For the existence of a wave, a source of oscillation and a material environment or field in which this wave propagates is necessary. Waves are of the most diverse nature, but they obey similar patterns.
By physical nature distinguish between:
By orientation of disturbances distinguish between:
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Longitudinal waves -
Particles are displaced along the direction of propagation; the presence of an elastic force in compression is necessary; can be distributed in any environment. Examples: sound waves  |
Shear waves -
Particles are displaced across the direction of propagation; can spread only in elastic media; the presence of an elastic force in shear is necessary; can only propagate in solid media (and at the interface between two media). Examples: elastic waves in a string, waves on water
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By the nature of dependence on time distinguish between:
Elastic waves - mechanical compensations (deformations) propagating in elastic medium... An elastic wave is called harmonic(sinusoidal) if the corresponding vibrations of the medium are harmonic.
Traveling waves - waves that carry energy in space.
By the shape of the wave surface : plane, spherical, cylindrical wave.
Wave front- the locus of the points to which the oscillations have reached a given moment 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 oscillation period
Amplitude of wave A - amplitude of particle oscillations in the wave
Wave speed v - the speed of propagation of disturbances in the environment
Wave period T - oscillation period
Wave frequency ν is the reciprocal of the period
Traveling wave equation
In the process of propagation of a traveling wave, 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 vibrate in the same place in space).
where v - speed , φ 0 - initial phase , ω – cyclic frequency , A- amplitude
Properties mechanical waves
1. Reflection of waves – 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 any obstacle on its way, then it can drastically change the nature of its behavior. For example, at the interface between two media with different mechanical properties, the wave is partially reflected and partially penetrates into the second medium.
2. Refraction of waves – during the propagation of mechanical waves, the phenomenon of refraction can also be observed: a change in the direction of propagation of mechanical waves when passing from one medium to another.
3. Wave diffraction – deviation of waves from rectilinear propagation, that is, they go 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 vibration amplitude
Interference and diffraction of mechanical waves.
A wave traveling along a rubber band or string is reflected from a fixed end; in this case, a wave appears traveling in the opposite direction.
When the waves are superimposed, the phenomenon of interference can be observed. The phenomenon of interference occurs when coherent waves are superimposed.
Coherent are calledwaveshaving the same frequency, constant phase difference, and oscillations occur in the same plane.
Interference is called a 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.
If the waves from sources A and B arrive at point C in the same phases, then the oscillations will intensify; if - in opposite phases, then a weakening of oscillations is observed. 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 in the path 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 = 2x 0.
Minimum condition
If the difference in the course 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:
Amplitude of the resulting fluctuation 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 wave bending around obstacles is calleddiffraction.
The relationship between the wavelength (λ) and the size of the obstacle (L) determines the behavior of the wave. Diffraction is most pronounced if the incident wavelength more sizes obstacles. 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 general property of waves of any nature, which always occurs, but the conditions for its observation are different.
A wave on the surface of the water propagates towards a sufficiently large obstacle, behind which a shadow is formed, i.e. no wave process is observed. This property is used when constructing breakwaters in ports. If the size of the obstacle is comparable to the wavelength, then there will be excitement behind the obstacle. Behind him, the wave propagates as if there were no obstacles at all, i.e. diffraction of the wave is observed.
Examples of the manifestation of diffraction ... Audibility of 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 when the direct and reflected waves are added, if they have the same frequency and amplitude.
Complex vibrations occur in a string fixed at both ends, which can be considered as a result of superposition ( superposition) two waves propagating in opposite directions and experiencing reflections and re-reflections at the ends. The vibrations of the strings attached 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, upon excitation of transverse vibrations, standing waves , and knots should be located in the places where the string is fastened. Therefore, the string is excited with noticeable intensity only those vibrations, half the wavelength of which fits the length of the string an integer number of times.
This implies the condition 
Wavelengths correspond to frequencies 
n = 1, 2, 3 ...Frequencies vn are called natural frequencies strings.
Harmonic vibrations with frequencies vn are called natural or normal vibrations ... They are also called harmonics. In general, the vibration of a string is a superposition of various harmonics.
