Topics of the USE codifier: mechanical waves, wavelength, sound.

mechanical waves - this is the process of propagation in space of oscillations of particles of an elastic medium (solid, liquid or gaseous).

The presence of elastic properties in the medium is a necessary condition for the propagation of waves: the deformation that occurs in any place, due to the interaction of neighboring particles, is successively transferred from one point of the medium to another. Different types of deformations will correspond to different types of waves.

Longitudinal and transverse waves.

The wave is called longitudinal, if the particles of the medium oscillate parallel to the direction of wave propagation. A longitudinal wave consists of alternating tensile and compressive strains. On fig. 1 shows a longitudinal wave, which is an oscillation of flat layers of the medium; the direction along which the layers oscillate coincides with the direction of wave propagation (i.e., perpendicular to the layers).

A wave is called transverse if the particles of the medium oscillate perpendicular to the direction of wave propagation. A transverse wave is caused by shear deformations of one layer of the medium relative to another. On fig. 2, each layer oscillates along itself, and the wave travels perpendicular to the layers.

Longitudinal waves can propagate in solids, liquids and gases: in all these media, an elastic reaction to compression occurs, as a result of which there will be compression and rarefaction running one after another.

However, liquids and gases, unlike solids, do not have elasticity with respect to the shear of the layers. Therefore, transverse waves can propagate in solids, but not inside liquids and gases*.

It is important to note that during the passage of the wave, the particles of the medium oscillate near constant equilibrium positions, i.e., on average, remain in their places. The wave thus
transfer of energy without transfer of matter.

The easiest to learn harmonic waves. They are caused by an external influence on the environment, changing according to the harmonic law. When a harmonic wave propagates, the particles of the medium perform harmonic oscillations with a frequency equal to the frequency of the external action. In the future, we will restrict ourselves to harmonic waves.

Let us consider the process of wave propagation in more detail. Let us assume that some particle of the medium (particle ) began to oscillate with a period . Acting on a neighboring particle, it will pull it along with it. The particle, in turn, will pull the particle along with it, etc. Thus, a wave will arise in which all particles will oscillate with a period.

However, particles have mass, i.e., they have inertia. It takes some time to change their speed. Consequently, the particle in its motion will lag somewhat behind the particle , the particle will lag behind the particle, etc. When the particle finishes the first oscillation after some time and starts the second, the particle , located at a certain distance from the particle, will start its first oscillation.

So, for a time equal to the period of particle oscillations, the perturbation of the medium propagates over a distance . This distance is called wavelength. The oscillations of the particle will be identical to the oscillations of the particle, the oscillations of the next particle will be identical to the oscillations of the particle, etc. The oscillations, as it were, reproduce themselves at a distance can be called spatial oscillation period; along with the time period, it is the most important characteristic of the wave process. In a longitudinal wave, the wavelength is equal to the distance between adjacent compressions or rarefactions (Fig. 1). In the transverse - the distance between adjacent humps or depressions (Fig. 2). In general, the wavelength is equal to the distance (along the direction of wave propagation) between two nearest particles of the medium, oscillating in the same way (ie, with a phase difference equal to ).

Wave propagation speed is the ratio of the wavelength to the period of oscillation of the particles of the medium:

The frequency of the wave is the frequency of particle oscillations:

From here we get the relationship of the wave speed, wavelength and frequency:

. (1)

Sound.

sound waves in a broad sense, any waves propagating in an elastic medium are called. In a narrow sense sound called sound waves in the frequency range from 16 Hz to 20 kHz, perceived by the human ear. Below this range is the area infrasound, above - area ultrasound.

The main characteristics of sound are volume and height.
The loudness of sound is determined by the amplitude of pressure fluctuations in the sound wave and is measured in special units - decibels(dB). So, the volume of 0 dB is the threshold of audibility, 10 dB is the ticking of a clock, 50 dB is a normal conversation, 80 dB is a scream, 130 dB is the upper limit of audibility (the so-called pain threshold).

Tone - this is the sound that a body makes, making harmonic vibrations (for example, a tuning fork or a string). The pitch is determined by the frequency of these oscillations: the higher the frequency, the higher the sound seems to us. So, by pulling the string, we increase the frequency of its oscillations and, accordingly, the pitch.

