Longitudinal waves are. Transverse and longitudinal waves
longitudinal wave. LONGITUDINAL WAVE, a wave in which the direction of the quantity characterizing it (for example, the displacement of oscillating particles of the medium) is parallel to the direction of propagation. Longitudinal waves include, in particular, plane (uniform) ... ... Illustrated Encyclopedic Dictionary
A wave whose vector quantity characterizes it (for example, for harmonic waves, the vector amplitude) is collinear to the direction of its propagation (for harmonic waves, to the wave vector It). To P. in. include, in particular, flat (homogeneous) sound. ... ... Physical Encyclopedia
A wave in which the direction of the vector quantity characterizing it (for example, the displacement of oscillating particles of the medium) is parallel to the direction of propagation. To P. in. include, in particular, sound, waves in gases and liquids. Longitudinal wave... Natural science. encyclopedic Dictionary
longitudinal wave- rarefaction compression wave 1. A wave in which the directions of oscillation of the particles of the medium coincide with the direction of wave propagation 2. A wave in which the particles of the medium oscillate in the direction of wave propagation Topics vibration EN longitudinal wave DE longitudinalwelle FR onde longitudinale … Technical Translator's Handbook
If an oscillating body (a source of oscillations) is placed in an elastic medium, then the particles of the medium adjacent to it will also begin to oscillate. The oscillation of these particles is transmitted (by elastic forces) to neighboring particles of the medium, etc. After some time, the oscillation will cover the entire environment. However, it will be performed with different phases: the farther the particle is located from the source of oscillations, the later it will begin to oscillate and the more its oscillation will lag in phase. Propagation of vibrations in a medium called. wave process or wave. Example: seismic waves, water waves. The direction of propagation of a wave (oscillation) is called beam.
The wave is called transverse, if the particles of the medium oscillate perpendicular to the beam. If they oscillate along the beam, then the wave is called longitudinal.
Longitudinal waves can arise in a medium with volume elasticity, i.e. in solids, liquids and gaseous bodies. transverse waves arise only in a medium with form elasticity (shear deformation), i.e. only in solids. The exception is waves on the surface of the water.
The main laws of the wave process are valid not only for mechanical waves of an elastic medium, but also for waves of any nature, in particular for waves electromagnetic field.
WAVE EQUATION. WAVE INTENSITY.
Let the oscillations of the source O be harmonic, i.e. x \u003d Asin t.
Then all the particles of the medium will also come into harmonic oscillation with the same frequency and amplitude, but with different phases. A sinusoidal wave will appear in the medium.
A wave graph is superficially similar to a harmonic wave graph, but essentially they are different. The oscillation graph is the dependence of the displacement of a given particle on time, the wave graph is the displacement of all particles of the medium from the distance to the source of oscillations at a given time. He is like snapshot of a wave.
We get the wave equation. Consider some particle C. It is obvious that if the particle O oscillates already t sec., then the particle C oscillates only (t - ) sec., where is the propagation time of oscillations from O to C. Then the oscillation equation for C will be
X \u003d Аsin (t - ) , but \u003d y / V,
where V - speed of wave propagation.
Then Х = Аsin(t – y/ V) is the wave equation (1)
Given that the wavelength V T= V/, from where V= /T, = 2/T =2 we get
X \u003d Asin2 (t / T - y / ) \u003d Asin2 (t - y / ) \u003d Asin (t -2y / ),
where k = 2/ is the wave number. If we change the coordinate axes, then
y(x,t) = Asin(t kx). The sign (+) indicates the opposite direction of propagation.
The distance over which the oscillation propagates in one period is called wavelength.
The wave motion propagation velocity is the phase propagation velocity (phase velocity). In a homogeneous medium, the speed is constant. When passing from one medium to another, the speed of wave propagation changes, because the elastic properties of the medium change, but the frequency of oscillations, as experience shows, remains unchanged. This means that at moving from one environment to another will change .
If we excited vibrations at any point of the medium, then the vibrations will be transmitted to all the surrounding points, i.e. a set of particles enclosed in a certain volume will oscillate. Spreading from the sources of oscillations, the wave process covers more and more new parts of space. Geometric locus of points to which oscillations reach a certain point in timet, called wave front.
