From studying waves I find that I can visualise longitudinal waves where the wave propagates in the direction of the displacement. However I don't understand what causes the propagation perpendicular to the displacement in transverse waves. How is a perpendicular displacement caused in adjacent particles?
This transverse plane wave is illustrated here
If we go to a medium, the "points " will be the molecules which are adjacent. An organized impulse is set up as a function of time, pulling down the one on the right and transferring an impulse delta(p) in the longitudinal direction and then the next molecule pulls up and passes the impulse.
A longitudinal wave is what you get when you stretch and compress a slinky or spring. The slinky can stay inside the cylinder, but you can see it compressing and expanding. Sound is longitudinal.
A transverse wave is what you get when you wave the end of a piece of string up and down. The bits go up and down, but the waveform travels along the string. Light is transverse, as are ocean waves.