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Any help with this will be much appreciated. I've scoured the web and searched through textbooks, but I don't have a definite answer to my question as of yet.

First, here's the background on my question:

If I have a standard mass on a spring set into motion, All the textbooks say that the standard solution to the differential equation which describes the motion of the mass is Something of the x=ACos(wt + phi) form, where w is the angular frequency and phi is a phase angle.

However, there is a general equation that describes sinusoidal oscillatory motion that is in the form x=Acos(kx-wt), where k is the angular wave number 2pi/lambda, and where lambda is the wavelength. I've seen this equation typically ascribed to transverse waves, but not to longitudinal ones. So what I'm stuck on is... It seems to me that there should be an analogous angular wavenumber value for longitudinal waves, but there I'm not so sure what the "wavelength" would be. On some websites I've seen them define the wavelength of a longitudinal wave to be from compression to compression or rarefaction to rarefaction, but for a mass on a spring Would that just be the "initial displacement"? Also, assuming that friction damps the oscillation, wouldn't I have an angular wavenumber that's changing in time, delta k? If so, How would I factor that into an equation that describes the motion using that X=ACos(kx-wt) form? (k here representing the angular wavenumber and A including the damping decay exponential factor).

Thank you for your thoughts on this!