Parameters of mechanical waves. Lesson summary "mechanical waves and their main characteristics"

DEFINITION

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/extension, 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 waves

DEFINITION

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. That's why transverse waves can only spread 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, in a circular pattern. 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- so called surface waves. They arise as a result of the action and force of surface tension.

Examples of problem solving

EXAMPLE 1

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

2. Wave front. Speed ​​and wavelength.

3. Plane wave equation.

4. Energy characteristics of the wave.

5. Some special types of waves.

6. The Doppler effect and its use in medicine.

7. Anisotropy during the propagation of surface waves. The 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) vibrations of its particles are excited, then, due to the interaction between particles, this vibration 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, 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 vibrate with the same frequency, equal to the frequency of vibration of the body. This frequency is called wave frequency.

Wave called the process of propagation mechanical vibrations in an elastic medium.

Wave frequency is the frequency of oscillations of the points of the medium in which the wave propagates.

The wave is associated with the transfer of oscillation energy from the source of oscillations to the peripheral parts of the medium. At the same time, in the environment there arise

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

According to frequency mechanical waves are divided into different ranges, which are indicated in the table. 2.1.

Table 2.1. Mechanical wave scale

Depending on the direction of particle oscillations relative 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, areas of compression and rarefaction alternate in the medium.

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

Transverse waves- waves, during the propagation of which the 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, therefore transverse waves are not formed in these media. The exception is waves on the surface of a liquid.

2.2. Wave front. Speed ​​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 medium will not reach another point instantly, but after some time. In this case, the medium is divided into two regions: a region whose points are already involved in oscillatory motion, and a region whose points are still in equilibrium. The surface separating these areas is called wave front.

Wave front - the geometric locus of the points to which the oscillation (perturbation of the medium) has reached at this moment.

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

The wave speed (v) is the speed at which its front moves.

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

The speed of propagation 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 movement of the particles of the medium involved in the wave process. For example, when a sound wave propagates in air, the average speed of vibration of its molecules is about 10 cm/s, and the speed of the sound wave at normal conditions about 330 m/s.

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

Flat is 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 virtually 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.

This, 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 in inverse proportion to the square of the distance.

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

Wavelength is 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 points in the medium, ω - cyclic frequency.

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

This definition of wavelength has an important geometric interpretation. Let's look at Fig. 2.1 a, which shows the displacements of points in 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 oscillation period, the wave front will move. Its positions are shown in Fig. 2.1, b points A 1 and B 1. From the figure it can be seen that the wavelength λ equal to the distance between adjacent points oscillating in the same phase, for example, the distance between two adjacent maxima or minima of a disturbance.

Rice. 2.1. Geometric interpretation of wavelength

2.3. Plane wave equation

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

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

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

Here we have used formulas for 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 in the medium at any time, is called plane wave equation. The argument for cosine is 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 the sum of the energies of the vibrational motion of all its particles. The energy of one particle with mass m 0 is found according to formula (1.21): E 0 = m 0 Α 2ω 2 /2. A unit volume of the medium contains n = p/m 0 particles (ρ - density of the medium). Therefore, a unit volume of the medium has energy w р = nЕ 0 = ρ Α 2ω 2 /2.

Volumetric energy density(\¥р) is the energy of vibrational motion of 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 a wave propagates, the energy imparted by the source is transferred to distant areas.

To quantitatively describe energy transfer, the following quantities are introduced.

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

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

It can be shown that the intensity of a wave is equal to the product of the speed of its propagation and the volumetric energy density

2.5. Some special varieties

waves

1. Shock waves. When sound waves propagate, the speed of particle vibration does not exceed several 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 electrical 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.

In the event of an explosion, products heated to high temperatures with high density, expand and compress a thin layer of surrounding air.

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

The shock wave can have significant energy. Yes, when nuclear explosion for the formation of a shock wave in environment about 50% of the total explosion energy is spent. The shock wave, reaching objects, can cause 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. These are, in particular, surface waves in liquids and elastic media, discovered by the English physicist W. Strutt (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 disturbances 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 a body with other media and quickly attenuate with distance from the boundary.

