Wave speed in water. Surface waves

Waves on surfaces water- there is a cumulative vibration of particles surface mass water under the influence external force: wind, tide, underwater earthquake, moving ship, etc. The line on which all the points of the top of one ridge lie is called the wave front (The wave front can only be depicted at a short distance by a straight line; usually it is a smooth curve.).

Rice. 19.8. Wave elements

Rice. 19.9. Structure of ordinary waves (top view)

Rice. 19.10. Wave parameters

Wave parameters (cross section):

h- height (As can be seen from Figure 19.9 (top view of waves) the height of the wave h along its front is not the same and ranges from hmin to hmax.); λ - length; - steepness; C - speed of movement; N°- angle between the velocity vector WITH and direction to N(north); τ is the period, i.e. the time during which the wave travels its length.

The wave parameters also include the shape of its cross section, for example:

We can distinguish a type of waves called “crush”, which is obtained when waves of approximately the same height meet, but coming from different directions. In the crowd big waves managing ships (including yachts) is difficult.

« Dead swell»has a smooth, flat (harmonic) waveform, usually long length(λ) and happens in calm weather. This is excitement due to inertia when there is no longer any wind. The dead swell may be waves, followed by a storm.

Waves have the following properties:

• reflected from obstacles (angle of incidence equal to angle reflections);
• overlap each other: reflected wave to the main one or from different sources;
• preservation of inertia for some time (the forces that caused the waves stopped acting, and the waves continue to run);
• waves caused by wind do not always move in the direction of the wind. The wind can change its direction, and the waves will move as before (inertia again);
• in shallow water, where the depth is less than the wavelength, the wave shape changes, its length (λ) decreases and the speed (s) and height (h) increase, but the period (τ) remains the same;
• floating algae, heavy rain, shallow ice, and spilled oil smooth out the waves.

During swimming on yacht The wave parameters (h and λ) are determined by eye. The value of τ can be measured by throwing a sheet of paper into the water and starting a stopwatch at the moment the sheet appears at the top of the ridge. The stopwatch is stopped at the 11th appearance of the leaf at the top of the ridge and the time t = 10τ is obtained. Knowing τ and λ, we can calculate the speed of the wave C=λ/τ.

Other calculation formulas give:

WITH m/s = 0.65 × τ s 2 (or C node = 3 × τ s)

C m/s = 1.2√λ m; λ m = 1.56 × τ s 2;

(during a storm ).

For inland waters, where the wave run-up is only a few kilometers and steep waves predominate, use the formula:

λ m = τ s 2.

The given formulas are approximate and valid for waves average size at the time of their observation.

Every yachtsman deals with wind and waves when sailing. All these components influence the progress of the yacht and can not only contribute to its progress, but also have harmful effect. The yachtsman’s task is to identify harmful factors and reduce their influence to a minimum if they cannot be avoided (for example, bypassed) and, at the same time, it is desirable to take full advantage of their beneficial components. This also occurs when sailing in rough water.

1. In oncoming waves, especially when the wave is steep and its length is 1 ÷ 1.5 times the length of the yacht, it is very important to choose smooth areas of water (this is possible! See the structure of the waves, top view) and not point the yacht exactly against the oncoming wave - it will a strong beat stopping the yacht. It is better to expose the cheekbone to the wave and let the yacht gently rise to the crest, and then fall off a little. Thus, the yacht will zigzag among the waves, choosing smooth areas, adducting and dodging sharp blows, and even accelerating, falling somewhat when leaving the crest into the hollow. The yacht's path will be somewhat longer, but the loss of time on the transition will be minimal.
2. A. In tail or side seas, sailing the yacht is a pleasure. The oncoming crest (it is better to meet it from the backstay) picks up the yacht and carries it forward with its slope and accelerates it. A feeling of flight arises, which can be prolonged by correctly choosing a place for the passage of the crest of the wave in front, on which acceleration can again be obtained, etc. Again yacht will follow an elongated zigzag path, but in this case, due to a significant increase in speed, the gain will be very noticeable.
   B. If the yacht’s progress is ahead of the waves, the direction of the yacht’s movement should be changed so that it does not rest against the next mountain of water, but would go slanting along it and would be picked up again by the wave. The lengthened path is more than compensated by the increased speed of the yacht. In all cases, when leaving the ridge, they fall away somewhat, and when ascending, they are brought up.

