§ 1.7. mechanical waves

The vibrations of a substance or field propagating in space are called a wave. Fluctuations of matter generate elastic waves (a special case is sound).

mechanical wave is the propagation of oscillations of the particles of the medium over time.

Waves in a continuous medium propagate due to the interaction between particles. If any particle comes into oscillatory motion, then, due to the elastic connection, this motion is transferred to neighboring particles, and the wave propagates. In this case, the oscillating particles themselves do not move with the wave, but hesitate around their equilibrium positions.

Longitudinal waves are waves in which the direction of particle oscillations x coincides with the direction of wave propagation . Longitudinal waves propagate in gases, liquids and solids.

P
opera waves- these are waves in which the direction of particle oscillations is perpendicular to the direction of wave propagation . Transverse waves propagate only in solid media.

Waves have two periodicity - in time and space. Periodicity in time means that each particle of the medium oscillates around its equilibrium position, and this movement is repeated with an oscillation period T. Periodicity in space means that the oscillatory motion of the particles of the medium is repeated at certain distances between them.

The periodicity of the wave process in space is characterized by a quantity called the wavelength and denoted .

The wavelength is the distance over which a wave propagates in a medium during one period of particle oscillation. .

From here
, where - particle oscillation period, - oscillation frequency, - speed of wave propagation, depending on the properties of the medium.

To how to write the wave equation? Let a piece of cord located at point O (the source of the wave) oscillate according to the cosine law

Let some point B be at a distance x from the source (point O). It takes time for a wave propagating with a speed v to reach it.
. This means that at point B, oscillations will begin later on
. I.e. After substituting into this equation the expressions for
and a number of mathematical transformations, we get

,
. Let's introduce the notation:
. Then. Due to the arbitrariness of the choice of point B, this equation will be the desired equation of a plane wave
.

The expression under the cosine sign is called the phase of the wave
.

E If two points are at different distances from the source of the wave, then their phases will be different. For example, the phases of points B and C, located at distances and from the source of the wave, will be respectively equal to

The phase difference of the oscillations occurring at point B and at point C will be denoted
and it will be equal

In such cases, it is said that between the oscillations occurring at points B and C there is a phase shift Δφ. It is said that oscillations at points B and C occur in phase if
. If a
, then the oscillations at points B and C occur in antiphase. In all other cases, there is simply a phase shift.

The concept of "wavelength" can be defined in another way:

Therefore, k is called the wave number.

We have introduced the notation
and showed that
. Then

.

Wavelength is the path traveled by a wave in one period of oscillation.

Let us define two important concepts in the wave theory.

wave surface is the locus of points in the medium that oscillate in the same phase. The wave surface can be drawn through any point of the medium, therefore, there are an infinite number of them.

Wave surfaces can be of any shape, and in the simplest case they are a set of planes (if the wave source is an infinite plane) parallel to each other, or a set of concentric spheres (if the wave source is a point).

wave front(wave front) - the locus of points to which fluctuations reach by the moment of time . The wave front separates the part of space involved in the wave process from the area where oscillations have not yet arisen. Therefore, the wave front is one of the wave surfaces. It separates two areas: 1 - which the wave reached by the time t, 2 - did not reach.

There is only one wave front at any moment of time, and it moves all the time, while the wave surfaces remain stationary (they pass through the equilibrium positions of particles oscillating in the same phase).

plane wave- this is a wave in which the wave surfaces (and the wave front) are parallel planes.

spherical wave is a wave whose wave surfaces are concentric spheres. Spherical wave equation:
.

Each point of the medium reached by two or more waves will take part in the oscillations caused by each wave separately. What will be the resulting vibration? It depends on a number of factors, in particular, on the properties of the medium. If the properties of the medium do not change due to the process of wave propagation, then the medium is called linear. Experience shows that waves propagate independently of each other in a linear medium. We will consider waves only in linear media. And what will be the fluctuation of the point, which reached two waves at the same time? To answer this question, it is necessary to understand how to find the amplitude and phase of the oscillation caused by this double action. To determine the amplitude and phase of the resulting oscillation, it is necessary to find the displacements caused by each wave, and then add them. How? Geometrically!

The principle of superposition (overlay) of waves: when several waves propagate in a linear medium, each of them propagates as if there were no other waves, and the resulting displacement of a particle of the medium at any time is equal to the geometric sum of the displacements that the particles receive, participating in each of the components of the wave processes.

