Pdf controllability for a wave equation with moving boundary. Controllability for a wave equation with moving boundary. Finally, boundary conditions must be imposed on the pde system. For example, xx 0 at x 0 and x l x since the wave functions cannot penetrate the wall. It will represent the fundamental equation of motion of a matter wave, which when solved subject to boundary conditions, will give us the wave function. In this section, we reduce maxwells equations to wave equations that apply to the electric and magnetic fields in this simpler category of scenarios.
This paper is concerned with longtime dynamics of semilinear wave equations defined on moving boundary domains. The approach we propose is based on a general transform method for solving boundary value problems for integrable nonlinear pde in two variables, that has been applied extensively to the study of linear parabolic and elliptic equations. If the medium is dispersive different frequencies travel at. Most of you have seen the derivation of the 1d wave equation from newtons and hookes law. This wave equation is very similar to the one for transverse waves on a string, which was. In the presence of absorption, the wave will diminish in size as it move. 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. As for the wave equation, we use the method of separation of variables.
A wave is disturbance of a continuous medium that propagates with a fixed shape at constant velocity. A water wave is an example of a surface wave, which is a combination of. Group velocity group velocity is the speed of propagation of a packet, or group, of waves. Setting ux,tfxgt gives 1 c2g dg dt 1 f d2f dx2 k, where k is some constant to be determined. Since the boundary is a function of the time variable the problem is intrinsically nonautonomous. First and second order linear wave equations 1 simple. The action of two kinds of reciprocal transformation on the moving.
If you were to clock a wave crest you would find that it moves at the phase speed, cp. Kurylev leningrad branch of the steklov mathematical institute lomi fontardca 27, leningrad, 191011 ussr abstractwe consider the problem of the wave field continuation and recovering of coefficients. When using a neumann boundary condition, one prescribes the gradient normal to the boundary of a variable at the boundary, e. Sufficient conditions are formulated which insure the exact distributedparameter controllability of the wave equation. Lecture 6 boundary conditions applied computational.
Wave equation in 1d part 1 derivation of the 1d wave equation. Pdf a remark on observability of the wave equation with. We study certain boundary value problems for the onedimensional wave equation posed in a timedependent domain. Wave equations with moving boundaries university of new. An example of pure stability for the wave equation with moving boundary. Here it is, in its onedimensional form for scalar i. Our quantum wave equation will play the same role in quantum mechanics as newtons second law does in classical mechanics. Solution of the wave equation by separation of variables. An example of pure stability for the wave equation with.
The boundary condition at x 0 leads to xx a 1sin k xx. We put this into the di erential equation for vand obtain after moving the 4v xx term to the left side x1 n1. The wave equation in one dimension later, we will derive the wave equation from maxwells equations. On the solution of the wave equation with moving boundaries core. In section 8, we reconstruct the so called weighted wave fields, i. So fxvt represents a rightward, or forward, propagating wave. In this paper we outline this derivation and show how the lie group of point transformations, admitted by the onedimensional wave equation, can be used to nd a general series solution for an associated initial moving boundary value problem. Control and stabilization for the wave equation, part iii. A boundary value problem has conditions specified at the extremes boundaries of the independent variable in the equation whereas an initial value problem has all of the conditions specified at the same value of the independent variable and that value is at the lower boundary of the domain, thus the term initial.
The model is adapted from the selective lumped mass slm numerical model. Pdf an example of pure stability for the wave equation. Recall that the wave equation for the continuous springmass system was given in eq. Solutions to pdes with boundary conditions and initial conditions. W e deal with the wave equation with assigned moving bound ary 0 boundary conditions are speci. Numeric solution of wave equations with moving boundaries p. Moving boundary problems for the harry dym equation and. The phase speed of a wave is defined as the sp eed at which the wave is moving. If we impose periodic boundary conditions on the solution and its first derivatives. The wave equation is a secondorder linear hyperbolic pde that describesthe propagation of a variety of waves, such as sound or water waves. Keep a fixed vertical scale by first calculating the maximum and minimum values of u over all times, and scale all plots to use those zaxis limits. Outline s generalized stefan problems s application to braneworld cosmology. The second step impositionof the boundary conditions if xixtit, i 1,2,3, all solve the wave equation 1, then p i aixixtit is also a solution for any choice of the constants ai. Moving boundary value problems for the wave equation.
Observability and controllability of the 1d wave equation. A new moving boundary shallow water wave equation numerical model is presented. Boundary value problems are similar to initial value problems. By mean of generalized fourier series and parsevals equality in weighted \l2\spaces, we derive a sharp energy estimate for the wave equation in a bounded interval with a moving endpoint. Under the hypothesis that the lateral boundary is timelike, the solution operator of the problem generates an evolution process ut. It arises in fields like acoustics, electromagnetics, and fluid dynamics historically, the problem of a vibrating string such as that of a musical. We investigate the controllability for a onedimensional wave equation in domains with moving boundary. Thus, if the right end is allowed to move freely and the left end is fixed. Note that at a given boundary, different types of boundary. Chapter 4 the wave equation another classical example of a hyperbolic pde is a wave equation. There are one way wave equations, and the general solution to the two way equation could. Then, we show the observability, in a sharp time, at each of the endpoints of the interval. Wave is unconstrained in plane orthogonal to wave direction, i.
Choosing which solution is a question of initial conditions and boundary values. Pdf stabilization of the wave equation with moving boundary. Create an animation to visualize the solution for all time steps. R, such that its intersections with hyperplanes x, s. This model characterizes small vibrations of a stretched elastic string when one of. They are due to the relevant problems of boundary control and appear to be some natural multidimensional analogs of the classical equations of gelfandlevitankrein. We consider the one dimensional wave equation where the domain available for the wave process is a function of time. Pdf we investigate the controllability for a onedimensional wave equation in domains with moving boundary. In the one dimensional wave equation, when c is a constant, it is interesting to observe that. Applying the boundary conditions case the general solution is. The mathematics of pdes and the wave equation mathtube. Wave equations for sourcefree and lossless regions. This equation determines the properties of most wave phenomena, not only light.
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