Absolutely everything in this world happens at some speed. Bodies don't move instantly, it takes time. Waves are no exception, no matter what medium they propagate.
Wave propagation speed
If you throw a stone into the water of the lake, then the waves that have arisen will not reach the shore immediately. It takes time for the waves to travel a certain distance, therefore, we can talk about the speed of wave propagation.
The wave speed depends on the properties of the medium in which it propagates. When passing from one medium to another, the speed of the waves changes. For example, if a vibrating iron sheet is thrust into water with its end, then the water will be covered with ripples of small waves, but the speed of their propagation will be less than in the iron sheet. This is easy to check even at home. Just don't cut yourself on the vibrating iron sheet ...
Wavelength
There is one more important characteristic, this is the wavelength. The wavelength is the distance that the wave propagates in one period of oscillatory movements. It is easier to understand this graphically.
If you draw a wave in the form of a picture or graph, then the wavelength will be the distance between any nearest crests or troughs of the wave, or between any other nearest points of the wave that are in the same phase.
Since the wavelength is the distance traveled by it, then you can find this value, like any other distance, by multiplying the speed of passage by a unit of time. Thus, the wavelength is directly proportional to the speed of wave propagation. To find the wavelength can be by the formula:
where λ is the wavelength, v is the wave speed, T is the oscillation period.
And given that the period of the oscillations is inversely proportional to the frequency of the same oscillations: T = 1⁄υ, we can derive relationship of wave propagation speed with vibration frequency:
v = λυ .
Oscillation frequency in different environments
The frequency of wave oscillations does not change when passing from one medium to another. So, for example, the frequency of forced oscillations coincides with the oscillation frequency of the source. The vibration frequency does not depend on the properties of the propagation medium. When passing from one medium to another, only the wavelength and the speed of its propagation change.
These formulas are valid for both shear and longitudinal waves. When longitudinal waves propagate, the wavelength will be the distance between the two nearest points with the same stretching or compression. It will also coincide with the distance traveled by the wave in one period of oscillation, so the formulas will be completely suitable in this case.
Longitudinal waves are waves in which oscillations of the particles of the medium occur along the direction of propagation of the wave process.
The appearance of the type of waves depends on the elastic properties of the medium in which the waves propagate.
In bodies in which elastic deformations of compression, tension and shear are possible, there can simultaneously be longitudinal and transverse waves - rigid bodies.
In gases and liquids - longitudinal waves, because they are not shear-elastic.
II. Wave characteristics. Wave equation.
Wavelength - the distance between the nearest points of the wave, oscillating in the same phases (l).
The period of the wave is the time of one complete oscillation of the points of the wave (T).
Wave frequency is the reciprocal of the period (ν).
During the time t = T, the wave propagates to a distance equal to l.
Having introduced the concepts of l and T, we can talk about the speed of wave propagation.
The speed of wave propagation depends on the environment:
a) on its density;
b) from elasticity.
where Е is Young's modulus;
G - shear modulus.
For solids E> G, therefore Vpr> Vper.
The propagation speed does not depend on:
a) from the shape of the pulse (i.e. how the compression changes over time);
b) on the amount of compression.
Let's try to mathematically express the process of wave propagation. The source of the waves is an oscillating system. Particles of the medium adjacent to it also vibrate.
Traveling wave equation 
The traveling wave equation determines the displacement of any point in the medium located at a distance ℓ from the vibrator at a given time.
Note also that the particles of the medium do not move after the wave, but only oscillate around the equilibrium position. The propagation speed of a wave is the speed of propagation of a disturbance that causes the particles to move from their equilibrium position.
To find the velocity of displacement in a wave of an oscillating particle of the medium, take the derivative of X in formula (2):
those. the velocity of particles in the wave changes according to the same law as the displacement, but is shifted in phase relative to the displacement by π / 2.
When the displacement reaches its maximum, the particle's velocity changes sign, i.e. vanishes momentarily.
Similarly, you can find the law of variation with time of particle acceleration:
Acceleration also changes according to the law of displacement, but is directed against the displacement, i.e. phase-shifted relative to the p-offset.
Graphs of displacement, velocity and acceleration of wave particles.
In addition to longitudinal and transverse waves propagating in continuous media, there are other types of wave processes:
surface waves , arise at the interface between two media with different densities.
Wave energy
The bulk density of wave energy in an elastic medium ( w) is defined as follows:
where is the total mechanical energy of the wave in the volume. From (8.11) it follows that the volumetric energy density of plane sinusoidal waves
So, the region of space participating in the wave process has an additional supply of energy. This energy is delivered from the source of vibrations to various points in the medium of the wave itself, therefore, the wave transfers energy.
Addition of harmonic vibrations directed along one straight line.