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

3. Excitation waves in active media.

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

In this case, 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 become excited only from a state of rest. Excitation waves in active media are called autowaves. Autowaves - These are self-sustaining waves in an active medium, maintaining their characteristics constant due to energy sources distributed in the medium.

The characteristics of an autowave - period, wavelength, propagation speed, amplitude and shape - in a 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 because the energy reserves are exhausted - all the grass has burned out.

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

Table 2.2. Comparison of autowaves and ordinary mechanical waves

2.6. The 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 of a change in the frequency of oscillations perceived by the observer due to the relative movement of the source of oscillations and the observer.

The effect is observed in acoustics and optics.

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

Disturbances created by source oscillations propagate through 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 period of oscillation of the source). The disturbances of the environment created at these moments of time reach the receiver at moments t" 1 and t" 2, respectively. In this case, the receiver records 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 each other, and the initial distance between them is equal to S. At the moment t 2 = T 0 this distance will become equal to S - (v И + v П)T 0 (Fig. 2.2).

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

This formula is valid for the case when the velocities 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 a “+” sign if it moves in the direction of the receiver, and with a “-” sign otherwise. For the receiver - similarly (Fig. 2.3).

Rice. 2.3. Selection of signs for the speeds of the source and receiver of waves

Let's consider one special case use of the Doppler effect in medicine. Let the ultrasound generator be combined with a receiver in the form of some technical system that is stationary relative to the medium. The generator emits ultrasound with a frequency ν 0, which propagates in the medium with a speed v. Towards a certain body is moving in a system with a speed vt. First the system performs the role source (v AND= 0), and the body is the role of the receiver (v Tl= v T). The wave is then reflected from the object and recorded by a stationary receiving device. In this case v И = v T, and v p = 0.

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

At approaching object to the sensor frequency of the reflected signal increases, and when removal - decreases.

By measuring the Doppler frequency shift, from formula (2.8) you can find the speed of movement 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 installation for measuring blood velocity is shown in Fig. 2.4.

Rice. 2.4. Installation diagram for measuring blood velocity: 1 - ultrasound source, 2 - ultrasound receiver

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

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

Solution. Using formula (2.9) we find:

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

2.7. Anisotropy during the propagation of surface waves. The 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 appears. This is expressed in the fact that the speed of propagation of a surface wave in mutually perpendicular directions - along the vertical (Y) and horizontal (X) axes of the body - differs.

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 -).

Numerical values ​​of the speed of surface waves in the skin and the degree of anisotropy are objective criteria for assessing various effects, including on the skin. 2. The effect 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.

Shock waves occur in tissues when they are exposed to high-intensity laser radiation. Often after this, scar (or other) changes begin to develop in the skin. This, for example, occurs in cosmetic procedures. Therefore, in order to reduce harmful effects shock waves, it is necessary to calculate the dosage of exposure in advance, taking into account the physical properties of both the radiation and the skin itself.

Rice. 2.5. Propagation of radial shock waves

Shock waves are used in radial shock wave therapy. In Fig. Figure 2.5 shows the propagation of radial shock waves from the applicator.

Such waves are created in devices equipped with a special compressor. The radial shock wave is generated by a pneumatic method. The piston located in the manipulator moves at high speed under the influence of a controlled impulse compressed air. When the piston strikes the applicator mounted in the manipulator, its kinetic energy is converted into mechanical energy of the area of ​​the body that was impacted. At the same time, to reduce losses when transmitting waves to air gap, located between the applicator and the skin, and a contact gel is used to ensure good shock wave conductivity. 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, signaling 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 distance of 2S: 2S = vt →S = vt/2 = 1090 m. Answer: S = 1090 m.

2. What minimum size objects whose position can be determined the bats using its 100,000 Hz sensor? 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:

λ 1= 330 m/s / 10 5 Hz = 3.3 mm. This is approximately the size of the insects that bats feed on;

λ 2= 1500 m/s / 10 5 Hz = 1.5 cm. A dolphin can detect a small fish.

