method, later developed by Considère & Athanassoula (1982, 1988) and Puerari & Dottori (1992), to study the spiral structure of several galaxies. This method can be applied to H II region distributions or to broad-band images, and basically consists in choosing a set of functions and calculating the Fourier coefficients of the intensity (or H II number) distribution. The most used set of functions are logarithmic spirals, mainly because they are mathematically easy to handle and because they closely resemble real arm shapes, so that with few coefficients a very good description of the galaxy is reached. Other base functions could also be considered, with the requirements that they be mathematically easy to handle, fast to compute and reasonably similar in form to observed spirals. In summary, Fourier analysis of spiral patterns in galaxies provides quantitative data on arm multiplicity, form and radial extent, and does not assume that observed spiral structures are logarithmic. It merely analyses the observed distribution into a superposition of such logarithmic spirals.

Numerical orbit models (Patsis et al. 1991, and references therein; Patsis et al. 1994) seem to indicate that, because of the stochasticity of stellar orbits near corotation, spiral arms end at corotation or at the 4:1 inner Lindblad resonance (ILR), depending on the strength of the perturbation. These authors also claim that spiral structure extending beyond corotation as far as the outer Lindblad resonance (OLR) would be typical of barred spirals with their bars ending at corotation; otherwise a conspicuous gap in the spiral arms would appear. However, under conditions extracted from observations, N-body numerical simulations produce persistent spiral arms (Thomasson et al. 1990) which end at the OLR. According to the spiral density wave theory, although corotation is a singular region, short trailing waves can exist beyond corotation out to the OLR, while inside corotation, both short and long trailing and leading waves can exist (see, for example, Lin & Lau 1979, and references therein). On the other hand, several observational features can be readily understood if it is assumed that they mark the corotation region, thus implying that spiral arms end at the OLR, including spurs, gaps and interarm star formation (Elmegreen et al. 1989), breaks, bifurcations, changes of pitch angle, lower star formation in the arms compared with interarm star formation (Cepa & Beckman 1990a,b) and changes of arm skewness (Paper III). These features are usually present one at a time, and the gaps predicted by Patsis et al. (1991) are not generally present. Anyway, numerical simulations, analytical solutions and observational evidence coincide in that corotation is a singular region where the behaviour of density waves is not easy to predict. In this paper (the last of a series of four) we use Fourier transforms to complete the analysis of the sample of S(b-c) and SB(b-c) galaxies described in del Río & Cepa (1998a, hereafter Paper II), to analyse arm behaviour in the neighbourhood of the resonances and corotation determined in Paper III, together with the relative importance of the different arm components in these zones. Also, we compare the results with those obtained using the method of EEM.

In Sect. 2 we give a brief description of the data and the mathematical methods. Section 3 presents the analysis of spiral structure. Section 4 concerns to detailed analysis of each galaxy, and finally Sect. 5 shows the conclusions.