Interactions described yachts with waves they quickly accustom the helmsman to automatic control. This is surprising, but true!

• Friction waves:
  
  
  • wind, formed as a result of the action of wind
  
  
  • deep
  
  


• Tidal waves.


• Gravitational waves:
  
  
  • gravitational waves in shallow water
  
  
  • gravitational waves on deep water
  
  
  • seismic waves (tsunamis) arising in the oceans as a result of an earthquake (or volcanic activity) and reaching heights of 10-30 m near the coast.
  
  
  • ship waves
  
  

Wind waves arise with the wind; when the wind stops, these waves in the form of a dead swell, gradually fading, continue to move in the same direction. Wind waves depend on the size of the water space open for wave acceleration, wind speed and time of action in one direction, as well as depth. As the depth decreases, the wave becomes steeper.
Wind waves are asymmetrical, their windward slope is gentle, their leeward slope is steep. Since the wind is on top part waves acts stronger than on the lower one, the crest of the wave crumbles, forming “lambs”. In the open sea, "lamblets" are formed in a wind that is called "fresh" (wind force 5 and a speed of 8.0-10.7 m/s, or 33 km/h).
Swell- excitement that continues after the wind has already died down, weakened or changed direction. A disturbance that spreads by inertia in complete calm is called a dead swell.
When waves from different directions meet in a certain area, a crush. The chaotic accumulation of waves formed when direct waves meet reflected ones is also crush.
When waves pass over banks, reefs and rocks, breakers.
The approach of waves onto the shore with an increase in height and steepness and subsequent capsizing is called surf.

The surf gets different character depending on which shore: shallow (having small angles of inclination and a large width of the underwater slope) or deep (having significant slopes of the underwater slope).

The overturning of the crest of a moving wave onto a steep bank forms reverse faults, having a large destructive force.

© Yuri Danilevsky: November storm. Sevastopol

When the surf occurs near a deep shore that rises steeply from the water, the wave breaks up only when it hits the shore. In this case, a reverse wave is formed, meeting the next one and reducing its impact force, and then a new wave comes in and hits the shore again.
Such wave impacts in case of large swell or strong excitement are often accompanied by surges of waves to a considerable height.

© Storm in Sevastopol, November 11, 2007

On the shores of the Black Sea, the wave impact force can reach 25 tons per 1 m 2.
When upturning, the wave receives enormous force. On Shetland Islands, north of Scotland, there are fragments of gneiss rocks weighing up to 6-13 tons, thrown by the surf to a height of up to 20 m above sea level.

The rapid movement of waves and swell onto the shore is called roll up.

Waves are regular when their crests are clearly visible, and irregular when the waves do not have clearly defined crests and are formed without any visible pattern.
Wave crests perpendicular to the wind direction in the open sea, lake, reservoir, but near the shore they take a position parallel to the coastline, running onto the banks.
The direction of wave propagation in the open sea is indicated on the surface of the water by a family of parallel stripes of foam - the traces of collapsing wave crests.