Thus, the wave front is the surface that separates the part of space already involved in the wave process from the area in which oscillations have not yet arisen. The locus of points that oscillate in the same phase is called. wave surface. Wave surfaces can be of various shapes. The simplest of them have the shape of a sphere or plane. Waves having such surfaces are called spherical or plane waves, respectively.
Often, when solving problems of wave propagation, it is necessary to construct a wave front for a certain moment of time using the wave front given for the initial moment of time. This can be done using Huygens principle, whose essence is as follows:
Let the wave front moving in a homogeneous medium occupy position 1 at a given time, Fig. 2.
It is required to find its position after a time interval t. According to Huygens, each point of the medium reached by the wave itself becomes a source of secondary waves(first position).
This means that a spherical wave begins to propagate from it, as from the center. To construct secondary waves, around each point of the initial front we describe spheres with a radius
y = Vt, where V – wave speed .
The secondary waves are mutually canceled in all directions, except for the directions of the initial front ( second proposition of Huygens' principle).
In other words, oscillations are preserved only on the outer envelope of the secondary waves. By constructing this envelope, we obtain the initial position 2 of the wave front.
Huygens' principle is also applicable to an inhomogeneous medium. In this case, the values V, a hence y are not the same in different directions.
Because the passage of a wave is accompanied by oscillations of the particles of the medium, then together with the wave the energy of oscillations also moves in space.
Wave intensity or energy flux density called. the ratio of the energy transferred by the wave through the area perpendicular to the beam to the duration of the transfer time and the size of the area.
We obtain an expression for the wave intensity.
Let 1 cm 3 of the medium contain n 0 particles of mass m. Then the energy of oscillation of the medium per unit volume is equal to
E \u003d n 0 m 2 A 2 / 2 \u003d 2 A 2 / 2, where \u003d n 0 m.
Obviously, in 1 s through an area of 1 cm 2 the energy contained in the volume of a rectangular parallelepiped with a base of 1 cm 2 and a height equal to V, hence the intensity
I=E V = V 2 A 2 /2.
Thus, the intensity of the wave is proportional to the density of the medium and speed, the square of the circular frequency and the square of the wave amplitude.
standing waves.
It is often necessary to observe the mutual superposition of waves, while the particles of the medium participate in several wave motions at once. Experience shows that in this case the displacement of each particle of the medium is the sum of its displacements corresponding to all superimposed waves. The overlap phenomenon is called summation of waves. One of the most important examples of such an addition is the superposition of two plane waves traveling in opposite directions with the same amplitude. In this case, the resulting offset is given by
Y(x,t) = Asin(t – kx) + Asin(t + kx) = 2Asin t coskx = B(x) sint.
We can observe such an addition when waves are reflected from obstacles. The wave incident on the barrier and the reflected wave running towards it, overlapping each other, give the resulting oscillation, called standing wave.
It can be seen from the standing wave equation that at each point of this wave, oscillations of the same frequency occur as in the counterpropagating waves, and the amplitude B depends on the x coordinate:
B(x) \u003d 2A cos kx \u003d 2Acos2x / .
At those points where 2x/ = n (n = 0,1,2,...), the amplitude AT reaches a maximum of 2A. These points are called antinodes of a standing wave.
The antinode coordinate is x n = n/2. At points where 2х/ = (n+1/2), the amplitude AT goes to zero. These points are called standing wave nodes. The points of the medium located at the nodes do not oscillate. Node coordinates are equal
X y = (n ½)/2.
From the formulas for the coordinates of nodes and antinodes, it follows that the distance between neighboring nodes (as well as neighboring antinodes) is equal to /2.
SOUND.
Human perceived sound is also a wave motion that occurs in the environment around us. The source of sound is always some vibrating body. This body sets in motion the surrounding air, in which they begin to spread longitudinal elastic waves. When these waves reach the ear, they cause the eardrum to vibrate and we feel the sound. . Mechanical waves, the effect of which on the ear causes the sensation of sound, are called sound waves. A person perceives f \u003d 20–16000 Hz. f< 20 Гц – infrasound, f > 16 kHz – ultrasound.
(Mountains, avalanches, sat down! Infrasound fear).
Elastic waves can propagate only in a medium where there is a connection between the individual particles of this medium, so sound cannot propagate in a vacuum. In the air V=330 m/s.