The speed of sound in different media is different: the more elastic the medium is, the faster sound propagates in it. In liquids, the speed of sound is greater than in gases, and in solids it is greater than in liquids.
For example, the speed of sound in air at is approximately 340 m / s (it is convenient to remember it as "a third of a kilometer per second") *. In water, sound propagates at a speed of about 1500 m/s, and in steel - about 5000 m/s.
notice, that frequency sound from a given source in all media is the same: the particles of the medium make forced oscillations with the frequency of the sound source. According to formula (1), we then conclude that when passing from one medium to another, along with the speed of sound, the length of the sound wave changes.

DEFINITION

Longitudinal wave- this is a wave, during the propagation of which the displacement of the particles of the medium occurs in the direction of the wave propagation (Fig. 1, a).

The cause of the occurrence of a longitudinal wave is compression / extension, i.e. the resistance of a medium to a change in its volume. In liquids or gases, such deformation is accompanied by rarefaction or compaction of the particles of the medium. Longitudinal waves can propagate in any media - solid, liquid and gaseous.

Examples of longitudinal waves are waves in an elastic rod or sound waves in gases.

transverse waves

DEFINITION

transverse wave- this is a wave, during the propagation of which the displacement of the particles of the medium occurs in the direction perpendicular to the propagation of the wave (Fig. 1b).

The cause of a transverse wave is the shear deformation of one layer of the medium relative to another. When a transverse wave propagates in a medium, ridges and troughs are formed. Liquids and gases, unlike solids, do not have elasticity with respect to layer shear, i.e. do not resist shape change. Therefore, transverse waves can propagate only in solids.

Examples of transverse waves are waves traveling along a stretched rope or along a string.

Waves on the surface of a liquid are neither longitudinal nor transverse. If you throw a float on the surface of the water, you can see that it moves, swaying on the waves, in a circular fashion. Thus, a wave on a liquid surface has both transverse and longitudinal components. On the surface of a liquid, waves of a special type can also occur - the so-called surface waves. They arise as a result of the action and force of surface tension.

Examples of problem solving

EXAMPLE 1

Exercise

Determine the direction of propagation of the transverse wave if the float at some point in time has the direction of velocity indicated in the figure.

Decision

Let's make a drawing. Let's draw the surface of the wave near the float after a certain time interval , considering that during this time the float went down, since it was directed down at the moment of time. Continuing the line to the right and to the left, we show the position of the wave at time . Comparing the position of the wave at the initial moment of time (solid line) and at the moment of time (dashed line), we conclude that the wave propagates to the left.

When in any place of a solid, liquid or gaseous medium, particle vibrations are excited, the result of the interaction of the atoms and molecules of the medium is the transmission of vibrations from one point to another with a finite speed.

Definition 1

Wave is the process of propagation of vibrations in the medium.

There are the following types of mechanical waves:

Definition 2

transverse wave: particles of the medium are displaced in a direction perpendicular to the direction of propagation of a mechanical wave.

Example: waves propagating along a string or a rubber band in tension (Figure 2.6.1);

Definition 3

Longitudinal wave: the particles of the medium are displaced in the direction of propagation of the mechanical wave.

Example: waves propagating in a gas or an elastic rod (Figure 2.6.2).

Interestingly, the waves on the liquid surface include both transverse and longitudinal components.

Remark 1

We point out an important clarification: when mechanical waves propagate, they transfer energy, form, but do not transfer mass, i.e. in both types of waves, there is no transfer of matter in the direction of wave propagation. While propagating, the particles of the medium oscillate around the equilibrium positions. In this case, as we have already said, waves transfer energy, namely, the energy of oscillations from one point of the medium to another.

Figure 2. 6. one . Propagation of a transverse wave along a rubber band in tension.

Figure 2. 6. 2. Propagation of a longitudinal wave along an elastic rod.

A characteristic feature of mechanical waves is their propagation in material media, unlike, for example, light waves, which can also propagate in a vacuum. For the occurrence of a mechanical wave impulse, a medium is needed that has the ability to store kinetic and potential energies: i.e. the medium must have inert and elastic properties. In real environments, these properties are distributed over the entire volume. For example, each small element of a solid body has mass and elasticity. The simplest one-dimensional model of such a body is a set of balls and springs (Figure 2.6.3).

Figure 2. 6. 3 . The simplest one-dimensional model of a rigid body.

