The process of propagation of mechanical vibrations in a medium. Lesson summary "Propagation of oscillations in a medium. Waves. Characteristics of waves." Wave process. Types of waves
Waves are the process of propagation of vibrations in space. Due to the presence of connections Mechanism – disturbance of distribution. Elastic (mechanical) waves Elastic connection forces act between the particles of the medium 1. Perpendicular to the direction of propagation of the wave - transverse waves. 2. 2. Along the direction of wave propagation - longitudinal. Transverse when elastic shear deformation. Longitudinal – elastic deformation of compression and tension.
Traveling wave Suppose the cross section of the rod is not deformed. It oscillates perpendicularly (shear) or longitudinally (tension - compression.) Z=0 Z (Attenuation is not taken into account) The geometric location of points oscillating in the same phase is called the wave surface Z
Wavelength The distance over which a wave travels during one period of particle oscillation. Let's substitute (5) into (4) => Traveling wave equation From (1) and (6) the phase lag of a point with coordinate z. The phase difference is the shortest distance between points oscillating in the same phases
1 2 Graphs (family) x=x(z) For transverse B they give: the magnitude, the sign of the displacement and give the configuration of particles at moment t For the longitudinal only the magnitude and sign for the inverse B
Relative deformation and stress in the medium during wave propagation. If z offset on x and z+ z on x+ x, then the absolute deformation is negative. z is equal to relative: x, and In the limit Relative deformation (shear-compression) Modulus of elasticity Stress (shear, compression stress)
Hooke's law is a mechanical measure of internal forces during deformation of a material. -elastic modulus k Shear modulus G (trans.) Young's modulus E (cont.) The strain components at a given point are linear and homogeneous functions of the stress components.
It is necessary to find the resultant F of the forces f 1 and f 2 and the mass of the section. Then the acceleration of the equation is found.
We express the length through and the increment of deformation by decolog. def. In the Taylor series near z Acceleration acquired by the rod d'Alembert's equation
Summary: 1. The only assumption 2. D'Alembert's equation is valid for the propagation of movement of any nature in a medium with linear fur. characteristic in the case of quasi-elastic waves. 3. The wave equation is satisfied by traveling waves 4. as well as in general by a periodic signal, the displacement in which is the Velocity of propagation of elastic waves
Equation (3.1) is convenient for calculating V for known and Velocity of elastic wave propagation in solid body Where: E – Young’s modulus G – shear modulus. Velocity of propagation of an elastic wave in a liquid In a liquid, waves are longitudinal Coef. Fluid compressibility
The speed of propagation of an elastic wave in a gas The heat exchange between the thickening and the discharge does not keep up - the process of propagation of the elastic wave is adiabatic. To calculate V, you need to find E based on 3.10 and the adiabatic equation From the Clapeyron-Mendeleev equation: It looks like the root-mean-square speed of molecules in a gas
Potential energy An elastic sample of length l is stretched by a force f. The entire sample has the same stress state - stress S - cross section. Under the influence of force f, elongation is formed. The specific energy stored per unit volume is the energy density. The work of tension of an elastic body = the total potential energy of elastic deformation accumulated in the body (4.2) obtained for a uniform stress state is also suitable for an inhomogeneous state (traveling waves), when V is so small that the stress state at its different points can be considered the same. (4.2) gives instantaneous values
Kinetic energy of a wave We consider a plane wave propagating in the z direction along a thin rod with cross section S. The section of the rod Sdz contains the energy of motion of particles in the propagating wave. , then we can assume that all particles, neg. dz, move with the same speeds Instantaneous value of kinetic energy density, expressed through the value (instantaneous) of vibrational speed
Ppot=Pkin For any point of the traveling wave, the instantaneous values of the density of potential and kinetic energy are equal to each other. Let us prove: Instantaneous value of the total energy density
Explicit dependence of the instantaneous value of energy density on coordinates and time See (4.5) According to (4.6) during the propagation of B, energy transfer occurs. The speed of energy transfer depends on the speed of displacement transfer, vibrational speed. particles and deformation in the medium, due to some connection between energy and these quantities. Oscillation frequency P = twice the oscillation frequency
Energy flux density (Umov vector) Energy through a given section per unit time Flux density - energy flux per unit time per unit area, perpendicular to the direction of transfer
The wave (acoustic resistance of the medium) equation allows us to establish a connection between the voltage in the medium during the passage of a wave and the speed that arises in the oscillating particle. The proportionality coefficient that connects the voltage value at a given point of the medium with the instantaneous value of the speed of this point is called the wave (sound or acoustic) impedance of the medium. Wave impedance is a very important characteristic of the medium: when a wave passes from one medium to another or when a wave is reflected from the boundary of two media , the value of the reflection and penetration coefficients is entirely determined by the ratio of the wave impedances of the adjacent media
Substance Velocity of wave propagation, Density, Acoustic resistance, Air Water Copper Mercury Rubber From (4) it follows that the ratio of the voltage in the medium performing a harmonic oscillation and the oscillatory velocity of particles remains unchanged in time: The invariance of the ratio of instantaneous values occurs only in a plane wave. Here the following relations are always valid for the amplitude and effective values of these quantities:
In an isotropic medium at a distance r from the source, pay attention to the following: 1. The oscillations of each point lag in phase from the oscillations of the previous point. Then the phase difference between them: 2. The surface of the wave (G.M.T., oscillating in the same phases) is determined by (2) and is a spherical surface. Such waves are called spherical.
