The formulas derived above are only suitable for waves at deep water. They are still quite accurate if the water depth is equal to half the wavelength. At shallower depths, water particles on the surface of the wave describe not circular trajectories, but elliptical ones, and the derived relationships are incorrect and actually take more complex look. However, for waves in very shallow water, as well as for very long waves in middle water the relationship between the length and speed of wave propagation again takes on a simpler form. In both of these cases, the vertical movements of water particles on the free surface are very small compared to the horizontal movements. Therefore, we can again assume that the waves have an approximately sinusoidal shape. Since the particle trajectories are very flattened ellipses, the effect of vertical acceleration on the pressure distribution can be neglected. Then at each vertical the pressure will change according to a static law.
Let on the surface of the water above flat bottom spreads at a speed from right to left, a “shaft” of water with width b, increasing the water level from h 1 to h 2 (Figure 4.4). Before the arrival of the swell, the water was at rest. The speed of her movement after increasing the level of shield. This speed does not coincide with the speed of the shaft; it is necessary in order to cause a lateral movement of the volume of water in the transition zone of width b to the right and thereby raise the water level.
Fig 4.4 P
The inclination of the shaft over its entire width is assumed to be constant and equal. Provided that the speed u is small enough that it can be neglected in comparison with the speed c of propagation of the shaft, the vertical speed of water in the area of the shaft will be equal to (Figure 4.5)
Continuity condition 3.4, applied to a single layer of water (in the direction perpendicular to the plane of Figure 4.4), has the form
u 1 l 1 = u 2 l 2 , (the integral disappeared due to the linearity of the areas under consideration),
here u 1 and u 2 are the average velocities in the cross sections l 1 and l 2 of the flow, respectively. l 1 and l 2 - linear quantities (lengths).
This equation, applied to this case, leads to the relation
h 2 u = bV, or h 2 u = c (h 2 -h 1). (4.9)
From 4.9 it is clear that the relationship between the speeds u and c does not depend on the width of the shaft.
Equation 4.9 remains true for a shaft with a non-rectilinear profile (provided that the angle b is small). This is easy to show by dividing such a shaft into a number of narrow shafts with straight profiles and adding up the continuity equations compiled for each individual shaft:
Where, provided that the difference h 2 - h 1 can be neglected and instead of h 2i in each case, substitute h 2, it turns out. This condition is valid under the already accepted assumption that the velocity u is small (see 4.9).
To the kinematic relation 4.9 should be added a dynamic relation derived from the following considerations:
A volume of water with width b in the area of the shaft is in accelerated motion, since the particles that make up this volume begin their movement on the right edge with zero speed, and on the left edge they have speeds w (Figure 4.4). An arbitrary particle of water is taken from the area inside the shaft. The time it takes for the shaft to pass over this particle is
therefore the particle acceleration
Next, the width of the shaft (its linear dimension in a plane perpendicular to the figure) is taken equal to one (Figure 4.6). This allows us to write the expression for the mass of the volume of water located in the shaft area as follows:
Where h m is the average water level in the shaft area. (4.11)
The pressure difference on both sides of the shaft at the same height is (according to the hydrostatic formula) , where is a constant for a given substance (water).
Therefore, the total pressure force acting on the considered volume of water in the horizontal direction is equal. Newton's second law (the basic equation of dynamics), taking into account 4.10 and 4.11, will be written as:
Where. (4.12)
So the shaft width was taken out of the equation. In the same way as was done for equation 4.9, it is proven that equation 4.12 is also applicable for a shaft with a different profile, provided that the difference h 2 - h 1 is small compared to h 2 and h 1 themselves.
So, there is a system of equations 4.9 and 4.12. Next, on the left side of equation 4.9, h 2 is replaced by h m (which, with a low shaft and, as a consequence, a small difference h 2 - h 1, is quite acceptable) and equation 4.12 is divided into equation 4.9:
After the reductions it turns out
The alternation of shafts with symmetrical angles of inclination (the so-called positive and negative shafts) leads to the formation of waves. The speed of propagation of such waves does not depend on their shape.
Long waves in shallow water travel at a speed called the critical speed.
If several low shafts follow each other on the water, each of which slightly increases the water level, then the speed of each subsequent shaft is slightly greater than the speed of the previous shaft, since the latter has already caused a slight increase in the depth h. In addition, each subsequent shaft no longer propagates in still water, but in water already moving in the direction of movement of the shaft at a speed of All this leads to the fact that subsequent shafts catch up with the previous ones, resulting in a steep shaft of finite height.