In this model, inert and elastic properties are separated. The balls have mass m, and springs - stiffness k . Such a simple model makes it possible to describe the propagation of longitudinal and transverse mechanical waves in a solid. When a longitudinal wave propagates, the balls are displaced along the chain, and the springs are stretched or compressed, which is a stretching or compression deformation. If such deformation occurs in a liquid or gaseous medium, it is accompanied by compaction or rarefaction.

Remark 2

A distinctive feature of longitudinal waves is that they are able to propagate in any medium: solid, liquid and gaseous.

If in the specified model of a rigid body one or several balls receive a displacement perpendicular to the entire chain, we can speak of the occurrence of a shear deformation. Springs that have received deformation as a result of displacement will tend to return the displaced particles to the equilibrium position, and the nearest undisplaced particles will begin to be influenced by elastic forces tending to deflect these particles from the equilibrium position. The result will be the appearance of a transverse wave in the direction along the chain.

In a liquid or gaseous medium, elastic shear deformation does not occur. Displacement of one liquid or gas layer at some distance relative to the neighboring layer will not lead to the appearance of tangential forces at the boundary between the layers. The forces that act on the boundary of a liquid and a solid, as well as the forces between adjacent layers of a fluid, are always directed along the normal to the boundary - these are pressure forces. The same can be said about the gaseous medium.

Remark 3

Thus, the appearance of transverse waves is impossible in liquid or gaseous media.

In terms of practical applications, simple harmonic or sine waves are of particular interest. They are characterized by particle oscillation amplitude A, frequency f and wavelength λ. Sinusoidal waves propagate in homogeneous media with some constant speed υ.

Let us write an expression showing the dependence of the displacement y (x, t) of the particles of the medium from the equilibrium position in a sinusoidal wave on the coordinate x on the O X axis along which the wave propagates, and on time t:

y (x, t) = A cos ω t - x υ = A cos ω t - k x .

In the above expression, k = ω υ is the so-called wave number, and ω = 2 π f is the circular frequency.

Figure 2. 6. 4 shows "snapshots" of a shear wave at time t and t + Δt. During the time interval Δ t the wave moves along the axis O X at a distance υ Δ t . Such waves are called traveling waves.

Figure 2. 6. 4 . "Snapshots" of a traveling sine wave at a moment in time t and t + ∆t.

Definition 4

Wavelengthλ is the distance between two adjacent points on the axis O X oscillating in the same phases.

The distance, the value of which is the wavelength λ, the wave travels in a period T. Thus, the formula for the wavelength is: λ = υ T, where υ is the wave propagation speed.

With the passage of time t, the coordinate changes x any point on the graph displaying the wave process (for example, point A in Figure 2 . 6 . 4), while the value of the expression ω t - k x remains unchanged. After a time Δ t point A will move along the axis O X some distance Δ x = υ Δ t . Thus:

ω t - k x = ω (t + ∆ t) - k (x + ∆ x) = c o n s t or ω ∆ t = k ∆ x .

From this expression it follows:

υ = ∆ x ∆ t = ω k or k = 2 π λ = ω υ .

It becomes obvious that a traveling sinusoidal wave has a double periodicity - in time and space. The time period is equal to the oscillation period T of the particles of the medium, and the spatial period is equal to the wavelength λ.

Definition 5

wave number k = 2 π λ is the spatial analogue of the circular frequency ω = - 2 π T .

Let us emphasize that the equation y (x, t) = A cos ω t + k x is a description of a sinusoidal wave propagating in the direction opposite to the direction of the axis O X, with the speed υ = - ω k .

When a traveling wave propagates, all particles of the medium oscillate harmonically with a certain frequency ω. This means that, as in a simple oscillatory process, the average potential energy, which is the reserve of a certain volume of the medium, is the average kinetic energy in the same volume, proportional to the square of the oscillation amplitude.

Remark 4

From the foregoing, we can conclude that when a traveling wave propagates, an energy flux appears that is proportional to the speed of the wave and the square of its amplitude.

Traveling waves move in a medium with certain velocities, which depend on the type of wave, inert and elastic properties of the medium.

The speed with which transverse waves propagate in a stretched string or rubber band depends on the linear mass μ (or mass per unit length) and the tension force T:

The speed with which longitudinal waves propagate in an infinite medium is calculated with the participation of such quantities as the density of the medium ρ (or the mass per unit volume) and the bulk modulus B(equal to the coefficient of proportionality between the change in pressure Δ p and the relative change in volume Δ V V , taken with the opposite sign):

∆ p = - B ∆ V V .