Standing wave equation :
At the points where the coordinates satisfy the condition 
(n= 1, 2, 3, ...), the total amplitude is equal to the maximum value - this is antinodes
standing wave. Coordinates of antinodes
: 
At points whose coordinates satisfy the condition
(n= 0, 1, 2, ...), the total oscillation amplitude is zero - This knots standing wave. Node coordinates: 
The formation of standing waves is observed with the interference of the traveling and reflected waves. 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 the node is more dense (b).
Considering traveling wave , then in the direction of its propagation energy is transferred oscillatory motion. When the same there is no standing wave of energy transfer since incident and reflected waves of the same amplitude carry the same energy in opposite directions.
Standing waves arise, for example, in a stretched string fixed at both ends when transverse vibrations are excited in it. Moreover, in the places of anchoring, nodes of a standing wave are located.
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 node is formed at the opposite end.
Experience shows that vibrations excited at any point in an elastic medium are transmitted over time to its other parts. So from a stone thrown into the calm water of the lake, waves diverge in circles, which eventually reach the shore. The heartbeat inside the chest can be felt on the wrist, which is used to measure the pulse. The listed examples are related to the propagation of mechanical waves.
- Mechanical wave called the process of propagation of vibrations in an elastic medium, which is accompanied by the transfer of energy from one point of the medium to another. Note that mechanical waves cannot propagate in a vacuum.
The source of a mechanical wave is an oscillating body. If the source oscillates sinusoidally, then the wave in the elastic medium will also have a sinusoidal shape. Oscillations caused anywhere in the elastic medium propagate in the medium at a certain speed, depending on the density and elastic properties of the medium.
We emphasize that when the wave propagates no substance transfer that is, the particles only vibrate near equilibrium positions. The average displacement of particles relative to the equilibrium position over a long period of time is equal to zero.
The main characteristics of the wave
Let's consider the main characteristics of the wave.
- "Wave front "- this is an imaginary surface, to which the wave disturbance has reached at a given moment in time.
- A line drawn perpendicular to the wavefront in the direction of wave propagation is called ray.
The beam indicates the direction of propagation of the wave.
Depending on the shape of the wave front, there are plane waves, spherical waves, etc.
IN flat wave wave surfaces are planes perpendicular to the direction of wave propagation. Plane waves can be obtained on the surface of the water in a flat bath by vibrating a flat bar (Fig. 1).
IN spherical wave wave surfaces are concentric spheres. A spherical wave can be created by a ball pulsating in a homogeneous elastic medium. Such a wave propagates at the same speed in all directions. The rays are the radii of the spheres (Fig. 2).
The main characteristics of the wave:
- amplitude (A) is the modulus of maximum displacement of points of the medium from equilibrium positions during oscillations;
- period (T) - full oscillation time (the oscillation period of the points of the medium is equal to the oscillation period of the wave source)
\ (T = \ dfrac (t) (N), \)
Where t- the period of time during which the N fluctuations;
- frequency(ν) is the number of complete oscillations performed at a given point per unit of time
\ ((\ rm \ nu) = \ dfrac (N) (t). \)
The frequency of the wave is determined by the oscillation frequency of the source;
- speed(υ) is the speed of movement of the wave crest (this is not the speed of particles!)
- wavelength(λ) is the smallest distance between two points, where oscillations occur in the same phase, i.e., this is the distance the wave propagates over a period of time equal to the oscillation period of the source
\ (\ lambda = \ upsilon \ cdot T. \)
To characterize the energy carried by waves, the concept is used wave intensity (I) defined as energy ( W) carried by the wave per unit time ( t= 1 c) through a surface with an area S= 1 m 2, located perpendicular to the direction of wave propagation:
\ (I = \ dfrac (W) (S \ cdot t). \)
In other words, intensity is the power carried by waves across a unit area surface perpendicular to the direction of wave propagation. The SI unit of intensity is watts per meter squared (1 W / m2).
Traveling wave equation
Consider oscillations of a wave source occurring with a cyclic frequency ω \ (\ left (\ omega = 2 \ pi \ cdot \ nu = \ dfrac (2 \ pi) (T) \ right) \) and an amplitude A:
\ (x (t) = A \ cdot \ sin \; (\ omega \ cdot t), \)
where x(t) is the displacement of the source from the equilibrium position.