So far we have only considered one-dimensional(1-d ) waves, that is, waves propagating in a string, in linear environment. No less familiar to us two-dimensional waves in the form of long mountain ridges and depressions on two-dimensional water surface. The next step in discussing waves we have to take is into the space of two ( 2-d ) and three ( 3-d ) measurements. Again, no new physical principles will be used; the task is simply description wave processes.
We will begin the discussion by returning to the simple situation with which this chapter began - single wave pulse . However, now it will not be a disturbance on the string, but splash on the surface of the reservoir. splash settles under its own weight, and the adjacent areas, experiencing increased pressure, rise, starting to propagate the wave. This process is depicted “in cross-section” in rice. 7-7(a). The further logic of considering the situation is exactly the same as what was already used when studying the effects that arise after a sharp blow to the central part of the string. But this time the wave can travel in everyone directions. Having no reason to prefer one direction over another, the wave propagates in all directions. The result is the familiar expanding circle of ripples on the surface of a still body of water, see below. rice. 7-7 (b).
We are well known and flat
waves on the surface of water - those waves whose crests form long, sometimes almost parallel, lines on the surface of the water. These are the same waves that periodically roll onto the shore. An interesting feature of this type of wave is the way it overcomes obstacles - for example, holes in a continuous wall breakwater. Drawing 7-8
illustrates this process. If the size of the hole is comparable to the wavelength, then each successive wave creates a burst within the hole, which, as in Fig. 7-7, serves as a source of circular ripples in the port water area. As a result, between the breakwater and the shore there are concentric
, “ring” waves.
This phenomenon is known as diffraction waves If the width of the hole in the breakwater is much greater than the wavelength, then this will not happen - the waves passing through the obstacle will retain their flat shape, except that slight distortions will appear at the edges of the wave
Like waves on the surface of water, there are also three-dimensional waves (3-d –waves) . Here the most familiar example is sound waves. The crest of a sound wave is an area thickening air molecules. Drawing similar to Fig. 7-7 for a three-dimensional case would represent an expanding wave in the shape of a sphere .
All waves have the property refraction
. This is an effect that occurs when a wave passes through the boundary of two media and enters a medium in which it moves more slowly. This effect is especially clear in the case of plane waves (see Fig. rice. 7-9). That part of the plane wave that found itself in the new, “slow” medium moves in it at a lower speed. But since this part of the wave inevitably remains associated with the wave in the “fast” medium, it front(the dotted line at the bottom of Fig. 7-9) should break, that is, approach the interface between the two media, as shown in Fig. 7-9.
If the change in the speed of wave propagation does not occur abruptly, but gradually, then the rotation of the wave front will also occur smoothly. This, by the way, explains the reason why the surf waves, no matter how they moved in open water, almost always parallel to the coastline. The fact is that as the thickness of the water layer decreases, the speed of waves on its surface decreases, therefore, near the coast, where the waves enter the shallow water area, they are slowing down. The gradual rotation of their front makes the waves almost parallel to the coastline.
What does the world tell Suvorov Sergey Georgievich
Waves on the surface of the water
Everyone knows that water waves are different. On the surface of the pond, a barely noticeable swell gently shakes the fisherman's plug, and in the vast expanses of the sea, huge water shafts rock ocean steamers. How do waves differ from each other?
Let's see how water waves arise.
Rice. 4. A device for rhythmically exciting waves on the surface of the water
To excite waves on water, we take the device shown in Fig. 4. When the motor A rotates the eccentric B, rod IN moves up and down, plunging into water to different depths. Circular waves diverge from it (Fig. 5).
They are a series of alternating ridges and depressions.
The distance between adjacent ridges (or troughs) is called wavelength and is usually denoted by the Greek letter ? (lambda) (Fig. 6).
Rice. 5. Waves created by a rhythmically oscillating rod; letter? wavelength indicated
Let us double the number of revolutions of the motor, and therefore the frequency of oscillation of the rod. Then the number of waves appearing during the same time will be twice as large. But the wavelength will be half as long.
The number of waves produced in one second is called frequency waves It is usually denoted by the Greek letter ? (nude).
Rice. 6. Cross section of a water wave. AB - amplitude a, BV - wavelength?