Waves generated on free surface water, set in motion the air in contact with them. In most cases, the mass of this air can be neglected compared to the mass of the liquid. Then the pressure on the free surface of the liquid will be equal to atmospheric pressure Observations show that with the simplest wave motion individual particles the free surface of the water is described by trajectories that approximately coincide with a circle. In a reference frame moving along with the waves at the speed of their propagation, the wave motion is obviously a steady motion (Fig. 80). Let the speed of wave propagation be equal to c, the radius of the circle described by a particle of water located on a free surface is equal, and the period of revolution of this particle along its trajectory is equal. Then in the indicated reference system the speed of the current at the wave crests will be equal to

and in the troughs of the waves

Since the difference in height between the highest and lowest positions of the points on the free surface is equal, applying Bernoulli’s equation to the streamline located on the free surface, we obtain:

or, after substituting instead and their values,

whence it follows that

The radius is not included in this formula, therefore, the speed of wave propagation does not depend on the height of the waves. When waves propagate, the wave crest moves over time through a distance called the wavelength, therefore,

Eliminating the period from equalities (60) and (61), we obtain:

Thus, for waves on the surface of water, their speed of propagation, unlike sound waves, strongly depends on the wavelength. Long waves travel faster than short ones. Waves of different lengths can overlap each other without noticeable mutual disturbance. In this case, the short waves seem to be lifted by the long waves, but then the long waves go forward, and the short waves remain behind them. Streamlines in a reference frame stationary relative to undisturbed water are shown in Fig. 81. From the location of the streamlines it is clear that the speed of water decreases very quickly with increasing depth, namely, in proportion to the decrease in value; therefore, at a depth equal to the wavelength, the speed is only the speed on the free surface.

Rice. 81. Streamlines of wave motion

The exact theory shows that formula (62) is valid only for low waves, regardless of their height. For high waves the speed c is actually several Furthermore the value given by formula (62). In addition, when high waves the trajectories of water particles located on the free surface are not closed: the water on the crest of the wave moves forward by longer distance, than to which it returns back in the trough of the wave (see. right side rice. 81). Consequently, with high waves, water is transferred forward.

For short waves important factor is, in addition to gravity, also surface tension. It seeks to smooth out wave surface, and therefore the speed of wave propagation increases. Theory shows that in this case the speed of wave propagation is equal to

where C is the capillary constant. For long waves the predominant role is played by the first term under the root sign, and for short waves, on the contrary, is the second term. For wavelength

the speed of propagation c has minimum value, equal

For water dynes/cm, therefore,

Waves whose length is longer are called gravitational, and waves whose length is shorter are called capillary.

The speed of propagation of the group should be distinguished from the speed of movement of the wave crests, called the phase speed (above we called it the speed of wave propagation and denoted by c)

waves, called group velocity and denoted by c. The easiest way to explain the meaning of this concept is by the example of movement resulting from the superposition of two waves having equal amplitudes, but slightly different in length. Let us have a sine wave

where A is amplitude, time, and some coefficients. When increasing by y or y, the sine takes on the same value, therefore, the quantity

is the wavelength, and the magnitude

there is a period of oscillation. If

i.e. if

then the argument of the sine does not depend on time, therefore the ordinate y does not depend on time. This means that the entire wave, without changing its shape, moves to the right with a speed

Let's superimpose a second wave on this wave

i.e. a wave with the same amplitude A, but with slightly different values. The resulting movement will be

At those points of the x axis at which the phases of both oscillations coincide, the amplitude is equal at the same points at which the phases of both oscillations

are opposite, the amplitude is zero. This phenomenon is called beating. Applying the well-known formula

we will get:

In this equality the term

represents a wave for which the coefficients are equal to the average values ​​of and, respectively, the Multiplier

which changes slowly for small values ​​of the differences, can be considered as a variable amplitude (Fig. 82).

Rice. 82. Beat

The group of waves ends at the point where the cosine is made equal to zero. The speed of movement of this point, called the group velocity c, based on considerations similar to the previous ones, is equal to

For long groups, i.e. for slow beats, with sufficient accuracy we can assume that

For waves arising under the influence of gravity, from formula (60) we have:

But, according to equality (65),

hence,

On the other hand, substituting the value from equality (64) into formula (62), we get:

From here, differentiating with respect to and bearing in mind equality (67), we find:

Thus, groups of waves propagate at speed c, equal to half phase speed, in other words, the crests in a group of waves move at a speed twice as fast as the group of waves itself; At the back end of the group new waves appear all the time, and at the front end of the group they disappear. This phenomenon is very easy to observe in waves caused by a stone falling into still water.