Thus, the propagation velocity of longitudinal waves in an infinite medium is determined by the formula:

Example 1

At a temperature of 20 ° C, the propagation velocity of longitudinal waves in water is υ ≈ 1480 m / s, in various grades of steel υ ≈ 5 - 6 km / s.

If we are talking about longitudinal waves propagating in elastic rods, the formula for the wave velocity contains not the compression modulus, but Young's modulus:

For steel difference E from B insignificantly, but for other materials it can be 20 - 30% or more.

Figure 2. 6. 5 . Model of longitudinal and transverse waves.

Suppose that a mechanical wave propagating in a certain medium encounters some obstacle on its way: in this case, the nature of its behavior will change dramatically. For example, at the interface between two media with different mechanical properties, the wave is partially reflected, and partially penetrates into the second medium. A wave running along a rubber band or string will be reflected from the fixed end, and a counter wave will arise. If both ends of the string are fixed, complex oscillations will appear, which are the result of the superimposition (superposition) of two waves propagating in opposite directions and experiencing reflections and re-reflections at the ends. This is how the strings of all stringed musical instruments “work”, fixed at both ends. A similar process occurs with the sound of wind instruments, in particular, organ pipes.

If the waves propagating along the string in opposite directions have a sinusoidal shape, then under certain conditions they form a standing wave.

Suppose a string of length l is fixed in such a way that one of its ends is located at the point x \u003d 0, and the other at the point x 1 \u003d L (Figure 2.6.6). There is tension in the string T.

Picture 2 . 6 . 6 . The emergence of a standing wave in a string fixed at both ends.

Two waves with the same frequency run simultaneously along the string in opposite directions:

• y 1 (x, t) = A cos (ω t + k x) is a wave propagating from right to left;
• y 2 (x, t) = A cos (ω t - k x) is a wave propagating from left to right.

The point x = 0 is one of the fixed ends of the string: at this point the incident wave y 1 creates a wave y 2 as a result of reflection. Reflecting from the fixed end, the reflected wave enters antiphase with the incident one. In accordance with the principle of superposition (which is an experimental fact), the vibrations created by counterpropagating waves at all points of the string are summed up. It follows from the above that the final fluctuation at each point is defined as the sum of the fluctuations caused by the waves y 1 and y 2 separately. Thus:

y \u003d y 1 (x, t) + y 2 (x, t) \u003d (- 2 A sin ω t) sin k x.

The above expression is a description of a standing wave. Let us introduce some concepts applicable to such a phenomenon as a standing wave.

Definition 6

Knots are points of immobility in a standing wave.

antinodes– points located between the nodes and oscillating with the maximum amplitude.

If we follow these definitions, for a standing wave to occur, both fixed ends of the string must be nodes. The above formula meets this condition at the left end (x = 0) . For the condition to be satisfied at the right end (x = L) , it is necessary that k L = n π , where n is any integer. From what has been said, we can conclude that a standing wave does not always appear in a string, but only when the length L string is equal to an integer number of half-wavelengths:

l = n λ n 2 or λ n = 2 l n (n = 1 , 2 , 3 , . . .) .

The set of values ​​λ n of wavelengths corresponds to the set of possible frequencies f

f n = υ λ n = n υ 2 l = n f 1 .

In this notation, υ = T μ is the speed with which transverse waves propagate along the string.

Definition 7

Each of the frequencies f n and the type of string vibration associated with it is called a normal mode. The lowest frequency f 1 is called the fundamental frequency, all others (f 2 , f 3 , ...) are called harmonics.

Figure 2. 6. 6 illustrates the normal mode for n = 2.

A standing wave has no energy flow. The energy of vibrations, "locked" in the segment of the string between two neighboring nodes, is not transferred to the rest of the string. In each such segment, a periodic (twice per period) T) conversion of kinetic energy into potential energy and vice versa, similar to an ordinary oscillatory system. However, there is a difference here: if a weight on a spring or a pendulum has a single natural frequency f 0 = ω 0 2 π , then the string is characterized by the presence of an infinite number of natural (resonant) frequencies f n . Figure 2. 6. 7 shows several variants of standing waves in a string fixed at both ends.

Figure 2. 6. 7. The first five normal vibration modes of a string fixed at both ends.

According to the principle of superposition, standing waves of different types (with different values n) are able to simultaneously be present in the vibrations of the string.

