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I cannot find this answer anywhere.

What I do know is that the stapes pounds on the round window of the cochlea and this causes the fluid to move inside the cochlea itself, which has the three chambers (the scala vestibuli, the scala media, and the scala tympani), where the basilar membrane between the scala media and scala tympani is what fluctuates with pressure. This leads to the sound processing in the scala media and the organ of Corti due to the shearing motion of the hair cells against the tectorial membrane, and so on and so forth.

But I don't understand the pathway of the pressure. A lot of resources show the pressure wave moving front-to-back (towards the apex), making the basilar membrane move simultaneously. But at the same time, they also show that the full pathway through the cochlea goes front-to-back through the scala vestibuli, turns around at the apex, and then goes back-to-front through the scala tympani where the basilar membrane is located. So how can the basilar membrane be moving front-to-back when it's stretching along the latter half of this cochlear pathway, way over on the other side of the scala media?

Therefore I am unclear if (1), (2), or both of the following are true:

(1): Does the pressure somehow permeate through Reissner's membrane (between the scala vestibuli and scala media), move through the scala media (without disturbing any hairs in the organ of Corti), and then push on the basilar membrane from above, which would allow for this front-to-back wave motion?

(2): Does the pressure move the basilar membrane from below during its "return trip" going back-to-front through the scala tympani?

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Short answer
The pressure wave through the scala vestibuli drives the basilar membrane response (BM). Your option (1) is correct, (2) is not. The pressure not really permeates or penetrates Reissner's membrane though; it is the pressure differential across the scala vestibuli and scala tympani that is the driving force for the BM movement.

Background
The most accepted way of viewing acoustic activation of the auditory system is the traveling wave theory (Fig. 1) as posed by the famous experiments of Georg von Békésy in the 1960's. These experiments showed, basically, that the basal part of the basilar membrane (BM) starts vibrating first (near the cochlear 'entrance' at the round window) and as the wave travels through the scala vestibuli, more apical parts get activated (Fig 2). This phenomenon explains the cochlear delay that's seen, i.e., that low frequencies tend to activate hair cells later than high-pitched sounds do (He and Ren 2013).

Reissner's membrane separates the scala vestibuli from the scala media. The traveling wave generates a pressure difference between the scala vestibuli and scala tympani. As you rightfully say, this pressure difference is established across Reissner's membrane and across the BM (Fig. 3). It is the movement of the BM that ultimately results in the sensation of sound (source: Ento Key). In fact, the movement of Reissner's membrane is comparable to that of the BM (Reichenbach et al., 2012). However, Reissner's membrane is not tuned and contains no neural elements involved in hearing. The stereocilia of the hair cells are embedded in the tectorial membrane. The shearing forces between the BM and tectorial membrane eventually result in mechanotransduction (Fig. 3). See Fig. 12.5 on this web page from Neuroscience online textbook for an informative video on the BM movement relative to the tectotial membrane.

The traveling wave is basically confined from base to apex. Because of the tuning of the basilar membrane, high frequencies resonate the basal part (because the BM is stiff and narrow) and low frequencies the apical part (because the BM is wide and floppy). Each time a resonant frequency is met, the BM starts vibrating maximally and basically the frequency cannot go further and its energy is dissipated. Thus, along the way, sound traveling through the scala vestibuli to the helicotrema (the boundary of the scala vestibulu and tympani) more and more energy is used up from the pressure wave. Thus, the basilar membrane is set in motion primarily by the pressure differences occurring in the scala vestibuli and by the time the traveling wave reaches the helicotrema no appreciable energy is left as all auditory frequencies have been filtered out. Remaining energy containing frequencies outside the audible range (roughly 20 - 20,000 Hz) continue to travel to the scala tympani (think of static pressure during a deep dive) and therefore the round window is necessary to act as a buffer (Robles & Rugggero, 2001).

However, there are other theories, notably one from Hermann von Helmholtz, a brilliant scientist of the 1800's that has revolutionized sensory sciences. It is called the 'sympathetic resonance' theory. I am not at all familiar with this theory, as it has been largely abandoned, at least as far as I know. The theory from Von Békésy assumes that the basilar membrane is a place-to-frequency converter that acts like a Fourier transformer, known as the place coding theory. Basically, the traveling wave dissects complex acoustic stimuli into different acoustic frequencies by menas of frequency tuning of the BM (Fig. 2). By contrast, Helmholtz assumed there are tiny tuned elements positioned in the cochlea that basically perform the Fourier transform (not shown here). If you wish to read more of the latter theory, I happily refer you to Bell, 2004).

traveling wave
Fig. 1. Traveling wave. source: NY University

traveling wave BM
Fig. 2. Basilar membrane movement. source: NY University

cross section cochlea
Fig. 3. cross section through cochlear duct. source: Psychology Wiki

References
- Bell, Plos Biology (2004); 2(10):e337
- He & Ren, Scientific Reports (2013); 3: 1874
- Purves et al. (eds), Neuroscience (2001) 2nd ed. Sunderland (MA): Sinauer Associates
- Reichenbach et al., Cell Rep (2012); 1(4): 374–84
- Robles & Rugggero, Phys Rev (2001); 81(3): 1306-43

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