Solution of wave equation, 2nd partial derivatives of time/position

In summary, the conversation discussed a solution of the wave equation using the Laplacian and Alembert's solution for 1D waves. They also mentioned the meaning of the second partial derivatives in respect to time and position, and how it can be used to derive the speed of an electromagnetic wave.
  • #1
mathnerd15
109
0
f(z,t)=\frac{A}{b(z-vt)^{2}+1}...

\frac{\partial^{2} f(z,t)v^{2} }{\partial z^2}=\frac{-2Abv^{2}}{[b(z-vt)^{2}+1]^{2}}+\frac{8Ab^{2}v^{2}(z-vt)^{2}}{[b(z-vt)+1]^{3}}=\frac{\partial^2 f}{\partial t^2}
\frac{-2Abv^{2}}{[b(z-vt)^{2}+1]^{2}}+\frac{8Ab^{2}v^{2}(z-vt)^{2}}{[b(z-vt)+1]^{3}}

this is a solution of the wave equation, but it can be written with the Laplacian. is this also a hyperbolic partial differential equation. Alembert derived the solution that 1D waves are the addition of right and left moving functions

what is the meaning of the 2nd partial derivatives in respect to time and position which differ by v^2? (I wrote this on online Latex editor, the differentiation is in the attachment)
thanks very much!
 

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  • #2
so you can derive the speed c from the Maxwell equations which for an electromagnetic wave is the Weber/Kohrausch ratio 1/(epsilono*muo)^(1/2)...
 
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1. What is the wave equation?

The wave equation is a mathematical formula that describes the behavior of waves in a given medium. It is a second-order partial differential equation that relates the second derivatives of the wave's position with respect to time and space.

2. How is the wave equation solved?

The general solution to the wave equation involves finding a function that satisfies the equation and its boundary conditions. This can be done through various mathematical techniques, such as separation of variables, Fourier transform, or numerical methods.

3. What do the 2nd partial derivatives of time and position represent?

The second partial derivative of time represents the acceleration of the wave, while the second partial derivative of position represents the curvature or change in direction of the wave. Together, they describe the dynamics of the wave as it propagates through a medium.

4. Why is the wave equation important?

The wave equation is a fundamental equation in physics and engineering, as it is used to model a wide range of phenomena, including sound, light, and electromagnetic waves. It also plays a crucial role in fields such as acoustics, optics, and signal processing.

5. What are some applications of the wave equation?

The wave equation has numerous practical applications, such as predicting the behavior of seismic waves in earthquakes, designing musical instruments, and creating images with medical ultrasound. It is also used in various technologies, including wireless communication, radar, and sonar.

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