Wavelength and wave propagation speed. Wavelength. Wave speed. Equation of harmonic traveling wave. Energy characteristics of the wave
Longitudinal waves are waves in which the 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 - solid bodies.
In gases and liquids - longitudinal waves, because they do not have shear elasticity.
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).
The wave frequency is the reciprocal of the period (ν).
During the time t = T, the wave propagates over a distance equal to l.
Introducing the concepts of l and T, we can talk about the speed of wave propagation.
Wave propagation speed depends on the medium:
a) on its density;
b) elasticity.
where E is Young's modulus;
G is the shear modulus.
For solids E > G, therefore Vpr > Vper.
The propagation speed does not depend on:
a) on the shape of the pulse (i.e. how the compression changes with time);
b) on the amount of compression.
Let's try to mathematically express the process of wave propagation. The wave source is an oscillating system. The particles of the medium adjacent to it also come into oscillation.
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.
We also note that the particles of the medium do not follow the wave, but only oscillate around the equilibrium position. The speed of wave propagation is the speed of propagation of the perturbation that causes the displacement of particles from the equilibrium position.
To find the displacement velocity in a wave of an oscillating particle of the medium, take the derivative of X in formula (2):
those. the particle velocity in the wave changes according to the same law as the displacement, but is shifted in phase with respect to the displacement by π/2.
When the displacement reaches its maximum, the particle velocity changes sign, i.e. momentarily vanishes.
Similarly, one can find the law of change of particle acceleration with time:
The acceleration also changes according to the displacement law, but is directed against the displacement, i.e. phase-shifted relative to the offset by p.
Graphs of the displacement, velocity and acceleration of the particles of the wave.
In addition to longitudinal and transverse waves propagating in continuous media, there are other types of wave processes:
surface waves , appear at the interface between two media with different densities.
wave energy
Volumetric wave energy density 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 volume energy density of plane sinusoidal waves
So, the region of space participating in the wave process has an additional energy reserve. This energy is delivered from the source of oscillations to various points in the medium of the wave itself, therefore, the wave carries energy.
Addition of harmonic oscillations directed along one straight line.
This implies the conclusion that the total movement is a harmonic oscillation having a given cyclic frequency
Addition of mutually perpendicular vibrations. COULD NOT REDUCE. SORRY
Let the material point simultaneously participate in two harmonic oscillations occurring with the same periods T in two mutually perpendicular directions. The rectangular coordinate system XOY can be associated with these directions by placing the origin at the point's equilibrium position. Let us denote the displacement of the point C along the axes OX and OY, respectively, through x and y. (Figure 7.7)
Let's consider several special cases.
A. The initial phases of the oscillations are the same. Let us choose the moment of the beginning of the countdown in such a way 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 for the trajectory of point C:
or
Consequently, as a result of the addition of two mutually perpendicular oscillations, point C oscillates 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 oscillates along a straight 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 have the form:
Divide the first equation by , the second - by : 
We square both equalities and add them. We obtain the following equation for the trajectory of the resulting motion of the oscillating point
| (7.16) |
The oscillating point C moves along an ellipse with semi-axes and . With equal amplitudes, the trajectory of the total motion will be a circle. In the general case, at , 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 the amplitudes, initial phases, and periods of the component oscillations.
|
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, oscillations gradually decay. To characterize 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 е = 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 a periodic force acts on an oscillatory system with losses, then forced vibrations , the nature of which to some extent repeats the changes in the external force. The frequency of forced oscillations does not depend on the parameters of the oscillatory system.
The property of a medium to conduct acoustic energy, including ultrasonic energy, is characterized by acoustic resistance. Acoustic impedance medium is expressed by the ratio of 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 speed (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 value at a given point in the medium in the presence of sound vibrations and the static pressure at the same point in their absence. In other words, sound pressure is a variable pressure in the medium due to acoustic vibrations. The maximum value of the variable acoustic pressure (pressure amplitude) can be calculated from the particle oscillation amplitude:
where P is the maximum acoustic pressure (pressure amplitude);
f - frequency;
c is the speed of propagation of ultrasound;
· - medium density;
· A is the amplitude of oscillations of the particles of the medium.