Let the cork float on the water. Under the influence of a traveling wave, it will oscillate. The ridge that approaches the cork will lift it up, and the depression that follows it will lower it down. In one second, the cork will raise as many crests (and lower as many troughs) as waves are formed during this time. And this number is the frequency of the wave ? . This means that the plug will oscillate with a frequency ? . Thus, by detecting the action of waves at any point in their propagation, we can determine their frequency.
Rice. 7. Scheme of connection between wavelength?, speed v and frequency?. From the figure it is clear that v = ??
For the sake of simplicity, we will assume that the waves do not decay. The frequency and length of continuous waves are related to each other simple law. In a second it is formed ? waves All these waves will fit within a certain segment (Fig. 7). The first wave formed at the beginning of the second will reach the end of this segment; it is located at a distance from the source, equal to length wave multiplied by the number of waves formed, that is, by the frequency ? . But the distance traveled by a wave in a second is the speed of the wave v. Thus,
? ? ? =v
The wavelength and speed of propagation of waves are often learned from experience, but then the frequency v can be determined from a calculation, namely:
? =v/?
Frequency and wavelength are their essential characteristics; These characteristics distinguish some waves from others.
In addition to frequency (or wavelength), waves also differ in the height of the crests (or the depth of the troughs). The height of the wave is measured from the horizontal level of the resting surface of the water. It is called the amplitude, or range of vibrations.
The amplitude of the oscillations is related to the energy carried by the wave. The greater the amplitude of the water wave (this also applies to vibrations of strings, soil, foundation, etc.), the greater the energy that is transmitted by the waves, and the greater is squared (if the amplitude is twice as large, then the energy is 4 times greater times, etc.).
Now we can tell how an ocean wave differs from a swell in a pond: wavelength, vibration frequency and amplitude.
And knowing what quantities characterize each wave, it will not be difficult to understand the nature of the interaction of waves with each other.
From the book The Newest Book of Facts. Volume 3 [Physics, chemistry and technology. History and archaeology. Miscellaneous] author Kondrashov Anatoly Pavlovich From the book The History of Candles author Faraday Michael From the book Nuclear Energy for Military Purposes author Smith Henry Dewolf From the book Drop author Geguzin Yakov Evseevich From the book Physics at every step author Perelman Yakov Isidorovich From the book Movement. Heat author Kitaygorodsky Alexander IsaakovichLECTURE II CANDLE. BRIGHTNESS OF THE FLAME. AIR IS REQUIRED FOR COMBUSTION. WATER FORMATION In the last lecture we looked at general properties and the location of the liquid part of the candle, as well as how this liquid gets to where combustion occurs. Are you convinced that when the candle
From the book For young physicists [Experiments and entertainment] author Perelman Yakov IsidorovichINSTALLATIONS FOR HEAVY WATER PILOT PLANT USING THE CENTRIFUGATION METHOD 9.36. The next two chapters describe three methods used for the industrial separation of uranium isotopes. They have highest value for the Project at present. At the beginning of work
From the book How to Understand the Complex Laws of Physics. 100 simple and exciting experiences for children and their parents author Dmitriev Alexander StanislavovichFIRST DROP OF MELT WATER
From the book Asteroid-Comet Hazard: Yesterday, Today, Tomorrow author Shustov Boris MikhailovichYou already know out of the blue that the air that surrounds us on all sides presses with considerable force on all things with which it comes into contact. The experiment that will now be described will show you even more clearly the existence of atmospheric pressure. Place it on a flat
From the book The Eye and the Sun author Vavilov Sergey IvanovichWaves moving along the surface Submariners do not know sea storms. During the most severe storms, calm reigns several meters below sea level. Sea waves is one example of wave motion that covers only the surface of the body. Sometimes it may seem that
From the author's book13. Dry out of the water Now you are convinced that the air surrounding us on all sides presses with considerable force on all things with which it comes into contact. The experience that we are about to describe will prove to you even more clearly the existence of this, as physicists say, “atmospheric
From the author's book10 Why the ocean does not freeze, or Freezing out clean water For the experiment we will need: a plastic jar, salt. Everyone talks about the environment. buzzword such. Usually they mean pollution of the world around us. Indeed, anything can be polluted.
