Message on the topic of mechanical waves. Lesson summary "mechanical waves and their main characteristics"

1. Mechanical waves, wave frequency. Longitudinal and transverse waves.

2. Wave front. Velocity and wavelength.

3. Equation of a plane wave.

4. Energy characteristics of the wave.

5. Some special types of waves.

6. Doppler effect and its use in medicine.

7. Anisotropy during the propagation of surface waves. Effect of shock waves on biological tissues.

8. Basic concepts and formulas.

9. Tasks.

2.1. Mechanical waves, wave frequency. Longitudinal and transverse waves

If in any place of an elastic medium (solid, liquid or gaseous) oscillations of its particles are excited, then due to the interaction between particles, this oscillation will begin to propagate in the medium from particle to particle with a certain speed v.

For example, if an oscillating body is placed in a liquid or gaseous medium, then the oscillatory motion of the body will be transmitted to the particles of the medium adjacent to it. They, in turn, involve neighboring particles in oscillatory motion, and so on. In this case, all points of the medium oscillate with the same frequency, equal to the frequency of the vibration of the body. This frequency is called wave frequency.

wave is the process of propagation of mechanical vibrations in an elastic medium.

wave frequency called the frequency of oscillations of the points of the medium in which the wave propagates.

The wave is associated with the transfer of vibration energy from the source of vibrations to the peripheral parts of the medium. At the same time, in the environment there are

periodic deformations that are carried by a wave from one point of the medium to another. The particles of the medium themselves do not move along with the wave, but oscillate around their equilibrium positions. Therefore, the propagation of the wave is not accompanied by the transfer of matter.

according to frequency mechanical waves are divided into different ranges, which are indicated in Table. 2.1.

Table 2.1. Scale of mechanical waves

Depending on the direction of particle oscillations in relation to the direction of wave propagation, longitudinal and transverse waves are distinguished.

Longitudinal waves- waves, during the propagation of which the particles of the medium oscillate along the same straight line along which the wave propagates. In this case, the areas of compression and rarefaction alternate in the medium.

Longitudinal mechanical waves can occur in all media (solid, liquid and gaseous).

transverse waves- waves, during the propagation of which particles oscillate perpendicular to the direction of propagation of the wave. In this case, periodic shear deformations occur in the medium.

In liquids and gases, elastic forces arise only during compression and do not arise during shear, so transverse waves do not form in these media. The exception is waves on the surface of a liquid.

2.2. wave front. Velocity and wavelength

In nature, there are no processes that propagate at an infinitely high speed, therefore, a disturbance created by an external influence at one point in the environment will reach another point not instantly, but after some time. In this case, the medium is divided into two regions: the region, the points of which are already involved in the oscillatory motion, and the region, the points of which are still in equilibrium. The surface separating these regions is called wave front.

Wave front - locus of points up to which present moment an oscillation (perturbation of the environment) has come.

When a wave propagates, its front moves at a certain speed, which is called the speed of the wave.

Wave speed (v) is the speed of movement of its front.

The speed of a wave depends on the properties of the medium and the type of wave: transverse and longitudinal waves in a solid propagate at different speeds.

The propagation velocity of all types of waves is determined under the condition of weak wave attenuation by the following expression:

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

The speed of a wave in a medium should not be confused with the speed of the particles of the medium involved in the wave process. For example, when a sound wave propagates in air average speed vibrations of its molecules of the order of 10 cm / s, and the speed sound wave under normal conditions about 330 m/s.

The wavefront shape determines the geometric type of the wave. The simplest types of waves on this basis are flat and spherical.

flat A wave is called a wave whose front is a plane perpendicular to the direction of propagation.

Plane waves arise, for example, in a closed piston cylinder with gas when the piston oscillates.

The amplitude of the plane wave remains practically unchanged. Its slight decrease with distance from the wave source is associated with the viscosity of the liquid or gaseous medium.

spherical called a wave whose front has the shape of a sphere.

Such, for example, is a wave caused in a liquid or gaseous medium by a pulsating spherical source.

The amplitude of a spherical wave decreases with distance from the source inversely proportional to the square of the distance.

To describe a number of wave phenomena, such as interference and diffraction, use a special characteristic called the wavelength.

