OF THE 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 one small parallelepiped oscillates with the maximal amplitude, while the other one does not oscillate, because it is situated in the node of the standing wave. Parallelepiped is just rotating.
Simulation of the waves on the water surface allows many physical phenomenon common for waves of different types (interference, diffraction, reflection, etc.) to be investigated and visualized. Let us consider circular wave on the water surface generated by small ball oscillating in the vertical direction. At distances large as compared to the diameter of the ball it can be considered as a point source of the wave. Any flat wave can be considered as a circular wave with the center situated in infinity. In general case the equation of the circular wave can be written as: s=A(r)cos(wt - kr), where w is the angle frequency of the wave, k = w/v, v is the speed of the waves propagation, r is the distance from the source to the point of observation, A(r) is amplitude of the wave, which depends on distance. I many cases we can neglect the attenuation of the wave and consider A as constant.
Next, we shall consider two small balls, which oscillate on the water surface. Every ball excites the wave, which interferes with the wave from the other ball. As a result we see on the water surface a typical interference pattern.
Let us derive the equation for this interference. The wave from every ball is described by the formula:
s1=A1cos(wt - kr1); s2=A2cos(wt - kr2);
where A1 and A2 are the amplitudes of the waves in the points of excitation (balls), r1 and r2 are the distances from ball 1 and ball 2 to the point of observation.
Because the difference D =r2-r1 is much less than each radius r1 and r2 we can consider r1» r2 and A = A1 » A2. Superposition of the waves s1 and s2 can be described as follows:
s=s1+s2+2Acos[ k(r2-r1)/2 ] cos[ wt-k(r1+r2)/2 ]
We can see from this equation that in the points where r2 - r1 = l (1/2+n) the water surface does not oscillate. These node points (lines) are clearly seen in the animated picture.
|Let us consider the point source of the circular surface wave (vibrating ball) situated near to the totally reflected wall. According to principle of Huygens (1678) the reflected wave coincides with the one generates by the fictitious source situated at the opposite side of the wall symmetrically to the real source of the circular wave. If the distance between the source and the wall is equal to the integer number of the half wavelengths, then on the right side of the source the circular wave will interfere in phase with the reflection from the wall increasing the wave crest. The interferometric pattern appeared in this case is given in animation.|
|Next animation also shows the interference of the circle wave with the wave reflected from the wall, but the distance between the source and the wall is equal to odd number of quarter wavelengths (integer number of half wavelengths plus the quarter of wavelength). In this case on the right side of the source the waves will interfere in anti-phase and we see wide valley where the amplitude of oscillation equals to zero.|
Next animation shows the diffraction of the circular wave on the narrow slit in the wall. On the right side of the wall we see the appearance of the new circular wave with the smaller amplitude. This corresponds to Huygens-Fresnel principle according to which every element of the wave front generates the secondary waves. The tangency to all these waves will coincide with the wave front in the next moment of time (the waves reflected back are not considered). According to this principle the wave on the right side of the wall can be considered as superposition of the secondary waves the centers of which are situated at slit. If the slit is narrow and the distance between the wall and source as well as between the wall and point of observation is smaller than the width of slit, then the slit will play a role of the new point source of the circular wave. Because the most of the power of the incident wave is absorbed by the wall, the amplitude of the new circular wave will be much smaller as compared to the incident one.