Wave equation the purpose of these lectures is to give a basic introduction to the study of linear wave equation. Waterwaves 5 wavetype cause period velocity sound sealife,ships 10. Note that the solution 2 can be obtained by other means, including fourier transforms. In addition, pdes need boundary conditions, give here as 4. The use of the standard adomian decomposition method for obtain ing the approximate solution of the linear homogeneous onedimensional wave equation is investigated. It, and its modifications, play fundamental roles in continuum mechanics, quantum mechanics, plasma physics, general relativity, geophysics, and many other scientific and technical disciplines.
The string has length its left and right hand ends are held. The electromagnetic wave from maxwells equations contd 2 2 t e e w w u u 2 2 2 t e e e o o w w x ph xe 0 using the vector identity becomes, in free space and. We shall discuss the basic properties of solutions to the wave equation 1. Solution of the wave equation by separation of variables the problem let ux,t denote the vertical displacement of a string from the x axis at position x and time t. Wave equations, examples and qualitative properties. This is the wave equation in one spatial dimension.
The schrodingers schrodingers equation is the basic equation of quantum mechanics. The solution 2 therefore merely translates the initial data at speed cas time progresses. Linear wave theory is the core theory of ocean surface waves used in ocean and coastal engineering and naval architecture. The regularity and exponential decay of solution for a linear wave equation associated with twopoint boundary conditions. The schrodingers schrodingers equation is the basic equation of quantum mechanics w is the wave function, i2. Second order linear partial differential equations part i.
Before considering our linear equation, we consider the following wave equation with timedependent damping. A linear partial di erential equation is nonhomogeneous if it contains a term that does not depend on the dependent variable. Pdf exact solutions to linear and nonlinear wave and. Numerical integration of linear and nonlinear wave equations. A better way of deriving the wave equation is to start from physical principles. The wave equation is the simplest example of a hyperbolic differential equation. The 1d wave equation for light waves 22 22 0 ee xt where.
The conservation of energy provides a straightforward way of showing that the solution to an ivp associated with the linear equation is unique. Deriving the 1d wave equation one way wave equations solution via characteristic curves solution via separation of variables helmholtz equation classi. Pdf the regularity and exponential decay of solution for a. The properties and behavior of its solution are largely dependent of its type, as classified below. It means that light beams can pass through each other without altering each other. The wave equation the heat equation the onedimensional wave equation separation of variables the twodimensional wave equation rectangular membrane continued since the wave equation is linear, the solution u can be written as a linear combination i. We demonstrate this for the wave equation next, while a similar procedure will be applied to establish uniqueness of solutions for the heat ivp in the next section.
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. For the equation to be of second order, a, b, and c cannot all be zero. Pdf on the solution of the nonlinear wave equation by the. But as well see, it is somewhat of a trivial dispersion relation, in the sense that there is no dispersion. Inevitably they involve partial derivatives, and so are partial di erential equations pdes. Since this pde contains a secondorder derivative in time, we need two initial conditions. May 14, 2012 quick argument to find solutions of wave equation derivation of general solution of the wave equation. Ex,t is the electric field is the magnetic permeability is the dielectric permittivity this is a linear, secondorder, homogeneous differential equation. Finite di erence methods for wave motion github pages. We will now exploit this to perform fourier analysis on the. We also investigate the decay properties of the global solutions to this problem by the.
Solution of the wave equation by separation of variables. We demonstrate this for the wave equation next, while a similar procedure will be applied to establish uniqueness. Notice that if uh is a solution to the homogeneous equation 1. The wave equation is often encountered in elasticity, aerodynamics, acoustics, and. But as well see, it is somewhat of a trivial dispersion relation, in. Solutions to pdes with boundary conditions and initial conditions. The condition 2 speci es the initial shape of the string, ix, and 3 expresses that the initial velocity of the string is zero. Wirth 6, 7, 8 classify it into four cases from the viewpoint of the global behavior of the solutions. This paper is concerned with the existence and the regularity of global solutions to the linear wave equation associated with twopoint type boundary conditions.
If f 1 x,t and f 2 x,t are solutions to the wave equation, then. Second order linear partial differential equations part iv. The wave equation is a secondorder linear hyperbolic pde that describesthe propagation of a variety of waves, such as sound or water waves. Exact solutions linear partial differential equations secondorder hyperbolic partial differential equations wave equation linear wave equation 2. Write down the solution of the wave equation utt uxx with ics u x, 0.
Thus, before tackling nonlinear wave equations, one must rst understand the theory of linear wave equations. First and second order linear wave equations 1 simple. The electromagnetic wave from maxwells equations contd 2 2 t e e w w u u 2 2 2 t e e e o o w w x ph xe 0 using the vector identity becomes, in free space and we are left with the wave equation 0 2 2 2 w w t e e p oh o. The kortewegde vries equation is nonlinear, which makes numerical solution important. Chapter 6 partial di erential equations most di erential equations of physics involve quantities depending on both space and time. It also means that waves can constructively or destructively interfere. The 1d wave equation hyperbolic prototype the 1dimensional wave equation is given by. The solutions to this equation can be built up from exponential functions. Equation 1 is known as the onedimensional wave equation. His solution takes on an especially simple form in the above case of zero initial velocity. If b2 4ac 0, then the equation is called hyperbolic. The assumption that one can add the waves together agrees with the linearity of the wave equation. Pdf on jul 10, 2015, majeed ahmed weli and others published exact solutions to linear and nonlinear wave and diffusion equations find, read and cite all the research you need on researchgate.
The most general solution has two unknown constants, which. The treatment is kept at a level that should be accessible to first year undergraduate students and does not require more than elementary calculus, probability and statistics. Illustrate the nature of the solution by sketching the uxpro. More speci cally, we will discuss the initial value, or cauchy, problem for both of the following.
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