For geotechnical practicioners, the wave equation is a phenomenon which is generally dealt with using a finite element or difference computer program. For mathematicians, it is a second order, hyperbolic partial differential equation. Although the computer programs are certainly a solution to these problems (and in the case of the nonlinear phenomena just about the only one) it is not bad to consider more “theoretical” types of solutions. This page presents some of them, along with practical applications. They will thus be of interest to both geotechnical researchers, practitioners and mathematicians. These links are either to html or pdf format files.

### Application of the Closed Form Solution for the Damped Wave Wave Equation to Piles

This paper presents the application of the closed form solution for the damped wave equation to piles. The wave equation in numerical solution has been used for many years, generally without even a simple closed form counterpart. In this paper the closed form solution for the damped wave equation will first be stated and related to an actual pile driven into the soil. Following this is a discussion of the boundary conditions: the hammer at the pile top and the soil response at the pile toe. To avoid spectral components in the Fourier series eigenvalues and to preserve orthogonality, a new strain based soil model to simulate radiation dampening from the pile toe is proposed. A solution to this equation which involves the solution of the semi-infinite pile using Laplace transform for the first part of the impact followed by a Fourier series solution for the remainder. Comparison with numerical methods for a sample case is also presented.

### Closed Form Solution of the Wave Equation for Piles

*Don C. Warrington, P.E.*

This thesis details the research into the one-dimensional wave equation as applied to piles used in the support of structures for civil works and driven using impact equipment. Since the 1950’s, numerical methods, both finite difference and finite element, have been used extensively for the analysis of piles during driving and are the most accepted method of analysis for the determination of driving stresses, dynamic and static resistance of piles. In this thesis the wave equation is solved in a relatively simple closed form without recourse to numerical methods. A review of past efforts to solve the wave equation in closed form is included. Problems that appear in previous related works are discussed and derived again, including the Prescott-Laura problem of the cable system stopped at one end and the solution of a hammer/cushion/cap/pile system for a semi-infinite pile. The latter is used to assist in the determination of a pile top force-time function that can be used to simulate the impact of the hammer on the pile. The basic equations, initial and boundary conditions are detailed, with the parameters adjusted to match actual soil dynamic behaviour while at the same time being a form convenient for closed form solution. To avoid difficulties due to spectral elements in the boundary conditions, a strain-based model of the radiation dampening in the pile toe was developed. The solution technique uses a Laplace transform of the semi-infinite pile problem for 0 < t < L/c (or for a time duration 0 < t < d, where d < L/c) and a Fourier series solution of the Sturm-Liouville problem thereafter. This solution is applied both to undamped and damped wave equations. The work includes comparison with existing numerical methods such as WEAP87, ANSYS, and Newmark’s method using Maple V.

### Deflections of Pile Toe Plates on Elastic Foundations

*Don C. Warrington, P.E.*

This paper is an analysis of pile toe plates that are assumed to interact with elastic foundations. A solution to the deflection and moment equations is derived and discovered to be in fact made up of Bessel functions with complex arguments. A solution based on the analysis of the series that make up the Bessel functions is performed. The solution is presented in the form of charts based on dimensionless parameters. A sample case is analysed and discussed.

### Direct Derivation of the Equation of Motion for an Undamped Oscillating System in Phase Angle Form

The equations of motion for linear vibrating systems are well known and widely used in both mechanical and electrical devices. However, when students are introduced to these, they are frequently presented with solutions which are either essentially underived or inadequately so.

This brief presentation will attempt to address this deficiency and hopefully show the derivation of the equation of motion for an undamped oscillating system in a more rigourous way.

Although this solution, strictly speaking, does not apply to a distributed system such as a drive pile, it can be applied to both the hammer/cushion system at the pile top or vibratory hammers.

*Soil Dynamics*

### Arnold Verruijt, Univerisity of Delft, The Netherlands

A complete treatment on this subject, whose coverage in the literature is woefully inadequate. Topics include the following:

- Vibrating Systems
- Theory of Consolidation
- Plane Waves in Porous Media
- Waves in Piles
- Earthquakes in Soft Layers
- Cylindrical Waves
- Spherical Waves
- Elastostatics of a Half Space
- Elastodynamics of a Half Space
- Foundation Vibrations
- Moving Loads on an Elastic Half Plane

An example of the figures in the book is at the top of the page.

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