At a certain point in the medium, the oscillations will not come instantly, but after a time interval determined by the wave speed and the distance from the source to the observation point. If the wave velocity in a given medium is equal to υ, then the time dependence t coordinates (offset) x oscillating point at a distance r from the source is described by the equation
\ (x (t, r) = A \ cdot \ sin \; \ omega \ cdot \ left (t- \ dfrac (r) (\ upsilon) \ right) = A \ cdot \ sin \; \ left (\ omega \ cdot tk \ cdot r \ right), \; \; \; (1) \)
where k-wave number \ (\ left (k = \ dfrac (\ omega) (\ upsilon) = \ dfrac (2 \ pi) (\ lambda) \ right), \; \; \; \ varphi = \ omega \ cdot tk \ cdot r \) - wave phase.
Expression (1) is called traveling wave equation.
A traveling wave can be observed in the following experiment: if one end of a rubber cord lying on a smooth horizontal table is fixed and, by slightly pulling the cord by hand, bring its other end into oscillatory motion in a direction perpendicular to the cord, then a wave will run along it.
Longitudinal and transverse waves
Distinguish between longitudinal and transverse waves.
- The wave is called transverse, if particles of the medium vibrate in a plane perpendicular to the direction of wave propagation.
Let us consider in more detail the process of the formation of transverse waves. Let us take as a model of a real cord a chain of balls (material points) connected to each other by elastic forces (Fig. 3, a). Figure 3 shows the process of shear wave propagation and shows the positions of the balls at successive intervals of time equal to a quarter of the period.
At the initial moment of time \ (\ left (t_1 = 0 \ right) \) all points are in equilibrium (Fig. 3, a). If you deflect the ball 1 from the equilibrium position perpendicular to the entire chain of balls, then 2 -th ball, elastically connected with 1 -th, will begin to move behind him. Due to the inertia of movement 2 -th ball will repeat the movements 1 th, but with a time lag. Ball 3 th, elastically associated with 2 -th, will start to move behind 2 th ball, but with an even greater delay.
After a quarter of a period \ (\ left (t_2 = \ dfrac (T) (4) \ right) \) oscillations propagate to 4 th ball, 1 -th ball will have time to deviate from its equilibrium position to a maximum distance equal to the vibration amplitude BUT(Fig. 3, b). After half a period \ (\ left (t_3 = \ dfrac (T) (2) \ right) \) 1 -th ball, moving down, will return to the equilibrium position, 4 -th will deviate from the equilibrium position by a distance equal to the amplitude of oscillations BUT(Fig. 3, c). The wave during this time reaches 7 th ball, etc.
After a period \ (\ left (t_5 = T \ right) \) 1 the th ball, having made a full oscillation, passes through the equilibrium position, and the oscillatory motion will propagate to 13 -th ball (Fig. 3, e). And further movement 1 -th ball begin to repeat, and more and more balls participate in the oscillatory motion (Fig. 3, e).
Examples of longitudinal waves are sound waves in air and liquid. Elastic waves in gases and liquids arise only when the medium is compressed or rarefied. Therefore, in such media, only longitudinal waves can propagate.
Waves can propagate not only in the medium, but also along the interface between the two media. Such waves are called surface waves... The well-known waves on the water surface are an example of this type of waves.
Literature
- Aksenovich L.A. Physics in high school: Theory. Tasks. Tests: Textbook. allowance for institutions providing the receipt of obs. environments, education / L. A. Aksenovich, N. N. Rakina, K. S. Farino; Ed. K. S. Farino. - Minsk: Adukatsya i vyhavanne, 2004. - P. 424-428.
- Zhilko, V.V. Physics: textbook. general education allowance for grade 11. shk. from rus. lang. training / V.V. Zhilko, L.G. Markovich. - Minsk: Nar. Asveta, 2009 .-- S. 25-29.
With waves of any origin, under certain conditions, four phenomena listed below can be observed, which we will consider using the example of sound waves in the air and waves on the surface of water.
Reflection of waves. Let's make an experiment with an audio frequency current generator to which a loudspeaker (speaker) is connected, as shown in Fig. "but". We will hear a whistling sound. At the other end of the table we will put a microphone connected to an oscilloscope. Since a sine wave with a small amplitude appears on the screen, it means that the microphone is picking up a weak sound.
