Direction of oscillations in longitudinal and transverse waves. Longitudinal mechanical waves can propagate in any media - solid, liquid and gaseous
Longitudinal wave– this is a wave, during the propagation of which the particles of the medium are displaced in the direction of propagation of the wave (Fig. 1, a).
The cause of the longitudinal wave is compression/tension deformation, i.e. resistance of the medium to changes 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 wave– this is a wave, during the propagation of which the particles of the medium are displaced in the direction perpendicular to the propagation of the wave (Fig. 1, b).
The cause of the transverse wave is the shear deformation of one layer of the medium relative to another. When a transverse wave propagates through a medium, ridges and troughs are formed. Liquids and gases, unlike solids, do not have elasticity with respect to the shear of layers, i.e. do not resist changing shape. Therefore, transverse waves can propagate only in solids.
Examples of transverse waves are waves traveling along a stretched rope or string.
Waves on the surface of a liquid are neither longitudinal nor transverse. If you throw a float onto the surface of the water, you can see that it moves, swaying on the waves, along a circular path. Thus, a wave on the surface of a liquid has both transverse and longitudinal components. Waves can also appear on the surface of a liquid special type- the so-called surface waves. They arise as a result of gravity and surface tension.
Fig.1. Longitudinal (a) and transverse (b) mechanical waves
Question 30
Wavelength.
Each wave travels at a certain speed. Under wave speed understand the speed of propagation of disturbance. For example, a blow to the end of a steel rod causes local compression in it, which then propagates along the rod at a speed of about 5 km/s.
The speed of the wave is determined by the properties of the medium in which the wave propagates. When a wave passes from one medium to another, its speed changes.
In addition to speed, important characteristic wave is the wavelength. Wavelength is the distance over which a wave travels in time equal to the period fluctuations in it.
Since the speed of a wave is a constant value (for a given medium), the distance traveled by the wave is equal to the product of the speed and the time of its propagation. Thus, to find the wavelength, you need to multiply the speed of the wave by the period of oscillation in it:
v - wave speed; T is the period of oscillation in the wave; λ (Greek letter "lambda") - wavelength.
By choosing the direction of wave propagation as the direction of the x axis and denoting by y the coordinate of the particles oscillating in the wave, we can construct wave chart. A graph of a sine wave (at a fixed time t) is shown in Figure 45. The distance between adjacent crests (or troughs) in this graph coincides with the wavelength λ.
Formula (22.1) expresses the relationship between wavelength and its speed and period. Considering that the period of oscillation in a wave is inversely proportional to the frequency, i.e. T = 1/ν, we can obtain a formula expressing the relationship between the wavelength and its speed and frequency:
The resulting formula shows that the speed of the wave is equal to the product of the wavelength and the frequency of oscillations in it.
The frequency of oscillations in the wave coincides with the frequency of oscillations of the source (since the oscillations of the particles of the medium are forced) and does not depend on the properties of the medium in which the wave propagates. When a wave passes from one medium to another, its frequency does not change, only the speed and wavelength change.
Question 30.1
Wave equation
To obtain the wave equation, that is, an analytical expression for a function of two variables S = f (t, x) , Let's imagine that at some point in space there arise harmonic vibrations with circular frequency w and the initial phase, equal to zero for simplicity (see Fig. 8). Offset at a point M: S m = A sin w t, Where A- amplitude. Since the particles of the medium filling the space are interconnected, vibrations from the point M spread along the axis X with speed v. After some time D t they reach the point N. If there is no attenuation in the medium, then the displacement at this point has the form: S N = A sin w(t- D t), i.e. oscillations are delayed by time D t relative to the point M. Since , then replacing an arbitrary segment MN coordinate X, we get wave equation in the form.
Mechanical waves
If vibrations of particles are excited in any place in a solid, liquid or gaseous medium, then due to the interaction of atoms and molecules of the medium, the vibrations begin to be transmitted from one point to another with a finite speed. The process of propagation of vibrations in a medium is called wave .
Mechanical waves there are different types. If particles of the medium in a wave are displaced in a direction perpendicular to the direction of propagation, then the wave is called transverse . An example of a wave of this kind can be waves running along a stretched rubber band (Fig. 2.6.1) or along a string.
If the displacement of particles of the medium occurs in the direction of propagation of the wave, then the wave is called longitudinal . Waves in an elastic rod (Fig. 2.6.2) or sound waves in a gas are examples of such waves.
Waves on the surface of a liquid have both transverse and longitudinal components.