3. The rays (directions of propagation of vibrational energy) in an isotropic medium are perpendicular to the surface of the wave and the rays form two orthogonal families ray surface wave 4. The length of a spherical wave is the shortest distance (along the ray) between two points oscillating in the same phases. 5. The amplitude of oscillations of points in the medium is a decreasing function of r, because the oscillation, as it moves away from the source, spreads to an increasing number of points; the intensity of the wave (energy flux density) decreases with distance from the source.
Dependence of the amplitude of oscillations on distance If there is no absorption in the medium: And from (3) it then follows that the amplitude of oscillations of particles is inversely proportional to the distance from the source
Let us accept the condition: the smallest distance from the source of oscillations at which the source can be considered a point and the amplitude of the wave is spherical at this distance then:
When a spherical wave propagates, there is a phase difference between the voltage oscillations in the medium (the relative deformation proportional to it) and the particle speed. Voltage fluctuation can be represented as the sum of two oscillations: One in the same phase as the speed and one out of phase by 900
Let us consider the wave regime in a system whose linear dimensions are equal to a small number of wavelengths. In this case, we almost always observe not the incident and reflected waves, but the result of their superposition. A standing wave is the result of the superposition of the incident and reflected waves. The medium is a string, air is a resonator
The wave propagates in the direction of the z axis. Let us accept the condition: total reflection takes place, i.e. reflection vibrational energy is not transmitted to the neighboring medium. In this case, the amplitude of the reflected wave = the amplitude of the incident wave. The superposition of these two waves gives: Sin α+ Sin β = 2 sin((α+β)/2)cos((α-β)/2) The resulting equation x= x(t, z) describes a new wave regime - a standing wave
Let's consider the graphs of the dependence x=x(z) M N M We see that two neighboring points oscillate in the same phases, but with different amplitudes N The amplitude of particles in a standing wave depends on the coordinates of the particles A=A(z)
Unlike a traveling wave, in which the amplitudes of oscillations of all points are the same, and the phases are different, in a standing wave the phases of neighboring points are the same, and the difference in their oscillations is determined by the difference in amplitude. For comparison, graphs of traveling and standing waves for close moments in time node
Characteristics standing waves 1. The amplitude of particle oscillations varies according to a cosine law (cm(4)). There are points at which the amplitude is zero. Such points are called nodes. There are points at which the amplitude reaches highest value These points are called antinodes. . 2. The distance between two adjacent nodes is equal to half the wavelength. The distance between adjacent antinodes is also equal to half the wavelength. The distance between adjacent nodes and antinodes is equal to a quarter of the wavelength
3. Oscillations of points between two nodes occur in the same phases. The phase of oscillations abruptly changes to the opposite when passing through node 4. Oscillatory speed: The velocity node takes place in the same place as the displacement node.
5. Standing stress wave: 5. 1 The coordinates of the stress nodes coincide with the coordinates of the displacement and velocity antinodes 5. 2 The stress wave was reflected with a change in phase to the opposite (reflection, see above)
6. A standing wave does not transfer energy. Indeed, the instantaneous value of the energy flux density depends on. products σx. From the previous fig. It can be seen that the instantaneous value of this product changes sign every quarter of a wave. The average value of the energy flux J is equal to zero ψ In a standing wave ψ = 90 o and J = 0 When deriving (4), the amplitudes of the incident and reflected waves were the same (with total reflection) With partial energy transfer, the maximum amplitude is not, as in (5) Such a wave carries energy transferred to the neighboring medium.