Now we place a board on top of the table, as shown in Fig. "B". As the amplitude on the oscilloscope screen has increased, the sound reaching the microphone has become louder. This and many other experiments make it possible to assert that mechanical waves of any origin have the ability to be reflected from the interface between two media.
Refraction of waves. Let's turn to the figure, which shows waves running on a coastal aground (top view). The sandy shore is shown in gray-yellow, and the deep part of the sea is in blue. There is a sandy shallow between them - shallow water.
Waves traveling through deep water travel in the direction of the red arrow. In the place of running aground, the wave refracts, that is, changes the direction of propagation. Therefore, the blue arrow indicating the new direction of wave propagation is located differently.
This and many other observations show that mechanical waves of any origin can be refracted when the propagation conditions change, for example, at the interface between two media.
Diffraction of waves. Translated from Latin "diffractus" means "broken". In physics diffraction refers to the deviation of waves from rectilinear propagation in the same medium, leading to them bending around obstacles.
Take a look now at another pattern of waves on the surface of the sea (view from the coast). Waves running towards us from afar are obscured by a large rock on the left, but at the same time they partially bend around it. The smaller rock on the right is not at all an obstacle to the waves: they completely go around it, spreading in the same direction.
Experiments show that diffraction is most clearly manifested if the incident wavelength is greater than the dimensions of the obstacle. Behind him, the wave spreads as if there were no obstacles.
Wave interference. We examined the phenomena associated with the propagation of a single wave: reflection, refraction and diffraction. Consider now superimposed propagation of two or more waves - interference phenomenon(from Latin "inter" - mutually and "ferio" - I strike). Let us study this phenomenon by experience.
We connect two speakers connected in parallel to the audio frequency current generator. The sound receiver, as in the first experiment, will be a microphone connected to an oscilloscope.
Let's start moving the microphone to the right. The oscilloscope will indicate that the sound is getting fainter and sometimes stronger, despite the fact that the microphone is farther away from the speakers. Bring the microphone back to the center line between the speakers, and then move it to the left, again moving away from the speakers. The oscilloscope will again show us the attenuation and the amplification of the sound.
This and many other experiments show that in space where several waves propagate, their interference can lead to the appearance of alternating regions with intensification and weakening of oscillations.
A mechanical or elastic wave is the process of propagation of vibrations in an elastic medium. For example, air begins to vibrate around a vibrating string or speaker cone - the string or speaker has become sources of a sound wave.
For a mechanical wave to appear, two conditions must be met - the presence of a wave source (it can be any oscillating body) and an elastic medium (gas, liquid, solid).
Let's 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.
Let's say the first ball vibrates with a frequency ω. Spring 1-2 is deformed, an elastic force arises in it, changing with frequency ω. Under the influence of an external periodically varying force, the second ball begins to perform forced vibrations. Since the forced vibrations always occur with the frequency of the external driving force, the vibration frequency of the second ball will coincide with the vibration frequency of the first. However, the forced vibrations of the second ball will occur with some phase delay relative to the external driving force. In other words, the second ball will start oscillating a little later than the first ball.
The vibrations of the second ball will cause the spring 2-3 to deform periodically, which will cause the third ball to vibrate, and so on. Thus, all balls in the chain will be alternately involved in vibrational motion with the vibration 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.
IN longitudinal wave particles oscillate along the direction of wave propagation.
The propagation of a longitudinal wave is associated with the occurrence of tension-compression deformation in the medium. In the stretched areas of the medium, a decrease in the density of the substance is observed - rarefaction. In the 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 thickening and rarefaction.
Tensile deformation - compression can occur in any elastic medium, so longitudinal waves can propagate in gases, liquids and solids. An example of a longitudinal wave is sound.
IN shear wave particles vibrate 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 type of deformation can only exist in solids, so shear waves can propagate exclusively in solids. An example of a shear wave is a seismic S-wave.
Surface waves arise 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. Particles of the medium describe, during their oscillations, elliptical trajectories in a plane perpendicular to the surface and passing through the direction of wave propagation. Examples of surface waves are water surface waves and seismic L - waves.
