10. Discussion 987 11. Conclusions 991 References 993A new method for analysing nonlinear and non-stationary data has been developed. The key part of the method is the 'empirical mode decomposition' method with which any complicated data set can be decomposed into a finite and often small number of 'intrinsic mode functions' that admit well-behaved Hilbert transforms. This decomposition method is adaptive, and, therefore, highly efficient. Since the decomposition is based on the local characteristic time scale of the data, it is applicable to nonlinear and non-stationary processes. With the Hilbert transform, the 'instrinic mode functions' yield instantaneous frequencies as functions of time that give sharp identifications of imbedded structures. The final presentation of the results is an energy-frequency-time distribution, designated as the Hilbert spectrum.In this method, the main conceptual innovations are the introduction of 'intrinsic mode functions' based on local properties of the signal, which makes the instantaneous frequency meaningful; and the introduction of the instantaneous frequencies for complicated data sets, which eliminate the need for spurious harmonics to represent nonlinear and non-stationary signals. Examples from the numerical results of the classical nonlinear equation systems and data representing natural phenomena are given to demonstrate the power of this new method. Classical nonlinear system data are especially interesting, for they serve to illustrate the roles played by the nonlinear and non-stationary effects in the energy-frequency-time distribution.

Based on theoretical analysis and laboratory data, we proposed a unified two-parameter wave spectral model as $\phi(n) = \frac{\beta g^2}{n^m n_0^{5-m}} {\rm exp} \left\{-\frac{m}{4}\left(\frac{n_0}{n}\right)^4\right\}$ with β and m as functions of the internal parameter, the significant slope η of the wave field which is defined as $\sect = \frac{(\overline{\zeta^2})^{\frac{}1{2}}}{\lambda_0},$ where $\overline{\zeta^2}$ is the mean squared surface elevation, and λ0, n0 are the wavelength and frequency of the waves at the spectral peak. This spectral model is independent of local wind. Because the spectral model depends only on internal parameters, it contains information about fluid-dynamical processes. For example, it maintains a variable bandwidth as a function of the significant slope which measures the nonlinearity of the wave field. And it also contains the exact total energy of the true spectrum. Comparisons of this spectral model with the JONSWAP model and field data show excellent agreements. Thus we established an alternative approach for spectral models. Future research efforts should concentrate on relating the internal parameters to the external environmental variables.

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Probability density function of the surface elevation of a nonlinear random wave field is obtained. The wave model is based on the Stokes expansion carried to the third order for both deep water waves and waves in finite depth. The amplitude and phase of the first‐order component of the Stokes wave are assumed to be Rayleigh and uniformly distributed and slowly varying, respectively. The probability density function for the deep water case was found to depend on two parameters: the root‐mean‐square surface elevation and the significant slope. For water of finite depth, an additional parameter, the nondimensional depth, is also required. An important difference between the present result and the Gram‐Charlier representation is that the present probability density functions are always nonnegative. It is also found that the ‘constant’ term in the Stokes expansion, usually neglected in deterministic studies, plays an important role in determining the details of the density function. The results compare well with laboratory and field experiment data.

This paper is concerned with the response of a rigid body subjected to base excitation. Using analytical and numerical methods of investigation, the results are presented in a series of graphs giving (1) the criteria for initiation of various modes (rest, slide, rock and slide-rock) of response and, using an ensemble of 75 real earthquake acceleration records, (2) the mean plus standard deviation of the distance of sliding relative to the base and (3) the probability of overturning during rocking. Based on the results in (3), two tables are prepared which give, corresponding to probability of overturning equal to 0.16 and 0.5, allowable peak base acceleration.

Traditionally, investigation of statistical properties of ocean waves has been limited largely to global quantities related to elevation and amplitude such as the power spectral and various probability density functions. Although these properties give valuable information about the wave field, the results cannot be related directly to any portion of the time data from which it was derived. We present a new approach using phase information to view and study the properties of frequency modulation, wave group structures, and wave breaking. We apply the method here to ocean wave time series data and identify a new type of wave group (containing the large “rogue” waves), but the method also has the capability of broad applications in the analysis of time series data in general.

A freestanding rigid body under the action of base excitation can move in many different ways. In this study, the sliding response of a body is considered. The body is placed on a horizontal base that undergoes a one-dimensional horizontal motion. In 1965, Newmark (1965) gave a simple formula to determine the sliding distance of a freestanding body subjected to a single rectangular acceleration pulse of short duration at the base. The objective of this study is to see if this formula can be applied to estimate the sliding displacement of a body under the action of real earthquakes. Newmark's formula calls for the maximum velocity of the base which information is usually not directly available. To make use of the response spectrum commonly available to the engineers, Newmark's formula is first re-derived and expressed in terms of the maximum displacement of the base, which can be determined from the absolute displacement response spectrum in low frequency range. An ensemble of 75 real earthquakes is then employed, the equation of sliding motion is solved numerically and the average of the maximum sliding displacement of the body relative to the base is computed. The computed displacement is then compared with that obtained by Newmark's formula. This is done for a body placed on the ground as well as on the floors of a building. It is shown that Newmark's formula can be used if an adjustment factor, as suggested in this study, is applied. Key words: earthquake, sliding, unanchored body

On the basis of the mapping method developed by Huang et al. (1983), an analytic expression for the non‐Gaussian joint probability density function of slope and elevation for nonlinear gravity waves is derived. Various conditional and marginal density functions are also obtained through the joint density function. The analytic results are compared with a series of carefully controlled laboratory observations, and good agreement is noted. Furthermore, the laboratory wind wave field observations indicate that the capillary or capillary‐gravity waves may not be the dominant components in determining the total roughness of the wave field. Thus, the analytic results, though derived specifically for the gravity waves, may have more general applications.