Parabolic equation development in the twentieth century

In the 20th century, an important contribution to the modeling of wave prop agation prediction is the Parabolic Equation (PE) approximation method. Thi s did not happen until the last quarter of the century. During that time pe riod, a number of contributions were made in the first 15 years, then, cont ributions were made continuously until the end of the 20th century. Contrib utions to the PE method are still going on. During the second half of the 1 970 when the PE was introduced to the field of underwater acoustics, its ma in purpose was to predict long-range, low-frequency acoustic propagations u nder range-dependent environments in fluid medium; thus, there were a numbe r of limitations. As time progresses these limitations were somewhat relaxe d 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 al ready reported in some detail in archived literatures; those contributions will be outlined. Surveying the contributions from 1995 to 1999 is a new ad dition to the already published survey papers. In recent years, the PE meth od has been used for many real applications; some results are included in t his paper. A discussion will be given at the end of this paper on looking a head how much more the PE can do in order to stimulate future research, dev elopment, and applications.