A wavefront is the locus of points to which the wave process has reached. The shape of the wavefront can be different. The most common are plane, spherical and cylindrical waves.
Please note - the wavefront is always located perpendicular direction of wave propagation! All points of the wavefront will start to oscillate in one phase.
To characterize the wave process, the following values are introduced:
1. Wave frequencyν is the vibration frequency of all particles in the wave.
2. Wave amplitude A is the vibration amplitude of the particles in the wave.
3. Wave speedυ is the distance covered by the wave process (disturbance) per unit time.
Pay attention - the speed of the wave and the speed of vibration of the particles in the wave are different concepts! The speed of the wave depends on two factors: the type of wave and the medium in which the wave propagates.
The general pattern is as follows: the velocity of a longitudinal wave in a solid is greater than in liquids, and the velocity in liquids, in turn, is greater than the velocity of a wave in gases.
It is not difficult to understand the physical reason for this pattern. The cause of wave propagation is the interaction of molecules. Naturally, the perturbation spreads faster in the environment 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 earthquake source is based on the difference in the velocities of longitudinal and transverse seismic waves.
The speed of a transverse wave in a tensioned cord or string is determined by the tensile force F and the mass of a unit of length μ:
4. Wavelengthλ is the minimum distance between points that oscillate in the same way.
For waves traveling along the surface of the water, the wavelength is easily defined as the distance between two adjacent humps or adjacent troughs.
For a longitudinal wave, the wavelength can be found as the distance between two adjacent condensations or expansions.
5. In the process of wave propagation, parts of the medium are involved in the oscillatory process. An 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 energy transfer to unexcited parts of the medium. To characterize the energy transfer process, we introduce wave intensity I.
When in some place of a solid, liquid or gaseous medium, the vibrations of particles are excited, the result of the interaction of atoms and molecules of the medium is the transfer of vibrations from one point to another with a finite speed.
Definition 1
Wave Is the process of propagation of vibrations in the environment.
There are the following types of mechanical waves:
Definition 2
Transverse wave: the particles of the medium are displaced in the direction perpendicular to the direction of propagation of the mechanical wave.
Example: waves propagating along a string or rubber band in tension (Figure 2. 6. 1);
Definition 3
Longitudinal wave: 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, waves on the surface of the liquid include both transverse and longitudinal components.
Remark 1
Let us point out an important clarification: when mechanical waves propagate, they transfer energy, shape, 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 vibrate around equilibrium positions. In this case, as we have already said, waves transfer energy, namely the energy of vibrations from one point of the medium to another.
Figure 2. 6. one . The propagation of a shear wave along a rubber band in tension.
Figure 2. 6. 2. Propagation of a longitudinal wave along an elastic bar.
A characteristic feature of mechanical waves is their propagation in material media, in contrast, for example, to light waves, which can propagate in emptiness. For the appearance of a mechanical wave impulse, a medium is required 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, every small element of a rigid 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 rigid body model.
In this model, inert and elastic properties are separated. Balls have mass m, and the springs are the 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 deformation of tension or compression. If such a deformation occurs in a liquid or gaseous medium, it is accompanied by compaction or vacuum.
Remark 2
A distinctive feature of longitudinal waves is that they are capable of propagating in any media: solid, liquid and gaseous.
If, in the specified model of a solid, one or more balls are displaced perpendicular to the entire chain, we can talk about the occurrence of 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 unbiased particles will begin to be influenced by elastic forces that tend to deflect these particles from the equilibrium position. The result will be the appearance of a shear wave in the direction along the chain.
In a liquid or gaseous medium, elastic shear deformation does not occur. The displacement of one layer of liquid or gas at a certain distance relative to the adjacent layer will not lead to the appearance of tangential forces at the boundary between the layers. The forces that act on the interface between the liquid and the solid, as well as the forces between adjacent layers of the liquid, are always directed along the normal to the boundary - these are pressure forces. The same can be said about a gaseous medium.
Remark 3
Thus, the appearance of transverse waves is impossible in liquid or gaseous media.
