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I have difficulty in connecting three concepts:

  1. Waves, defined as perturbations of a field (in all generality) that move in space.
  2. Functions of space and time coordinates of the following type: $ F(\vec{x},t)=f(\vec{x}-\vec{v}t) $ and their superposition.
  3. Solutions of the d'Alembert equation: $\nabla^2F=\frac{1}{v^2}\frac{\partial^2 F}{\partial^2 t}$

Which are the inclusion relations between these three objects? Are other differential equations (such as Klein-Gordon or Schroedinger) interconnected into this scheme?

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The inclusion relations can be represented in the following way: $$ 1. \supset 3. \supset 2. $$ The definition of a wave as perturbations of a field that move in space is not entirely accurate but contains the main feature common to all kinds of waves, from linear to shock waves. It is not entirely accurate because, on one side, it should be modified to include stationary waves. On the other side, there are time-dependent perturbations evolving in the space that are not waves (for example, the field of local temperature in a cooling solid). A slightly better definition could probably be perturbations of a field that evolve according to causal, local dynamics. But I do not want to focus too much on the most general definition of a wave. In the following, I'll explain the inclusion relations.

$(1. \supset 3.)$

D'Alambert's equation is an explicit differential equation controlling the dynamical behavior of a perturbation. It is a non-dispersive linear equation. Therefore, non-linear waves are excluded, as well as dispersive waves, i.e., waves with a wavelength-dependent frequency.

$(3. \supset 2.)$

Solutions of the d'Alembert equation corresponding to a one-dimensional propagation can be written as $$ F(x,t) = f(x-vt)+g(x+vt) $$ with $f$ and $g$ arbitrary functions. However, not every solution of d'Alambert's equation has such a form corresponding to a rigidly moving wave profile. For example, a spherical solution of the wave equation has the form $$ F({\bf r},t) = \frac{f(r-vt)}{r}+\frac{g(r+vt)}{r} $$ where the shape of the wave depends on the radial distance.

Solutions of other differential equations, such as Klein-Gordon or Schroedinger equations, different from the d'Alembert equation, are examples of linear dispersive waves.

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  • $\begingroup$ Really helpful! Another sub-question: Is the opposite inclusion $3.\subset 2.$ true for one-dimensional d'Alembert equation? $\endgroup$ Jul 6, 2023 at 12:45
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    $\begingroup$ @Antonio19932806 In the one-dimensional case, $f(x-vt)$ is certainly solution of d'Alembert's equation. However, there are also solutions of a different form: $g(x+vt)$ $\endgroup$ Jul 6, 2023 at 15:57
  • $\begingroup$ The all important initial conditions at t=0 are not mentioned here; a) the initial displacement of the field and b) the initial speed of the initial displacement of the field. $\endgroup$
    – user45664
    Jul 6, 2023 at 17:40
  • $\begingroup$ @user45664 the initial conditions are important, and absolutely essential to find an explicit solution. But they are a byproduct of the kind of pde. In the differentiable case, it is the hyperbolic character of the pde establishing at the same time the wave-like character of the time evolution and the kind of initial conditions required to have a well-posed mathematical problem. I did want to be too technical, but the condition of a causal, local dynamics was a kind of generalization of the concept of hyperbolic pde. $\endgroup$ Jul 6, 2023 at 17:56

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