1. Sound vibrations and their propagation Sound is longitudinal elastic vibrations of the air, ear, brain, sensation of sound. Perceived from 16 Hz to 20,000 Hz. associated with human physiology. f>20000 Hz – ultrasound; f
Sound impressions: - height - depends on frequency - timbre - overtones - volume A: 1st 2nd 440 880 3rd 1760 Hz. Hearing threshold - min intensity of the wave that causes a sound sensation Most audible 1000 -4000 Hz hearing threshold For other f it lies higher
Pain threshold: intensity Subjective characteristic – volume level L – log relative intensity of a given sound I to some I 0 – initial. Volume level unit – bel (B); B/10 - decibel The ratio of intensity I 1 and I 2 can be expressed in d. B 20 d. B - decrease by 100 30 d. B - decrease by 1000 40 d. B - decrease by 10000, etc. Whisper - 30 d B Creek - 80 d. B 10 102 103 104 105
2. Velocity of propagation of elastic waves in gas. Speed of propagation of control waves in a continuous medium By definition for an elastic rod Young's modulus Density of the medium For volume of volumetric deformations Pos. infinite. small d. P and d. V. Took away d. P decrease d. V (negative) Let's rewrite (2) The sound of oscillations occurs so quickly that the heat exchange between the thickening and rarefaction does not have time to occur - that is, it occurs adiabatically
(Austrian Christian Doppler (1803 -1853)) The Doppler effect is a change in the frequency of oscillations propagating in a medium, which occurs when the receiver or source of oscillations moves relative to this medium. V – speed of propagating oscillations in the medium U – speed of the source relative to the medium v – speed of the receiver relative to the medium approaching n and (+) (V, U) moving away n and (-) (V, U)
Src="http://present5.com/presentation/-29884334_94992875/image-50.jpg" alt="II. The receiver moves relative to the medium with speed v; the source is stationary; U=0 V v v>0"> II. Приёмник движется относительно среды со скоростью v; источник неподвижен; U=0 V v v>0 приближается П (U=0) И v 0 , то мимо приёмника за единицу времени пройдёт большее число волн. Волны идут мимо прибора со скоростью: Т. е. Частота воспринятых колебаний больше числа испущенных в 2) Если v!}
III. The source is moving, the receiver is at rest (U = U; v = 0) And U P (v = 0) Since V depends on the medium, then for T oscillations are distributed, regardless of the movement of the source; 1. U>0 But! during this time the source will travel the path u. T As a result, the perceived will change, because now (for u>0) 2. for U
IV. The source and receiver move simultaneously (U=0; v=0) Due to the movement of the source Due to the movement of the receiver Due to both reasons: If v and U are directed at an angle, then their components should be taken on the straight line connecting the source and receiver.
Interference of waves If waves from the source of oscillations reach the receiver in two different ways, then the receiver will oscillate under the simultaneous influence of both waves, and the addition of oscillations of the same frequencies will take place. With the same directions of the component oscillations, the amplitude and energy of the resulting oscillation: When adding identically directed oscillations of equal frequencies, the energy of the resulting oscillation is not equal to the sum of the energies of the component oscillations that occur separately. Interference of waves - strengthening or weakening of the energy of the resulting oscillation depending on the phase difference of the component oscillations. When adding mutually perpendicular ones there are no interference oscillations, because at any energy
The receiver under the influence of one first wave would make oscillations following the equation: a under the influence of the second wave - the equation The difference in the phases of oscillations of the receiver under the influence of one and the other oscillations: The difference in the distances that the waves travel from the sources to the receiver is called the difference in the wave paths Interference amplification, according to (1), holds under the condition hence
Similarly, for interference attenuation it is necessary: Thus: Interference amplification occurs if the path difference of the rays is equal to an integer number of wavelengths or an even number of half-wavelengths Interference attenuation occurs if the path difference of the rays is equal to an odd number of half-wavelengths
Reflection of waves Penetration of waves through the boundary Condition: the wave propagates along the z axis, perpendicular to the interface between two media. Wave impedance of the first medium (the supply and reflected waves propagate in it) Wave impedance of the second medium (the wave penetrating through the interface propagates in it) Ratio of the wave impedances of the media Amplitudes of oscillations of particles of the incident, reflected and refracted waves, respectively Amplitudes of the oscillatory velocity of particles Amplitudes of the stresses of the medium caused by incident, reflected and transmitted waves, respectively Reflection coefficient Penetration coefficient
Since the area on which the wave falls is equal to the area from which it is reflected, the ratio of energy fluxes can be replaced by the ratio of energy flux densities (Umov vectors) Since the incident and reflected waves propagate in the same medium, then: Incident and transmitted through waves propagate in different media, therefore:
Incident wave Reflected wave Wave that penetrated the second medium Displacement wave Wave of oscillatory velocities Stress wave It is necessary to pay attention to the appearance of additional (compared to the incident wave) phase angles and, taking into account a possible change in the phase of the wave upon reflection and penetration into the second medium. At the interface between two media, the continuity condition is satisfied: in nature there are no infinitely large differences in displacements, vibrational velocities of particles and stresses
Let us accept z=0 at the boundary, then: Because the stress wave must be reflected from the boundary in a phase opposite to the velocity wave. If we substitute the + sign in (10), then it will turn out to be incompatible with (9) From (10) after substitution it follows: By (9) brackets in l. h. and p.h. equations (11) are equal, therefore, which does not correspond to the condition. From (9) and (10), valid at any time, we can obtain:
Using the introduced notations, equations (12) – (15) can be represented in the form: The system of equations makes it possible to determine
1. definition Subtracting (19) from (17) we obtain: To determine the sign, add (16) and (18) The wave penetrates the second medium without changing the phase, i.e., with respect to the refractive phase, the wave is a continuation of the previous one.
