Analytical studies of day-to-day behaviour of the travelling and quasi-stationary ultra-long waves in the 1967/68 winter stratosphere are made by means of the time-filter method [Iwashima and Yamamoto (1971)]. A brief description of the method of analysis and a criterion of numerical reliability for treating erroneous data are provided in the first place. Daily variations of zonal mean temperature and zonal mean wind which characterize the sudden warming are firstly depicted. Secondly from the

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... nalysis of the total ultra-long waves with wavenumbers one, two and three, such a few characteristic features as amplitude-decay of the wavenumber one and simultaneous amplification of the wavenumber two at the sudden warming, which has been suggested by Teweles (1963) , etc., are confirmed again. Thirdly, in order to show the process of applying the time-filter method, the wavenumber two is analyzed somewhat in detail, because of its predominance during the period of sudden warming. It is found out that the amplification of several fluctuating components of the quasi-stationary part is accompanied with the sudden warming. The travelling part is classified into the westsward and eastwardmoving modes. Finally, the travelling and quasi-stationary parts of the ultra-long waves with the wavenumbers one, two and three are described in meridional-and vertical-time sections. The travelling part of wavenumber one predominates during the warming stage and rapidly decays at the mature stage of the warming. The former stage may correspond to the "third stage of the sudden warming" by Miyakoda (1963) or the "migratory-stage" termed by Hirota (1967). While the quasi-stationary parts of wavenumber one and both parts of wavenumber two amplify with the warming and decay afterward. Taking account of the results of the former observational and theoretical studies [Hirota (1968 ), Matsuno (1971 , etc.], we may infer the following close relations: i) the non-linear interaction between the travelling part of wavenumber one and that of wavenumber two, and ii) the interaction between the quasi-stationary part and the travelling one of the respective wavenumber, as Murakami (1960) showed the energetical relation between the stationary disturbance and the transient eddy.