Wavelength called the distance over which its front moves in a time equal to the period of oscillation of the particles of the medium:

Here v- wave speed, T - oscillation period, ν - frequency of oscillations of medium points, ω - cyclic frequency.

Since the speed of wave propagation depends on the properties of the medium, the wavelength λ when moving from one medium to another, it changes, while the frequency ν stays the same.

This definition of wavelength has an important geometric interpretation. Consider Fig. 2.1a, which shows the displacements of the points of the medium at some point in time. The position of the wave front is marked by points A and B.

After a time T equal to one period of oscillation, the wave front will move. Its positions are shown in Fig. 2.1, b points A 1 and B 1. It can be seen from the figure that the wavelength λ is equal to the distance between adjacent points oscillating in the same phase, for example, the distance between two adjacent maxima or minima of the perturbation.

Rice. 2.1. Geometric interpretation of the wavelength

2.3. Plane wave equation

The wave arises as a result of periodic external influences on the medium. Consider the distribution flat wave created by harmonic oscillations of the source:

where x and - displacement of the source, A - amplitude of oscillations, ω - circular frequency of oscillations.

If some point of the medium is removed from the source at a distance s, and the wave speed is equal to v, then the perturbation created by the source will reach this point in time τ = s/v. Therefore, the phase of the oscillations at the considered point at the time t will be the same as the phase of the source oscillations at the time (t - s/v), and the amplitude of the oscillations will remain practically unchanged. As a result, the fluctuations of this point will be determined by the equation

Here we have used the formulas for the circular frequency (ω = 2π/T) and wavelength (λ = v T).

Substituting this expression into the original formula, we get

Equation (2.2), which determines the displacement of any point of the medium at any time, is called plane wave equation. The argument at cosine is the magnitude φ = ωt - 2 π s /λ - called wave phase.

2.4. Energy characteristics of the wave

The medium in which the wave propagates has mechanical energy, which is made up of the energies of the oscillatory motion of all its particles. The energy of one particle with mass m 0 is found by formula (1.21): E 0 = m 0 Α 2 w 2/2. The volume unit of the medium contains n = p/m 0 particles (ρ is the density of the medium). Therefore, a unit volume of the medium has the energy w р = nЕ 0 = ρ Α 2 w 2 /2.

Bulk energy density(\¥ p) - the energy of the oscillatory motion of the particles of the medium contained in a unit of its volume:

where ρ is the density of the medium, A is the amplitude of particle oscillations, ω is the frequency of the wave.

As the wave propagates, the energy imparted by the source is transferred to distant regions.

For a quantitative description of the energy transfer, the following quantities are introduced.

Energy flow(Ф) - a value equal to the energy carried by the wave through a given surface per unit time:

Wave intensity or energy flux density (I) - a value equal to the energy flux carried by a wave through a single area perpendicular to the direction of wave propagation:

It can be shown that the wave intensity is equal to the product of its propagation velocity and the volume energy density

2.5. Some special varieties

waves

1. shock waves. When sound waves propagate, the particle oscillation velocity does not exceed a few cm/s, i.e. it is hundreds of times less than the wave speed. Under strong disturbances (explosion, movement of bodies at supersonic speed, powerful electric discharge), the speed of oscillating particles of the medium can become comparable to the speed of sound. This creates an effect called a shock wave.

During an explosion, products with high density, heated to high temperatures, expand and compress thin layer ambient air.

shock wave - a thin transition region propagating at supersonic speed, in which there is an abrupt increase in pressure, density, and velocity of matter.

The shock wave can have significant energy. So, in a nuclear explosion, the formation of a shock wave in environment about 50% of the total energy of the explosion is expended. The shock wave, reaching objects, is capable of causing destruction.

2. surface waves. Along with body waves in continuous media in the presence of extended boundaries, there can be waves localized near the boundaries, which play the role of waveguides. Such, in particular, are surface waves in a liquid and an elastic medium, discovered by the English physicist W. Strett (Lord Rayleigh) in the 90s of the 19th century. In the ideal case, Rayleigh waves propagate along the boundary of the half-space, decaying exponentially in the transverse direction. As a result, surface waves localize the energy of perturbations created on the surface in a relatively narrow near-surface layer.

surface waves - waves that propagate along the free surface of a body or along the boundary of the body with other media and decay rapidly with distance from the boundary.