Figure 2. 6. eight . Model of normal modes of a string.

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You can imagine what mechanical waves are by throwing a stone into the water. The circles that appear on it and are alternating troughs and ridges are an example of mechanical waves. What is their essence? Mechanical waves are the process of propagation of vibrations in elastic media.

Waves on liquid surfaces

Such mechanical waves exist due to the influence of intermolecular forces and gravity on the particles of the liquid. People have been studying this phenomenon for a long time. The most notable are the ocean and sea waves. As the wind speed increases, they change and their height increases. The shape of the waves themselves also becomes more complicated. In the ocean, they can reach frightening proportions. One of the most obvious examples of force is the tsunami, sweeping away everything in its path.

Energy of sea and ocean waves

Reaching the shore, sea waves increase with a sharp change in depth. They sometimes reach a height of several meters. At such moments, a colossal mass of water is transferred to coastal obstacles, which are quickly destroyed under its influence. The strength of the surf sometimes reaches grandiose values.

elastic waves

In mechanics, not only oscillations on the surface of a liquid are studied, but also the so-called elastic waves. These are perturbations that propagate in different media under the action of elastic forces in them. Such a perturbation is any deviation of the particles of a given medium from the equilibrium position. A good example of elastic waves is a long rope or rubber tube attached at one end to something. If you pull it tight, and then create a disturbance at its second (unfixed) end with a lateral sharp movement, you can see how it “runs” along the entire length of the rope to the support and is reflected back.

The initial perturbation leads to the appearance of a wave in the medium. It is caused by the action of some foreign body, which in physics is called the source of the wave. It can be the hand of a person swinging a rope, or a pebble thrown into the water. In the case when the action of the source is short-lived, a single wave often appears in the medium. When the “disturber” makes long waves, they begin to appear one after another.

Conditions for the occurrence of mechanical waves

Such oscillations are not always formed. A necessary condition for their appearance is the occurrence at the moment of disturbance of the medium of forces preventing it, in particular, elasticity. They tend to bring neighboring particles closer together when they move apart and push them away from each other when they approach each other. The forces of elasticity, acting on the particles remote from the source of perturbation, begin to bring them out of balance. Over time, all particles of the medium are involved in one oscillatory motion. The propagation of such oscillations is a wave.

Mechanical waves in an elastic medium

In an elastic wave, there are 2 types of motion simultaneously: particle oscillations and perturbation propagation. A longitudinal wave is a mechanical wave whose particles oscillate along the direction of its propagation. A transverse wave is a wave whose medium particles oscillate across the direction of its propagation.

Properties of mechanical waves

Perturbations in a longitudinal wave are rarefaction and compression, and in a transverse wave they are shifts (displacements) of some layers of the medium relative to others. The compression deformation is accompanied by the appearance of elastic forces. In this case, it is associated with the appearance of elastic forces exclusively in solids. In gaseous and liquid media, the shift of the layers of these media is not accompanied by the appearance of the mentioned force. Due to their properties, longitudinal waves are able to propagate in any medium, and transverse waves - only in solid ones.

Features of waves on the surface of liquids

Waves on the surface of a liquid are neither longitudinal nor transverse. They have a more complex, so-called longitudinal-transverse character. In this case, the fluid particles move in a circle or along elongated ellipses. particles on the surface of the liquid, and especially with large fluctuations, are accompanied by their slow but continuous movement in the direction of wave propagation. It is these properties of mechanical waves in the water that cause the appearance of various seafood on the shore.

Frequency of mechanical waves

If in an elastic medium (liquid, solid, gaseous) vibration of its particles is excited, then due to the interaction between them, it will propagate with a speed u. So, if an oscillating body is in a gaseous or liquid medium, then its movement will begin to be transmitted to all particles adjacent to it. They will involve the next ones in the process and so on. In this case, absolutely all points of the medium will begin to oscillate with the same frequency, equal to the frequency of the oscillating body. It is the frequency of the wave. In other words, this quantity can be characterized as points in the medium where the wave propagates.

It may not be immediately clear how this process occurs. Mechanical waves are associated with the transfer of energy of oscillatory motion from its source to the periphery of the medium. As a result, so-called periodic deformations arise, which are carried by the wave from one point to another. In this case, the particles of the medium themselves do not move along with the wave. They oscillate near their equilibrium position. That is why the propagation of a mechanical wave is not accompanied by the transfer of matter from one place to another. Mechanical waves have different frequencies. Therefore, they were divided into ranges and created a special scale. Frequency is measured in hertz (Hz).

