I What defines a wave according to the wave equation?

[FAS1998]

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?

 

[BvU]

Perhaps (for the 1D case): " A distortion that propagates without changing shape "

 

[fresh_42]

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.

 

[robphy]

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

 

[jtbell]

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.