In many cases (for example, in the classic wave equation), the equation describing the wave is linear. So I would say you just need D'Alembert's solution formula. This equation is obtained for a special case of wave called simple harmonic wave but it is equally true for other periodic or non-periodic waves. This is one of the most important equations of physics. In the section “Wave Propagation on a TEM Transmission Line,” 1 we found that the potential and current along a transverse electromagnetic (TEM) transmission line satisfy the same wave equations that we have developed in this section, having a complex-valued propagation constant \(\gamma=\alpha+j\beta\), and the same physical interpretation of \(\beta\) as the phase propagation constant.

Optimize your recovery, training and performance. Numerical Solution of the Wave Dispersion Equation. Equation (3) is known as the wave dispersion equation.

$$ \sin(kx-\omega t) = \frac{ e^{i(kx-\omega t)} - e^{-i(kx-\omega t)}}{2i} $$ But we wouldn't normally proceed by replacing sin by this expression. \eqref{11} is called linear wave equation which gives total description of wave motion. Sign up to join this community.

In order to solve this problem from first principles it is first necessary to solve the wave dispersion equation for [math]k=2 \pi / L[/math] in any depth [math]h[/math]. You will get the same wave equation for the wave travelling in negative x-direction. In order for a periodic process to be a wave, it must be a function of both time and one position coordinate, thus enabling it to travel. When solving wave electromagnetics problems with either the RF or Wave Optics modules, we use the finite element method to solve the governing Maxwell’s equations.

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It is a three-dimensional form of the wave equation .

In this blog post, we will look at the various modeling, meshing, solving, and postprocessing … Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top Home ; Questions ; Tags ; Users ; Unanswered ; Wave equation PDE with inhomogeneous boundary. This is not a "wave equation", it is an "oscillation equation".

The above equation Eq.

When this is true, the superposition principle can be applied. A "mathematical wave", being the solution of "the wave equation" (or perhaps a higher-order variant of it), might agree intimately with the data obtained from a physical wave, but it is an inherently different object from the physical wave which it describes.

The theorem you propose seems to refer to bounded domains rather than to the problem on the whole space.

$\begingroup$ I think it follows from the fact that the wave equation has a finite velocity of propagation.

In QM we don't worry about having a complex solution because the observable is the squared modulus, … The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It only takes a minute to sign up.

Both the sin form and the exponential form are mathematically valid solutions to the wave equation, so the only question is their physical validity.