Waves in earth's crust(seismic waves). The penetration depth of surface waves is several wavelengths. At a depth equal to the wavelength λ, the volumetric energy density of the wave is approximately 0.05 of its volumetric density at the surface. The displacement amplitude rapidly decreases with distance from the surface and practically disappears at a depth of several wavelengths.

3. Waves of excitation in active environments.

An actively excitable, or active, environment is a continuous environment consisting of a large number of elements, each of which has an energy reserve.

Moreover, each element can be in one of three states: 1 - excitation, 2 - refractoriness (non-excitability for a certain time after excitation), 3 - rest. Elements can go into excitation only from a state of rest. Excitation waves in active media are called autowaves. Autowaves - these are self-sustaining waves in an active medium, keeping their characteristics constant due to energy sources distributed in the medium.

The characteristics of an autowave - period, wavelength, propagation velocity, amplitude and shape - in the steady state depend only on the local properties of the medium and do not depend on the initial conditions. In table. 2.2 shows the similarities and differences between autowaves and ordinary mechanical waves.

Autowaves can be compared with the spread of fire in the steppe. The flame spreads over an area with distributed energy reserves (dry grass). Each subsequent element (dry blade of grass) is ignited from the previous one. And thus the front of the excitation wave (flame) propagates through the active medium (dry grass). When two fires meet, the flame disappears, as the energy reserves are exhausted - all the grass is burned out.

The description of the processes of propagation of autowaves in active media is used in the study of the propagation of action potentials along nerve and muscle fibers.

Table 2.2. Comparison of autowaves and ordinary mechanical waves

2.6. Doppler effect and its use in medicine

Christian Doppler (1803-1853) - Austrian physicist, mathematician, astronomer, director of the world's first physical institute.

Doppler effect consists in changing the frequency of oscillations perceived by the observer, due to the relative motion of the source of oscillations and the observer.

The effect is observed in acoustics and optics.

We obtain a formula describing the Doppler effect for the case when the source and receiver of the wave move relative to the medium along one straight line with velocities v I and v P, respectively. Source commits harmonic vibrations with frequency ν 0 relative to its equilibrium position. The wave created by these oscillations propagates in the medium at a speed v. Let us find out what frequency of oscillations will fix in this case receiver.

Disturbances created by source oscillations propagate in the medium and reach the receiver. Consider one complete oscillation of the source, which begins at time t 1 = 0

and ends at the moment t 2 = T 0 (T 0 is the source oscillation period). The disturbances of the medium created at these moments of time reach the receiver at the moments t" 1 and t" 2, respectively. In this case, the receiver captures oscillations with a period and frequency:

Let's find the moments t" 1 and t" 2 for the case when the source and receiver are moving towards to each other, and the initial distance between them is equal to S. At the moment t 2 \u003d T 0, this distance will become equal to S - (v I + v P) T 0, (Fig. 2.2).

Rice. 2.2. Mutual position of the source and receiver at the moments t 1 and t 2

This formula is valid for the case when the speeds v and and v p are directed towards each other. In general, when moving

source and receiver along one straight line, the formula for the Doppler effect takes the form

For the source, the speed v And is taken with the “+” sign if it moves in the direction of the receiver, and with the “-” sign otherwise. For the receiver - similarly (Fig. 2.3).

Rice. 2.3. Choice of signs for the velocities of the source and receiver of waves

Consider one special case use of the Doppler effect in medicine. Let the ultrasound generator be combined with the receiver in the form of some technical system that is stationary relative to the medium. The generator emits ultrasound having a frequency ν 0 , which propagates in the medium with a speed v. Towards system with a speed v t moves some body. First, the system performs the role source (v AND= 0), and the body is the role of the receiver (vTl= v T). Then the wave is reflected from the object and fixed by a fixed receiving device. In this case, v AND = v T, and v p \u003d 0.

Applying formula (2.7) twice, we obtain the formula for the frequency fixed by the system after reflection of the emitted signal:

At approach object to the sensor frequency of the reflected signal increases and at removal - decreases.

By measuring the Doppler frequency shift, from formula (2.8) we can find the speed of the reflecting body:

The sign "+" corresponds to the movement of the body towards the emitter.

The Doppler effect is used to determine the speed of blood flow, the speed of movement of the valves and walls of the heart (Doppler echocardiography) and other organs. A diagram of the corresponding setup for measuring blood velocity is shown in Fig. 2.4.