Hence it follows that the total motion is a harmonic vibration with a given cyclic frequency
Addition of mutually perpendicular vibrations. COULD NOT REDUCE. EXCUSE ME
Let a material point simultaneously participate in two harmonic oscillations, occurring with the same periods T in two mutually perpendicular directions. You can associate a rectangular XOY coordinate system with these directions by positioning the origin at the equilibrium position of the point. Let's denote the displacement of point C along the OX and OY axes, respectively, through x and y. (Figure 7.7)
Let's consider several special cases.
A. The initial phases of oscillation are the same. Let us choose the moment of the beginning of the timing so that the initial phases of both oscillations are equal to zero. Then the displacements along the OX and OY axes can be expressed by the equations:
Dividing these equalities term by term, we obtain the equations of the trajectory of point C:
or
Consequently, as a result of the addition of two mutually perpendicular vibrations, point C vibrates along a straight line segment passing through the origin (Fig. 7.7).
B. The initial phase difference is equal to π The oscillation equations in this case have the form:
Point trajectory equation
| (7.15) |
Consequently, point C vibrates along a line segment passing through the origin, but lying in other quadrants than in the first case. The amplitude A of the resulting oscillations in both considered cases is equal to
B. The initial phase difference is.
The oscillation equations are as follows:
Divide the first equation by, the second by: 
Let's square both equalities and add. We obtain the following equation of the trajectory of the resulting motion of the oscillating point
| (7.16) |
Oscillating point C moves along an ellipse with semiaxes and. For equal amplitudes, the trajectory of the total motion will be a circle. In the general case, for, but multiple, i.e. When adding mutually perpendicular oscillations, the oscillating point moves along curves called Lissajous figures. The configuration of these curves depends on the ratio of amplitudes, initial phases and periods of the oscillation components.
|
or for pressure loss: 
Physical parameters of sound
Oscillatory speed measured in m / s or cm / s. In terms of energy, real oscillatory systems are characterized by a change in energy due to its partial expenditure on work against friction forces and radiation into the surrounding space. In an elastic medium, the vibrations gradually damp. For characteristics damped oscillations damping factor (S), logarithmic decrement (D) and quality factor (Q) are used.
The damping factor reflects the rate at which the amplitude decays over time. If we denote the time during which the amplitude decreases by e = 2.718 times, through, then:
The decrease in amplitude in one cycle is characterized by a logarithmic decrement. The logarithmic decrement is equal to the ratio of the oscillation period to the decay time:
If an oscillatory system with losses is acted upon by a periodic force, then forced vibrations , the nature of which, to one degree or another, repeats the changes in the external force. The frequency of the forced vibrations does not depend on the parameters of the vibrating system.
The property of a medium to conduct acoustic energy, including ultrasonic energy, is characterized by acoustic resistance. Acoustic resistance the medium is expressed by the ratio of the sound density to the volumetric velocity of ultrasonic waves. Numerically, the specific acoustic resistance of the medium (Z) is found as the product of the density of the medium () by the velocity (c) of propagation of ultrasonic waves in it.
Specific acoustic impedance is measured in pascal-second on the meter(Pa s / m)
Sound or acoustic pressure in a medium is the difference between the instantaneous pressure at a given point in the medium in the presence of sound vibrations and static pressure at the same point in the absence of them. In other words, the sound pressure is the variable pressure in the medium caused by acoustic vibrations. The maximum value of the alternating acoustic pressure (pressure amplitude) can be calculated from the vibration amplitude of the particles:
where P is the maximum acoustic pressure (pressure amplitude);
F - frequency;
· S - the speed of propagation of ultrasound;
· - density of the medium;
· A - the amplitude of vibration of the particles of the medium.
To express the sound pressure in SI units, Pascal is used (Pa The amplitude value of acceleration (a) is determined by the expression:
If the traveling ultrasonic waves collide with an obstacle, it experiences not only alternating pressure, but also constant pressure. The areas of thickening and rarefaction of the medium arising during the passage of ultrasonic waves create additional pressure changes in the medium in relation to the external pressure surrounding it.
Ultrasound - high frequency elastic waves, which are devoted to special sections of science and technology. The human ear perceives elastic waves propagating in the medium with a frequency of up to approximately 16,000 vibrations per second (Hz); vibrations with a higher frequency represent ultrasound (out of earshot). Usually, the ultrasonic range is considered to be the frequency range from 20,000 to several billion hertz.
Application of ultrasound
Diagnostic use of ultrasound in medicine ( Ultrasound)
Main article: Ultrasound procedure
Due to the good propagation of ultrasound in human soft tissues, its relative harmlessness compared to x-rays and ease of use versus magnetic resonance imaging ultrasound is widely used to visualize the state of human internal organs, especially in abdominal cavity and pelvic cavity.
Questions.
1. What is called a wavelength?
The wavelength is the distance between the two nearest points oscillating in the same phases.
2. What letter represents the wavelength?
The wavelength is indicated by the Greek letter λ (lambda).
3. How long does it take for the oscillatory process to travel over a distance equal to the wavelength?
The oscillatory process extends over a distance equal to the wavelength λ for the period of the full oscillation T.