Answer:λ 1= 3.3 mm; λ 2= 1.5 cm.

3. First, a person sees a flash of lightning, and 8 seconds later he hears a clap of thunder. At what distance from him did the lightning flash?

Solution

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

4. Two sound waves have same characteristics, except that the wavelength of one is twice that of the other. Which one carries more 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: the one with the shorter wavelength; 4 times.

5. A sound wave with a frequency of 262 Hz travels through 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;

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

6. Estimate the upper limit (frequency) of ultrasound in air if its propagation speed 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 the two nearest concentrations (or rarefactions) of molecules. Since the distance between the condensations cannot possibly be smaller sizes molecules, then obviously the limiting case should be considered 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 emits a signal with a frequency ν 0 = 800 Hz. Sound speed v= 340 m/s. What frequency signal will the driver of the second car hear: a) before the cars meet; b) after the cars meet?

8. As a train passes by, you hear the frequency of its whistle change from ν 1 = 1000 Hz (as it approaches) to ν 2 = 800 Hz (as the train moves 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, we obtain a system of equations with two unknowns:

Solution

Let v- wind speed, and it blows from a person (receiver) to the sound source. They are motionless relative to the earth, but relative to air environment both are moving to the right with speed u.

Using formula (2.7), we obtain the sound frequency. perceived by a person. It is unchanged:

Answer: the frequency will not change.

You can imagine what mechanical waves are by throwing a stone into the water. The circles that appear on it and are alternating depressions and ridges are an example of mechanical waves. What is their essence? Mechanical waves are the process of propagation of vibrations in elastic media.

Waves on liquid surfaces

Such mechanical waves exist due to the influence of intermolecular interaction forces and gravity on liquid particles. People have been studying this phenomenon for a long time. The most notable are the ocean and sea waves. As the wind speed increases, they change and their height increases. The shape of the waves themselves also becomes more complex. In the ocean they can reach frightening proportions. One of the most obvious examples of force is a tsunami that sweeps away everything in its path.

Energy of sea and ocean waves

Reaching the shore, sea waves increase with a sharp change in depth. They sometimes reach a height of several meters. At such moments, a colossal mass of water is transferred to coastal obstacles, which are quickly destroyed under its influence. The strength of the surf sometimes reaches enormous levels.

Elastic waves

In mechanics, they study not only vibrations on the surface of a liquid, but also so-called elastic waves. These are disturbances that propagate in different media under the influence of elastic forces in them. Such a disturbance represents any deviation of particles of a given medium from the equilibrium position. A clear example elastic waves are a long rope or rubber tube attached at one end to something. If you pull it tightly, and then create a disturbance at the second (unsecured) end with a sharp lateral movement, you can see how it “runs” along the entire length of the rope to the support and is reflected back.

The initial disturbance leads to the appearance of a wave in the medium. It is caused by the action of some foreign body, which in physics is called a wave source. It could be the hand of a person swinging a rope, or a pebble thrown into the water. In the case when the action of the source is short-term, a single wave often appears in the medium. When the “disturber” makes long waves, they begin to appear one after another.

Conditions for the occurrence of mechanical waves

This kind of oscillation does not always occur. A necessary condition for their appearance is the appearance at the moment of disturbance of the environment of forces preventing it, in particular, elasticity. They tend to bring neighboring particles closer together when they move apart, and push them away from each other when they approach each other. Elastic forces, acting on particles remote from the source of disturbance, begin to unbalance them. Over time, all particles of the medium are involved in one oscillatory movement. The propagation of such oscillations is a wave.

Mechanical waves in an elastic medium

In an elastic wave, there are 2 types of motion simultaneously: particle oscillations and the propagation of disturbances. A mechanical wave is called longitudinal, the particles of which oscillate along the direction of its propagation. A transverse wave is a wave whose medium particles oscillate across the direction of its propagation.