Rice. 2.4. Scheme of an installation for measuring blood velocity: 1 - ultrasound source, 2 - ultrasound receiver

The device consists of two piezocrystals, one of which is used to generate ultrasonic vibrations (inverse piezoelectric effect), and the second - to receive ultrasound (direct piezoelectric effect) scattered by blood.

Example. Determine the speed of blood flow in the artery, if the counter reflection of ultrasound (ν 0 = 100 kHz = 100,000 Hz, v \u003d 1500 m / s) a Doppler frequency shift occurs from erythrocytes ν D = 40 Hz.

Solution. By formula (2.9) we find:

v 0 = v D v /2v0 = 40x 1500/(2x 100,000) = 0.3 m/s.

2.7. Anisotropy during the propagation of surface waves. Effect of shock waves on biological tissues

1. Anisotropy of surface wave propagation. When studying the mechanical properties of the skin using surface waves at a frequency of 5-6 kHz (not to be confused with ultrasound), acoustic anisotropy of the skin is manifested. This is expressed in the fact that the propagation velocities of the surface wave in mutually perpendicular directions - along the vertical (Y) and horizontal (X) axes of the body - differ.

To quantify the severity of acoustic anisotropy, the mechanical anisotropy coefficient is used, which is calculated by the formula:

where v y- speed along the vertical axis, v x- along the horizontal axis.

The anisotropy coefficient is taken as positive (K+) if v y> v x at v y < v x the coefficient is taken as negative (K -). The numerical values ​​of the velocity of surface waves in the skin and the degree of anisotropy are objective criteria for assessing various effects, including those on the skin.

2. Action of shock waves on biological tissues. In many cases of impact on biological tissues (organs), it is necessary to take into account the resulting shock waves.

So, for example, a shock wave occurs when a blunt object hits the head. Therefore, when designing protective helmets, care is taken to dampen the shock wave and protect the back of the head in a frontal impact. This purpose is served by the internal tape in the helmet, which at first glance seems to be necessary only for ventilation.

Shock waves arise in tissues when exposed to high-intensity laser radiation. Often after that, cicatricial (or other) changes begin to develop in the skin. This is the case, for example, in cosmetic procedures. Therefore, in order to reduce harmful effect shock waves, it is necessary to pre-calculate the dosage of exposure, taking into account the physical properties of both radiation and the skin itself.

Rice. 2.5. Propagation of Radial Shock Waves

Shock waves are used in radial shock wave therapy. On fig. 2.5 shows the propagation of radial shock waves from the applicator.

Such waves are created in devices equipped with a special compressor. A radial shock wave is generated pneumatic method. The piston, located in the manipulator, moves at high speed under the influence of a controlled pulse of compressed air. When the piston hits the applicator installed in the manipulator, its kinetic energy is converted into mechanical energy of the area of ​​the body that was affected. At the same time, a contact gel is used to reduce losses during the transmission of waves in the air gap located between the applicator and the skin, and to ensure good conductivity of shock waves. Normal operating mode: frequency 6-10 Hz, operating pressure 250 kPa, number of pulses per session - up to 2000.

1. On the ship, a siren is turned on, giving signals in the fog, and after t = 6.6 s, an echo is heard. How far away is the reflective surface? speed of sound in air v= 330 m/s.

Solution

In time t, sound travels a path 2S: 2S = vt →S = vt/2 = 1090 m. Answer: S = 1090 m.

2. What minimum size objects whose position can be determined the bats with your sensor, which has a frequency of 100,000 Hz? What is the minimum size of objects that dolphins can detect using a frequency of 100,000 Hz?

Solution

The minimum dimensions of an object are equal to the wavelength:

λ1\u003d 330 m / s / 10 5 Hz \u003d 3.3 mm. This is roughly the size of the insects that bats feed on;

λ2\u003d 1500 m / s / 10 5 Hz \u003d 1.5 cm. A dolphin can detect a small fish.

Answer:λ1= 3.3 mm; λ2= 1.5 cm.

3. First, a person sees a flash of lightning, and after 8 seconds after that he hears a thunderclap. At what distance did the lightning flash from him?

Solution

S \u003d v star t \u003d 330 x 8 = 2640 m. Answer: 2640 m

4. Two sound waves have the same characteristics, except that one has twice the wavelength of the other. Which one carries the most energy? How many times?