Pascal (Pa) is used to express sound pressure in SI units. The amplitude value of acceleration (a) is given by:
If traveling ultrasonic waves collide with an obstacle, it experiences not only a variable pressure, but also a constant one. The areas of thickening and rarefaction of the medium that arise during the passage of ultrasonic waves create additional pressure changes in the medium in relation to the external pressure surrounding it.
Ultrasound - elastic waves of high frequency, 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 oscillations per second (Hz); vibrations with a higher frequency represent ultrasound (beyond hearing). Usually, the ultrasonic range is considered to be a frequency band from 20,000 to several billion hertz.
Application of ultrasound
Diagnostic application 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 compared to magnetic resonance imaging ultrasound is widely used to visualize the state of human internal organs, especially in abdominal cavity and pelvic cavity.
1. Mechanical waves, wave frequency. Longitudinal and transverse waves.
2. Wave front. Velocity and wavelength.
3. Equation of a plane wave.
4. Energy characteristics of the wave.
5. Some special types of waves.
6. Doppler effect and its use in medicine.
7. Anisotropy during the propagation of surface waves. Effect of shock waves on biological tissues.
8. Basic concepts and formulas.
9. Tasks.
2.1. Mechanical waves, wave frequency. Longitudinal and transverse waves
If in any place of an elastic medium (solid, liquid or gaseous) oscillations of its particles are excited, then due to the interaction between particles, this oscillation will begin to propagate in the medium from particle to particle with a certain speed v.
For example, if an oscillating body is placed in a liquid or gaseous medium, then the oscillatory motion of the body will be transmitted to the particles of the medium adjacent to it. They, in turn, involve neighboring particles in oscillatory motion, and so on. In this case, all points of the medium oscillate with the same frequency, equal to the frequency of the vibration of the body. This frequency is called wave frequency.
wave is the process of propagation of mechanical vibrations in an elastic medium.
wave frequency called the frequency of oscillations of the points of the medium in which the wave propagates.
The wave is associated with the transfer of vibration energy from the source of vibrations to the peripheral parts of the medium. At the same time, in the environment there are
periodic deformations that are carried by a wave from one point of the medium to another. The particles of the medium themselves do not move along with the wave, but oscillate around their equilibrium positions. Therefore, the propagation of the wave is not accompanied by the transfer of matter.
In accordance with the frequency, mechanical waves are divided into different ranges, which are indicated in Table. 2.1.
Table 2.1. Scale of mechanical waves
Depending on the direction of particle oscillations in relation to the direction of wave propagation, longitudinal and transverse waves are distinguished.
Longitudinal waves- waves, during the propagation of which the particles of the medium oscillate along the same straight line along which the wave propagates. In this case, the areas of compression and rarefaction alternate in the medium.
Longitudinal mechanical waves can occur in all media (solid, liquid and gaseous).
transverse waves- waves, during the propagation of which particles oscillate perpendicular to the direction of propagation of the wave. In this case, periodic shear deformations occur in the medium.
In liquids and gases, elastic forces arise only during compression and do not arise during shear, so transverse waves do not form in these media. The exception is waves on the surface of a liquid.
2.2. wave front. Velocity and wavelength
In nature, there are no processes that propagate at an infinitely high speed, therefore, a disturbance created by an external influence at one point in the environment will reach another point not instantly, but after some time. In this case, the medium is divided into two regions: the region, the points of which are already involved in the oscillatory motion, and the region, the points of which are still in equilibrium. The surface separating these regions is called wave front.
Wave front - the locus of points to which the oscillation (perturbation of the medium) has reached a given moment.
When a wave propagates, its front moves at a certain speed, which is called the speed of the wave.