Basic formulas

Mechanical waves, whose calculation formulas are quite simple, are an interesting object for study. The wave speed (υ) is the speed of its front movement (the geometric place of all points to which the oscillation of the medium has reached at a given moment):

where ρ is the density of the medium, G is the modulus of elasticity.

When calculating, one should not confuse the speed of a mechanical wave in a medium with the speed of movement of the particles of the medium that are involved in So, for example, a sound wave in air propagates with an average vibrational speed of its molecules of 10 m/s, while the speed of a sound wave in normal conditions is 330 m/s.

The wave front can be of different types, the simplest of which are:

Spherical - caused by fluctuations in a gaseous or liquid medium. In this case, the wave amplitude decreases with distance from the source in inverse proportion to the square of the distance.

Flat - is a plane that is perpendicular to the direction of wave propagation. It occurs, for example, in a closed piston cylinder when it oscillates. A plane wave is characterized by an almost constant amplitude. Its slight decrease with distance from the disturbance source is associated with the degree of viscosity of the gaseous or liquid medium.

Wavelength

Under understand the distance over which its front will move in a time that is equal to the period of oscillation of the particles of the medium:

λ = υT = υ/v = 2πυ/ ω,

where T is the oscillation period, υ is the wave speed, ω is the cyclic frequency, ν is the oscillation frequency of the medium points.

Since the propagation velocity of a mechanical wave is completely dependent on the properties of the medium, its length λ changes during the transition from one medium to another. In this case, the oscillation frequency ν always remains the same. Mechanical and similar in that during their propagation, energy is transferred, but no matter is transferred.

The existence of a wave requires a source of oscillation and a material medium or field in which this wave propagates. Waves are of the most diverse nature, but they obey similar patterns.

By physical nature distinguish:

According to the orientation of disturbances distinguish:

Longitudinal waves - The displacement of particles occurs along the direction of propagation; it is necessary to have an elastic force in the medium during compression; can be distributed in any environment. Examples: sound waves

Transverse waves - The displacement of particles occurs across the direction of propagation; can propagate only in elastic media; it is necessary to have a shear elastic force in the medium; can propagate only in solid media (and at the boundary of two media). Examples: elastic waves in a string, waves on water

According to the nature of the dependence on time distinguish:

elastic waves - mechanical displacements (deformations) propagating in an elastic medium. The elastic wave is called harmonic(sinusoidal) if the vibrations of the medium corresponding to it are harmonic.

running waves - Waves that carry energy in space.

According to the shape of the wave surface : plane, spherical, cylindrical wave.

wave front- the locus of points, to which the oscillations have reached a given point in time.

wave surface- locus of points oscillating in one phase.

Wave characteristics

Wavelength λ - the distance over which the wave propagates in a time equal to the period of oscillation

Wave amplitude A - amplitude of oscillations of particles in a wave

Wave speed v - speed of propagation of perturbations in the medium

Wave period T - oscillation period

Wave frequency ν - the reciprocal of the period

Traveling wave equation

During the propagation of a traveling wave, the disturbances of the medium reach the next points in space, while the wave transfers energy and momentum, but does not transfer matter (the particles of the medium continue to oscillate in the same place in space).

where v- speed , φ 0 - initial phase , ω – cyclic frequency , A– amplitude

Properties of mechanical waves

1. wave reflection – mechanical waves of any origin have the ability to be reflected from the interface between two media. If a mechanical wave propagating in a medium encounters an obstacle in its path, it can dramatically change the nature of its behavior. For example, at the interface between two media with different mechanical properties, a wave is partially reflected and partially penetrates into the second medium.

2. Refraction of waves – during the propagation of mechanical waves, one can also observe the phenomenon of refraction: a change in the direction of propagation of mechanical waves during the transition from one medium to another.

3. Wave diffraction – deviation of waves from rectilinear propagation, that is, their bending around obstacles.

4. Wave interference – addition of two waves. In a space where several waves propagate, their interference leads to the appearance of regions with the minimum and maximum values ​​of the oscillation amplitude

Interference and diffraction of mechanical waves.

A wave running along a rubber band or string is reflected from a fixed end; this creates a wave traveling in the opposite direction.