5. The distance between which points is equal to the length of the longitudinal wave shown in Figure 69?
The length of the longitudinal wave in Figure 69 is equal to the distance between points 1 and 2 (maximum of the wave) and 3 and 4 (minimum of the wave).
Exercises.
1.What is the speed of the wave propagating in the ocean if the wavelength is 270 m and the oscillation period is 13.5 s?
.jpg)
2. Determine the wavelength at 200 Hz if the wave velocity is 340 m / s.
.jpg)
3. The boat sways on waves propagating at a speed of 1.5 m / s. The distance between the two nearest wave crests is 6 m. Determine the period of vibration of the boat.
1. Mechanical waves, wave frequency. Longitudinal and transverse waves.
2. Wave front. Speed 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. The action 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 at 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 adjacent particles of the medium. 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 the frequency of the wave.
Wave the process of propagation of mechanical vibrations in an elastic medium is called.
By the frequency of the wave the frequency of oscillations of points of the medium in which the wave propagates is called.
The wave is associated with the transfer of vibration energy from the vibration source to the peripheral regions of the medium. In this case, in the environment there are
periodic deformations, which are transferred by a wave from one point of 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 the transfer of matter.
In accordance with the frequency, mechanical waves are divided into different ranges, which are indicated in table. 2.1.
Table 2.1. Mechanical wave scale
Depending on the direction of vibration of particles 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 vibrate along the same straight line along which the wave propagates. In this case, the areas of compression and discharge alternate in the medium.
Longitudinal mechanical waves can occur in all environments (solid, liquid and gaseous).
Transverse waves- waves, during the propagation of which the particles vibrate perpendicular to the direction of wave propagation. 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 in these media are not formed. The exception is waves on the surface of the liquid.
2.2. Wave front. Speed and wavelength
There are no processes in nature that propagate at an infinitely high speed, therefore, a disturbance created by an external influence at one point in the environment will not reach another point instantly, but after some time. In this case, the environment is divided into two areas: the area, the points of which are already involved in the oscillatory motion, and the area, the points of which are still in equilibrium. The surface separating these areas is called wave front.
Wave front - the locus of points to which the oscillation (disturbance of the environment) has reached this moment.
When a wave propagates, its front moves, moving at a certain speed, which is called the speed of the wave.
The speed of the wave (v) is the speed of movement of its front.
The wave speed depends on the properties of the medium and the type of wave: transverse and longitudinal waves in a solid propagate at different speeds.
The speed of propagation of all types of waves is determined under the condition of weak attenuation of the wave by the following expression:
where G is the effective modulus of elasticity, ρ is the density of the medium.
The speed of the wave in the 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 vibration velocity of its molecules is about 10 cm / s, and the speed of a sound wave under normal conditions is about 330 m / s.
The shape of the wavefront determines the geometric type of the wave. The simplest types of waves for this feature are flat and spherical.
Flat a wave is called a wave in which the front is a plane perpendicular to the direction of propagation.
Plane waves arise, for example, in a closed piston gas cylinder when the piston vibrates.
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 is called a wave in which the 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 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 called the distance to which its front moves in a time equal to the oscillation period of the particles of the medium:
Here v- wave speed, T - oscillation period, ν - frequency of oscillations of points of the medium, ω - cyclic frequency.
Since the wave propagation speed depends on the properties of the medium, the wavelength λ when passing from one environment to another changes, while the frequency ν remains the same.
This definition of wavelength has an important geometric interpretation. Consider fig. 2.1 a, which shows the displacements of the points of the medium at a certain moment 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 by points A 1 and B 1. The figure shows that the wavelength λ is equal to the distance between adjacent points oscillating in the same phase, for example, the distance between two adjacent maximums or minimums of the disturbance.
Rice. 2.1. Geometric interpretation of wavelength
2.3. Plane Wave Equation
The wave occurs 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 the oscillations, ω is the circular frequency of the oscillations.
If some point of the medium is at a distance s from the source, and the wave velocity is v, then the disturbance created by the source will reach this point in time τ = s / v. Therefore, the phase of oscillations at the point under consideration 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 fluctuations of this point will be determined by the equation
Here we 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 in the medium at any time, is called the equation of a plane wave. The argument for the cosine is the value φ = ωt - 2 π s /λ - called phase of the wave.
2.4. Energy characteristics of the wave
The medium in which the wave propagates has mechanical energy, which consists of the energies of the vibrational motion of all its particles. The energy of one particle with mass m 0 is found by the formula (1.21): E 0 = m 0 Α 2 ω 2/2. The unit volume of the medium contains n = p/ m 0 particles (ρ is the density of the medium). Therefore, a unit volume of the medium has energy w p = nЕ 0 = ρ Α 2 ω 2 /2.
Bulk energy density(\ ¥ p) is the energy of the vibrational 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 vibration amplitude of the particles, and ω is the wave frequency.