Wave speed (v) is the speed of movement of its front.
The speed of a wave depends on the properties of the medium and the type of wave: transverse and longitudinal waves in a solid propagate at different speeds.
The propagation velocity of all types of waves is determined under the condition of weak wave attenuation by the following expression:
where G is the effective modulus of elasticity, ρ is the density of the medium.
The speed of a wave in a medium should not be confused with the speed of the particles of the medium involved in the wave process. For example, when a sound wave propagates in air, 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 wavefront shape determines the geometric type of the wave. The simplest types of waves on this basis are flat and spherical.
flat A wave is called a wave whose front is a plane perpendicular to the direction of propagation.
Plane waves arise, for example, in a closed piston cylinder with gas when the piston oscillates.
The amplitude of the plane wave remains practically unchanged. Its slight decrease with distance from the wave source is associated with the viscosity of the liquid or gaseous medium.
spherical called a wave whose front has the shape of a sphere.
Such, for example, is a wave caused in a liquid or gaseous medium by a pulsating spherical source.
The amplitude of a spherical wave decreases with distance from the source inversely proportional to the square of the distance.
To describe a number of wave phenomena, such as interference and diffraction, use a special characteristic called the wavelength.
Wavelength called the distance over which its front moves in a time equal to the period of oscillation of the particles of the medium:
Here v- wave speed, T - oscillation period, ν - frequency of oscillations of medium points, ω - cyclic frequency.
Since the speed of wave propagation depends on the properties of the medium, the wavelength λ when moving from one medium to another, it changes, while the frequency ν stays the same.
This definition of wavelength has an important geometric interpretation. Consider Fig. 2.1a, which shows the displacements of the points of the medium at some point in time. The position of the wave front is marked by points A and B.
After a time T equal to one period of oscillation, the wave front will move. Its positions are shown in Fig. 2.1, b points A 1 and B 1. It can be seen from the figure that the wavelength λ is equal to the distance between adjacent points oscillating in the same phase, for example, the distance between two adjacent maxima or minima of the perturbation.
Rice. 2.1. Geometric interpretation of the wavelength
2.3. Plane wave equation
The wave arises as a result of periodic external influences on the medium. Consider the distribution flat wave created by harmonic oscillations of the source:
where x and - displacement of the source, A - amplitude of oscillations, ω - circular frequency of oscillations.
If some point of the medium is removed from the source at a distance s, and the wave speed is equal to v, then the perturbation created by the source will reach this point in time τ = s/v. Therefore, the phase of the oscillations at the considered point at the time t will be the same as the phase of the source oscillations at the time (t - s/v), and the amplitude of the oscillations will remain practically unchanged. As a result, the fluctuations of this point will be determined by the equation
Here we have used the formulas for the circular frequency (ω
= 2π/T) and wavelength (λ
= v T).
Substituting this expression into the original formula, we get
Equation (2.2), which determines the displacement of any point of the medium at any time, is called plane wave equation. The argument at cosine is the magnitude φ = ωt - 2 π s /λ - called wave phase.
2.4. Energy characteristics of the wave
The medium in which the wave propagates has mechanical energy, which is made up of the energies of the oscillatory motion of all its particles. The energy of one particle with mass m 0 is found by formula (1.21): E 0 = m 0 Α 2 w 2/2. The volume unit of the medium contains n = p/m 0 particles (ρ is the density of the medium). Therefore, a unit volume of the medium has the energy w р = nЕ 0 = ρ Α 2 w 2 /2.
Bulk energy density(\¥ p) - the energy of the oscillatory motion of the particles of the medium contained in a unit of its volume:
where ρ is the density of the medium, A is the amplitude of particle oscillations, ω is the frequency of the wave.
As the wave propagates, the energy imparted by the source is transferred to distant regions.
For a quantitative description of the energy transfer, the following quantities are introduced.
