Solution of wave equation
[DOC File]Partial Differential Equations
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Plane wave. 5.2. Solution of the wave equation in spherical coordinates. For spherical symmetry, such as emission from a point source in free space, the problem becomes one dimensional, with the radial coordinate as the only variable, Recall that the Fourier transform pair is defined in this case as.
Solution of the wave equation
The solution of the wave equation by separation of variables proceeds in a manner similar to the solution of other partial differential equations. We postulate a solution that is the product of two functions, X(x) a function of x only and T(t) a function of time only. With this assumption, our solution becomes. u(x,t) = X(x)T(t)
[DOC File]Relativity4 - Department of Physics
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∂2y/∂t2 = v2 ∂2y / ∂x2, with v = (T/μ)1/2 being the speed of the waves, as we shall see. Any equation of this form is called the wave equation. Modifications of it are usually called “so-and-so’s wave equation”.
[DOC File]Physics 406 - St. Bonaventure University
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Another solution of the wave equation is given by d’Alembert as u(x,t) = [f*(x (ct) + f*(x + ct)] + (7.3-2) where f* and g* denote the odd extension of f(x) and g(x). The reason why the odd extension is used can be deduced from the Fourier solution of (7.3-1) with g(x) = 0.
[DOC File]5
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Plugging into the wave equation yields: Thus, the solution to the wave equation that is a consequence of Maxwell’s equations in vacuum is a sinusoidally varying function for both the electric and magnetic fields. It is a traveling wave solution, which becomes more apparent if we write the solution in this form:
[DOC File]The Wave Equation:
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to a frequency (i.e. has units of 1/time or Hz). (ii) Use separation of variables to οΎ…nd the normal modes of the damped Wave Equation (1) subject to the BCs. u (0; t) = 0 = u (l; t) (8) Impose a restriction on the parameters c, l, k which will guarantee that all solutions are oscillatory in time.
[DOC File]Solutions to Problems for the 1-D Wave Equation
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Thus, the above form satisfies the wave equation. Thus, the simple wave equation in 1-D has two solutions which propagate undeformed in opposite directions with increasing t with a velocity . This is one of the fundamental properties of waves. The solution can be written as disturbances that propagate at well-defined velocities.
[DOC File]Purdue University
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The one-dimensional wave equation is . a. Solution. The wave equation has solutions of the form , , and . These are all traveling harmonic waves, where the wave number is and the angular frequency is . (f is the frequency in Hz.) We’ll concentrate on the complex exponential form: . Then the derivatives are.
[DOC File]California State University, Northridge
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By Galilean invariance, this is also the wave equation in the first inertial frame: (2) A solution to this equation is of the form (3) where A, k, and are constants. We want a real solution, so take. Note that initial conditions determine : . Let the properties of the wave be given: let be the wavelength and f.
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