All of the above applies not only to waves on the surface of water, but also to any other waves whose phase velocity depends on the wavelength.

Another type of wave group is the waves that appear on the surface of the water when a ship moves. A wave pattern very similar to ship waves can easily be obtained if the surface of deep, resting water is forced to move with constant speed point source of pressure disturbance. The resulting movement can be studied mathematically. According to the calculations of V. Thomson (lord Kelvin), Ekman and others, the wave system shown in Fig. is obtained. 83, on which wave crests are indicated by inclined lines. This wave system moves along with the source of disturbance. Length transverse waves based on formula (62) is equal to

where c is the speed of movement of the source of disturbance. When a ship moves, two systems of such waves are formed - one near the bow, the other near the stern of the ship, and the waves of both systems interfere with each other.

Rice. 83. System of waves formed when uniform motion on the water surface of the source of pressure disturbance

The group velocity of capillary waves, as can be easily shown by calculations similar to those made for gravitational waves, greater than the phase velocity, namely, in the limiting case of very small waves, 1.5 times. Consequently, if the source of disturbance moves at a constant speed, then groups of waves are ahead of it. Near the line of a fishing rod lowered into a river, the flow speed of which is more than 23.3 cm/sec, capillary waves are formed upstream, and gravity waves are formed downstream, and the latter have approximately the same shape as in Fig. 83, and the first ones diverge upstream in the form of arcs of circles. At speeds of movement of the source of disturbance less than 23.3 cm/sec, waves are not formed.

On the contact surface of two liquids various densities, located one above the other, waves can also occur. If both liquids are motionless and their densities are equal, then the theoretical calculation gives the value for the phase velocity of waves

If the upper fluid flows at a speed relative to the lower one, then the theory shows that the resulting waves are stable only if their length is sufficiently large. Short waves, just as was shown in § 7 for the movement of two liquid flows along the interface, are unstable, which leads to mixing of both liquids in intermediate zone; this mixing restores the stability of the flow. As the speed increases, the boundary between instability and stability moves towards waves with longer wavelengths. Waves of this kind can also arise in the atmosphere at the boundary of two layers of air of different densities moving relative to each other; Sometimes these waves are made visible by the formation of so-called wavy clouds.

When air moves over the surface of the water, waves are also formed. However, the theory of such waves, based on the assumption of the absence of friction, leads to results that contradict

reality. So, for example, calculations by V. Thomson showed that the minimum wind speed required for the formation of waves on the surface of the water should be a round number, and waves appear that have a minimum propagation speed cm/sec and a wavelength cm (at higher wind speeds, of course, , waves with longer length). Meanwhile, in reality, for the formation of waves, a wind with a speed is sufficient. According to Jeffrey's research, this is explained by the fact that due to friction, the pressure distribution on the surface of the wave becomes asymmetrical, and therefore the wind, if its speed is greater than the phase speed of the waves, does work on the crest of each wave. Motzfeld, by measuring the pressure distribution on the surface of model water waves, found that the resistance that air provides to the movement of waves is proportional to one and a half degrees of inclination of the wave surface at the inflection point relative to the horizon, as well as the square of the difference between the wind speed and the phase speed of the waves. Further, Motzfeld found by calculation that the inclination of the wave surface at the inflection point, depending on the phase velocity c, is greatest at

This speed c corresponds, based on formula (62), to a wave of length

If we take into account surface tension, which Motzfeld did not take into account, then the calculation shows that, in full accordance with observations, a wind with a speed slightly exceeding 23.3 cm/sec is sufficient to cause a slight disturbance on the surface of the water.