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) is a value equal to the energy carried by the wave through a given surface per unit of time:
Wave intensity or energy flux density (I) - a value equal to the energy flux carried by the wave through a unit 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 bulk energy density
2.5. Some special varieties
waves
1. Shock waves. During the propagation of sound waves, the vibration velocity of particles does not exceed several cm / s, i.e. it is hundreds of times less than the speed of the wave. Under strong disturbances (explosion, motion of bodies with supersonic speed, powerful electric discharge), the speed of oscillating particles of the medium can become comparable to the speed of sound. This produces an effect called a shock wave.
During an explosion, high-density products heated to high temperatures expand and compress a thin layer of ambient air.
Shock wave - a thin transition region propagating with supersonic speed, in which there is an abrupt increase in pressure, density and speed of motion of matter.
The shock wave can be very energetic. So, in a nuclear explosion, about 50% of the entire explosion energy is spent on the formation of a shock wave in the environment. 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, exponentially decaying 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 the body or along the boundary of the body with other media and quickly decay with distance from the boundary.
An example of such waves is waves in the earth's crust (seismic waves). The penetration depth of surface waves is several wavelengths. At a depth equal to the wavelength λ, the bulk density of the wave energy is approximately 0.05 of its bulk density at the surface. The amplitude of the displacement rapidly 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, medium is a continuous medium consisting of a large number of elements, each of which has a store of energy.
Moreover, each element can be in one of three states: 1 - excitement, 2 - refractoriness (non-excitability for a certain time after excitation), 3 - rest. Elements can only go into excitement from a state of rest. Excitation waves in active media are called autowaves. Autowaves - these are self-sustaining waves in an active medium that retain 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. Table 2.2 shows the similarities and differences between autowaves and ordinary mechanical waves.
Autowaves can be compared with the spread of a fire in the steppe. The flame spreads over an area with distributed energy reserves (over dry grass). Each subsequent element (dry blade of grass) is ignited from the previous one. And thus the front of the excitation wave (flame) spreads through the active medium (dry grass). When two fires meet, the flame disappears, since the energy reserves are exhausted - all the grass is burnt out.
The 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 conventional 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 physics institute.
Doppler effect consists in changing the frequency of vibrations perceived by the observer due to the relative motion of the source of vibrations 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 one straight line with the velocities v И and v П, respectively. Source performs harmonic oscillations with a frequency ν 0 relative to its equilibrium position. The wave created by these vibrations propagates through the medium at a speed v. Let us find out what vibration frequency will be fixed in this case receiver.
Disturbances created by oscillations of the source 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 oscillation period of the source). The disturbances in the environment created at these times reach the receiver at times 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 the 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 times t 1 and t 2
This formula is valid for the case when the velocities v and v p are directed towards each other. In general, when driving
source and receiver along one straight line, the formula for the Doppler effect takes the form
For the source, the velocity 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. Selecting signs for wave source and receiver velocities
Let us consider one particular case of the use of the Doppler effect in medicine. Let the ultrasound generator be combined with the receiver in the form of some technical system, which is motionless relative to the medium. The generator emits ultrasound having a frequency ν 0, which propagates in a medium with a speed v. Towards system with a speed v t some body is moving. First, the system plays the role source (v AND= 0), and the body is the role of the receiver (v Tl= v T). Then the wave is reflected from the object and is fixed by a stationary receiver. In this case, v И = v T, and v p = 0.
Applying formula (2.7) twice, we obtain the formula for the frequency fixed by the system after the reflection of the emitted signal:
At approaching object to the sensor frequency of the reflected signal increases, and at removal - decreases.
By measuring the Doppler frequency shift, from formula (2.8), you 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 schematic diagram of the corresponding setup 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 (reverse piezoelectric effect), and the other is used to receive ultrasound (direct piezoelectric effect) scattered by blood.
Example... Determine the speed of blood flow in the artery, if with the opposite reflection of ultrasound (ν 0 = 100 kHz = 100,000 Hz, v = 1500 m / s) Doppler frequency shift occurs from erythrocytes ν D = 40 Hz.
Solution. By formula (2.9) we find:
v 0 = v D v /2v 0 = 40x 1500/(2x 100,000) = 0.3 m / s.
2.7. Anisotropy in the Propagation of Surface Waves. The action of shock waves on biological tissues
1. Anisotropy of propagation of surface waves. When studying the mechanical properties of the skin using surface waves at a frequency of 5-6 kHz (not to be confused with ultrasound), the acoustic anisotropy of the skin appears. This is expressed in the fact that the speed of propagation of the surface wave in mutually perpendicular directions - along the vertical (Y) and horizontal (X) axes of the body - differ.
To quantify the degree of acoustic anisotropy, the mechanical anisotropy coefficient is used, which is calculated by the formula:
where v at- 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 severity of anisotropy are objective criteria for assessing various effects, including those on the skin.
2. The 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.
For example, a shock wave occurs when a blunt object is struck on the head. Therefore, when designing protective helmets, care is taken to damp the shock wave and protect the back of the head in case of a frontal impact. The inner tape in the helmet serves this purpose, which at first glance seems to be necessary only for ventilation.