Properties of mechanical waves

Disturbances in a longitudinal wave represent rarefaction and compression, and in a transverse wave they represent shifts (displacements) of some layers of the medium relative to others. Compressive deformation is accompanied by the appearance of elastic forces. In this case, it is associated with the appearance of elastic forces exclusively in solids. In gaseous and liquid media, the shift of the layers of these media is not accompanied by the appearance of the mentioned force. Due to their properties, longitudinal waves can propagate in any media, while transverse waves can propagate exclusively in solid media.

Features of waves on the surface of liquids

Waves on the surface of a liquid are neither longitudinal nor transverse. They have a more complex, so-called longitudinal-transverse character. In this case, liquid particles move in a circle or along elongated ellipses. particles on the surface of the liquid, and especially with large vibrations, are accompanied by their slow but continuous movement in the direction of propagation of the wave. It is these properties of mechanical waves in water that cause the appearance of various seafood on the shore.

Mechanical wave frequency

If vibration of its particles is excited in an elastic medium (liquid, solid, gaseous), then due to the interaction between them it will propagate with speed u. So, if there is an oscillating body in a gaseous or liquid medium, then its motion will begin to be transmitted to all particles adjacent to it. They will involve the next ones in the process and so on. In this case, absolutely all points of the medium will begin to oscillate at the same frequency, equal to the frequency of the oscillating body. This is the frequency of the wave. In other words, this quantity can be characterized as points in the medium where the wave propagates.

It may not be immediately clear how this process occurs. Mechanical waves are associated with the transfer of energy of vibrational motion from its source to the periphery of the medium. During this process, so-called periodic deformations arise, transferred by a wave from one point to another. In this case, the particles of the medium themselves do not move along with the wave. They oscillate near their equilibrium position. That is why the propagation of a mechanical wave is not accompanied by the transfer of matter from one place to another. Mechanical waves have different frequencies. Therefore, they were divided into ranges and a special scale was created. Frequency is measured in Hertz (Hz).

Basic formulas

Mechanical waves, the calculation formulas for which are quite simple, are interesting object for studying. The speed of the wave (υ) is the speed of movement of its front (the geometric location of all points to which the vibration of the medium has reached at a given moment):

where ρ is the density of the medium, G is the elastic modulus.

When calculating, you should not confuse the speed of a mechanical wave in a medium with the speed of movement of particles of the medium that are involved in the process. So, for example, a sound wave in air propagates with an average vibration speed of its molecules of 10 m/s, while the speed of a sound wave in normal conditions is 330 m/s.

The wave front happens different types, the simplest of which are:

Spherical - caused by vibrations in a gaseous or liquid medium. The amplitude of the wave decreases with distance from the source in inverse proportion to the square of the distance.

Flat - is a plane that is perpendicular to the direction of wave propagation. It occurs, for example, in a closed piston cylinder when it performs oscillatory movements. Plane wave characterized by an almost constant amplitude. Its slight decrease with distance from the source of disturbance is associated with the degree of viscosity of the gaseous or liquid medium.

Wavelength

By is meant the distance to which its front will be moved in a time that is equal to the period of oscillation of the particles of the medium:

λ = υT = υ/v = 2πυ/ ω,

where T is the period of oscillation, υ is the wave speed, ω is the cyclic frequency, ν is the frequency of oscillation of points in the medium.

Since the speed of propagation of a mechanical wave is completely dependent on the properties of the medium, its length λ changes during the transition from one medium to another. In this case, the oscillation frequency ν always remains the same. Mechanical and similar in that during their propagation, energy is transferred, but substance is not transferred.

When vibrations of particles are excited in any place in a solid, liquid or gaseous medium, 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 a medium.

The following types of mechanical waves are distinguished:

Definition 2

Transverse wave: particles of the medium are displaced in a 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 a liquid include both transverse and longitudinal components.

Note 1

Let us point out an important clarification: when mechanical waves propagate, they transfer energy and 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. As they spread, the particles of the medium oscillate around their equilibrium positions. In this case, as we have already said, waves transfer energy, namely the energy of vibrations from one point in the medium to another.