When waves are superimposed, the phenomenon of interference can be observed. The phenomenon of interference occurs when coherent waves are superimposed.

coherent calledwaveshaving the same frequencies, a constant phase difference, and the oscillations occur in the same plane.

interference called the time-constant phenomenon of mutual amplification and attenuation of oscillations at different points of the medium as a result of the superposition of coherent waves.

The result of the superposition of waves depends on the phases in which the oscillations are superimposed on each other.

If waves from sources A and B arrive at point C in the same phases, then the oscillations will increase; if it is in opposite phases, then there is a weakening of the oscillations. As a result, a stable pattern of alternating regions of enhanced and weakened oscillations is formed in space.

Maximum and minimum conditions

If the oscillations of points A and B coincide in phase and have equal amplitudes, then it is obvious that the resulting displacement at point C depends on the difference between the paths of the two waves.

Maximum conditions

If the difference between the paths of these waves is equal to an integer number of waves (i.e., an even number of half-waves) Δd = kλ , where k= 0, 1, 2, ..., then an interference maximum is formed at the point of superposition of these waves.

Maximum condition :

A = 2x0.

Minimum condition

If the path difference of these waves is equal to an odd number of half-waves, then this means that the waves from points A and B will come to point C in antiphase and cancel each other out.

Minimum condition:

The amplitude of the resulting oscillation A = 0.

If Δd is not equal to an integer number of half-waves, then 0< А < 2х 0 .

Diffraction of waves.

The phenomenon of deviation from rectilinear propagation and rounding of obstacles by waves is calleddiffraction.

The relationship between the wavelength (λ) and the size of the obstacle (L) determines the behavior of the wave. Diffraction is most clearly manifested if the length of the incident wave is greater than the dimensions of the obstacle. Experiments show that diffraction always exists, but becomes noticeable under the condition d<<λ , where d is the size of the obstacle.

Diffraction is a common property of waves of any nature, which always occurs, but the conditions for its observation are different.

A wave on the water surface propagates towards a sufficiently large obstacle, behind which a shadow is formed, i.e. no wave process is observed. This property is used in the construction of breakwaters in ports. If the size of the obstacle is comparable to the wavelength, then there will be a wave behind the obstacle. Behind him, the wave propagates as if there was no obstacle at all, i.e. wave diffraction is observed.

Examples of the manifestation of diffraction . Hearing a loud conversation around the corner of the house, sounds in the forest, waves on the surface of the water.

standing waves

standing waves are formed by adding the direct and reflected waves if they have the same frequency and amplitude.

In a string fixed at both ends, complex vibrations arise, which can be considered as the result of superposition ( superpositions) two waves propagating in opposite directions and experiencing reflections and re-reflections at the ends. Vibrations of strings fixed at both ends create the sounds of all stringed musical instruments. A very similar phenomenon occurs with the sound of wind instruments, including organ pipes.

string vibrations. In a stretched string fixed at both ends, when transverse vibrations are excited, standing waves , and knots should be located in the places where the string is fixed. Therefore, the string is excited with noticeable intensity only such vibrations, half of the wavelength of which fits on the length of the string an integer number of times.

This implies the condition

Wavelengths correspond to frequencies

n = 1, 2, 3...Frequencies vn called natural frequencies strings.

Harmonic vibrations with frequencies vn called own or normal vibrations . They are also called harmonics. In general, the vibration of a string is a superposition of different harmonics.

Standing wave equation :

At points where the coordinates satisfy the condition (n= 1, 2, 3, ...), the total amplitude is equal to the maximum value - this antinodes standing wave. Antinode coordinates :

At points whose coordinates satisfy the condition (n= 0, 1, 2,…), the total oscillation amplitude is equal to zero – This nodes standing wave. Node coordinates:

The formation of standing waves is observed when the traveling and reflected waves interfere. At the boundary where the wave is reflected, an antinode is obtained if the medium from which the reflection occurs is less dense (a), and a knot is obtained if it is more dense (b).

If we consider traveling wave , then in the direction of its propagation energy is transferred oscillatory movement. When same there is no standing wave of energy transfer , because incident and reflected waves of the same amplitude carry the same energy in opposite directions.

Standing waves arise, for example, in a string stretched at both ends when transverse vibrations are excited in it. Moreover, in the places of fixings, there are nodes of a standing wave.

If a standing wave is established in an air column that is open at one end (sound wave), then an antinode is formed at the open end, and a knot is formed at the opposite end.