In order to cause a sound sensation, a wave must have a certain minimum intensity, which is called
hearing threshold. It is different for different people and strongly depends on f. The human ear is most sensitive to f = 1000 - 4000 Hz. In this frequency range I 0 = 10 -16 W.
A sound of very high intensity also does not cause an auditory sensation, but only creates a sensation of pain and pressure in the ear. The minimum value of the sound intensity, the excess of which causes pain, called. pain threshold. The values of different thresholds are different for different frequencies, Fig.1.
pain threshold
Audible area
Fig.1. hearing threshold
First audible sound quality is volume. A change in the volume of sound is caused by a change in the amplitude of the oscillations. This happens because the energy carried by the wave is proportional to the square of the amplitude (E ~ A 2).
Second sound quality is the height of his tone. A sound corresponding to a strictly defined frequency of vibrations is called. tone. The higher the frequency of the sound, the higher the tone. You can get sounds of various tones using a tuning fork.
Third sound quality is timbre. In life, we often recognize a familiar person by voice, not yet seeing him. We easily distinguish the sounds of the violin from the sounds of the piano, although they may be of the same tone. The quality of sound, which allows you to determine the source of its formation, called. timbre. The timbre of different sound sources is not the same. This is explained by the formation of additional standing waves in the sound source itself, which give additional tones. Additional tones of the sound source, higher than the main tone, called higher harmonic tones or overtones.
Each sound source has a certain number of overtones. They give the sound its characteristic shade - timbre.
Noise differs from musical sound only in that it contains vibrations of various frequencies with different amplitudes.
At the interface between two media, sound waves undergo partial or total reflection. The return of a sound wave after reflection is called. echo. The phenomenon of reflection of sound waves is widely used in acoustics. The relatively weak attenuation of ultrasonic waves in water made it possible to use them for sonar - detection of objects and determination of distances from the sound source to objects. Sonar (echo sounder) - measures the depth and relief of the seabed, the distance to the iceberg, schools of fish, etc. Examples: robotics, ultrasound.
t=2 l /V from where l= tv/2. l
impulse
ultrasound source
If an oscillating body (a source of oscillations) is placed in an elastic medium, then the particles of the medium adjacent to it will also begin to oscillate. The oscillation of these particles is transmitted (by elastic forces) to neighboring particles of the medium, etc. After some time, the oscillation will cover the entire environment. However, it will be performed with different phases: the farther the particle is located from the source of oscillations, the later it will begin to oscillate and the more its oscillation will lag in phase. Propagation of vibrations in a medium called. wave process or wave. Example: seismic waves, water waves. The direction of propagation of a wave (oscillation) is called beam.
The wave is called transverse, if the particles of the medium oscillate perpendicular to the beam. If they oscillate along the beam, then the wave is called longitudinal.
Longitudinal waves can arise in a medium with volume elasticity, i.e. in solids, liquids and gases. transverse waves arise only in a medium with form elasticity (shear deformation), i.e. only in solids. The exception is waves on the surface of the water.
The main laws of the wave process are valid not only for mechanical waves of an elastic medium, but also for waves of any nature, in particular for waves of an electromagnetic field.
WAVE EQUATION. WAVE INTENSITY.
Let the oscillations of the source O be harmonic, i.e. x \u003d Asin t.
Then all the particles of the medium will also come into harmonic oscillation with the same frequency and amplitude, but with different phases. A sinusoidal wave will appear in the medium.
A wave graph is superficially similar to a harmonic wave graph, but essentially they are different. The oscillation graph is the dependence of the displacement of a given particle on time, the wave graph is the displacement of all particles of the medium from the distance to the source of oscillations at a given time. He is like snapshot of a wave.
We get the wave equation. Consider some particle C. It is obvious that if the particle O oscillates already t sec., then the particle C oscillates only (t - ) sec., where is the propagation time of oscillations from O to C. Then the oscillation equation for C will be
X \u003d Аsin (t - ) , but \u003d y / V,
where V - speed of wave propagation.
Then Х = Аsin(t – y/ V) is the wave equation (1)
Given that the wavelength V T= V/, from where V= /T, = 2/T =2 we get
X \u003d Asin2 (t / T - y / ) \u003d Asin2 (t - y / ) \u003d Asin (t -2y / ),
where k = 2/ is the wave number. If we change the coordinate axes, then
y(x,t) = Asin(t kx). The sign (+) indicates the opposite direction of propagation.