From the author's book17 Standing wave, or Storm in a glass of water For the experiment we will need: a large plastic bowl (you can take a wide plastic bottle with the neck cut off), mixer. Since we started about ropes, let's think about what laws of physics can be studied using a rope. Liquids
From the author's book8.3. The release of jets of water and tsunamis caused by impacts Seas and oceans cover most of the Earth's surface, so the likelihood of asteroids and comets hitting the water surface is higher than on land. Waves in the water in the near impact zone. Waves caused by falling meteoroids
From the author's book8.4. Vulnerable objects on the Earth's surface As human civilization develops, more and more new aspects of asteroid danger appear. Currently, high hydroelectric dams, large chemical plants, powerful
Any local violation of the horizontal surface of the liquid leads to the appearance of waves that propagate over the surface and quickly attenuate with depth. The occurrence of waves occurs due to the combined action of gravity and inertial force (gravitational hydrodynamic waves) or surface tension and inertial force (capillary waves).
Let us present a number of results on the hydrodynamics of surface waves of a liquid, which we will need in the future. The problem can be significantly simplified if we consider the liquid to be ideal; taking into account dissipation is necessary mainly for capillary and short gravitational waves.
Assuming the displacements of liquid particles to be small, we can limit ourselves to a linear problem and neglect the nonlinear term in the Euler equation, which corresponds to the smallness of the wave amplitude compared to its length X. Then, for an incompressible liquid, the wave motion on its surface without taking into account surface tension forces is determined by such a system of equations for the potential ( Let us remind you that:
Directed vertically upward and corresponds to the undisturbed surface of the liquid).
For an unbounded liquid surface, the depth of which is significantly greater than the wavelength, one can seek a solution to the problem in the form of a plane inhomogeneous wave propagating in the positive x direction and damping with depth:
where is the wave frequency and wave number, where is the phase velocity. Substituting this value of the potential into equation (6.1), and also taking into account that the solutions make sense for , we obtain the expression for the potential:
and satisfying the boundary condition on the surface of the liquid, the dispersion equation
Thus, the group velocity of propagation of a gravitational wave
whereas the phase velocity of such a wave is
As can be seen, gravitational waves have dispersion; As the wavelength increases, their phase velocity increases.
An interesting question is what is the distribution of velocities of liquid particles in a wave; it is found by differentiating the potential (6.3) with respect to x.
Rice. 1.4. Dispersion curve for gravity-capillary waves on the surface of deep water in a region where both g and a are significant.
Consideration shows that liquid particles in a wave describe motion approximately in a circle (around their equilibrium points), the radius of which decreases exponentially with depth. At a depth equal to one wavelength, its amplitude is approximately 535 times less than near the surface. The results presented apply to waves in deep water, when where h is the depth of the liquid. If the opposite case occurs (for example, waves propagate in a channel of finite but small depth), then
As you can see, such waves do not have dispersion.
Taking into account the Laplace capillary force due to surface tension 0,
i.e., unlike gravitational waves, the speed of capillary waves increases with decreasing wavelength. The combined action of gravity and surface tension is determined by the following dispersion equation (deep water):
In Fig. Figure 1.4 shows the dependence of the phase velocity of wave propagation on the surface of a liquid on the wavelength for water according to expression (6.9). From this figure it is clear that at cm there is a minimum speed of surface waves, which are mixed gravity-capillary waves.
The results presented were for one-dimensional linear waves in the absence of dissipation. In addition, it was believed that the waves were regular and propagated in one direction. The waves that arise when a ship moves in calm water or when approaching a shallow shore really represent
regular disturbances. Waves on the surface of a liquid that arise under the influence of wind are predominantly random - they move in different directions and have different frequencies and amplitudes; This is exactly the picture we see when we are on a ship on the open sea in windy weather.
The attenuation of gravitational waves with wavelengths longer than a meter is small, but it is still significantly greater than what follows from the linear theory. This discrepancy is obviously caused by processes associated with nonlinearity in the propagation of gravitational and capillary waves. Thus, if a single wave propagates in shallow water with phase velocity , then such a wave does not have dispersion. As it propagates, its profile becomes steeper due to the fact that the upper particles of the medium, for which the depth h is greater than for the lower particles, will move with greater speed, according to (6.7), and the wave will begin to be overwhelmed; when approaching the shore, a wave crashes onto him. The sweeping effect is also enhanced because as the depth h decreases, the amplitude of the wave increases according to the law of conservation of the energy flume; the energy density increases due to the decrease cross section layer of water. With growth, nonlinear effects become even stronger. The process of “steepening” of waves during their propagation also occurs in deep water due to the nonlinearity of the equations of motion. The theory of nonlinear waves on the surface of a liquid has received great development in Lately, although the first work in this direction was done at the end of the last century.