1. definition Subtracting (18) from (16) we obtain: 1. When reflected from a medium with lower acoustic resistance, the displacement wave and the wave of vibrational velocities of particles do not change phase; the stress wave changes phase by 2. When reflected from a medium with high acoustic resistance, displacements and the wave of vibrational velocities of particles change the phase by stress does not change the phase of the wave; wave
1. Definition of R Expressing from (16) and substituting it into (18), we obtain: The reflection coefficients from the boundary of these two media are the same for both the wave incident on the boundary from the first medium and for the wave incident on the boundary from the second medium 1 Definition of T Expressing from (16) and substituting it into (18), we obtain: According to the law of conservation of energy, the energy flux of the incident wave is equal to the sum of the energy fluxes of the waves reflected and penetrating into the second medium. Therefore there must be equality:
Huygens' principle Each point on the wave surface should be considered as an independent source of elementary spherical waves. Wave surface at the moment of time. Method of finding the position and shape of the wave surface after a period of time after the initial moment: From each point on the wave surface, given at the moment of time, it is necessary to draw in the direction of the direction of propagation hemispheres with a radius The common envelope of all these hemispheres is the desired wave surface. Wave surface at a moment in time;
1. Refraction of a plane wave through a flat interface between two media From the consideration of triangles ABD and AED: Law of refraction: The ratio of the sin angle of incidence to the sin angle of refraction for these two media is a constant value equal to the ratio of the speed of wave propagation in the first medium to the speed of wave propagation in second Wednesday. - relative refractive index of the second medium relative to the first
In the case of elastic waves: In the case of electromagnetic waves: The refractive index of the medium relative to vacuum, where it takes the form: For all non-ferromagnetic media, the magnetic permeability is practically equal to unity, therefore: When a wave passes from one medium to another, the oscillation frequency does not change. Since propagation speeds in different media are different, the wavelength changes when passing from one medium to another.
Maxwell, James Clerk D.C. Maxwell (1831 -1879) - great English scientist, creator of the theory of electromagnetism. In 1860-1865 Maxwell created the theory electromagnetic field, which he formulated in the form of a system of equations (Maxwell’s equations). Maxwell's equations form the basis of both electrical and radio engineering and the theory of any electromagnetic phenomena in any media. In 1861, he discovered that light is a type of electromagnetic wave. Maxwell also created the kinetic theory of gases (1859) and derived a relation for the distribution of gas particles by speed, called the Maxwell distribution.
Generalization of the laws of electromagnetism. MAXWELL'S equations (1867) 1. Experimental laws. I. Coulomb's law Gauss's theorem II. Law of Conservation of Charge The total charge of an electrically neutral system remains constant III. Ampere's law Lorentz force (magn) Faraday's law IV. Bio-Savarra. Laplace? Magnetic circulation theorem. fields
Application to Maxwell's equations in differential form Stokes and Ostrogradsky-Gauss theorems T. Stokes T. Ostrogradsky Gauss where
Electromagnetic waves EMV Frequency scale. Hz Range name Gamma rays X-rays Ultraviolet radiation Visible light Infrared radiation Microwaves Television and FM Broadcasting Radio waves Wavelength, cm
Electromagnetic waves Wave equation EMW (D'Alembert) Maxwell's equations for a plane-polarized wave are reduced to: D'Alembert's equation EMW
Electromagnetic waves Electromagnetic waves speed Previously, for elastic vibrations it was shown: For a traveling wave, v is the phase velocity. Comparing (7) and (5), (6) we see:
Electromagnetic waves In the case of a plane-polarized monochromatic wave, equations (5), (6) correspond to the solution: Problem: establish a connection between E and H in phase and magnitude Problem According to (4) synphase Id. issue (12) (i.e. at any coordinates and at any moment) It is possible only with B running electromagnetic waves E and H oscillating in the same phases
Chapter 2. WAVES
Wave process. Types of waves
Solid, liquid and gaseous bodies can be considered as media consisting of individual particles interacting with each other. If we excite oscillations of particles in a local region of the medium, then due to interaction forces, forced oscillations of neighboring particles will arise, which, in turn, will cause oscillations of particles associated with them, etc. Thus, vibrations excited at any point in the medium will propagate through it at a certain speed, depending on the properties of the medium. How further away is the particle from the source of vibrations, so later it will begin to oscillate. In other words, the phase of oscillation of particles of the medium depends on the distance to the source.