In terms of practical application, simple harmonic or sinusoidal waves are of particular interest. They are characterized by the particle vibration amplitude A, frequency f and wavelength λ. Sinusoidal waves propagate in homogeneous media with a certain 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 wavenumber, and ω = 2 π f is the circular frequency.
Figure 2. 6. 4 shows “snapshots” of the shear wave at time t and t + Δt. For a period of time Δ t, the wave moves along the O X axis 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 + Δ 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 during the period T. Thus, the formula for the wavelength has the form: λ = υ T, where υ is the wave propagation speed.
Over time t, the coordinate changes x of 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 for 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 the specified expression follows:
υ = ∆ x ∆ t = ω k or k = 2 π λ = ω υ.
It becomes obvious that a traveling sine 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 sine wave propagating in the direction opposite to the direction of the axis O X, with a speed υ = - ω k.
When a traveling wave propagates, all particles of the medium vibrate harmoniously with a certain frequency ω. This means that, as in a simple oscillatory process, the average potential energy, which is a 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 above, we can conclude that when a traveling wave propagates, an energy flow appears that is proportional to the speed of the wave and the square of its amplitude.
Traveling waves move in a medium at certain speeds, depending 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 the mass per unit length) and the tensile force T:
The speed with which longitudinal waves propagate in an unbounded medium is calculated using such quantities as the density of the medium ρ (or mass per unit volume) and the modulus of all-round compression B(equal to the coefficient of proportionality between the pressure change Δ p and the relative volume change Δ V V, taken with the opposite sign):
∆ p = - B ∆ V V.
Thus, the speed of propagation of longitudinal waves in an infinite medium is determined by the formula:
Example 1
At a temperature of 20 ° C, the speed of propagation of longitudinal waves in water is υ ≈ 1480 m / s, in various grades of steel υ ≈ 5 - 6 k m / s.
If we are talking about longitudinal waves propagating in elastic rods, the writing of the formula for the wave velocity contains not the modulus of all-round compression, but Young's modulus:
For steel the difference E from B insignificantly, but for other materials it can be 20 - 30% or more.
Figure 2. 6. five . Model of longitudinal and shear waves.
Suppose that a mechanical wave propagating in a certain medium meets 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 will partially be reflected and partially penetrate into the second medium. A wave traveling along a rubber band or string will bounce off the fixed end, and a counter-wave will appear. If the string has both ends fixed, complex vibrations will appear, which are the result of the superposition (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 = 0, and the other - at the point x 1 = L (Figure 2. 6. 6). There is tension in the string T.
Drawing 2 . 6 . 6 . The emergence of a standing wave in the string, fixed at both ends.
Two waves simultaneously run along the string in opposite directions with the same frequency:
- y 1 (x, t) = A cos (ω t + k x) - wave propagating from right to left;
- y 2 (x, t) = A cos (ω t - k x) is a wave propagating from left to right.
Point x = 0 is one of the fixed ends of the string: at this point, the incident wave y 1 as a result of reflection creates a wave y 2. Reflecting from the fixed end, the reflected wave enters into antiphase with the incident one. In accordance with the principle of superposition (which is an experimental fact), the oscillations created by the counterpropagating waves at all points of the string are summed up. It follows from what has been said that the total fluctuation at each point is determined as the sum of the fluctuations caused by the waves y 1 and y 2 separately. Thus:
y = y 1 (x, t) + y 2 (x, t) = (- 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
Nodes- points of immobility in a standing wave.
Thickness- points located between nodes and oscillating with maximum amplitude.
If you follow these definitions, for a standing wave to occur, both fixed ends of the string must be knots. 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 in a string does not always appear, but only when the length L strings are equal to an integer number of half-wavelengths:
l = n λ n 2 or λ n = 2 l n (n = 1, 2, 3,...).
A set of λ n wavelength values corresponds to a 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 associated type of vibration of the string is called the 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 vibration energy, "locked" in a string segment between two adjacent nodes, is not transferred to the rest of the string. In each such segment, a periodic (twice in a period T) transformation of kinetic energy into potential energy and vice versa, similar to a conventional 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 the string, fixed at both ends.
According to the principle of superposition, standing waves of various types (with different values n) are able to simultaneously be present in the vibrations of the string.
Figure 2. 6. eight . Normal string mode model.
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