In both transverse and longitudinal waves, there is no transfer of matter in the direction of wave propagation. In the process of propagation, particles of the medium only oscillate around equilibrium positions. However, waves transfer vibrational energy from one point in the medium to another.
Characteristic feature mechanical waves is that they propagate in material media (solid, liquid or gaseous). There are waves that can propagate in vacuum (for example, light waves). Mechanical waves necessarily require a medium that has the ability to store kinetic and potential energy. Therefore, the environment must have inert and elastic properties. In real environments, these properties are distributed throughout the entire volume. For example, any small element of a solid body has mass and elasticity. In the simplest one-dimensional model a solid body can be represented as a collection of balls and springs (Fig. 2.6.3).
Longitudinal mechanical waves can propagate in any media - solid, liquid and gaseous.
If in a one-dimensional model of a solid body one or more balls are displaced in a direction perpendicular to the chain, then deformation will occur shift. The springs, deformed by such a displacement, will tend to return the displaced particles to the equilibrium position. In this case, elastic forces will act on the nearest undisplaced particles, tending to deflect them from the equilibrium position. As a result, a transverse wave will run along the chain.
In liquids and gases, elastic shear deformation does not occur. If one layer of liquid or gas is displaced a certain distance relative to the adjacent layer, then no tangential forces will appear at the boundary between the layers. The forces acting at the boundary of a liquid and a solid, as well as the forces between adjacent layers of liquid, are always directed normal to the boundary - these are pressure forces. The same applies to gaseous media. Hence, transverse waves cannot exist in liquid or gaseous media.
Of significant practical interest are simple harmonic or sine waves . They are characterized amplitudeA particle vibrations, frequencyf And wavelengthλ. Sinusoidal waves propagate in homogeneous media with some constant speed υ.
Bias y (x, t) particles of the medium from the equilibrium position in a sinusoidal wave depends on the coordinate x on the axis OX, along which the wave propagates, and on time t in law.
1. Wave - propagation of vibrations from point to point from particle to particle. For a wave to occur in a medium, deformation is necessary, since without it there will be no elastic force.
2. What is wave speed?
2. Wave speed - the speed of propagation of vibrations in space.
3. How are speed, wavelength and frequency of oscillations of particles in a wave related to each other?
3. The speed of the wave is equal to the product of the wavelength and the oscillation frequency of the particles in the wave.
4. How are speed, wavelength and period of oscillation of particles in a wave related to each other?
4. The speed of the wave is equal to the wavelength divided by the period of oscillation in the wave.
5. What wave is called longitudinal? Transverse?
5. Transverse wave - a wave propagating in a direction perpendicular to the direction of oscillation of particles in the wave; longitudinal wave- a wave propagating in a direction coinciding with the direction of oscillations of particles in the wave.
6. In what media can transverse waves arise and propagate? Longitudinal waves?
6. Transverse waves can arise and propagate only in solid media, since the occurrence of a transverse wave requires shear deformation, and this is possible only in solids. Longitudinal waves can arise and propagate in any medium (solid, liquid, gaseous), since compression or tension deformation is necessary for the occurrence of a longitudinal wave.
Disturbances propagating in space, moving away from the place of their origin, are called waves.
Elastic waves- these are disturbances that propagate in solid, liquid and gaseous media due to the action of elastic forces in them.
These environments themselves are called elastic. Perturbation of an elastic medium is any deviation of the particles of this medium from their equilibrium position.
Take, for example, a long rope (or rubber tube) and attach one of its ends to the wall. Having pulled the rope tightly, with a sharp lateral movement of the hand we will create a short-term disturbance at its loose end. We will see that this disturbance will run along the rope and, reaching the wall, will be reflected back.
The initial disturbance of the medium, leading to the appearance of a wave in it, is caused by the action of some foreign body which is called wave source. This could be the hand of a person hitting the rope, a pebble falling into the water, etc. If the action of the source is short-term, then a so-called single wave. If the source of the wave performs a long oscillatory motion, then the waves in the medium begin to move one after another. A similar picture can be seen by placing a vibrating plate with a tip lowered into the water over a bath of water.
A necessary condition the occurrence of an elastic wave is the appearance at the moment of the disturbance of elastic forces that interfere with this disturbance. These forces tend to bring neighboring particles of the medium closer together when they move apart, and move them away when they come closer. Acting on particles of the medium that are increasingly distant from the source, elastic forces begin to remove them from their equilibrium position. Gradually, all particles of the medium, one after another, are involved in oscillatory motion. The propagation of these vibrations manifests itself in the form of a wave.