The process of propagation of vibrations in a certain medium is called a wave process or wave.
The particles of the medium in which the wave propagates undergo oscillatory motion around their equilibrium positions. When distributed waves particles of the medium are not transported by the wave. Along with the wave vibrational motion and its energy are transferred from particle to particle of the medium. Thus, the main property of waves, regardless of their nature, is the transfer of energy without transfer of matter.
The following types of waves occur in nature and technology: gravity-capillary waves(waves on the surface of the liquid), elastic waves(propagation of mechanical disturbances in an elastic medium) and electromagnetic(propagation of electromagnetic disturbances in the medium).
There are elastic waves longitudinal And transverse. In longitudinal waves particles of the medium oscillate in the direction of wave propagation, in transverse - in planes perpendicular to the direction of wave propagation(Fig. 2.1.1, a; b).
VIBRATIONS, WAVES, SOUND
Harmonic sin or cos.
1. Offset (s)
2. Amplitude (A)- maximum displacement.
3. Period (T)
4. Line frequency (v) . v = 1/T.
ω= 2πv .
6. Oscillation phase (φ) φ = ωt + φ 0
1. Available
2. Decaying
3. Forced
4. Self-oscillations
s = Asin ωt
Then the total energy is:
longitudinal
: λ=υT, λ=υv
: S = A sinωt
s = Asin (ωt-2πх/λ) 2πх/λ = φ 0
W = (mω 2 A 2)/2
ε = W 0 /V
Where W o = εV
ε = n 0 W = n 0 mω 2 A 2 /2 , But n o m = p , Then ε = (pω 2 A 2)/2
Ps=W 0 /t (W)
J=Ps/s = W 0 /st (W)
J=Ps/s (W/m2)
logarithmic. J (s) = LgJ/J 0 (W/m 2)
sound pressure.
objective subjective.
Pitch
timbre
Volume Weber-Fechner:
E=kLg J/J 0
1. Audiometry
2. Auscultation
3. Percussion
Laws of reflection
A medium in which the speed of propagation of light is the same at all points is called an optically homogeneous medium. The boundary of two media is the surface separating two optically inhomogeneous media. The angle α between the incident ray and the perpendicular restored to the boundary of the two media at the point of incidence is called the angle of incidence. The angle β between the reflected beam and the perpendicular to the interface between the two media at the point of incidence is called the reflection angle.
I law: The incident ray, the perpendicular one restored to the interface between two media at the point of incidence, and the reflected ray lie in the same plane.
II law: The angle of incidence is equal to the angle of reflection: α = β
I law: The incident ray, perpendicular, restored to the interface between two media at the point of incidence, and the refracted ray lie in the same plane.
I I law: The ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant value for these two media and is called the refractive index of the second medium relative to the first:
sinα/sinγ = const = n 21
Lenses
A lens is a transparent body bounded by two spherical surfaces and differing in refractive index from the surrounding medium.
The straight line passing through the centers of the spherical surfaces delimiting the lens (SS") is called the main optical axis.
The point of intersection of the main optical axis with the refractive plane is called the optical center of the lens (O). Any straight line passing through the optical center of the lens is called the optical axis (AA). Rays parallel to the main optical axis, after refraction in the lens, are collected at one point, called the main focus of the lens (F). The point where the optical axis intersects the focal plane is called subfocus (F").
Such lenses are called collecting. A parallel beam of rays after refraction in a lens can be scattered, then at one point called imaginary focus, the continuations of these rays will gather. Such lenses are called scattering.
The plane perpendicular to the main optical axis and passing through the main focus of the lens is called the focal plane.
In converging lenses, the image depends on the position of the object. If the object is located between the optical center of the lens and the main focus, then the image will be virtual, straight and enlarged.
