D'Alembert's Solution – Domain of dependence

[FluidStu]

When considering the Wave equation subject to initial conditions as follows…

…then D'Alembert's solution is given by (where c is wave speed):

I'd like to understand physically how this formula allows us to know the value of u (where u is the height of the wave, say) at some point (x₀,t₀). With relation to the diagram below, I understand that the height u is constant along the characteristic lines, and that moving along these characteristic lines in the x-t plane is how the wave crest (at height u) would propagate forwards/backwards in time.

So, referring to D'Alembert's formula above, says that the height of the wave is given by:

• The average height u of the wave between the two possible starting positions of the wave (the two intersections of x-axis).
• The average rate of increase of the height of the wave between the two possible starting points.

Could someone please explain physically how D'Alembert's solution allows us to know the height of the wave at some point (x₀,t₀)? I can't see how the two terms described in bullets above relate to the height of the wave at some point (x₀,t₀).

The region between the two possible starting points is called the domain of dependence – meaning that the height of the wave at the point (x₀,t₀) depends only on initial data specified within this range. Why is there a domain of dependence? Why do conditions in the region between these two points influence the height at (x₀,t₀),? It would seem that the height at (x₀,t₀) would depend only on the conditions at the two starting points.

 

[Orodruin]

First of all, note that the only dependence on the region in between the points is only in terms of the time derivative of the height, not the height itself. For the height itself, it only depends on the values at the points.

That the dependence is on a region for the time derivative is due to the fact that you are studying the wave equation in one dimension. For the one-dimensional wave equation, the height does not relax back to the original zero level as a wave front passes. You can understand this physically. Let us say that the wave is a transversal wave on an infinite string. If you give the wave some transversal momentum, in a small region - then this transversal momentum is conserved because there is nothing that changes the total transversal momentum. If the string relaxed back to its original state after the wave has passed - then the overall transversal momentum of the string would be zero and so this cannot happen. This can also be seen in the Green's function of the one-dimensional wave equation, which is essentially a step function.

You have a similar behaviour in two dimensions, where the Green's function is never zero in the region that the wave has passed, but instead goes to zero as $1/\sqrt{t}$. Only in three dimensions does the Green's function become a wave front that leaves the system at zero displacement after passing.

 

[bosque]

Maybe in three dimensions with an impulsively excited sphere there is also a non zero displacement:
see
https://www.researchgate.net/publication/316994209_Making_waves_the_geometric_derivation_of_Huygens%27_Principle_for_wave_propagation_and_the_problem_of_the_wake