Energy flow(Ф) - a value equal to the energy carried by the wave through a given surface per unit time:
Wave intensity or energy flux density (I) - a value equal to the energy flux carried by a wave through a single area perpendicular to the direction of wave propagation:
It can be shown that the wave intensity is equal to the product of its propagation velocity and the volume energy density
2.5. Some special varieties
waves
1. shock waves. When sound waves propagate, the particle oscillation velocity does not exceed a few cm/s, i.e. it is hundreds of times less than the wave speed. Under strong disturbances (explosion, movement of bodies at supersonic speed, powerful electric discharge), the speed of oscillating particles of the medium can become comparable to the speed of sound. This creates an effect called a shock wave.
During an explosion, high-density products heated to high temperatures expand and compress a thin layer of ambient air.
shock wave - a thin transition region propagating at supersonic speed, in which there is an abrupt increase in pressure, density, and velocity of matter.
The shock wave can have significant energy. So, in a nuclear explosion, about 50% of the total energy of the explosion 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, decaying exponentially in the transverse direction. As a result, surface waves localize the energy of perturbations created on the surface in a relatively narrow near-surface layer.
surface waves - waves that propagate along the free surface of a body or along the boundary of the body with other media and decay rapidly with distance from the boundary.
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 volumetric energy density of the wave is approximately 0.05 of its volumetric density at the surface. The displacement amplitude rapidly decreases with distance from the surface and practically disappears at a depth of several wavelengths.
3. 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 an energy reserve.
Moreover, each element can be in one of three states: 1 - excitation, 2 - refractoriness (non-excitability for a certain time after excitation), 3 - rest. Elements can go into excitation only from a state of rest. Excitation waves in active media are called autowaves. Autowaves - these are self-sustaining waves in an active medium, keeping their characteristics constant due to energy sources distributed in the medium.
The characteristics of an autowave - period, wavelength, propagation velocity, amplitude and shape - in the steady state depend only on the local properties of the medium and do not depend on the initial conditions. In table. 2.2 shows the similarities and differences between autowaves and ordinary mechanical waves.
Autowaves can be compared with the spread of fire in the steppe. The flame spreads over an area with distributed energy reserves (dry grass). Each subsequent element (dry blade of grass) is ignited from the previous one. And thus the front of the excitation wave (flame) propagates through the active medium (dry grass). When two fires meet, the flame disappears, as the energy reserves are exhausted - all the grass is burned out.
The description of the processes of propagation of autowaves in active media is used in the study of the propagation of action potentials along nerve and muscle fibers.
Table 2.2. Comparison of autowaves and ordinary mechanical waves
2.6. Doppler effect and its use in medicine
Christian Doppler (1803-1853) - Austrian physicist, mathematician, astronomer, director of the world's first physical institute.
Doppler effect consists in changing the frequency of oscillations perceived by the observer, due to the relative motion of the source of oscillations and the observer.
The effect is observed in acoustics and optics.
We obtain a formula describing the Doppler effect for the case when the source and receiver of the wave move relative to the medium along one straight line with velocities v I and v P, respectively. Source performs harmonic oscillations with frequency ν 0 relative to its equilibrium position. The wave created by these oscillations propagates in the medium at a speed v. Let us find out what frequency of oscillations will fix in this case receiver.
Disturbances created by source oscillations propagate in the medium and reach the receiver. Consider one complete oscillation of the source, which begins at time t 1 = 0
and ends at the moment t 2 = T 0 (T 0 is the source oscillation period). The disturbances of the medium created at these moments of time reach the receiver at the moments t" 1 and t" 2, respectively. In this case, the receiver captures oscillations with a period and frequency:
Let's find the moments t" 1 and t" 2 for the case when the source and receiver are moving towards to each other, and the initial distance between them is equal to S. At the moment t 2 \u003d T 0, this distance will become equal to S - (v I + v P) T 0, (Fig. 2.2).
