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thegirl
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I don't know if this is a silly question? Am I missing simple math? How does a wave depending on amplitude and frequency make it's equation a second order differential equation?
blue_leaf77 said:Actually it's the other way around, every system which is described by a second order type differential equation in space and time supports the propagation of waves.
Mh, I'd say, it should be a hyperbolic partial differential equation, because, e.g., the Laplace equation doesn't describe waves. The sign pattern in the corresponding differential operator of the linear case is important, i.e., for waves it should be a D'Alembertian rather than a Laplacian to lead to wave propgation solutions.blue_leaf77 said:Actually it's the other way around, every system which is described by a second order type differential equation in space and time supports the propagation of waves.
For mechanical waves, may be. But Maxwell equations are no approximation.nasu said:because we make a lot of approximations
I can agree with you, this shows that the EM waves are an example of a truly sinusiodal disturbance in nature provided they propagate in free space.nasu said:But this is a good (interesting) point.
nasu said:Actually what we call usually "wave equation" it has that form because we make a lot of approximations.
nasu said:Is maybe the lowest order that allow for a wave solution.
The wave equation is a second order differential equation because it represents the relationship between the second derivative of a wave (which describes the acceleration of the wave) and the wave itself. This relationship is necessary to accurately model the behavior of many types of waves, such as sound waves and electromagnetic waves.
The wave equation can be derived from the basic principles of wave motion, such as the conservation of energy and the relationship between velocity, frequency, and wavelength. It can also be derived using mathematical methods, such as separation of variables or the method of characteristics.
The wave equation has numerous applications in physics, engineering, and other fields. It is used to model many types of waves, including sound waves, light waves, and water waves. It is also used in the study of vibrations, oscillations, and other phenomena.
The wave equation is important in science because it provides a fundamental understanding of how waves behave and interact with their surroundings. It is also used to solve many practical problems, such as predicting the behavior of seismic waves in earthquakes or designing antennas for communication systems.
The wave equation is closely related to other equations in physics, such as the heat equation and the diffusion equation. These equations all describe the behavior of physical quantities over time and space, and can often be solved using similar mathematical techniques.