If there are several waves, they interact with each other nonlinearly; The principle of superposition for waves of finite amplitude is no longer observed. The conditions for nonlinear interaction of gravitational waves, due to their dispersion properties, are different interesting features, which we do not have the opportunity to dwell on here. Let us only note that the actually existing interaction of random waves of finite amplitude, in principle, explains much greater attenuation of waves on the surface than is predicted by the linear theory. The absorption mechanism operates due to nonlinear interaction; energy from the region of small wave numbers (long waves) is pumped into the region of ever shorter wavelengths and, finally, into the capillary region of the spectrum, where it is ultimately dissipated due to viscosity, turning into heat.
In ch. 3 we will deal with nonlinear sound waves and will return to questions of the interaction of waves on the surface of a liquid.
The next interesting type of wave, which everyone has undoubtedly seen and which is usually used as an example of waves in elementary courses, is waves on the surface of water. You will soon see that it is difficult to come up with a more unfortunate example, because they are not at all similar to either sound or light; all the difficulties that can be found in the waves have gathered here. Let's start with long waves in deep water. If we consider the ocean to be infinitely deep and some disturbances occur on its surface, then waves will arise. Generally speaking, any disturbances are possible, but sinusoidal motion with very little disturbance produces waves reminiscent of ordinary smooth ocean waves moving towards the shore. Water, of course , on average remains in place, but the waves themselves move. What kind of movement is this - transverse or longitudinal? It can be neither one nor the other: neither transverse nor longitudinal. Although in each given place humps alternate with depressions, it does not may be an upward and downward movement simply due to the law of conservation of the amount of water. Where should the water go from the depression? After all, it is practically incompressible. The speed of compression waves, that is, sound in water, is many times greater: we are not considering them now So, for us now water is incompressible, so when a depression forms, water from this place can only move sideways.This is how it actually turns out: particles of water near the surface will move approximately in a circle. One day, when you are basking on the water, lying on a circle, and such a smooth shaft comes, look at the neighboring objects and you will see that they are moving in circles. So the picture turns out to be unexpected: here we are dealing with a mixture of longitudinal and transverse waves. As the depth increases, the circles become smaller until at a sufficient depth there is nothing left of them (Fig. 51.9).
It is very interesting to determine the speed of such waves. It must be some combination of the density of the water, the acceleration of gravity, which in in this case is the restoring force, and possibly the wavelength and depth. If we consider the case of infinite depth, then the speed will no longer depend on it. But whatever formula for the phase velocity of waves we take, it must contain these quantities in such a combination as to give the correct dimension. Having tried many in various ways, we will find that there is only one combination g
and λ can give us the dimension of speed, namely √(gλ),
which does not include density at all. In fact, this formula for phase velocity is not entirely accurate, and full analysis dynamics, which we will not go into, shows that everything will really work out the way we did, except √(2
π),
i.e.
Interestingly, long waves travel faster than short ones. So when a motorboat passing in the distance creates waves, after a certain period of time they will reach the shore, but at first there will be rare splashes, since the long waves come first. Then the incoming waves become shorter and shorter, because the speed drops as the square root of the wavelength.
“This is not true,” someone might object, “after all, in order to make such a statement, we must look at group speed". That's right, of course. The formula for phase velocity doesn't tell us what comes first; Only the group velocity can tell us this. So we should get the group velocity and we can show that it is equal to half the phase velocity. To do this, you just need to remember that the phase velocity behaves as the square root of the wavelength. The group velocity behaves in the same way, i.e., as the square root of the wavelength. But how can the group velocity be half the phase velocity? Look at a group of waves caused by a passing boat and follow a particular crest. You will find that he runs with the group, but gradually becomes smaller and smaller, and when he reaches the front line, he dies completely. But in a mysterious and incomprehensible way, a weak wave rises to replace it from the rear front and it becomes stronger and stronger. In short, waves move through the group, while the group itself moves twice as slow as these waves.