Solution

The intensity of the wave is directly proportional to the square of the frequency (2.6) and inversely proportional to the square of the wavelength (ω = 2πv/λ ). Answer: one with a shorter wavelength; 4 times.

5. A sound wave having a frequency of 262 Hz propagates in air at a speed of 345 m/s. a) What is its wavelength? b) How long does it take for the phase at a given point in space to change by 90°? c) What is the phase difference (in degrees) between points 6.4 cm apart?

Solution

a) λ =v /ν = 345/262 = 1.32 m;

in) Δφ = 360°s/λ= 360 x 0.064/1.32 = 17.5°. Answer: a) λ = 1.32 m; b) t = T/4; in) Δφ = 17.5°.

6. Estimate the upper limit (frequency) of ultrasound in air, if the speed of its propagation is known v= 330 m/s. Assume that air molecules have a size of the order of d = 10 -10 m.

Solution

In air, a mechanical wave is longitudinal and the wavelength corresponds to the distance between two nearest concentrations (or discharges) of molecules. Since the distance between the clusters cannot be smaller sizes molecules, then d = λ. From these considerations, we have ν =v /λ = 3,3x 10 12 Hz. Answer:ν = 3,3x 10 12 Hz.

7. Two cars are moving towards each other with speeds v 1 = 20 m/s and v 2 = 10 m/s. The first machine gives a signal with a frequency ν 0 = 800 Hz. Sound speed v= 340 m/s. What frequency will the driver of the second car hear: a) before the cars meet; b) after the meeting of the cars?

8. As the train passes by, you hear how the frequency of its whistle changes from ν 1 = 1000 Hz (when approaching) to ν 2 = 800 Hz (when the train is moving away). What is the speed of the train?

Solution

This problem differs from the previous ones in that we do not know the speed of the sound source - the train - and the frequency of its signal ν 0 is unknown. Therefore, a system of equations with two unknowns is obtained:

Solution

Let v is the speed of the wind, and it blows from the person (receiver) to the source of the sound. Relative to the ground, they are motionless, and relative to the air, both move to the right with a speed u.

By formula (2.7) we obtain the sound frequency. perceived by man. She is unchanged:

Answer: frequency will not change.

Wave– the process of propagation of oscillations in an elastic medium.

mechanical wave– mechanical disturbances propagating in space and carrying energy.

Wave types:

longitudinal - particles of the medium oscillate in the direction of wave propagation - in all elastic media;

x

oscillation direction

points of the environment

transverse - the particles of the medium oscillate perpendicular to the direction of wave propagation - on the surface of the liquid.

X

Types of mechanical waves:

elastic waves - propagation of elastic deformations;

waves on the surface of a liquid.

Wave characteristics:

Let A oscillate according to the law:
.

Then B oscillates with a delay by an angle
, where
, i.e.

Wave energy.

is the total energy of one particle. If particlesN, then where - epsilon, V - volume.

Epsilon– energy per unit volume of the wave – volumetric energy density.

The wave energy flux is equal to the ratio of the energy transferred by waves through a certain surface to the time during which this transfer is carried out:
, watt; 1 watt = 1J/s.

Energy Flux Density - Wave Intensity- energy flow through a unit area - a value equal to the average energy transferred by a wave per unit time per unit area of ​​the cross section.

[W/m2]

.

Umov vector– vector I showing the direction of wave propagation and equal to the flow wave energy passing through a unit area perpendicular to this direction:

.

Physical characteristics of the wave:

Vibrational:

1. amplitude

Wave:

1. wavelength

   wave speed

   intensity

Complex oscillations (relaxation) - different from sinusoidal.

Fourier transform- any complex periodic function can be represented as the sum of several simple (harmonic) functions, the periods of which are multiples of the period of the complex function - this is harmonic analysis. Occurs in parsers. The result is the harmonic spectrum of a complex oscillation:

BUT

0

Sound - vibrations and waves that act on the human ear and cause an auditory sensation.

Sound vibrations and waves are a special case of mechanical vibrations and waves. Types of sounds:

tones- sound, which is a periodic process:

1. simple - harmonic - tuning fork

   complex - anharmonic - speech, music

A complex tone can be decomposed into simple ones. The lowest frequency of such decomposition is the fundamental tone, the remaining harmonics (overtones) have frequencies equal to 2 and others. A set of frequencies indicating their relative intensity is the acoustic spectrum.