Shock waves occur 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 the harmful effects of shock waves, it is necessary to calculate in advance the dosage of exposure, taking into account the physical properties of both radiation and the skin itself.
Rice. 2.5. Radial shock wave propagation
Shock waves are used in radial shock wave therapy. In fig. 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 pneumatically. The piston in the manipulator moves at high speed under the influence of a controlled impulse of compressed air. When the piston strikes the applicator installed in the manipulator, its kinetic energy is converted into mechanical energy of the affected area of the body. At the same time, a contact gel is used to reduce losses during transmission of waves in the air gap 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 is the reflective surface? Speed of sound in air v= 330 m / s.
Solution
In time t, the sound travels the path 2S: 2S = vt → S = vt / 2 = 1090 m. Answer: S = 1090 m.
2. What is the minimum size of objects that bats can detect with their 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 the object are equal to the wavelength:
λ 1= 330 m / s / 10 5 Hz = 3.3 mm. This is roughly the size of the insects that bats feed on;
λ 2= 1500 m / s / 10 5 Hz = 1.5 cm. A dolphin can spot a small fish.
Answer:λ 1= 3.3 mm; λ 2= 1.5 cm.
3. First, a person sees a flash of lightning, and after 8 seconds he hears a thunderclap. How far from him did the lightning flash?
Solution
S = v stars t = 330 x 8 = 2640 m. Answer: 2640 m.
4. Two sound waves have the same characteristics, except that the wavelength of one is twice that of the other. Which one carries the most energy? How many times?
Solution
The wave intensity 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 propagates in the 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 separated by 6.4 cm from each other?
Solution
but) λ = v /ν = 345/262 = 1.32 m;
in) Δφ = 360 ° s / λ = 360 x 0.064 / 1.32 = 17.5 °. Answer: but) λ = 1.32 m; b) t = T / 4; in) Δφ = 17.5 °.
6. Estimate the upper limit (frequency) of ultrasound in air, if the velocity of its propagation is known v= 330 m / s. Consider that air molecules have a size of the order of d = 10 -10 m.
Solution
In air, the mechanical wave is longitudinal and the wavelength corresponds to the distance between the two nearest concentrations (or discharges) of molecules. Since the distance between the clusters can in no way be less than the size of the molecules, then the obviously 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 beeps at a frequency ν 0 = 800 Hz. Sound speed v= 340 m / s. What signal frequency will the driver of the second car hear: a) before the cars meet; b) after meeting cars?
8.
As the train passes by, you hear the frequency of its whistle change from ν 1 = 1000 Hz (when approaching) to ν 2 = 800 Hz (when the train is receding). 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 be v- wind speed, and it blows from the person (receiver) to the sound source. They are motionless relative to the ground, and both move to the right with a speed u relative to the air environment.
By formula (2.7) we get the sound frequency. perceived by a person. It is unchanged:
Answer: the frequency will not change.
Suppose that the oscillating point is in the medium, all particles
which are related. Then the energy of its vibrations can be transmitted to the environment -
points, causing them to wobble.
The phenomenon of propagation of an oscillation in a medium is called a wave.
We note right away that when oscillations propagate in a medium, i.e., in a wave, I oscillate -
the spreading particles do not move with the spreading oscillatory process, but oscillate about their equilibrium positions. Therefore, the main property of all waves, regardless of their nature, is the transfer of energy without transfer of mass of matter.
Longitudinal and transverse waves
If the vibrations of the particles are perpendicular to the direction of propagation of the vibration -
niy, then the wave is called transverse; rice. 1, here is the acceleration, is the displacement, is the amplitude -
there - a period of fluctuations.
If the particles vibrate along the same straight line along which the
oscillation, then we call the wave longitudinal; rice. 2, where is the acceleration, is the displacement,
Amplitude, is the period of oscillation.
Elastic media and their properties
Whether waves propagating in a medium are longitudinal or transverse
- depends on the elastic properties of the medium.
If, when one layer of the medium is displaced with respect to another layer, elastic forces arise that tend to return the displaced layer to the equilibrium position, then transverse waves can propagate in the medium. A solid body serves as such a medium.
If elastic forces do not arise in the medium when the parallel layers shift relative to each other, then transverse waves cannot be formed. For example, liquid and gas represent media in which shear waves do not propagate. The latter does not apply to the surface of a liquid, in which transverse waves of a more complex nature can also propagate: in them, particles move in closed circles -
trajectories.
If elastic forces arise in the medium during compression or tension deformation, then longitudinal waves can propagate in the medium.
Only longitudinal waves propagate in liquid and gas.