An important concept of wave theory is the concept coherence - coordinated flow in time and space of several oscillatory or wave processes. If the phase difference of the waves arriving at the observation point does not depend on time, then such waves are called coherent. Obviously, only waves having the same frequency can be coherent.

R Let's consider what will be the result of adding two coherent waves coming to some point in space (observation point) B. In order to simplify mathematical calculations, we will assume that the waves emitted by sources S 1 and S 2 have the same amplitude and initial phases equal to zero. At the point of observation (at point B), the waves coming from the sources S 1 and S 2 will cause oscillations of the particles of the medium:
and
. The resulting fluctuation at point B is found as a sum.

Usually, the amplitude and phase of the resulting oscillation that occurs at the point of observation is found using the method of vector diagrams, representing each oscillation as a vector rotating with an angular velocity ω. The length of the vector is equal to the amplitude of the oscillation. Initially, this vector forms an angle with the chosen direction equal to the initial phase of oscillations. Then the amplitude of the resulting oscillation is determined by the formula.

For our case of adding two oscillations with amplitudes
,
and phases
,

.

Therefore, the amplitude of the oscillations that occur at point B depends on what is the path difference
traversed by each wave separately from the source to the observation point (
is the path difference between the waves arriving at the observation point). Interference minima or maxima can be observed at those points for which
. And this is the equation of a hyperbola with foci at the points S 1 and S 2 .

At those points in space for which
, the amplitude of the resulting oscillations will be maximum and equal to
. As
, then the oscillation amplitude will be maximum at those points for which.

at those points in space for which
, the amplitude of the resulting oscillations will be minimal and equal to
.oscillation amplitude will be minimal at those points for which .

The phenomenon of energy redistribution resulting from the addition of a finite number of coherent waves is called interference.

The phenomenon of waves bending around obstacles is called diffraction.

Sometimes diffraction is called any deviation of wave propagation near obstacles from the laws of geometric optics (if the dimensions of the obstacles are commensurate with the wavelength).

B
Due to diffraction, waves can enter the region of a geometric shadow, go around obstacles, penetrate through small holes in screens, etc. How to explain the hit of waves in the area of ​​geometric shadow? The phenomenon of diffraction can be explained using the Huygens principle: each point that a wave reaches is a source of secondary waves (in a homogeneous spherical medium), and the envelope of these waves sets the position of the wave front at the next moment in time.

Insert from light interference to see what might come in handy

wave called the process of propagation of vibrations in space.

wave surface is the locus of points at which oscillations occur in the same phase.

wave front called the locus of points to which the wave reaches a certain point in time t. The wave front separates the part of space involved in the wave process from the area where oscillations have not yet arisen.

For a point source, the wave front is a spherical surface centered at the source location S. 1, 2, 3 - wave surfaces; 1 - wave front. The equation of a spherical wave propagating along the beam emanating from the source: . Here - wave propagation speed, - wavelength; BUT- oscillation amplitude; - circular (cyclic) oscillation frequency; - displacement from the equilibrium position of a point located at a distance r from a point source at time t.

plane wave is a wave with a flat wave front. The equation of a plane wave propagating along the positive direction of the axis y:
, where x- displacement from the equilibrium position of a point located at a distance y from the source at time t.

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.

wave process- the process of energy transfer without the transfer of matter.

mechanical wave- perturbation propagating in an elastic medium.

The presence of an elastic medium is a necessary condition for the propagation of mechanical waves.

The transfer of energy and momentum in the medium occurs as a result of the interaction between neighboring particles of the medium.

Waves are longitudinal and transverse.

Longitudinal mechanical wave - a wave in which the movement of particles of the medium occurs in the direction of wave propagation. Transverse mechanical wave - a wave in which the particles of the medium move perpendicular to the direction of wave propagation.

Longitudinal waves can propagate in any medium. Transverse waves do not occur in gases and liquids, since they

there are no fixed positions of particles.

Periodic external action causes periodic waves.

harmonic wave- a wave generated by harmonic vibrations of the particles of the medium.

Wavelength- the distance over which the wave propagates during the period of oscillation of its source:

mechanical wave speed- velocity of perturbation propagation in the medium. Polarization is the ordering of the directions of oscillations of particles in a medium.

Plane of polarization- the plane in which the particles of the medium vibrate in the wave. A linearly polarized mechanical wave is a wave whose particles oscillate along a certain direction (line).

Polarizer- a device that emits a wave of a certain polarization.

standing wave- a wave formed as a result of the superposition of two harmonic waves propagating towards each other and having the same period, amplitude and polarization.

Antinodes of a standing wave- the position of the points with the maximum amplitude of oscillations.

