The definition of energy that is delivered by a wave

From the classical point of view, if we have any wave $u(t,x,y,z)$ satisfying the classical wave equation, i.e $$\frac{\partial^2 u }{ v^2 \partial t^2 } = \nabla^2 u ,$$

where $v$ is called the speed of that wave, what is the definition of energy that is delivered by that wave in a unit time $t$?

I mean abstractly (without specifying what kind of a wave that I'm talking about), what is the abstract definition of energy that the wave $u(t,x,y,z)$ delivers in a unit time $t$ ?

Answer

I'll give you an intuitive answer and, if it makes sense to you, you'll be able to formalize it yourself.

In the absence of obstacles, all energy of a spreading wave is used to energize the medium it is spreading through. So, if you know how much new medium is covered by the wave in a unit of time and can calculate the energy of the newly energized medium, you'll get your answer, i.e., "the energy delivered by the wave in a unit of time".

Once that new segment of medium has been energized, it will retain acquired energy forever, while passing more energy farther along.

For instance, if we have a 2D wave, spreading from a point source as an expanding circle, the wave will energize a ring of width υ per unit of time or width λ per wave period.

In this example the front of the wave is constantly expanding, therefore the power density and the amplitude of the wave will be decreasing with distance from the source.

A simple one dimensional example is presented here. In this example, the wave is spreading along a string and is energizing one λ of the string length per period. Naturally, in a one dimensional case the power density and the amplitude remain constant, which makes it easier to calculate.

If a wave encounters an obstacle, which could be anything that would interfere with its natural propagation (wall, antenna, etc.), part of the wave energy will be absorbed by that obstacle and then we could talk about energy or power delivered by that wave to that obstacle.