In solids, longitudinal waves can propagate along with transverse -
The propagation speed of longitudinal waves is inversely proportional to the square root of the elasticity coefficient of the medium and its density:
since approximately - Young's modulus of the medium, then (1) can be replaced by the following:
The speed of propagation of shear waves depends on the shear modulus:
(3)
Wavelength, phase velocity, wave surface, wave front
The distance over which a certain phase of the oscillation propagates in one
the period of the oscillation is called the wavelength, the wavelength is denoted by a letter.
|
|
In fig. 3 graphically interpreted the relationship between the displacement of the particles of the medium participating in the wave - a new process, and the distance of these particles, for example, a particle, from the source of oscillations for some fixed moment in time. Reduced gra - fik is a graph of a harmonic shear wave that propagates at a speed along the direction - lance distribution. Fig. 3, it is clear that the wavelength is the smallest distance between points that oscillate in the same phase. Although, the given graph is similar to the accordion graph - fluctuations, but they are different in essence: if |
the graph of the wave determines the dependence of the displacement of all particles of the medium on the distance to the source of oscillations at a given time, then the graph of oscillations - the dependence of the shift -
shifts of a given particle over time.
The wave propagation speed means its phase velocity, that is, the propagation velocity of a given phase of the oscillation; for example, at the time point, Fig. 1, Fig. 3 had some initial phase, that is, it got out of the equilibrium position; then after a period of time the same initial phase was acquired by a point located at a distance from the point. Consequently, the initial phase for a time equal to the period spread over a distance. Hence, for the phase velocity by -
we get the definition:
Imagine that the point from which the vibrations go (the center of the vibration) vibrates in a continuous medium. Oscillations spread from the center in all directions.
The locus of the points to which the oscillation has reached at some point in time is called the wave front.
It is also possible to select in the environment the locus of points vibrating in one -
new phases; this set of points forms a surface of identical phases or a wave -
howling surface. Obviously, the wave front is a special case of the wave front -
surface.
The shape of the wave front determines the types of waves, for example, a plane wave is a wave whose front represents a plane, etc.
The directions in which the vibrations propagate are called rays. In from -
in a tropic environment, the rays are normal to the wave front; with a spherical wave front, the rays on -
are directed along the radii.
Traveling sine wave equation
Let us find out how it is possible to analytically characterize the wave process,
rice. 3. Let us denote by the displacement of a point from the equilibrium position. The wave process will be known if we know what value it has at each moment of time for each point of the straight line along which the wave propagates.
Let the oscillations at the point in Fig. 3 occur according to the law:
(5)
here is the amplitude of oscillations; - circular frequency; - time counted from the moment of the beginning of oscillations.
Take an arbitrary point in the direction lying from the origin of coordinates -
nat at a distance. Oscillations, propagating from a point with a phase velocity (4), will reach a point after a time interval
Consequently, the point will begin to fluctuate a time later than the point. If the waves do not damp, then its displacement from the equilibrium position will be
(7)
where is the time counted from the moment when the point began to oscillate, which is related to time as follows: 
because the point began to fluctuate a period of time later; substituting this value into (7), we obtain
or, using (6) here, we have
This expression (8) gives the displacement as a function of time and distance of a point from the center of oscillations; it represents the desired wave equation, propagating -
running along, Fig. 3.
|
|
Formula (8) is the equation of a plane wave propagating along Indeed, in this case, any plane, Fig. 4, perpendicular to the direction, will represent on top of - ness of the same phases, and, therefore, all points of this plane have at the same time the same displacement, it is defined only by the distance at which the plane lies from the origin. A wave in the opposite direction than that of wave (8) has the form: Expression (8) can be transformed if we use relation (4), according to which you can enter the wave number: |
where is the wavelength,
or, if instead of the circular frequency, you enter the usual frequency, also called linear -
frequency, then
Let us consider the example of a wave, Fig. 3, consequences following from equation (8):
a) the wave process is a doubly periodic process: the argument of the cosine in (8) depends on two variables - time and coordinate; that is, the wave has a double periodicity: in space and in time;
b) for a given moment in time, equation (8) gives the distribution of the displacement of particles as a function of their distance from the origin;
c) particles vibrating under the influence of a traveling wave at a given time are located along the cosine;
d) a given particle, characterized by a certain value, performs harmonic oscillatory motion in time:
e) the value is constant for a given point and represents the initial phase of oscillations at this point;
f) two points, characterized by distances and from the origin, have a phase difference:
it is seen from (15) that two points spaced apart from each other at a distance equal to the wavelength, i.e., for which 
, have a phase difference; and they also have for each given moment of time the same in magnitude and direction -
displacement; such two points are said to oscillate in the same phase;
for points spaced apart from each other at a distance 
, that is, separated from each other by half a wave, the phase difference according to (15), is equal to; such points oscillate in opposite phases - for each given moment, they have displacements that are the same in absolute value, but different in sign: if one point is deflected upward, then the other is downward, and vice versa.
In an elastic medium, waves of a different type are possible than traveling waves (8), for example, spherical waves, in which the dependence of the displacement on coordinates and time has the form:
In a spherical wave, the amplitude decreases in inverse proportion to the distance from the oscillation source.
