**1.2 Application of wavelet analysis for study of marine riser**

The authors have investigated VIV characteristics of a circular cylinder with forced oscillation tests in still water (Ikoma & Masuda et al., 2006, 2007). As these results, VIV behaviors have been classified to the four power spectrum pattern. However an actual orbit of the model cylinder was different even if the spectrum pattern was same. Therefore detail of VIV characteristics and behaviors cannot be understood from only a power spectrum with the FFT analysis of a time history of vibrations. In addition, a vibration phenomenon of a marine riser etc. is a non-steady problem in practice so that fluid velocity in the ocean and oscillation of an upper structure such like a production platform are an unsteady phenomenon. Therefore vibration characteristics such like VIV varies to time table.

The Hilbert transform was applied to analysis of cylinder vibration with VIV (Khalak & Wiliamson, 1999). In there, it is described that phase deviation occurs in region entering into VIV lock-in. The Hilbert transform was useful in order to analysis of marine riser vibrations and examined frequency characteristics which vary to time table.

The wavelet transform is applied to analysis of vibration problems with VIV of a rigid circular cylinder which cross-flow vibration is allowed due to vortex shedding in this study. The wavelet analysis is possible to do the time-frequency analysis as same as the Hilbert transform analysis. Objectives of this study are: 1) to examine possibility of application of the wavelet transform to VIV analysis and 2) to discuss VIV characteristics from results of the wavelet analysis. In 2010, the wavelet analysis and the Hilbert transform were also applied to the estimation of riser behaviors (Shi et al., 2010).

This chapter introduces application of the wavelet analysis in the ocean engineering field using results of VIV characteristics. From the model experiment, relationship between the orbit pattern of vibration of the model cylinder and a contour pattern of the wavelet is considered. As a result, effectiveness of the wavelet analysis in order to understand VIV detail is given.