Rice. 2.2. Mutual position of the source and receiver at the moments t 1 and t 2
This formula is valid for the case when the speeds v and and v p are directed towards each other. In general, when moving
source and receiver along one straight line, the formula for the Doppler effect takes the form
For the source, the speed v And is taken with the “+” sign if it moves in the direction of the receiver, and with the “-” sign otherwise. For the receiver - similarly (Fig. 2.3).
Rice. 2.3. Choice of signs for the velocities of the source and receiver of waves
Consider one particular case of using the Doppler effect in medicine. Let the ultrasound generator be combined with the receiver in the form of some technical system that is stationary relative to the medium. The generator emits ultrasound having a frequency ν 0 , which propagates in the medium with a speed v. Towards system with a speed v t moves some body. First, the system performs the role source (v AND= 0), and the body is the role of the receiver (vTl= v T). Then the wave is reflected from the object and fixed by a fixed receiving device. In this case, v AND = v T, and v p \u003d 0.
Applying formula (2.7) twice, we obtain the formula for the frequency fixed by the system after reflection of the emitted signal:
At approach object to the sensor frequency of the reflected signal increases and at removal - decreases.
By measuring the Doppler frequency shift, from formula (2.8) we can find the speed of the reflecting body:
The sign "+" corresponds to the movement of the body towards the emitter.
The Doppler effect is used to determine the speed of blood flow, the speed of movement of the valves and walls of the heart (Doppler echocardiography) and other organs. A diagram of the corresponding setup for measuring blood velocity is shown in Fig. 2.4.
Rice. 2.4. Scheme of an installation for measuring blood velocity: 1 - ultrasound source, 2 - ultrasound receiver
The device consists of two piezocrystals, one of which is used to generate ultrasonic vibrations (inverse piezoelectric effect), and the second - to receive ultrasound (direct piezoelectric effect) scattered by blood.
Example. Determine the speed of blood flow in the artery, if the counter reflection of ultrasound (ν 0 = 100 kHz = 100,000 Hz, v \u003d 1500 m / s) a Doppler frequency shift occurs from erythrocytes ν D = 40 Hz.
Solution. By formula (2.9) we find:
v 0 = v D v /2v0 = 40x 1500/(2x 100,000) = 0.3 m/s.
2.7. Anisotropy during the propagation of surface waves. Effect of shock waves on biological tissues
1. Anisotropy of surface wave propagation. When studying the mechanical properties of the skin using surface waves at a frequency of 5-6 kHz (not to be confused with ultrasound), acoustic anisotropy of the skin is manifested. This is expressed in the fact that the propagation velocities of the surface wave in mutually perpendicular directions - along the vertical (Y) and horizontal (X) axes of the body - differ.
To quantify the severity of acoustic anisotropy, the mechanical anisotropy coefficient is used, which is calculated by the formula:
where v y- speed along the vertical axis, v x- along the horizontal axis.
The anisotropy coefficient is taken as positive (K+) if v y> v x at v y < v x the coefficient is taken as negative (K -). The numerical values of the velocity of surface waves in the skin and the degree of anisotropy are objective criteria for evaluating various effects, including those on the skin.
2. Action of shock waves on biological tissues. In many cases of impact on biological tissues (organs), it is necessary to take into account the resulting shock waves.
So, for example, a shock wave occurs when a blunt object hits the head. Therefore, when designing protective helmets, care is taken to dampen the shock wave and protect the back of the head in a frontal impact. This purpose is served by the internal tape in the helmet, which at first glance seems to be necessary only for ventilation.
Shock waves arise in tissues when exposed to high-intensity laser radiation. Often after that, cicatricial (or other) changes begin to develop in the skin. This is the case, for example, in cosmetic procedures. Therefore, in order to reduce the harmful effects of shock waves, it is necessary to pre-calculate the dosage of exposure, taking into account the physical properties of both radiation and the skin itself.
Rice. 2.5. Propagation of Radial Shock Waves
Shock waves are used in radial shock wave therapy. On fig. 2.5 shows the propagation of radial shock waves from the applicator.