6. Wave energy
The energy of the part of the medium in which the traveling wave propagates (8):
consists of kinetic energy and potential energy. Let the volume of the area of the medium be; let us designate its mass through and the speed of displacement of its particles through, then the kinetic energy
noticing that, where is the density of the medium, and finding an expression for the velocity based on (8)
we rewrite expression (17) in the form:
(19)
As is known, the potential energy of a section of a rigid body subjected to relative deformation is equal to
(20)
where is the elastic modulus or Young's modulus; - change in the length of a rigid body due to the action on its ends of forces equal in magnitude; - cross-sectional area.
We rewrite (20), introducing the coefficient of elasticity and dividing, and multiplying the right
part of it on, so
.
If the relative deformation is represented, using infinitesimal, in the form, where is the elementary difference in the displacements of particles spaced apart from each other by,
. (21)
Defining an expression for based on (8):
we write (21) in the form:
(22)
Comparing (19) and (22), we see that both kinetic energy and potential energy change in one phase, i.e., in phase and synchronously, they reach a maximum and a minimum. This is how the energy of the wave section differs significantly from the vibration energy
the point where at the maximum - kinetic energy - the potential has a minimum, and vice versa. When a single point vibrates, the total oscillation energy remains constant, and since the main property of all waves, regardless of their nature, is energy transfer without mass transfer of matter, the total energy of the part of the medium in which the wave propagates does not remain constant.
Let us add the right-hand sides (19) and (22), and calculate the total energy of a medium element with a volume:
Since, according to (1), the phase velocity of wave propagation in an elastic medium
then (23) is transformed as follows
Thus, the energy of the wave section is proportional to the square of the amplitude, the square of the cyclic frequency and the density of the medium.
The energy flux density vector is the Umov vector.
Let us introduce into consideration the energy density or the bulk energy density of an elastic wave
where is the volume of wave formation.
We see that the energy density, like the energy itself, is a variable quantity, but since the average value of the square of the sine over the period is equal, then, in accordance with (25), the average value of the energy density
, (26)
|
|
with unchanged parameters wavy - for an isotropic medium will be constant if there is no absorption in the medium. Due to the fact that energy (24) does not remain localized in a given volume, and the change - occurs in a medium, it is possible to introduce into consideration the concept of energy flow. Under the flow of energy through the top - we mean the value, the number - equal to the amount of energy, passing - passing through it in a unit of time. Take a surface perpendicular to the direction of the wave velocity; then through this surface in a time equal to the period, an amount of energy equal to the energy will flow, |
enclosed in a column of cross-section and length, Fig. five; this amount of energy is equal to the average value of the energy density taken over the period and multiplied by the volume of the column, hence
(27)
The average energy flow (average power) is obtained by dividing this expression by the time during which the energy flows through the surface
(28)
or, using (26), we find
(29)
The amount of energy flowing per unit of time through a unit of surface is called the flux density. By this definition, applying (28), we obtain
Thus, it is a vector, the direction of which is determined by the direction of the phase velocity and coincides with the direction of wave propagation.
This vector was first introduced into the theory of waves by a Russian professor
NA Umov and is called the Umov vector.
Take a point source of oscillations and draw a sphere of radius centered at the source. The wave and the energy that is associated with it will propagate along the radii,
that is, perpendicular to the surface of the sphere. During the period, energy will flow through the surface of the sphere, equal to, where is the flow of energy through the sphere. Flux density
we get if this energy is divided by the size of the surface of the sphere and time:
Since in the absence of absorption of oscillations in the medium and in a steady wave process, the average energy flux is constant and does not depend on what radius of conduction -
Given a sphere, then (31) shows that the average flux density is inversely proportional to the square of the distance from a point source.
Usually, the energy of vibrational motion in a medium is partially transferred into internal -
nude energy.
The total amount of energy that the wave will carry will depend on the distance traveled by it from the source: the farther from the source the wave surface is, the less energy it has. Since, according to (24), the energy is proportional to the square of the amplitude, the amplitude also decreases as the wave propagates. Let us assume that when the layer passes through the thickness, the relative decrease in the amplitude is proportional, i.e., we write
,
where is a constant depending on the nature of the environment.
The last equality can be rewritten
.
If the differentials of two quantities are equal to each other, then the quantities themselves differ from each other by an additive constant value, whence
The constant is determined from the initial conditions, which at the value is equal to, where is the amplitude of oscillations in the wave source, should be equal, thus:
(32)
The equation of a plane wave in a medium with absorption based on (32) will be
Let us now determine the decrease in the wave energy with distance. Let us denote - the average energy density at, and through - the average energy density at a distance, then using relations (26) and (32), we find
(34)
denote by and rewrite (34) as
The quantity is called the absorption coefficient.
8. Wave equation
One more relation can be obtained from the wave equation (8), which we will need further. Taking the second derivatives of with respect to the variables and, we obtain
whence follows
Equation (36) was obtained by differentiating (8). Conversely, it can be shown that a purely periodic wave, to which the cosine (8) corresponds, satisfies the differential
to the social equation (36). It is called the wave equation, since it has been established that (36) also satisfies a number of other functions describing the propagation of a wave disturbance of an arbitrary shape with velocity.
