The sound wave is not a "by-product." It is precisely the phenomenon you are describing. And while we are discussing equations and distribution functions, you might as well get the equation right: wave phenomena arise when the equations of motion are hyperbolic PDEs. Such systems involve the Laplacian, not just the gradient. Indeed, the physics of such systems are typically studied as a whole, in terms of... wave phenomena.
Further, not all waves require a medium. Light and gravitational waves are prominent examples. There is nothing to reduce these phenomena to, except for the fields themselves, whose form is dictated by... a wavelike solution.
While we are at it, let me disabuse you of your explanation of the sound speed. Turns out that the sound speed is a thermodynamic quantity; it is the speed at which small wavelike perturbations propagate. To properly derive the sound speed, one must linearize the Euler equations, and then adopt a thermodynamic equation of state, from which the sound speed is derived. It is, emphatically, not the speed at which molecules move.
Finally, it is obvious that you did not understand what I was driving at with respect to color. I agree wholeheartedly that we have no true color, within our minds, with which to perceive, say, soft x-rays. But this issue has to do with our own neurobiology, not fundamental physics. Comparing the two is what is problematic. There is little reason to suspect that our mental limitations have anything to do with anything but evolutionary necessity. Such limitations are categorically different than, say, the speed of light.
Further, not all waves require a medium. Light and gravitational waves are prominent examples. There is nothing to reduce these phenomena to, except for the fields themselves, whose form is dictated by... a wavelike solution.
While we are at it, let me disabuse you of your explanation of the sound speed. Turns out that the sound speed is a thermodynamic quantity; it is the speed at which small wavelike perturbations propagate. To properly derive the sound speed, one must linearize the Euler equations, and then adopt a thermodynamic equation of state, from which the sound speed is derived. It is, emphatically, not the speed at which molecules move.
Finally, it is obvious that you did not understand what I was driving at with respect to color. I agree wholeheartedly that we have no true color, within our minds, with which to perceive, say, soft x-rays. But this issue has to do with our own neurobiology, not fundamental physics. Comparing the two is what is problematic. There is little reason to suspect that our mental limitations have anything to do with anything but evolutionary necessity. Such limitations are categorically different than, say, the speed of light.