Parabolic equation development in the twentieth century

In the 20th century, an important contribution to the modeling of wave propagation prediction is the Parabolic Equation (PE) approximation method. This did not happen until the last quarter of the century. During that time period, a number of contributions were made in the first 15 years, then, contributions were made continuously until the end of the 20th century. Contributions to the PE method are still going on. During the second half of the 1970 when the PE was introduced to the field of underwater acoustics, its main purpose was to predict long-range, low-frequency acoustic propagations under range-dependent environments in fluid medium; thus, there were a number of limitations. As time progresses these limitations were somewhat relaxed a great deal gradually due to many useful contributions. Up to this date, all contributions are aimed at the enhancement of the capability of the PE method. A few review papers of the PE have been presented in open archived literatures. This paper gives a complete survey of what has been done from the start until the end of 1999. Contributions up to the year 1994 were already reported in some detail in archived literatures; those contributions will be outlined. Surveying the contributions from 1995 to 1999 is a new addition to the already published survey papers. In recent years, the PE method has been used for many real applications; some results are included in this paper. A discussion will be given at the end of this paper on looking ahead how much more the PE can do in order to stimulate future research, development, and applications.