Method of characteristics wave equation pdf

You can start with wikipedia, couranthilbert methods of mathematics physics vol. The theory of characteristics gives dalemberts solution in a systematic fashion. Well be looking primarily at equations in two variables, but there is an extension to higher. The wave equation is a secondorder linear hyperbolic pde that describesthe propagation of a variety of waves, such as sound or water waves. The cipmoc constrained interpolation profile method of characteristics is proposed to solve the tide wave equations with large time step size. Hence the characteristic equation is the two families of solutions characteristics are and. We start by looking at the case when u is a function of only two variables as. Although pdes are inherently more complicated that odes, many of the ideas from the previous chapters in. Characteristic curves suppose z is given along a curve c in the x,y plane. The equation of motion is replaced by an equation describing uniform flow. Method of characteristics we nish the introductory part of this material by discussing the solutions of some rst order pdes, more specically the equations we obtained from the advection model. Computational grid computer solution by method of characteristics.

The method of characteristics is a well known analytical procedure for transforming a set of hyperbolic pdes into a set of odes. The odes may subsequently be transformed into a set of difference equations through numerical integration and interpolation. Pdf 30 years of experience with the wave equation solution. Verbeek, vmsprofound, tyler, tx, usa this paper will describe a brief history of the method of characteristics as an accurate solution of the wave equation, and the authors. Solving the wave equation using method of characteristics. Typically, it applies to firstorder equations, although more generally the method of characteristics is valid for any hyperbolic partial differential equation. We can also use the method of characteristics in the nonhomogeneous case. The bottom topography and bottom friction, which are very important factor for the tidal wave model, are included to the equation of riemann invariants as the source term. Pdf cip method of characteristics for the solution of. In mathematics, the method of characteristics is a technique for solving partial differential equations. Cip method of characteristics for the solution of tide wave. Through each point a on c, we can continue the solution along. The method of characteristics applied to quasilinear pdes. The method of characteristics moc approach has been favored by textbook authors 2,3 because it is explicit and thus easy to set up and solve.

This elementary ideas from ode theory is the basis of the method of characteristics moc which applies to general quasilinear pdes. Chapter 4 the wave equation another classical example of a hyperbolic pde is a wave equation. The method of characteristics is an important method for hyperbolic pdes which applies to both linear and nonlinear equations. We shall discuss the basic properties of solutions to the wave equation 1. Its derivation was much more elegant than the method in sec. To see this, we write the wave equation in the form 14 by setting by the chain rule, and division by gives as stated before. The method ofcharacteristics solves the firstorder wave eqnation 12.

This is the rate at which the solution will propagate along the characteristics. Method of characteristics in this section, we describe a general technique for solving. The goal of the method of characteristics, when applied to this equation, is to change coordinates from x, t to a new coordinate system in which the pde becomes an ordinary differential equation ode along certain curves in the xt plane. For a firstorder pde partial differential equation, the method of characteristics discovers curves called characteristic curves or just characteristics along which the pde becomes an ordinary differential equation ode. Method of characteristics for linear equations wikiwaves. The particular interpolation and integration scheme is a matter of preference and the form of the resulting model is, in general, nonlinear. Dalemberts solution compiled 3 march 2014 in this lecture we discuss the one dimensional wave equation. Well be looking primarily at equations in two variables, but there is an extension to higher dimensions. One method of solution is so simple that it is often overlooked. We will now exploit this to perform fourier analysis on the. Pdf cip method of characteristics for the solution of tide. It, and its modifications, play fundamental roles in continuum mechanics, quantum mechanics, plasma physics, general relativity, geophysics, and many other scientific and technical disciplines. Cip method of characteristics for the solution of tide wave equations article pdf available in advances in mathematical physics 2018 june 2018 with 143 reads how we measure reads.

When the elasticity k is constant, this reduces to usual two term wave equation u tt c2u xx where the velocity c p k. It also tends to preserve wave behavior that is very important for fast, brief, transient calculations that are required for pipeline surge calculations. Method of characteristics an overview sciencedirect topics. The cipmoc constrained interpolation profilemethod of characteristics is proposed to solve the tide wave equations with large time step size. Illustrate the nature of the solution by sketching the uxpro. Since the integral is a function of v, say, the solution is of the form in terms of x and t, by 2, we thus have 4 this is known as dalemberts solution1 of the wave equation 1. The characteristic equation for z will always be a linear ode. The method of characteristics for linear and quasilinear. I am having a lot of trouble understanding the method of characteristics to solve the wave equation. The parameter a has dimensions of distance divided by time and is called the speed of propagation along the characteristic. The wave equation is the simplest example of a hyperbolic differential equation. This is the heart of the course and many of the standard theorems for these three equations will be covered in this part of the course.

The wave equation is an important secondorder linear partial differential equation for the description of wavesas they occur in classical physicssuch as mechanical waves e. Method of characteristics kinematic shock dynamic versus kinematic waves approximations of dynamic waves 1 1. Cip method of characteristics for the solution of tide. The solution of the linear wave equation can be obtained as a special case of the nonlinear wave equation 1. Chapter 6 partial di erential equations most di erential equations of physics involve quantities depending on both. The wave now breaks in two places and is multivalued in both hatched regions. Characteristics of firstorder partial differential equation.

It arises in fields like acoustics, electromagnetics, and fluid dynamics historically, the problem of a vibrating string such as that of a musical. Ii method of characteristics 19 example 1 solve zx. In fact, i have a final exam tomorrow and i cant seem to get a question from a previous assignment. Wave equations, examples and qualitative properties. We begin with linear equations and work our way through the semilinear, quasilinear, and fully nonlinear cases. Analytic solutions of partial di erential equations. Kinematicwave models are used extensively by the u.

Once the ode is found, it can be solved along the characteristic curves and transformed into a solution for. The theory developed will to a large extent be based on representation. Wave equation in 1d part 1 derivation of the 1d wave equation vibrations of an elastic string solution by separation of variables three steps to a solution several worked examples travelling waves more on this in a later lecture dalemberts insightful solution to the 1d wave equation. Write down the solution of the wave equation utt uxx with ics u x, 0 f x and ut x, 0 0 using dalemberts formula. In this section the method for integrating numerically the. The idea of the method of characteristics is to reduce the pde to an ode by.