Since the group and phase velocities are not equal to each other, the waves caused by a moving object will no longer be simply conical, but much more complex and interesting. You can see this in FIG. 51.10, which shows the waves caused by a boat moving through the water. Note that they are not at all like what we got for sound (where speed is independent of wavelength), where the wave front was simply a cone spreading out to the sides. Instead, we got waves behind the moving object, the front of which is perpendicular to its movement, and also small waves moving at other angles from the sides. This whole picture of wave motion as a whole can be very beautifully recreated, knowing only that the phase velocity is proportional to square root from wavelength. The trick is that the wave pattern is stationary relative to the boat (moving at constant speed); all other types of waves will lag behind it.
So far we have considered long waves for which the restoring force was gravity. But when the waves become very short, the main restoring force is capillary attraction, i.e., surface tension energy. For surface tension waves, the phase velocity is equal to
Where T is surface tension, and ρ is density. Here everything is the other way around: the shorter the wavelength, the greater turns out to be the phase velocity. If both gravity and capillary force act, as is usually the case, then we get the combination
Where k=
2
π/λ
— wave number. As you can see, the speed of waves on water is really quite a thing. complex. In fig. Figure 51.11 shows phase velocity as a function of wavelength. It is large for very short waves, large for very long waves, but between them there is a certain minimum propagation speed. Based on this formula, the group velocity can also be calculated: it turns out to be equal to 3/2 the phase velocity for ripples and 1
/
2
phase velocity for gravity waves. To the left of the minimum the group velocity is greater than the phase velocity, and to the right the group velocity is less. Several interesting phenomena are associated with this fact. Since the group velocity quickly increases with decreasing wavelength, then if we create some kind of disturbance, waves of the corresponding length will arise, which travel at a minimum speed, and short and very long waves will run ahead of them at a higher speed. In any body of water you can easily see very short waves, but long waves are more difficult to observe.
Thus, we have seen that the ripples that are so often used to illustrate simple waves are actually much more complex and interesting: they do not have a sharp wave front, as is the case with simple waves like sound or light. The main wave that rushes forward consists of small ripples. Thanks to dispersion, a sharp disturbance of the water surface does not lead to a sharp wave. Very small waves still come first. In any case, when an object moves through water at a certain speed, a very complex picture arises, since different waves move with at different speeds. Taking a trough of water, you can easily demonstrate that small capillary waves will be the fastest, followed by larger ones. In addition, by tilting the trough, you can see that where the depth is less, the speed is less. If the wave moves at some angle to the line of maximum inclination, then it turns towards this line. In this way, you can demonstrate many different things and come to the conclusion that waves in the water are a much more complex thing than waves in the air.
Long wave speed c in a circular motion water decreases in shallow places and increases in deep places. Thus, when the wave goes to the shore, where the depth is shallower, it slows down. But where the water is deeper, the wave moves faster, so again we are faced with a shock wave mechanism. However, this time, since the wave is not so simple, its shock front is much more distorted: the wave “bends over itself” in the most familiar way for us (Fig. 51.12). This is exactly what we see when a wave hits the shore: it reveals all the inherent difficulties of nature. No one has yet been able to calculate the shape of the wave at the moment it breaks. This is very easy to do when the waves are small, but when they get big it gets too complicated.
An interesting property of capillary waves can be observed when a surface is disturbed by a moving object. From the point of view of the object itself, the water flows past it, and the waves that end up staying with it will always be the waves that have just the right speed to stay on the water with the object. Likewise, if you place an object in a stream that washes over it, the wave pattern will be stationary and just the right wavelength to move at the same speed as the water. But if the group velocity is less than the phase velocity, then the disturbance goes along the flow back, because the group velocity is not sufficient to catch up with the flow. If the group velocity is greater than the phase velocity, then the wave pattern will appear in front of the object. If you closely monitor an object floating in a stream, you will notice small ripples in front of it, and long waves behind it.
Other interesting phenomena of this kind can be observed in flowing liquid. If, for example, you quickly pour milk from a bottle, you will notice how the stream of milk is intersected by many intersecting lines. These are waves caused by disturbance at the edges of the bottle; they are very similar to waves caused by an object floating in a stream. But now this effect occurs on both sides, so you get a picture of intersecting lines.
So, we have become acquainted with some interesting properties of waves, with various complications depending on phase speed and wavelength, as well as the dependence of wave speed on depth, etc.; all this leads to very complex and therefore interesting natural phenomena.
