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The wave equation is $\frac{\partial^2 \chi}{\partial t^2} = c^2 \frac{\partial^2 \chi}{\partial x^2}$. I'll be understanding it in terms of sound.

The wave equation is solved by many periodic functions like $\chi(x, t) = \sin(kx - \omega t)$.

However, there are valid physical phenomena which do not solve the wave equation.

Consider an infinitely long tube. The tube is full of air which is accelerating uniformly in the $+x$ direction. The displacement of that air vs time looks like $\chi(x, t) = t^2$. Plugging this function into the wave equation we get $2 \ne c^2 \times 0$.

What can be concluded from the fact that this function fails to solve the wave equation?

My guess is that its failure to solve the equation means that the case of uniformly accelerating air violates one of the assumptions made in deriving the wave equation for sound. However, I haven't been able to figure out how (I'm reading Feynman's lecture on sound and the wave equation).

I've found a few somewhat related questions and not quite been satisfied by the answers:

jrpear
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Simply put, air trapped in a cylinder does not spontaneously accelerate uniformly to the right. That is, unless it is subjected to external forces other than the forces due to pressure gradients within the air. If it is subjected to other external forces, then the wave equation does not accurately describe how the air moves.

AXensen
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Sure enough, this is because one of the assumptions made in the derivation was violated. I missed it several times.

Feynman's derivation assumes that the force on a particular volume of air is $F_{net} = P(x + \Delta x, t) - P(x, t)$. If you have some other force causing the uniform acceleration, this assumption does not hold.

So one of many answers to "What does it mean when a function doesn't solve the wave equation?" is:

Assuming the wave equation for the domain you're interested in was derived from some universally true laws and some conditionally approximately true assumptions, a physically possible system will not satisfy the wave equation when the system violates an assumption made in the derivation.

jrpear
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If a function does not satisfy the wave equation, it means that if that function describes a real phenomena, waves cannot be used to describe that phenomena.

Many properties of waves, such as the fact that waves propagate at a specific speed, can be derived from $\frac{\partial^2 \chi}{\partial t^2} = c^2 \frac{\partial^2 \chi}{\partial x^2}$, without having to dive into the particulars of how that specific function behaves. If we flip your question around, if we can show that a function solves the wave equation, then we can make a large number of statements about the wave without having to go through all of the math ourselves.

For example, any function which obeys the wave equation also exhibits superposition. This means that we can break any complicated problem involving that function up into simpler waves that sum up to provide the final wave. This is incredibly powerful. We can do all sorts of things using this information, such as showing that a system is stable. We could, of course, prove its stability directly using that particular function. But that may take a ton of math that we can skip.

It also provides negative information. You can make statements in the form of "no function can have this property if it solves the wave equation." This can be useful to avoid getting stuck diving into rabbit holes in the same way that the statement "It is impossible to make a perpetual motion machine" saves us from analyzing a billion "novel" ways to make perpetual motion. The more formal version of that would be to say "It is impossible to make a perpetual motion machine using systems that obey conservation of energy and have entropic behaviors." It says you need to do something different to permit perpetual motion (and so far we are unaware of any way to make it a reality, though some things like superconductors get close)

Cort Ammon
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