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); ? ?.?. where w is angle frequency of the wave, t -is time, Dj is the phase shift of oscillations from ball to ball. |
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 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. |
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 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) |
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 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. |
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 The 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. |
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.
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.
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 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. | |
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 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. |
Doppler effect describes the frequency shift of the signal in relation to the relative motion of a source and an observer. The wave generated by a source that moves away from an observer/receiver appears to him to be of lower frequency than the wave generated by a stationary source, or generated by a source moving toward the observer. The frequency of the signal detected by the receiver moving toward the still source is higher, compared to the frequency detected by the still receiver, or a receiver moving away from the source. This theory was formulated by Austrian physicist Ch.Doppler in 1842.
SHOCK WAVE is a thin transitive area propagating with supersonic speed in which there is a sharp increase of density, pressure and speeds of substance. Shock waves arise at explosions, detonation, supersonic movements of bodies, powerful electric discharges etc.
Let's consider an object which moves with speed of sound (the plane, for example). At each moment of time it will radiate a circular sound wave. Because the speed of source coincides with speed of sound, emitted waves superpose each other and there is a jump of pressure (front of a shock wave) ahead of source as shown in animation.
If the source moves with speed less than speed of sound, then the shock wave is not generated. Ordinary sound waves propagate forward in the direction of source motion, overtaking it, and back. Due to Doppler effect the still receiver located ahead a source detects a sound with the greater frequency, than emitted. If detector is behind of source, then the received frequency will be lower then emitted. In animation the rings show the surfaces of equal phase in the sound wave generated by a source. We see, that ahead of a source the density of such lines is more than behind of it. Because the speed of sound relatively the media is equal in all directions, the frequency of sound ahead a source and and behind of it will be different.
Let's consider a case when the source of a sound moves in the environment with supersonic speed (in animation the source moves with double speed of sound). In this case sound waves can not overtake a source and for this reason there are no sound waves before a source, they appear only behind of it. Sound waves arising behind a source superpose with each other and form in space a conic surface of high pressure. This surface is a shock wave. It is easy to show, that a sine of the angle formed by front of a shock wave with a direction of movement of a source is equal to the ratio of speed of sound to speed of source (i.e. it is the reciprocal of Mach number).
Let us consider the source of wave moving toward the observer with velocity
v and emitting the impulses with period
T. At the moment
t=0 the length between the source and the observer is
L. The first impulse will reach the observer at time
t=
L/
u, where
u in the velocity of the waves. The second impulse will be sent to observer at time
T. At this time the distance between the source and observer will be
L1=
L-
vT, so the second impulse will reach the observer at time
t1=
T+(
L-
vT)/
u. As a result the observer will detect the impulses with period
Tdop=t1-t= T(1- v/u)
And the frequency fdop registered by the observer equals:
fdop=f / (1-v/u) (source is moving toward the stationary observer)
where f is the frequency emitted by the source. We see from this formula that when the source is moving toward the observer the frequency is increased. The value of this increase is called Doppler shift. On the contrary, when source moves away from the receiver the frequency is diminished as follows:
fdop=f / (1+v/u) (source is moving away from the stationary observer)
In case when the source is moving and the observer is still, the Doppler shift appears because the wavelength is changes.
Next, we shall consider the case when observer moves and the source of the wave is still. In this case the wavelength is not changed and Doppler frequency shift appears because the velocity
w of the wave relatively the observer is changed:
w =
u +
v (observer is moving toward the stationary source)
w =
u -
v (observer is moving away from the stationary source)
Because
fdop=
w/
l , initial
f=
u/
l0 and
l =
l0 we find that
fdop=
f(1+
v/
u) (observer moves toward the stationary source)
fdop=
f(1-v/
u) (observer is moving away stationary source)
We can see from these conclusions that for acoustic waves the frequency shift will be different depending on what is moving: source or observer. On the contrary, in the case of electro-magnetic wave the Doppler shift depends only on the relative motion of source and receiver.
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