INTERFERENCE BETWEEN TWO LINEAR WAVES

Let us consider waves which appear on the water surface when long cylinder oscillates harmonically in the point with coordinate x=0. In this point the height z of water surface is described by the formula:

z = Acos(wt)

where A is amplitude of the cylinder oscillation, w = 2pf, f is the frequency of oscillation, t is time.

Any point of the water surface will oscillate with the same amplitude as cylinder, but this oscillation is shifted by phase, which depends upon the distance from the vibrating cylinder:

z = Acos(wt - kx)

where k = w/v, v  is the speed of the waves propagation.

In general case the amplitude A of the wave will attenuate with the distance because of internal friction between molecules in the water.

Next, let us consider two cylinders, which oscillate with the same frequency. The distance between cylinders is d. In this case the amplitude of oscillation at any point of the water surface can be found as superposition of two waves:

z = Acos(wt - kx) + Acos(wt + k(x - d))

The constant k in these two cosine functions has different signs because the waves from the different cylinders propagated in opposite directions.

Result of the superposition follows:

z = 2Acos(wt - kd/2)cos(kx - kd/2)

This equation describes the interference of two linear waves that propagate in opposite directions. We can see from this formula that there are points on the water surface where the waves interfere destructively and no oscillation is observed (nodes) and there are points where the waves interfere constructively and the water surface oscillates with the double amplitude 2A (antinodes). The nodes appear at the points where cos(kx-kd/2)=0, i.e. at the points x=l /2 (1/2+n)+d/2, where n is the integer number and l is the wavelength. This means that the distance between the nodes is the half of the wavelength. The same is with the maximums of the interference pattern. They appear at the points where cos(kx-kd/2)=± 1, i.e. at the points x= nl /2+d/2. Knowing the frequency at which we generate the waves and measuring the distance between the nodes (with the aid of microscope, for example), we can find the velocity of the waves on the water surface and then we can calculate many valuable parameters of the water (or other liquid).

Animation shows the interference of two linear waves propagating in the opposite directions. We can see that small red ball oscillates with the maximal amplitude, while the small red parallelepiped does not oscillate, because it is situated in the node of the standing wave. Parallelepiped is just rotating.