The distance over which the oscillation propagates in one period is called wavelength.
The wave motion propagation velocity is the phase propagation velocity (phase velocity). In a homogeneous medium, the speed is constant. When passing from one medium to another, the speed of wave propagation changes, because the elastic properties of the medium change, but the frequency of oscillations, as experience shows, remains unchanged. This means that at moving from one environment to another will change .
If we excited vibrations at any point of the medium, then the vibrations will be transmitted to all the surrounding points, i.e. a set of particles enclosed in a certain volume will oscillate. Spreading from the sources of oscillations, the wave process covers more and more new parts of space. Geometric locus of points to which oscillations reach a certain point in timet, called wave front.
Thus, the wave front is the surface that separates the part of space already involved in the wave process from the area in which oscillations have not yet arisen. The locus of points that oscillate in the same phase is called. wave surface. Wave surfaces can be of various shapes. The simplest of them have the shape of a sphere or plane. Waves having such surfaces are called spherical or plane waves, respectively.
Often, when solving problems of wave propagation, it is necessary to construct a wave front for a certain moment of time using the wave front given for the initial moment of time. This can be done using Huygens principle, whose essence is as follows:
Let the wave front moving in a homogeneous medium occupy position 1 at a given time, Fig. 2.
It is required to find its position after a time interval t. According to Huygens, each point of the medium reached by the wave itself becomes a source of secondary waves(first position).
This means that a spherical wave begins to propagate from it, as from the center. To construct secondary waves, around each point of the initial front we describe spheres with a radius
y = Vt, where V – wave speed .
The secondary waves are mutually canceled in all directions, except for the directions of the initial front ( second proposition of Huygens' principle).
In other words, oscillations are preserved only on the outer envelope of the secondary waves. By constructing this envelope, we obtain the initial position 2 of the wave front.
Huygens' principle is also applicable to an inhomogeneous medium. In this case, the values V, a hence y are not the same in different directions.
Because the passage of a wave is accompanied by oscillations of the particles of the medium, then together with the wave the energy of oscillations also moves in space.
Wave intensity or energy flux density called. the ratio of the energy transferred by the wave through the area perpendicular to the beam to the duration of the transfer time and the size of the area.
We obtain an expression for the wave intensity.
Let 1 cm 3 of the medium contain n 0 particles of mass m. Then the energy of oscillation of the medium per unit volume is equal to
E \u003d n 0 m 2 A 2 / 2 \u003d 2 A 2 / 2, where \u003d n 0 m.
Obviously, in 1 s through an area of 1 cm 2 the energy contained in the volume of a rectangular parallelepiped with a base of 1 cm 2 and a height equal to V, hence the intensity
I=E V = V 2 A 2 /2.
Thus, the intensity of the wave is proportional to the density of the medium and speed, the square of the circular frequency and the square of the wave amplitude.
standing waves.
It is often necessary to observe the mutual superposition of waves, while the particles of the medium participate in several wave motions at once. Experience shows that in this case the displacement of each particle of the medium is the sum of its displacements corresponding to all superimposed waves. The overlap phenomenon is called summation of waves. One of the most important examples of such an addition is the superposition of two plane waves traveling in opposite directions with the same amplitude. In this case, the resulting offset is given by
Y(x,t) = Asin(t – kx) + Asin(t + kx) = 2Asin t coskx = B(x) sint.
We can observe such an addition when waves are reflected from obstacles. The wave incident on the barrier and the reflected wave running towards it, overlapping each other, give the resulting oscillation, called standing wave.
It can be seen from the standing wave equation that at each point of this wave, oscillations of the same frequency occur as in the counterpropagating waves, and the amplitude B depends on the x coordinate:
B(x) \u003d 2A cos kx \u003d 2Acos2x / .
At those points where 2x/ = n (n = 0,1,2,...), the amplitude AT reaches a maximum of 2A. These points are called antinodes of a standing wave.
The antinode coordinate is x n = n/2. At points where 2х/ = (n+1/2), the amplitude AT goes to zero. These points are called standing wave nodes. The points of the medium located at the nodes do not oscillate. Node coordinates are equal
X y = (n ½)/2.