Knots of a standing wave- non-moving points of the wave, the oscillation amplitude of which is equal to zero.

On the length l of a string fixed at the ends, an integer n half-waves of transverse standing waves fit:

Such waves are called oscillation modes.

The oscillation mode for an arbitrary integer n > 1 is called the nth harmonic or the nth overtone. The oscillation mode for n = 1 is called the first harmonic or fundamental oscillation mode. Sound waves are elastic waves in the medium that cause auditory sensations in a person.

The frequency of oscillations corresponding to sound waves lies in the range from 16 Hz to 20 kHz.

The speed of propagation of sound waves is determined by the rate of transfer of interaction between particles. The speed of sound in a solid v p, as a rule, is greater than the speed of sound in a liquid v l, which, in turn, exceeds the speed of sound in a gas v g.

Sound signals are classified by pitch, timbre and loudness. The pitch of the sound is determined by the frequency of the source of sound vibrations. The higher the oscillation frequency, the higher the sound; vibrations of low frequencies correspond to low sounds. The timbre of sound is determined by the form of sound vibrations. The difference in the shape of vibrations having the same period is associated with different relative amplitudes of the fundamental mode and overtone. Sound volume is characterized by the level of sound intensity. Sound intensity - the energy of sound waves incident on an area of ​​1 m 2 in 1 s.

Mechanicalwave in physics, this is the phenomenon of the propagation of perturbations, accompanied by the transfer of energy of an oscillating body from one point to another without transporting matter, in some elastic medium.

A medium in which there is an elastic interaction between molecules (liquid, gas or solid) is a prerequisite for the occurrence of mechanical disturbances. They are possible only when the molecules of a substance collide with each other, transferring energy. One example of such perturbations is sound (acoustic wave). Sound can travel in air, water, or solids, but not in a vacuum.

To create a mechanical wave, some initial energy is needed, which will bring the medium out of equilibrium. This energy will then be transmitted by the wave. For example, a stone thrown into a small amount of water creates a wave on the surface. A loud scream creates an acoustic wave.

The main types of mechanical waves:

• Sound;
• On the surface of the water;
• Earthquakes;
• seismic waves.

Mechanical waves have peaks and troughs, like all oscillatory motions. Their main characteristics are:

• Frequency. This is the number of oscillations per second. Units of measurement in SI: [ν] = [Hz] = [s -1].
• Wavelength. The distance between adjacent peaks or troughs. [λ] = [m].
• Amplitude. The greatest deviation of the medium point from the equilibrium position. [X max] = [m].
• Speed. This is the distance that a wave travels in a second. [V] = [m/s].

Wavelength

The wavelength is the distance between points closest to each other, oscillating in the same phases.

Waves propagate in space. The direction of their propagation is called beam and denoted by a line perpendicular to the wave surface. And their speed is calculated by the formula:

The boundary of the wave surface, which separates the part of the medium in which oscillations are already occurring, from the part of the medium in which oscillations have not yet begun, - wavefront.

Longitudinal and transverse waves

One of the ways to classify the mechanical type of waves is to determine the direction of movement of individual particles of the medium in a wave in relation to the direction of its propagation.

Depending on the direction of movement of particles in waves, there are:

1. transversewaves. The particles of the medium in this type of waves oscillate at right angles to the wave beam. A ripple in a pond or the vibrating strings of a guitar can help visualize transverse waves. This type of oscillation cannot propagate in a liquid or gas medium, because the particles of these media move randomly and it is impossible to organize their movement perpendicular to the direction of wave propagation. The transverse type of waves moves much more slowly than the longitudinal.
2. Longitudinalwaves. The particles of the medium oscillate in the same direction as the wave propagates. Some waves of this type are called compression or compression waves. Longitudinal oscillations of a spring - periodic compressions and extensions - provide a good visualization of such waves. Longitudinal waves are the fastest waves of the mechanical type. Sound waves in air, tsunamis and ultrasound are longitudinal. These include a certain type of seismic waves propagating underground and in water.

A mechanical or elastic wave is the process of propagation of oscillations in an elastic medium. For example, air begins to oscillate around a vibrating string or speaker cone - the string or speaker has become sources of a sound wave.

For the occurrence of a mechanical wave, two conditions must be met - the presence of a wave source (it can be any oscillating body) and an elastic medium (gas, liquid, solid).

Find out the cause of the wave. Why do the particles of the medium surrounding any oscillating body also come into oscillatory motion?