In any elastic medium, two types of motion simultaneously exist: oscillations of particles of the medium and propagation of disturbance. A wave in which particles of the medium oscillate along the direction of its propagation is called longitudinal, and a wave in which particles of the medium oscillate across the direction of its propagation is called transverse.
Longitudinal wave.
A wave in which oscillations occur along the direction of propagation of the wave is called longitudinal.
In an elastic longitudinal wave, disturbances represent compression and rarefaction of the medium. Compressive deformation is accompanied by the appearance of elastic forces in any medium. Therefore, longitudinal waves can propagate in all media (liquid, solid, and gaseous).
An example of the propagation of a longitudinal elastic wave is shown in the figure A And b higher. The left end of a long spring suspended by threads is struck with the hand. The impact brings several turns closer together, and an elastic force arises, under the influence of which these turns begin to diverge. Continuing to move by inertia, they will continue to diverge, bypassing the equilibrium position and forming a vacuum in this place (Figure b). With rhythmic action, the coils at the end of the spring will either approach or move away from each other, i.e., oscillate around their equilibrium position. These vibrations will gradually be transmitted from coil to coil along the entire spring. Condensations and rarefaction of turns, or an elastic wave, will spread along the spring.
Transverse wave.
Waves in which oscillations occur perpendicular to the direction of their propagation are called transverse. In a transverse elastic wave, disturbances represent displacements (shifts) of some layers of the medium relative to others.
Shear deformation leads to the appearance of elastic forces only in solids: the shift of layers in gases and liquids is not accompanied by the appearance of elastic forces. Therefore, transverse waves can only propagate in solids.
Plane wave.
Plane wave is a wave whose direction of propagation is the same at all points in space.
The amplitude of particle oscillations in a spherical wave necessarily decreases with distance from the source. The energy emitted by the source is evenly distributed over the surface of the sphere, the radius of which continuously increases as the wave propagates. The spherical wave equation is:
.
Unlike a plane wave, where s m = A- the amplitude of the wave is a constant value; in a spherical wave it decreases with distance from the center of the wave.
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 propagation of the wave (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 set in a 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, you can notice how successive condensations and rarefactions of its turns will spread throughout the spring, running one after another. In Figure 15.6, the dots show the position of the spring coils at rest, and then the positions of the spring coils at successive intervals equal to a quarter of the period.
Thus, the longitudinal wave in the case under consideration represents alternating condensations (Сг) and rarefaction (Once) coils of spring.
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 caused by the deformation of the medium. It can be shown that the volumetric energy density for a plane traveling harmonic wave is S = Acos(ω(t-) + φ 0) where r = dm/dV is the density of the medium, i.e. periodically changes from 0 to rA2w2 during the time p/w = T/2. Average energy density over a period of time p/w = T/2
To characterize energy transfer, the concept of energy flux density vector is introduced - the Umov vector. Let's derive an expression for it. If energy DW is transferred through the area DS^, perpendicular to the direction of wave propagation, during time Dt, then the energy flux density Fig. 2 where DV = DS^ uDt is the volume of an elementary cylinder isolated in the medium. Since the energy transfer rate or group velocity is a vector, the energy flux density can be represented as a vector, W/m2 (18)
This vector was introduced by Moscow University professor 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 flow transferred through.
When they talk about intensity, they mean physical meaning vector—energy flow. The intensity of the wave is proportional to the square of the amplitude.
The Poynting vector S can be defined through the cross product of two vectors:
(in the GHS system),
(in SI system),
Where E And H are the electric and magnetic field strength vectors, respectively.
(in complex form)
Where E And H are the vectors of the complex amplitude of the electric and magnetic fields, respectively.
This vector is modulo equal 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 components tangential to the interface between two media E And H continuous (see boundary conditions), then the vector S continuous at the boundary of two media.
Standing wave - oscillations in distributed oscillatory systems with a characteristic arrangement of alternating maxima (antinodes) and minima (nodes) of amplitude. In practice, such a wave occurs during reflections from obstacles and inhomogeneities as a result of the superposition of the reflected wave on the incident one. At the same time, it is extremely important has the frequency, phase and attenuation coefficient of the wave at the place of reflection.
Examples of a standing wave are string vibrations, air vibrations in an organ pipe; in nature - Schumann waves.
A purely standing wave, strictly speaking, can exist only in the absence of losses in the medium and complete reflection of the waves from the boundary. Usually, except standing waves, there are also traveling waves in the medium that supply energy to the places of its absorption or emission.
A Rubens tube is used to demonstrate standing waves in gas.