Such waves are created in devices equipped with a special compressor. The radial shock wave is generated pneumatically. The piston, located in the manipulator, moves at high speed under the influence of a controlled pulse of compressed air. When the piston hits the applicator installed in the manipulator, its kinetic energy is converted into mechanical energy of the area of the body that was affected. In this case, to reduce losses during the transmission of waves in the air gap located between the applicator and the skin, and to ensure good conductivity of shock waves, a contact gel is used. Normal operating mode: frequency 6-10 Hz, operating pressure 250 kPa, number of pulses per session - up to 2000.
1. On the ship, a siren is turned on, giving signals in the fog, and after t = 6.6 s, an echo is heard. How far away is the reflective surface? speed of sound in air v= 330 m/s.
Solution
In time t, sound travels a path 2S: 2S = vt →S = vt/2 = 1090 m. Answer: S = 1090 m.
2. What is the minimum size of objects that bats can locate with their sensor, which has a frequency of 100,000 Hz? What is the minimum size of objects that dolphins can detect using a frequency of 100,000 Hz?
Solution
The minimum dimensions of an object are equal to the wavelength:
λ1\u003d 330 m / s / 10 5 Hz \u003d 3.3 mm. This is roughly the size of the insects that bats feed on;
λ2\u003d 1500 m / s / 10 5 Hz \u003d 1.5 cm. A dolphin can detect a small fish.
Answer:λ1= 3.3 mm; λ2= 1.5 cm.
3. First, a person sees a flash of lightning, and after 8 seconds after that he hears a thunderclap. At what distance did the lightning flash from him?
Solution
S \u003d v star t \u003d 330 x 8 = 2640 m. Answer: 2640 m
4. Two sound waves have the same characteristics, except that one has twice the wavelength of the other. Which one carries the most energy? How many times?
Solution
The intensity of the wave is directly proportional to the square of the frequency (2.6) and inversely proportional to the square of the wavelength (ω = 2πv/λ ). Answer: one with a shorter wavelength; 4 times.
5. A sound wave having a frequency of 262 Hz propagates in air at a speed of 345 m/s. a) What is its wavelength? b) How long does it take for the phase at a given point in space to change by 90°? c) What is the phase difference (in degrees) between points 6.4 cm apart?
Solution
a) λ =v /ν = 345/262 = 1.32 m;
in) Δφ = 360°s/λ= 360 x 0.064/1.32 = 17.5°. Answer: a) λ = 1.32 m; b) t = T/4; in) Δφ = 17.5°.
6. Estimate the upper limit (frequency) of ultrasound in air if the speed of its propagation is known v= 330 m/s. Assume that air molecules have a size of the order of d = 10 -10 m.
Solution
In air, a mechanical wave is longitudinal and the wavelength corresponds to the distance between two nearest concentrations (or discharges) of molecules. Since the distance between clumps 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 gives a signal with a frequency ν 0 = 800 Hz. Sound speed v= 340 m/s. What frequency will the driver of the second car hear: a) before the cars meet; b) after the meeting of the cars?
8.
When a train passes by, you hear how the frequency of its whistle changes from ν 1 = 1000 Hz (when approaching) to ν 2 = 800 Hz (when the train is moving away). What is the speed of the train?
Solution
This problem differs from the previous ones in that we do not know the speed of the sound source - the train - and the frequency of its signal ν 0 is unknown. Therefore, a system of equations with two unknowns is obtained:
Solution
Let v is the speed of the wind, and it blows from the person (receiver) to the source of the sound. Relative to the ground, they are motionless, and relative to the air, both move to the right with a speed u.
By formula (2.7) we obtain the sound frequency. perceived by man. She is unchanged:
Answer: frequency will not change.
>>Physics: Velocity and wavelength
Each wave propagates at a certain speed. Under wave speed understand the propagation speed of the disturbance. For example, a blow to the end of a steel rod causes local compression in it, which then propagates along the rod at a speed of about 5 km/s.
