It is the transverse wave that is involved in sideways movement. I think the simple theory says that the fundamental mode of transverse oscillation (which you see when you whack the stick on the side and watch it wiggle back and forth with two nodes) gives the speed of the wave (after taking the right ratio of length divided by period) and only the part of the cue that can be reached at that speed while the tip is on the ball can give resistance to sideways motion.But I doubt that a transverse wave has much to do with it. ...
The speed of the transverse wave is much, much smaller than the speed of the longitudinal wave (compression wave) that puts energy into the cue ball. You can get a feel for this by noting how much force is required on the side of the stick to bend it a quarter inch and then how much is required on the tip to compress the stick a quarter inch.
Since the transverse wave is slow, the tip "sees" only a small part of the cue stick while the tip is on the ball in the sideways direction. By contrast, the entire stick has time to compress during tip-ball contact because the longitudinal wave is very fast. According to the CRC Handbook, the speed of sound in maple is over 4000 meters/second along the fibers while ash is about 10% higher. That alone defines the speed of the compression wave, pretty nearly.
For the transverse wave the speed is a more complicated function of the cross section of the stick -- a thinner stick should have a lower resonant frequency and so a lower speed of sound for the wave.
Note that if you had a cue stick so long -- say 40 meters -- that the compression wave could not get to the end of the butt during tip contact, the cue ball would "see" only part of the weight of the stick.
Just as the cue stick "rings" sideways after an off-center hit (at about 40 cycles per second, IIRC), the stick also rings lengthwise after a center-ball hit with the tip going toward and away from the departing cue ball. This is visible in the high-speed videos on the Russian site because their equipment is fast enough and close enough to see the very small longitudinal oscillations.