From the formulas for the coordinates of nodes and antinodes, it follows that the distance between neighboring nodes (as well as neighboring antinodes) is equal to /2.
SOUND.
Human perceived sound is also a wave motion that occurs in the environment around us. The source of sound is always some vibrating body. This body sets in motion the surrounding air, in which they begin to spread longitudinal elastic waves. When these waves reach the ear, they cause the eardrum to vibrate and we feel the sound. . Mechanical waves, the effect of which on the ear causes the sensation of sound, are called sound waves. A person perceives f \u003d 20–16000 Hz. f< 20 Гц – infrasound, f > 16 kHz – ultrasound.
(Mountains, avalanches, sat down! Infrasound fear).
Elastic waves can propagate only in a medium where there is a connection between the individual particles of this medium, so sound cannot propagate in a vacuum. In the air V=330 m/s.
In order to cause a sound sensation, a wave must have a certain minimum intensity, which is called
hearing threshold. It is different for different people and strongly depends on f. The human ear is most sensitive to f = 1000 - 4000 Hz. In this frequency range I 0 = 10 -16 W.
A sound of very high intensity also does not cause an auditory sensation, but only creates a sensation of pain and pressure in the ear. The minimum value of the sound intensity, the excess of which causes pain, called. pain threshold. The values of different thresholds are different for different frequencies, Fig.1.
pain threshold
Audible area
Fig.1. hearing threshold
First audible sound quality is volume. A change in the volume of sound is caused by a change in the amplitude of the oscillations. This happens because the energy carried by the wave is proportional to the square of the amplitude (E ~ A 2).
Second sound quality is the height of his tone. A sound corresponding to a strictly defined frequency of vibrations is called. tone. The higher the frequency of the sound, the higher the tone. You can get sounds of various tones using a tuning fork.
Third sound quality is timbre. In life, we often recognize a familiar person by voice, not yet seeing him. We easily distinguish the sounds of the violin from the sounds of the piano, although they may be of the same tone. The quality of sound, which allows you to determine the source of its formation, called. timbre. The timbre of different sound sources is not the same. This is explained by the formation of additional standing waves in the sound source itself, which give additional tones. Additional tones of the sound source, higher than the main tone, called higher harmonic tones or overtones.
Each sound source has a certain number of overtones. They give the sound its characteristic shade - timbre.
Noise differs from musical sound only in that it contains vibrations of various frequencies with different amplitudes.
At the interface between two media, sound waves undergo partial or total reflection. The return of a sound wave after reflection is called. echo. The phenomenon of reflection of sound waves is widely used in acoustics. The relatively weak attenuation of ultrasonic waves in water made it possible to use them for sonar - detection of objects and determination of distances from the sound source to objects. Sonar (echo sounder) - measures the depth and relief of the seabed, the distance to the iceberg, schools of fish, etc. Examples: robotics, ultrasound.
t=2 l /V from where l= tv/2. l
impulse
ultrasound source
> Longitudinal waves
Sometimes they are called compression waves. oscillate in the direction of propagation.
Learning task
- Determine the properties and examples of the longitudinal wave type.
Key Points
- The oscillations of longitudinal waves are carried out in the direction of propagation, but they are too small and have equilibrium positions, so they do not displace the mass.
- This type can be considered as impulses transporting energy along the propagation axis.
- They can also be perceived as pressure waves with characteristic compression and rarefaction.
Terms
- Rarefaction is a decrease in the density of a material (primarily for a liquid).
- Longitudinal - in the direction of the length of the axis.
- Compression is an increase in density.
Example
The sound wave is the best. It accommodates the impulses resulting from air compression.
Longitudinal waves
The direction of vibration coincides with the direction of movement. That is, the movement of the medium is located in the same direction as the wave movement. Some longitudinal waves are also called compressional. If you want to experiment, then just get a Slinky toy (spring) and hold it at both ends. At the moment of compression and weakening, the impulse will move to the end.
The compressed Slinky is an example of a longitudinal wave. It propagates in the same direction as the vibrations
Longitudinal (as well as transverse) do not displace the mass. The difference is that each particle in the medium through which a longitudinal wave propagates will oscillate along the propagation axis. If you think about the Slinky, then the coils oscillate in points, but will not move along the length of the spring. Do not forget that it is not mass that is transported here, but energy in the form of momentum.
