Telling apart a right traveling non-periodic signal from a left-traveling non-periodic signal, with a background periodic signal I have a time-signal sampled over a one-dimensional spatial domain $x$. I know it is made of 3 components:

a right traveling non-periodic wave (characteristic time width $T_{R,NP}$, mostly 0 outside this width). This is prescribed at the left end of the domain.
a left-traveling non-periodic wave (characteristic time width $T_{L,NP}$, mostly 0 outside this width). This is prescribed at the right end of the domain
a periodic signal (right traveling in my case, of period $T_P$)


While traveling, all these signals slightly change shape due to non-linearities. Again changes are not drastic and fairly small. The periodic signal starts as a single harmonic but then more harmonics show up, which have small energy and the peak period is still $T_P$. Here is an example of the signal at a specific location (purple), together with my best guesses (I drew them by hand) of the two non-periodic signals. Two signals were prescribed on the left and traveled right, and one was prescribed on the right and travel left.

Which is the best way to tell apart the 3 components at different locations of my domain? In particular:

Is there a method that can take advantage of the known information about the direction of travel?
Sometimes (but not always) $T_{L,NP} &gt; T_{R,NP} &gt; T_P$. Would this help? I would prefer a general algorithm that does not rely on this information, but if a general algorithm does not exist, solving for these special cases would still be something.
If you can sample at any point in time, this shouldn't be that difficult.  The first step is to collect as series of frames in time.
In each frame you should be able to find a best fit sinusoid, or set of harmonic sinusoids, to your periodic signal.  Since this signal is assumed to have the same parameters across time, except the phase values which would need to be time adjusted, you can find an &quot;average&quot; value across all your frames for the periodic signal.
Subtract the best fit periodic signal from your frames.  This should leave the other two behind.  From your drawing, it appears that you have regions where one hump will b e zero, and others where the other hump is zero.  Those are the locations to read your humps at.
For more accuracy, once you have your hump signals estimated, go back and subtract them from your frames.  This should leave the periodic signal behind.  Once again, find the best fit sinusoid assembly.  This time your results should be more accurate.
Rinse and repeat.