Some waves in nature are neither transverse nor longitudinal, but a combination of the two.
When a water wave travels on the surface of deep water, water molecules at the surface move in nearly circular paths,  as shown in the following java applet.

The water surface is drawn as a series of crests and troughs.

Click and drag the left mouse button will create a rectangulararea.
The wavelength of the wave is determined from the width of the area.

The amplitude of the wave is determined from the height of the area.

Click the right mouse button toggle the animation. (pause /resume)
Click the mouse button twice quickly to change the direction of the wave.

Notice that the disturbance has both tranverse and longitudinal components.
As the wave passes, water molecules at the troughs move in the direction of the wave,
and molecules at the crests move in the opposite direction.

There is no Net displacement of a water molecule  after the passage of any number of complete wave.