In some cases, such waves act as pressure waves. Sound is a prime example. They are formed when a medium (most often, air) is compressed. Longitudinal sound waves - alternating pressure deviation from balanced pressure, which leads to local areas of compression and rarefaction.
Matter in the medium is periodically displaced sound wave and oscillate. To produce sound, you need to compress air particles to a certain amount. This is how transverse waves are formed. The ears react sensitively to different pressures and translate waves into tones.
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There are longitudinal and transverse waves. The wave is called transverse, if the particles of the medium oscillate in a direction perpendicular to the direction of wave propagation (Fig. 15.3). A transverse wave propagates, for example, along a stretched horizontal rubber cord, one of the ends of which is fixed, and the other is brought into vertical oscillatory motion.
The wave is called longitudinal, if the particles of the medium oscillate in the direction of wave propagation (Fig. 15.5).
A longitudinal wave can be observed on a long soft spring of large diameter. By hitting one of the ends of the spring, one can notice how successive condensations and rarefaction of its coils will spread along the spring, running one after another. In Figure 15.6, the dots show the position of the coils of the spring at rest, and then the positions of the coils of the spring at successive intervals of time equal to a quarter of the period.
Thus, the longitudinal wave in the case under consideration is an alternating cluster (Sg) and rarefaction (Once) spring coils.
Traveling wave energy. Energy flux density vector
The elastic medium in which the wave propagates has both kinetic energy oscillatory motion particles and potential energy, due to the deformation of the medium. It can be shown that the volumetric energy density for a plane traveling harmonic wave S=Acos(ω(t-)+φ 0)
where r=dm/dV is the density of the medium, i.e. periodically changes from 0 to rА2w2 during the time p/w=Т/2. The average value of the energy density over a period of time p / w \u003d T / 2
To characterize the energy transfer, the concept of the energy flux density vector is introduced - the Umov vector. Let's derive an expression for it. If the energy DW is transferred through the area DS^ perpendicular to the direction of wave propagation during the time Dt, then the energy flux density Fig. 2
where DV=DS^ uDt is the volume of an elementary cylinder selected in the environment. Since the energy transfer rate or group velocity is a vector, the energy flux density can also be represented as a vector, W/m2 (18) This vector was introduced by N.A. Umov in 1874. The average value of its modulus is called the intensity of the wave (19) For a harmonic wave u=v , therefore, for such a wave in formulas (17)-(19) u can be replaced by v. The intensity is determined by the energy flux density - this vector coincides with the direction in which the energy is transferred and is equal to the energy flux transferred through……………..
When they talk about intensity, they mean the physical meaning of the vector - the flow of energy. The intensity of the wave is proportional to the square of the amplitude.
Vector Pointing(also vector Umov- Pointing) - vector energy flux density of the electromagnetic field, one of the components energy-momentum tensor of the electromagnetic field. The Poynting vector S can be defined in terms of vector product two vectors:
(in the GHS system),
(in SI system),
where E and H- tension vectors electric and magnetic fields, respectively.
(in complex form)
,
where E and H- vectors complex amplitude of the electric and magnetic fields, respectively.
This vector is equal in absolute value to the amount of energy transferred through a unit area normal to S, per unit of time. By its direction, the vector determines the direction of energy transfer.
Since the components tangential to the interface between two media E and H continuous (see border conditions), then the vector S is continuous at the boundary of two media.
standing wave - fluctuations in distributed oscillatory systems with a characteristic arrangement of alternating maxima ( antinodes) and minima ( nodes)amplitude. In practice, such a wave occurs when reflections from obstacles and inhomogeneities as a result of the superimposition of the reflected wave on the incident one. At the same time, it is extremely important frequency, phase and the attenuation coefficient of the wave at the place of reflection.
An example of a standing wave is fluctuations of the string, vibrations of air in the organ pipe ; in nature - Schumann waves.
A purely standing wave, strictly speaking, can exist only in the absence of losses in the medium and total reflection of waves from the boundary. Usually, in addition to standing waves, the medium also contains traveling waves, bringing energy to the places of its absorption or emission.
To demonstrate standing waves in a gas, use rubens pipe.