9. Huygens' principle
Each point reached by the wave serves as the center of the secondary waves, and the envelope of these waves gives the position of the wave front at the next moment in time.
This is the essence of the Huygens principle, which is illustrated in the following figures:
|
Rice. 6 A small hole in an obstacle is a source of new waves |
Rice. 7 Construction of Huygens for a plane wave |
|
Rice. 8 Construction of Huygens for a spherical wave propagating - Xia from the center |
Huygens' principle is a geometric principle - cyp. It does not essentially touch upon the question of the amplitude and, consequently, of the intensity of waves propagating behind the obstacle. |
Group speed
Rayleigh showed for the first time that, along with the phase velocity of waves, it makes sense
introduce the concept of another speed, called group speed. Group velocity refers to the case of propagation of waves of a complex non-cosine nature in a medium, where the phase velocity of propagation of cosine waves depends on their frequency.
The dependence of the phase velocity on their frequency or wavelength is called wave dispersion.
Let us imagine on the water surface a wave in the form of a single hump or soliton, Fig. 9, spreading in a certain direction. According to the Fourier method, this is so complicated -
This vibration can be decomposed into a group of purely harmonic vibrations. If all harmonic vibrations propagate over the water surface at the same speed -
the complex oscillations formed by them will propagate at the same speed -
nie. But, if the velocities of individual cosine waves are different, then the phase differences between them are constantly changing, and the hump resulting from their addition, continuously changes its shape and moves at a speed that does not coincide with the phase velocity of any of the waves added.
Any segment of the cosine, Fig. 10, can also, by Fourier's theorem, be decomposed into an infinite set of ideal cosine waves that are unlimited in time. Thus, any real wave is a superposition - a group - of infinite cosine waves, and the velocity of its propagation in a dispersive medium is different from the phase velocity of the waves. This speed of propagation of real waves in dispersing -
medium and is called group velocity. Only in a medium devoid of dispersion, a real wave propagates at a speed that coincides with the phase speed of those cosine waves, the addition of which it is formed.
Suppose that a group of waves consists of two waves that differ little in length:
a) waves with wavelength propagating with speed;
b) waves with wavelength 
propagating with speed
The relative position of both waves for a certain moment of time is shown in Fig. 11. a. The humps of both waves converge at a point; in one place the maximum of the resulting fluctuations is located. Let, then the second wave overtakes the first. After a certain period of time, she will overtake her by a segment; as a result of which the humps of both waves will already be folded at the point, Fig. 11.b, that is, the place of the maximum of the resulting complex oscillation will be shifted back by a segment equal to. Hence, the speed of propagation of the maximum of the resulting oscillations relative to the medium will be less than the speed of propagation of the first wave by an amount. This speed of propagation of the maximum of the complex oscillation is the group speed; denoting it through, we have, that is, the more pronounced the dependence of the wave propagation speed on their length, called dispersion.
If 
, then short wavelengths overtake longer ones; this case is called anomalous variance.
The principle of superposition of waves
When several waves of small amplitude propagate in a medium, performing -
That is, discovered by Leonardo da Vinci, the principle of superposition: the vibration of each particle of the medium is defined as the sum of independent vibrations that these particles would perform during the propagation of each wave separately. The superposition principle is violated only for waves with very large amplitudes, for example, in nonlinear optics. Waves characterized by the same frequency and constant, independent of time, phase difference are called coherent; for example, for example, cosine -
nye or sine waves with the same frequency.
Interference is called the addition of coherent waves, as a result of which there is a time-stable amplification of oscillations at some points and its weakening at others. In this case, there is a redistribution of the vibration energy between neighboring regions of the medium. Interference of waves only occurs if they are coherent.
Standing waves
A special example of the result of the interference of two waves are as follows
called standing waves, formed as a result of the superposition of two opposite flat waves with the same amplitudes.
|
Addition of two waves propagating in opposite directions |
Suppose that two plane waves with the same propagation amplitudes are nay - one in a positive direction - phenomenon, fig. 12, the other - by negation - body. If the origin of coordinates is taken at such a point - where the counterpropagating waves have the same displacement directions, that is, they have the same phases, and choose the timing so that the initial phases of the eye are Elastic waves in elastic environment standing waves... 2. Study the method of determining the speed of propagation ... to the direction of propagation waves. Elastic transverse waves can occur only in such environments that possess ... Application of sound waves (1)Abstract >> PhysicsMechanical vibrations, radiation and propagation of sound ( elastic) waves in environment, methods are being developed for measuring the characteristics of sound ... the laws of radiation, propagation and reception elastic hesitation and waves in different environments and systems; conditionally her ... Physics course answersCheat Sheet >> Physics... elastic strength. T = 2π root of m / k (s) - period, k - coefficient elasticity, m is the mass of the cargo. No. 9. Waves in elastic environment... Length waves... Intensity waves... Speed waves Waves ... |