The formulas derived above are only suitable for waves in 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 elliptical rather than circular trajectories, and the relationship between the length and speed of wave propagation is more complex than for waves in deep water. However, for waves at

very shallow water, as well as for very long waves on middle water The dependence just indicated takes on a simpler form again. In both recent cases vertical movements of water particles on the free surface are very insignificant compared to horizontal movements. Therefore, we can again assume that the waves have an approximately sinusoidal shape. Since (particle trajectories are very flattened ellipses, the influence of vertical acceleration on the pressure distribution can be neglected. Then at each vertical the pressure will change according to a static law, and differences in the heights of the liquid will determine almost only horizontal accelerations. We will limit ourselves here to calculations only for the case the movement of the “shaft” of water, shown in Fig. 84. These calculations are very simple and will be used by us in the future to study the propagation of pressure disturbances in a compressible medium (see § 2 of Chapter IV).

Rice. 84. Shaft on the surface of the water

Suppose that on the surface of the water above a flat bottom, a shaft with a width increasing the water level from to propagates at a speed c from right to left. Let us assume that before the arrival of the shaft the water was at rest. The speed of its movement after the level rises will be denoted by This speed, which does not at all coincide with the speed c of the propagation of the shaft, is necessary in order to cause a lateral movement of the volume of water in transition zone width to the right and thereby raise the water level from a height to a height. We assume for simplicity that the inclination of the shaft is constant across its entire width, therefore it is equal to Then, provided that the speed is small enough to be neglected in comparison with the speed c of propagation shaft, the vertical speed of the rise of water in the area of ​​the shaft will be equal to and the height difference must also be small; therefore, this equation is applicable only to low shafts, and therefore the just mentioned condition is completely justified.

The kinematic relation (72) should be accompanied by a dynamic relation, which can be easily derived as follows. A volume of water with a width 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 a speed of zero, and on the left edge they have velocities. Let's take some particle of water in the area of ​​the shaft. The time during which the shaft passes over this particle is obviously equal to

therefore the acceleration of the particle will be

The volume of water in the shaft area, if its thickness in the direction perpendicular to the plane of the figure is taken equal to one, has a mass where In addition, each subsequent swell does not propagate in still water, but in water already moving to the right with speed. This leads to the fact that subsequent swells catch up with the previous ones, resulting in a steep swell of finite height.

The study of the propagation of a shaft of finite height can be carried out using the momentum theorem in exactly the same way as was done in § 13 when considering the sudden expansion of a flow. In order for the movement of water during the propagation of the shaft to be considered as steady, the calculation should be carried out in a reference system moving with the shaft. The speed of propagation of the shaft of final height is greater than

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, nothing new physical principles will not 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 adjacent areas, testing high blood 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. Interesting feature This type of wave is the way in which 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 condensation 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 plane wave, which finds itself in a new, “slow” environment, 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 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.

Arising and propagating along the free surface of a liquid or at the interface of two immiscible liquids. V. on p.zh. are formed under the influence external influence, as a result of which the surface of the liquid is removed from equilibrium state(for example, when a stone falls). In this case, forces arise that restore balance: forces surface tension and heaviness. Depending on the nature of the restoring forces of V. on the line. are divided into: capillary waves, if surface tension forces predominate, and gravitational waves, if gravity forces predominate. In the case when gravity and surface tension forces act together, the waves are called gravitational-capillary. The influence of surface tension forces is most significant at short wavelengths, and gravity forces at long wavelengths.

Speed With spread of V. to p. depends on the wavelength λ. As the wavelength increases, the propagation speed of gravitational-capillary waves first decreases to a certain minimum value

and then increases again (σ - surface tension, g - acceleration due to gravity, ρ - fluid density). The value c 1 corresponds to the wavelength

For λ > λ 1, the propagation speed depends primarily on gravity, and for λ cm.

The reasons for the occurrence of gravitational waves: the attraction of liquid by the Sun and the Moon (see Ebbs and flows), the movement of bodies near or on the surface of water (ship waves), the action of a system of impulsive pressures on the surface of the liquid (wind waves, the initial deviation of a certain section of the surface from equilibrium position, for example, a local rise in level during an underwater explosion). The most common in nature are wind waves (see also Sea waves).

Big Soviet encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .

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