The speed of a wave is determined by the properties of the medium in which this wave propagates. When a wave passes from one medium to another, its speed changes.
In addition to speed, an important characteristic of a wave is its wavelength. Wavelength called the distance over which a wave propagates in a time equal to the period of oscillations in it.
The direction of the spread of the war
Since the speed of the wave is a constant value (for a given medium), the distance traveled by the wave is equal to the product of the speed and the time of its propagation. In this way, to find the wavelength, you need to multiply the speed of the wave by the period of oscillation in it:
By choosing the direction of wave propagation for the direction of the x axis and denoting by y the coordinate of the particles oscillating in the wave, we can construct wave chart. A sine wave graph (for a fixed time t) is shown in Figure 45. 
The distance between adjacent crests (or troughs) on this graph is the same as the wavelength.
Formula (22.1) expresses the relationship of the wavelength with its speed and period. Considering that the period of oscillations in a wave is inversely proportional to the frequency, i.e. T=1/ v, you can get a formula expressing the relationship of the wavelength with its speed and frequency:
The resulting formula shows that the speed of a wave is equal to the product of the wavelength and the frequency of oscillations in it.
The frequency of oscillations in the wave coincides with the frequency of oscillations of the source (since the oscillations of the particles of the medium are forced) and does not depend on the properties of the medium in which the wave propagates. When a wave passes from one medium to another, its frequency does not change, only the speed and wavelength change.
??? 1. What is meant by wave speed? 2. What is the wavelength? 3. How is the wavelength related to the speed and period of oscillations in a wave? 4. How is the wavelength related to the speed and frequency of oscillations in a wave? 5. Which of the following wave characteristics change when a wave passes from one medium to another: a) frequency; b) period; c) speed; d) wavelength?
Experimental task . Pour water into the tub and, by rhythmically touching the water with your finger (or a ruler), create waves on its surface. Using different oscillation frequencies (for example, touching the water once and twice per second), pay attention to the distance between adjacent wave crests. At what frequency is the wavelength longer?
S.V. Gromov, N.A. Motherland, Physics Grade 8
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To begin with, let's remember that mechanical wave is an oscillation that propagates over time in an elastic medium. Since this is an oscillation, the wave will have all the characteristics that correspond to the oscillation: amplitude, oscillation period and frequency.
In addition, the wave has its own special characteristics. One of these characteristics is wavelength. Wavelength is denoted by the Greek letter (lambda, or they say "lambda") and is measured in meters. We list the characteristics of the wave:
What is a wavelength?
Wavelength - this is the smallest distance between particles that oscillate with the same phase.
Rice. 1. Wavelength, wave amplitude
It is more difficult to talk about the wavelength in a longitudinal wave, because it is much more difficult to observe particles that make the same vibrations there. But there is also a characteristic wavelength, which determines the distance between two particles making the same oscillation, oscillation with the same phase.
Also, the wavelength can be called the distance traveled by the wave in one period of particle oscillation (Fig. 2).
Rice. 2. Wavelength
The next characteristic is the speed of wave propagation (or simply the speed of the wave). Wave speed It is denoted in the same way as any other speed by a letter and is measured in. How to clearly explain what is the speed of the wave? The easiest way to do this is with a transverse wave as an example.
transverse wave is a wave in which perturbations are oriented perpendicular to the direction of its propagation (Fig. 3).
Rice. 3. Shear wave
Imagine a seagull flying over the crest of a wave. Its flight speed over the crest will be the speed of the wave itself (Fig. 4).
Rice. 4. To the determination of the wave speed
Wave speed depends on what is the density of the medium, what are the forces of interaction between the particles of this medium. Let's write down the relationship between the wave speed, wavelength and wave period: .