If the object is between focus and double focus, the image is real, inverse, magnified.
If the object is between double and triple focus and beyond, the image is real, inverse, reduced.
Diverging lenses always give a virtual, direct and reduced image.
The distance from the optical center of the lens to the main focus is called focal length F. The reciprocal of the focal length is called optical power lenses: D =1/F
The optical power of a lens is measured in diopters (Dopters). One diopter is the optical power of a lens whose focal length is 1 m . For converging lenses it is positive, for diverging lenses it is negative. In practice, to determine the focal length and optical power of a lens, the thin lens formula is used: D = 1/F = 1/d +1/f ,
where d is the distance from the object to the lens, f is the distance from the lens to the image.
Images produced with a single lens tend to be different from the object itself. In this case, they talk about image distortion. Spherical aberration occurs because the edges of the lens deflect rays more than the central part.
As a result, the image of a luminous point on the screen appears as a blurry spot, and the image of an extended object becomes blurry and not sharp. To eliminate spherical aberration, centered optical systems consisting of collecting and diverging lenses are used. A system of lenses having a common main optical axis is called centered. .
Chromatic aberration due to the dispersion of light, since the lens can be represented as a prism. In this case, the focal length for rays of different wavelengths is not the same.
Therefore, when an object is illuminated with complex, for example, white light, a point on the screen will be visible in the form of a colored spot, and the image of an extended object will also be colored and blurred. Chromatic aberration can be eliminated by combining converging and diverging lenses made from different types of glass with different relative dispersions. Such lens systems are called achromats. Reason astigmatism is the unequal refraction of rays in different meridional planes of the lens. There are two types of astigmatism. The first, so-called oblique ray astigmatism, occurs in lenses that have a spherical surface shape, but the rays strike the lens at a significant angle to the main optical axis. In this case, the rays in mutually perpendicular planes are refracted unequally and the point on the screen will be visible as a line, while the shape of an extended object will be distorted, for example, a square will be visible as a rectangle.
The second type of astigmatism, correct, occurs when the surface of the lens deviates from spherical, when different meridional planes have an unequal radius of curvature, i.e. the shape of the surface in this plane is not spherical. Astigmatism of oblique rays is eliminated by turning the lens towards the imaged object. Correct astigmatism is eliminated by selecting the radii of curvature and optical powers of the refractive surfaces. These are most often cylindrical lenses. An optical system corrected for astigmatism in addition to spherical and chromatic aberrations is called anastigmata.
Optical system of the eye
The human eye is a unique optical device that occupies a special place in optics. This is explained, firstly, by the fact that many optical instruments are designed for visual perception, and secondly, by the human eye of an animal), as a biological system improved in the process of evolution, brings some ideas for the design and improvement of optical systems. The eye can be represented as a centered optical system formed by the cornea (P), the anterior chamber fluid (K) and the lens (X), limited in front by the air environment, and behind by the vitreous body. The main optical axis (MA) passes through the optical centers of the cornea and lens. In addition, the visual axis of the eye (30) is also distinguished, which determines the direction of greatest light sensitivity and passes through the centers of the lens and macula (G). The angle between the main optical and visual axes is about 5". The main refraction of light occurs at the outer border of the cornea, the optical power of which is approximately 40 diopters, the lens - about 20 diopters, and the entire eye - about 60 diopters. The adaptation of the eye to clearly seeing objects at different distances is called accommodation. In an adult healthy person, when an object approaches the eye to a distance of 25 cm, accommodation occurs without tension and, thanks to the habit of examining objects in the hands, the eye most often accommodates exactly this distance, called the distance of best vision. To characterize the resolution of the eye, the smallest angle of view is used at which the human eye can still distinguish two points on an object. In medicine, the resolution of the eye is assessed by visual acuity. One is taken as the norm for visual acuity; in this case, the smallest visual angle is 1 ".
VIBRATIONS, WAVES, SOUND
Any deviations physical body or a parameter of its state, now in one direction, now in the other direction from the equilibrium position is called oscillatory motion or simply oscillation.
Oscillatory motion is called periodic if the values physical quantities, changing during the oscillation process, are repeated at regular intervals.
Harmonic vibrations that occur according to the law are called sin or cos.
s = Asin (ωt +φ 0), s = Acos (ωt +φ 0)
They occur under the action of quasi-elastic forces, i.e. forces proportional to displacement
The main characteristics of vibrations are:
1. Offset (s)- this is the distance by which the oscillating system deviates at a given time from the equilibrium position.
2. Amplitude (A)- maximum displacement.
3. Period (T)- time of one complete oscillation.
4. Line frequency (v)- this is the number of oscillations per unit of time, measured in Hz - this is one oscillation per second . v = 1/T.