The simplest model of a one-dimensional elastic medium is a chain of balls connected by springs. Balls are models of molecules, the springs connecting them model the forces of interaction between molecules.

Suppose the first ball oscillates with a frequency ω. Spring 1-2 is deformed, an elastic force arises in it, which changes with frequency ω. Under the action of an external periodically changing force, the second ball begins to perform forced oscillations. Since forced oscillations always occur at the frequency of the external driving force, the oscillation frequency of the second ball will coincide with the oscillation frequency of the first. However, the forced oscillations of the second ball will occur with some phase delay relative to the external driving force. In other words, the second ball will begin to oscillate somewhat later than the first ball.

The vibrations of the second ball will cause a periodically changing deformation of the spring 2-3, which will make the third ball oscillate, and so on. Thus, all the balls in the chain will alternately be involved in an oscillatory motion with the oscillation frequency of the first ball.

Obviously, the cause of wave propagation in an elastic medium is the presence of interaction between molecules. The oscillation frequency of all particles in the wave is the same and coincides with the oscillation frequency of the wave source.

According to the nature of particle oscillations in a wave, waves are divided into transverse, longitudinal and surface waves.

AT longitudinal wave particles oscillate along the direction of wave propagation.

The propagation of a longitudinal wave is associated with the occurrence of tensile-compressive deformation in the medium. In the stretched areas of the medium, a decrease in the density of the substance is observed - rarefaction. In compressed areas of the medium, on the contrary, there is an increase in the density of the substance - the so-called thickening. For this reason, a longitudinal wave is a movement in space of areas of condensation and rarefaction.

Tensile-compressive deformation can occur in any elastic medium, so longitudinal waves can propagate in gases, liquids and solids. An example of a longitudinal wave is sound.

AT shear wave particles oscillate perpendicular to the direction of wave propagation.

The propagation of a transverse wave is associated with the occurrence of shear deformation in the medium. This kind of deformation can only exist in solids, so transverse waves can only propagate in solids. An example of a shear wave is the seismic S-wave.

surface waves occur at the interface between two media. Oscillating particles of the medium have both transverse, perpendicular to the surface, and longitudinal components of the displacement vector. During their oscillations, the particles of the medium describe elliptical trajectories in a plane perpendicular to the surface and passing through the direction of wave propagation. An example of surface waves are waves on the water surface and seismic L - waves.

The wave front is the locus of points reached by the wave process. The shape of the wave front can be different. The most common are plane, spherical and cylindrical waves.

Note that the wavefront is always located perpendicular direction of the wave! All points of the wavefront will begin to oscillate in one phase.

To characterize the wave process, the following quantities are introduced:

1. Wave frequencyν is the oscillation frequency of all the particles in the wave.

2. Wave amplitude A is the oscillation amplitude of the particles in the wave.

3. Wave speedυ is the distance over which the wave process (perturbation) propagates per unit time.

Pay attention - the speed of the wave and the speed of oscillation of particles in the wave are different concepts! The speed of a wave depends on two factors: the type of wave and the medium in which the wave propagates.

The general pattern is as follows: the speed of a longitudinal wave in a solid is greater than in liquids, and the speed in liquids, in turn, is greater than the speed of a wave in gases.

It is not difficult to understand the physical reason for this regularity. The cause of wave propagation is the interaction of molecules. Naturally, the perturbation propagates faster in the medium where the interaction of molecules is stronger.

In the same medium, the regularity is different - the speed of the longitudinal wave is greater than the speed of the transverse wave.

For example, the speed of a longitudinal wave in a solid, where E is the elastic modulus (Young's modulus) of the substance, ρ is the density of the substance.

Shear wave velocity in a solid, where N is the shear modulus. Since for all substances , then . One of the methods for determining the distance to the source of an earthquake is based on the difference in the velocities of longitudinal and transverse seismic waves.

The speed of a transverse wave in a stretched cord or string is determined by the tension force F and the mass per unit length μ:

4. Wavelengthλ is the minimum distance between points that oscillate equally.

For waves traveling on the surface of water, the wavelength is easily defined as the distance between two adjacent humps or adjacent depressions.

For a longitudinal wave, the wavelength can be found as the distance between two adjacent concentrations or rarefactions.

5. In the process of wave propagation, sections of the medium are involved in an oscillatory process. The oscillating medium, firstly, moves, therefore, it has kinetic energy. Secondly, the medium through which the wave runs is deformed, therefore, it has potential energy. It is easy to see that wave propagation is associated with the transfer of energy to unexcited parts of the medium. To characterize the energy transfer process, we introduce wave intensity I.