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In physics we deal with the waves of different nature: mechanical, electro-magnetic, etc. In spite of different physical nature of these waves they have many common features. Waves, where the parameter of interest (displacement, mechanical stress, etc.) oscillates along the axis of the wave propagation, are called longitudinal waves. If oscillation occurs perpendicularly to the direction of the wave propagation, then such a wave is called transverse wave (electro-magnetic waves, for example, are transverse one). 

If the particle of medium are interacting by means of elastic forces appeared during the wave propagation, then the waves are called elastic. For example, the wave in a metal rod are elastic one.  The first animation shows the propagation of elastic wave in a grid which consists of the balls connected each other by springs. Every ball oscillates harmonically in longitudinal direction, which coincides with the direction of the wave propagation. Amplitude of oscillation is the same for every ball and equals to A, while the phase of oscillation increases with the number of the ball by  Dj i.e.

x0=Asin(wt);  x1=Asin(wt+Dj); x2=Asin(wt+2Dj); x3=Asin(wt+3Dj);  ่ ๒.ไ.

where w is angle frequency of the wave, t -is time,  Dj is the phase shift of oscillations from ball to ball.

The oscillation in transverse wave occurs in the direction perpendicular to the direction of the wave propagation. As in the case of transverse wave every ball in longitudinal wave oscillates with the same amplitude and the phase of oscillation increases linearly from ball to ball: 

y0=Bsin(wt);  y1=Bsin(wt+Dj); y2=Bsin(wt+2Dj); y3=Bsin(wt+3Dj);  etc.

In general case the equation of the wave propagation can be written as: z = Acos(wt - kx), where z is the displacement of the particle from position of equilibrium, x is coordinate in the axis of the wave propagation, k = w / v, v  is the velocity of the wave. Knowing the frequency and velocity of the wave we can calculate the phase shift between the nearest particles (balls): Dj = (w / v)a, where a is the distance between the balls in the lattice.

The next animation shows superposition of transverse and longitudinal waves of equal amplitudes shifted by phase at p/2. As a result every ball moves in a circle. This motion can be described by the equation:

x=Acos(wt+j0); y=Asin(wt+j0)

The molecules on the water surface move under the action of surface tension and gravity. Next animation simulates the wave motion of the molecules in the surface layer of water (or other liquid). If the amplitude of this wave is small, then every molecule moves in a circle path. The radiuses of these circles are diminishing with depth, so the balls in bottom part of animation are still. 
L-wave.gifThe waves on the surface of the water are neither longitudinal nor transverse. We can see in animation that red ball, which simulates the molecule of the water surface, moves in a circle path. So, the wave on the water surface is the superposition of transverse and longitudinal motions of the molecules.
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