5. Cyclic or circular frequency (ω). It is related to linear frequency by the following relationship: ω= 2πv .
6. Oscillation phase (φ) characterizes the state of an oscillating system at any time: φ = ωt + φ 0 , φ 0 - initial phase of oscillation.
The oscillatory process can be represented graphically in the form of an expanded or vector diagram.
The expanded diagram is a graph of a sine or cosine wave, from which you can determine the displacement of the oscillating system at any time.
However, any complex oscillation can be represented as a sum of harmonics. This position is determined by a special diagnostic method - spectral analysis.
The set of harmonic components into which a complex vibration is decomposed is called the harmonic spectrum of this vibration.
Oscillations are divided into the following main types:
1. Available- these are ideal vibrations that do not exist in nature, but help to understand the essence of other types of vibrations and determine the properties of a real oscillatory system. They occur with their own frequency, which depends only on the properties of the oscillating system itself. We will denote the natural frequency and period by v 0 and T o.
2. Decaying- these are oscillations, the amplitude of which decreases over time, but the frequency does not change and is close to its own. Energy is supplied to the system once. The decrease in amplitude per unit time is characterized by the damping coefficient β = r / 2m, where r is the friction coefficient, m is the mass of the oscillating system. The decrease in amplitude over a period is characterized by a logarithmic attenuation decrement δ = βТ. The logarithmic attenuation decrement is the logarithm of the ratio of two adjacent amplitudes: δ = log (At / A t + T).
3. Forced- These are vibrations that occur under the influence of a periodically changing external force. They occur with the frequency of a compelling force. The phenomenon of a sharp increase in the amplitude of oscillations as the frequency of the driving force approaches the natural frequency of the system is called resonance. This increase will depend on the amplitude of the driving force, the mass of the system and the damping coefficient.
4. Self-oscillations undamped oscillations that exist in any system in the absence of a variable external influence are called, and the systems themselves are called self-oscillatory. The amplitude and frequency of self-oscillations depend on the properties of the self-oscillating system itself. The self-oscillatory system consists of three main elements: 1) the oscillatory system itself; 2) source of energy; 3) feedback mechanism. A striking example of such a system in biology is the heart.
Let us determine the energy of a body of mass m performing free harmonic oscillations with amplitude A and cyclic frequency ω.
s = Asin ωt
Total energy consists of potential and kinetic energy:
Wn=ks 2 /2=(kA 2 /2)sin 2 ωt, where k=mω
W=mυ 2 /2, taking into account that υ=ds/dt=Aωcosωt
we get Wk=(mω 2 A2/2)*cos 2 ωt
Then the total energy is:
W=(mω 2 A 2 /2)(sin 2 ωt+cos 2 ωt)=(mω 2 A 2)/2
The process of propagation of vibrations in space is called wave motion or simply a wave.
There are two types of waves: mechanical and electromagnetic. Mechanical waves propagate only in elastic media. Mechanical waves are divided into two types: transverse and longitudinal.
If the particles oscillate perpendicular to the direction of propagation of the wave, then it is called transverse.
If the vibrations of the particles coincide with the direction of propagation of the wave, then it is called longitudinal
Let's consider the main characteristics of wave motion. These include:
1. All parameters of the oscillatory process (s, A, v, ω, T, φ).
2. Additional parameters characterizing only wave motion:
a) Phase velocity (υ) is the speed at which vibrations propagate in space.
b) Wavelength (λ) is the shortest distance between two particles of wave space oscillating in the same phases or the distance over which a wave propagates during one period. Characteristics are related to each other : λ=υT, λ=υv
The oscillatory motion of any particle of wave space is determined by the wave equation. Let oscillations occur at point O according to the law : S = A sinωt
Then at an arbitrary point C the law of oscillations is: s c = sinω (t-∆t), where ∆t=x/υ=x/λv, xc=Asin(2πv t-(2πvx/λx))
s = Asin (ωt-2πх/λ) is the wave equation. It determines the law of oscillation at any point in wave space 2πх/λ = φ 0 is called the initial phase of oscillation at an arbitrary point in space.
3. Energy characteristics of the wave:
A. Energy of vibration of one particle: W = (mω 2 A 2)/2
b. The energy of vibration of all particles contained in a unit volume of wave space is called volumetric energy density: ε = W 0 /V
Where W o = εV is the total energy of all vibrating particles in any volume.