Speed can be defined as the ratio of the wavelength, the distance traveled by the wave in one period, to the period of oscillation of the particles of the medium in which the wave propagates. In addition, remember that the period is related to the frequency as follows:
Then we get a relation that relates the speed, wavelength and frequency of oscillations: .
We know that a wave arises as a result of the action of external forces. It is important to note that when a wave passes from one medium to another, its characteristics change: the speed of the wave, the wavelength. But the oscillation frequency remains the same.
Bibliography
- Sokolovich Yu.A., Bogdanova G.S. Physics: a reference book with examples of problem solving. - 2nd edition redistribution. - X .: Vesta: publishing house "Ranok", 2005. - 464 p.
- Peryshkin A.V., Gutnik E.M., Physics. Grade 9: textbook for general education. institutions / A.V. Peryshkin, E.M. Gutnik. - 14th ed., stereotype. - M.: Bustard, 2009. - 300 p.
- Internet portal "eduspb" ()
- Internet portal "eduspb" ()
- Internet portal "class-fizika.narod.ru" ()
Homework
During the lesson, you will be able to independently study the topic “Wavelength. Wave propagation speed. In this lesson, you will learn about the special characteristics of waves. First of all, you will learn what a wavelength is. We will look at its definition, how it is labeled and measured. Then we will also look at the propagation speed of the wave in detail.
To begin with, let's remember that mechanical wave is an oscillation that propagates over time in an elastic medium. Since this is an oscillation, the wave will have all the characteristics that correspond to the oscillation: amplitude, oscillation period and frequency.
In addition, the wave has its own special characteristics. One of these characteristics is wavelength. Wavelength is denoted by the Greek letter (lambda, or they say "lambda") and is measured in meters. We list the characteristics of the wave:
What is a wavelength?
Wavelength - this is the smallest distance between particles that oscillate with the same phase.
Rice. 1. Wavelength, wave amplitude
It is more difficult to talk about the wavelength in a longitudinal wave, because it is much more difficult to observe particles that make the same vibrations there. But there is also a characteristic wavelength, which determines the distance between two particles making the same oscillation, oscillation with the same phase.
Also, the wavelength can be called the distance traveled by the wave in one period of particle oscillation (Fig. 2).
Rice. 2. Wavelength
The next characteristic is the speed of wave propagation (or simply the speed of the wave). Wave speed It is denoted in the same way as any other speed by a letter and is measured in. How to clearly explain what is the speed of the wave? The easiest way to do this is with a transverse wave as an example.
transverse wave is a wave in which perturbations are oriented perpendicular to the direction of its propagation (Fig. 3).
Rice. 3. Shear wave
Imagine a seagull flying over the crest of a wave. Its flight speed over the crest will be the speed of the wave itself (Fig. 4).
Rice. 4. To the determination of the wave speed
Wave speed depends on what is the density of the medium, what are the forces of interaction between the particles of this medium. Let's write down the relationship between the wave speed, wavelength and wave period: .
Speed can be defined as the ratio of the wavelength, the distance traveled by the wave in one period, to the period of oscillation of the particles of the medium in which the wave propagates. In addition, remember that the period is related to the frequency as follows:
Then we get a relation that relates the speed, wavelength and frequency of oscillations: .
We know that a wave arises as a result of the action of external forces. It is important to note that when a wave passes from one medium to another, its characteristics change: the speed of the wave, the wavelength. But the oscillation frequency remains the same.
Bibliography
- Sokolovich Yu.A., Bogdanova G.S. Physics: a reference book with examples of problem solving. - 2nd edition redistribution. - X .: Vesta: publishing house "Ranok", 2005. - 464 p.
- Peryshkin A.V., Gutnik E.M., Physics. Grade 9: textbook for general education. institutions / A.V. Peryshkin, E.M. Gutnik. - 14th ed., stereotype. - M.: Bustard, 2009. - 300 p.
- Internet portal "eduspb" ()
- Internet portal "eduspb" ()
- Internet portal "class-fizika.narod.ru" ()
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