• As we saw in the page What is a wave, a wave does not carry along the particles of the medium in which it moves, but only their energy. The particles of the medium merely oscillate around equilibrium positions, and do not follow the "crest" of the wave in its motion.
• The wave, however, generally moves energy from one place to another in space, and therefore has a definite direction of propagation.
• Given the direction of propagation of the wave, the particles of the medium can oscillate in different ways with respect to this direction, depending on the forces acting on them.
• Waves can therefore be classified according to the way in which the particles of the medium move with respect to the direction of propagation of the wave.

Let us examine the possible cases through some examples.

Examples in pictures

The simplest case: the chain in a plane

• In these two pictures a chain made of balls connected by small strings or springs is set into oscillation. For clarity, only the balls are shown, and not the elements connecting them.
• Let us first consider the motion in two dimensions only (that is, in the plane of the screen). It is clear that two different types of motion are possible:
• in both figures the wave moves from left to right, but the balls oscillate in completely different ways. In the first case they oscillate horizontally, in the second vertically. To highlight their motion, we have painted two reference balls red.

Longitudinal wave	Transverse wave
The balls oscillate in the same direction as the propagation of the wave	The balls oscillate in the direction perpendicular to that of the propagation of the wave

More difficult: a solid in a plane

• Let us now see what happens when, instead of a small chain, we study a solid made of a set of balls (atoms?) in the plane of the screen. Again, we consider only oscillations taking place in the plane of the screen.
• In the following pictures we have highlighted the motion of the wave with a blue dot, following one of its crests, while a red dot highlights four atoms chosen as reference.
• In this case too we have the two types of waves that we saw in the chain: in the first case the atoms oscillate longitudinally to the right and to the left, while in the second they oscillate transversely up and down.

Longitudinal wave	Transverse wave
The atoms in a horizontal plane of the solid stay in the horizontal plane and oscillate back and forth in the same direction as the propagation of the wave	The atoms in a vertical plane of the solid stay in the vertical plane and oscillate in the direction perpendicular to the direction of propagation of the wave

However, it is now easier to notice that it is also possible to have waves that are neither transverse nor longitudinal: the atoms can also move along circles (or ellipses). This type of wave is particularly evident when it reaches the surface of a solid or a liquid.

More and more difficult: surface waves

• The next two figures show waves that particularly concern the surface of a medium; indeed, we see that going deeper the oscillations of the particles become smaller and smaller: the waves die out.
• These waves are neither purely longitudinal nor purely transverse: the particles move in small ellipses, or circles.

Surface wave (Rayleigh wave)	Gravity wave at the surface of a liquid
This type of wave shakes the Earth's surface during earthquakes, together with the longitudinal and transverse waves illustrated above. Note that the particles rotate "backwards" with respect to the direction of propagation of the wave.	These are typically the waves we see on the surface of the sea. Note that the particles rotate "forwards" with respect to the direction of propagation of the wave.

Conclusions

1. We have seen that waves can be classified as longitudinal or transverse depending on whether the oscillation of the particles of the medium takes place in the same direction in which the wave propagates, or in the perpendicular direction.
2. We have seen that there are also more complex motions, which can be thought of as combinations of longitudinal and transverse motions. See the following animations, the result of numerical simulations on model structures made of masses and ideal elastic elements.

transverse wave	surface wave
A purely transverse wave.	A complex wave containing both a transverse and a longitudinal component. The longitudinal component leads the transverse one by a quarter of a period, and the two compressions (orange colour on the connections between the masses) can be seen travelling one after the other.

What happens in gases?

• In a gas, the attractive forces between the molecules are much weaker than those binding the molecules of a solid, and the same holds for surface forces, which are relevant in a liquid.
• Consequently, while both transverse and longitudinal waves can propagate in a solid, in a gas there are only longitudinal, or compression, waves.
• Sound waves are precisely mechanical compression waves.

The following figures represent disordered particles, as the molecules in a gas are.

Longitudinal waves: possible in a gas	Transverse waves: impossible in a gas
Sound waves are compression waves, and propagate both in solids and in gases	The transverse waves represented here can propagate in a non-crystalline solid (in which the atoms are not arranged in a lattice, but are disordered), but not in a gas, because the transverse forces (technically, the shear modulus) are too small

What happens when there are no particles?

• In the case of mechanical waves, like those shown in the examples above, the wave always needs a "supporting" material medium. Indeed, the wave is nothing but the propagation of mechanical energy through the particles of the medium.
• Electromagnetic waves such as light and radio waves, however, can also propagate in vacuum, and there their oscillation does not correspond to the oscillation of any atom or molecule.
• Nevertheless, electromagnetic waves can still be called transverse, because the quantities that oscillate, namely the electric and magnetic fields, always oscillate perpendicularly to the direction of propagation of the wave.

In the case of electromagnetic waves too, it is possible to combine transverse waves of different kinds to obtain oscillations that, while remaining transverse, do not take place along a single direction (for example circular or elliptical ones). This is called polarization.