What defines a wave according to the wave equation?

In summary, this system seems to be described by a wave if it follows the wave equation, but it doesn't seem to require the functions to be periodic or cyclical in any sense.
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FAS1998
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If I’m not mistaken, a system can be described as a wave if it follows the wave equation.

On Wikipedia, the general solution for the one-dimensional wave equation is written as u(x,t) = F(x - ct) + G(x + ct).

I don’t see the connection between this solution and what I understand waves to be. Don’t waves need to by cyclical? This general solution doesn’t seem to require the functions to be periodic or cyclical in any sense. What would be a good description of a wave according to the wave equation?
 
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Perhaps (for the 1D case): " A distortion that propagates without changing shape "
 
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FAS1998 said:
If I’m not mistaken, a system can be described as a wave if it follows the wave equation.

On Wikipedia, the general solution for the one-dimensional wave equation is written as u(x,t) = F(x - ct) + G(x + ct).

I don’t see the connection between this solution and what I understand waves to be. Don’t waves need to by cyclical? This general solution doesn’t seem to require the functions to be periodic or cyclical in any sense. What would be a good description of a wave according to the wave equation?
Differentiation turns multiplication into addition, the linear approximation: ##d(f\cdot g) = f\cdot dg + g\cdot df##. Integration does the opposite and turns addition into multiplication, which is why we use the exponential function as template for solutions: ##\exp(x+y)=\exp(x) \cdot \exp(y)##. This is roughly what's going on.

Now the wave equation is ##y''+y=0##. If we set ##y=ae^{cx}## then we get ##ac^2+a=0##. We are not interested in the solution ##a=0##, so we have ##c=\pm i## instead, and ##y=ae^{\pm i cx}## which is a circle in the complex plane and ##y## thus a cyclic function.
 
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FAS1998 said:
On Wikipedia, the general solution for the one-dimensional wave equation is written as u(x,t) = F(x - ct) + G(x + ct).

I don’t see the connection between this solution and what I understand waves to be. Don’t waves need to by cyclical? This general solution doesn’t seem to require the functions to be periodic or cyclical in any sense.

Did you look further down to
https://en.wikipedia.org/wiki/Wave_equation#Plane_wave_eigenmodes
 
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FAS1998 said:
Don’t waves need to by cyclical?
No, they don't need to be periodic (the usual term for what I assume you mean by "cyclical.").

Consider the disturbance that you produce on the surface of water in a pond or bathtub when you poke your finger into the water and withdraw it quickly, once. This is described by the same differential wave equation as the periodic disturbance that you get when you poke your finger into the water and withdraw it, many times at regular intervals.
 
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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 relates the wave's amplitude, frequency, and wavelength to the properties of the medium, such as its density and elasticity.

2. What is the significance of the wave equation?

The wave equation is a fundamental equation in physics, as it helps us understand and predict the behavior of various types of waves, including sound, light, and water waves. It also has practical applications in fields such as engineering, acoustics, and optics.

3. How is the wave equation derived?

The wave equation is derived from the laws of motion and conservation of energy, specifically the second law of motion (F=ma) and Hooke's law (F=-kx). By applying these laws to a small element of a wave, we can derive the wave equation.

4. What are the assumptions made in the wave equation?

The wave equation assumes that the medium in which the wave travels is homogeneous (uniform), isotropic (has the same properties in all directions), and continuous. It also assumes that the wave is small in amplitude and that there are no external forces acting on the medium.

5. Can the wave equation be applied to all types of waves?

No, the wave equation only applies to waves that travel through a medium. It cannot be used to describe electromagnetic waves, which do not require a medium to propagate. However, it can be modified to describe other types of waves, such as seismic waves, by incorporating additional factors specific to those waves.

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