Figure 2. 6. 1 . Propagation of a transverse wave along a 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, in contrast, for example, to light waves, which can propagate in emptiness. For the occurrence of a mechanical wave impulse, a medium is required that has the ability to store kinetic and potential energy: i.e. the medium must have inert and elastic properties. In real environments, these properties are distributed throughout the entire volume. For example, each small element solid inherent mass and elasticity. The simplest one-dimensional model of such a body is a collection of balls and springs (Figure 2, 6, 3).

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

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

Note 2

A distinctive feature of longitudinal waves is that they can propagate in any media: solid, liquid and gaseous.

If in the specified model of a solid body one or more balls receive a displacement perpendicular to the entire chain, we can talk about the occurrence of shear deformation. Springs that have become deformed 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. The displacement of one layer of liquid or gas by 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 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 can be said about a gaseous medium.

Note 3

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

In respect of practical application Of particular interest are simple harmonic or sine waves. They are characterized by the amplitude A of particle vibrations, 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 particles of the medium from the equilibrium position in a sine 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 transverse wave at time t and t + Δt. Over a period of time Δt, the wave moves along the O X axis to a distance υ Δt. Such waves are called traveling waves.

Figure 2. 6. 4 . "Snapshots" of a traveling sine wave at a moment in time t and t + Δt.

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 wavelength formula has the form: λ = υ T, where υ is the speed of propagation of the wave.

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 time Δt, point A will move along the axis O X to 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 this expression it 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 a 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 speed υ = - ω k.

When a traveling wave propagates, all particles of the medium oscillate harmoniously 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.

Note 4

From the above we can conclude that when a traveling wave propagates, an energy flow appears 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 mass per unit length) and the tension force T:

The speed with which longitudinal waves propagate in an unbounded medium is calculated with the participation of such quantities as the density of the medium ρ (or mass per unit volume) and the modulus of compression 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 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 types of steel υ ≈ 5 – 6 km/s.

If we're talking about O longitudinal waves, which are distributed in elastic rods, the formula for the wave speed contains not the modulus of uniform compression, but the Young’s modulus:

For steel the difference E from B insignificant, 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, which has spread 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 will be partially reflected and partially penetrated 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 appear. If both ends of the string are 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 strings “work” musical instruments, fixed at both ends. A similar process occurs with the sound of wind instruments, in particular organ pipes.

If waves propagating along a string in counter 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 point x = 0, and the other at point x 1 = L (Figure 2. 6. 6). There is tension in the string T.

Drawing 2 . 6 . 6 . The appearance of a standing wave in a string fixed at both ends.

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

• 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) – 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 vibrations created by counter-propagating waves at all points of the string are summed up. From the above it follows that the final oscillation at each point is determined as the sum of oscillations caused by 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 given 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.

Antinodes– points located between nodes and oscillating with maximum amplitude.

If we follow these definitions, for a standing wave to occur, both fixed ends of the string must be nodes. The formula stated earlier 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 the above we can conclude that a standing wave in a string does not always appear, but only when the length L string is equal to an integer number of half-wave lengths:

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

A set of wavelength values ​​λ n 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 string vibration is called a normal mode. The smallest frequency f 1 is called the fundamental frequency, all others (f 2, f 3, ...) are called harmonics.

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

A standing wave has no energy flow. The vibration energy “locked” in a section of string between two adjacent nodes is not transferred to the rest of the string. In each such segment there is a periodic (twice per period) T) conversion of kinetic energy into potential energy and vice versa, similar to a conventional oscillatory system. However, there is a difference here: if a load 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. In Figure 2. 6. Figure 7 shows several variants of standing waves in a string fixed at both ends.

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

According to the principle of superposition standing waves various types(With different meanings n) are capable of simultaneously being present in the vibrations of the string.

Figure 2. 6. 8 . Model of normal modes of a string.

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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 emptiness (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 a certain constant speed v.

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.