Noise - sound with a complex non-repeating time dependence (rustle, creak, applause). The spectrum is continuous.

Physical characteristics of sound:

Hearing sensation characteristics:

Height is determined by the frequency of the sound wave. The higher the frequency, the higher the tone. The sound of greater intensity is lower.

Timbre– determined by the acoustic spectrum. The more tones, the richer the spectrum.

Volume- characterizes the level of auditory sensation. Depends on sound intensity and frequency. Psychophysical Weber-Fechner law: if you increase irritation in geometric progression(in the same number of times), then the feeling of this irritation will increase in arithmetic progression(by the same amount).

, where E is loudness (measured in phons);
- intensity level (measured in bels). 1 bel - change in intensity level, which corresponds to a change in sound intensity by 10 times. K - proportionality coefficient, depends on frequency and intensity.

The relationship between loudness and intensity of sound is equal loudness curves, built on experimental data (they create a sound with a frequency of 1 kHz, change the intensity until an auditory sensation arises similar to the sensation of the volume of the sound under study). Knowing the intensity and frequency, you can find the background.

Audiometry- a method for measuring hearing acuity. The instrument is an audiometer. The resulting curve is an audiogram. The threshold of hearing sensation at different frequencies is determined and compared.

Noise meter - noise level measurement.

In the clinic: auscultation - stethoscope / phonendoscope. A phonendoscope is a hollow capsule with a membrane and rubber tubes.

Phonocardiography - graphic registration of backgrounds and heart murmurs.

Percussion.

Ultrasound– mechanical vibrations and waves with a frequency above 20 kHz up to 20 MHz. Ultrasound emitters are electromechanical emitters based on the piezoelectric effect ( alternating current to the electrodes, between which - quartz).

The wavelength of ultrasound is less than the wavelength of sound: 1.4 m - sound in water (1 kHz), 1.4 mm - ultrasound in water (1 MHz). Ultrasound is well reflected at the border of the bone-periosteum-muscle. Ultrasound will not penetrate into the human body if it is not lubricated with oil (air layer). The speed of propagation of ultrasound depends on the environment. Physical processes: microvibrations, destruction of biomacromolecules, restructuring and damage of biological membranes, thermal effect, destruction of cells and microorganisms, cavitation. In the clinic: diagnostics (encephalograph, cardiograph, ultrasound), physiotherapy (800 kHz), ultrasonic scalpel, pharmaceutical industry, osteosynthesis, sterilization.

infrasound– waves with a frequency less than 20 Hz. Adverse action - resonance in the body.

vibrations. Beneficial and harmful action. Massage. vibration disease.

Doppler effect– change in the frequency of the waves perceived by the observer (wave receiver) due to the relative motion of the wave source and the observer.

Case 1: N approaches I.

Case 2: And approaches N.

Case 3: approach and distance of I and H from each other:

System: ultrasonic generator - receiver - is motionless relative to the medium. The object is moving. It receives ultrasound with a frequency
, reflects it, sending it to the receiver, which receives an ultrasonic wave with a frequency
. Frequency difference - doppler frequency shift:
. It is used to determine the speed of blood flow, the speed of movement of the valves.

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

Definition 1

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

There are the following types of mechanical waves:

Definition 2

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

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

Definition 3

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

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

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

Remark 1

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

Figure 2. 6. one . Spreading shear wave along the rubber band in tension.

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

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

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

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

Remark 2

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

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

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

Remark 3

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

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

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

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

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

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

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

Definition 4

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

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

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

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

From this expression it follows:

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

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

Definition 5

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

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

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

Remark 4

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

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

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

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

∆ p = - B ∆ V V .

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

Example 1

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

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

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

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

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

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

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

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

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

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

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

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

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

Definition 6

Knots are points of immobility in a standing wave.

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

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

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

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

f n = υ λ n = n υ 2 l = n f 1 .

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

Definition 7

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

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

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

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

According to the superposition principle, 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 . Model of normal modes of a string.

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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 type 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.

Please note that the speed of the wave and the speed of oscillation of the 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. Wavelength λ - 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. 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 the transfer of energy to unexcited parts of the medium. To characterize the energy transfer process, we introduce wave intensity I.