If n 0 is the concentration of particles, then ε = n 0 W = n 0 mω 2 A 2 /2 , But n o m = p , Then ε = (pω 2 A 2)/2
The energy of vibration is constantly transferred to other particles in the direction of propagation of the wave.
A value numerically equal to the average value of energy transferred by a wave per unit time through a certain surface perpendicular to the direction of propagation of the wave is called the energy flow through this surface.
Ps=W 0 /t (W)
The energy flux per unit surface area is called the energy flux density or wave intensity.
J=Ps/s = W 0 /st (W)
A special case of mechanical waves are sound waves:
Sound waves are vibrations of particles propagating in elastic media in the form of longitudinal waves with a frequency from 16 to 20,000 Hz.
For sound waves, the same characteristics apply as for any wave process, but there are some specifics.
1. The intensity of a sound wave is called sound power. J=Ps/s (W/m2)
For this value, special units of measurement are adopted: Bels (B) and decibels (dB). The sound intensity scale, expressed in B or dB, is called logarithmic. To convert from the SI system to the logarithmic scale, the following formula is used: J (s) = LgJ/J 0 (W/m 2)
where J o = 10 -12 W/m 2 is a certain threshold intensity.
2. To describe sound waves, a quantity called sound pressure.
Sound or acoustic pressure is the additional pressure (excessive above the average ambient pressure) in the places of greatest concentration of particles in a sound wave.
In the SI system it is measured in Pa, and the non-systemic unit is 1 acoustic bar = 10 -1 Pa.
3. The shape of vibrations of particles in a sound wave, which is determined by the harmonic spectrum of sound vibrations (∆v), is also important.
All of the listed physical characteristics of sound are called objective, i.e. independent of our perception. They are determined using physical instruments. Our hearing aid is able to differentiate (distinguish) sounds in pitch, timbre and volume. These characteristics of the auditory sensation are called subjective. A change in the perception of sound by ear is always associated with a change in the physical parameters of the sound wave.
Pitch is determined mainly by the frequency of vibrations in the sound wave and slightly depends on the strength of the sound. The higher the frequency, the higher the pitch of the sound. In this regard, the range of sounds perceived by the hearing aid is divided into octaves: 1- (16-32) Hz; 2 -(32-64)Hz; 3-(64-128) Hz; etc., 10 octaves in total.
If the vibrations of particles in a sound wave are harmonic, then such a tone of sound is called simple or pure. Such sounds are produced by a tuning fork and a sound generator.
If the vibrations are not harmonic, but periodic, then such a tone of sound is called complex. .
If complex sound vibrations do not periodically change their intensity, frequency and phase, then such sound is usually called noise.
Complex tones of the same pitch, in which the vibration shape is different, are perceived differently by a person (for example, the same note on different musical instruments). This difference in perception is called timbre sound. It is determined by the frequency spectrum of harmonic vibrations that make up a complex sound.
Volume The perception of sound depends mainly on the strength of the sound, as well as on the frequency. This dependence is determined by the psychophysical law Weber-Fechner:
As the sound intensity increases in geometric progression (J, J 2, J 3,...), the sensation of loudness at the same frequency increases in arithmetic progression (E, 2E, ZE,...).
E=kLg J/J 0
where k is a coefficient depending on the sound frequency. Loudness is measured in the same way as sound intensity in Bels (B) and decibels (dB). The dB volume is called background (F), in contrast to the dB sound intensity. Conventionally, it is believed that for a frequency of 1000 Hz, the volume and sound intensity scales completely coincide, i.e. k = 1.
The use of sound methods in diagnostics
1. Audiometry- a method of measuring hearing acuity by perceiving sounds standardized in frequency and intensity.
2. Auscultation- listening to sounds arising from the work of various organs (heart, lungs, blood vessels and etc.)
3. Percussion- listening to the sound of individual parts of the body when they are tapped.
Ultrasound is the process of propagation of vibrations in a high-pressure medium in the form of longitudinal waves with a frequency of over 20 kHz.
Ultrasound is obtained using special devices based on the phenomena of magnetostriction - at low frequencies and the inverse piezoelectric effect - at high frequencies.
Vibrations excited at any point in the medium (solid, liquid or gaseous) propagate in it at a finite speed, depending on the properties of the medium, being transmitted from one point of the medium to another. The further a particle of the medium is located from the source of oscillation, the later it will begin to oscillate. In other words, the entrained particles will be out of phase with the particles that entrain them.
When studying the propagation of vibrations, the discrete (molecular) structure of the medium is not taken into account. The medium is considered as continuous, i.e. continuously distributed in space and having elastic properties.
So, an oscillating body placed in an elastic medium is a source of vibrations spreading from it in all directions. The process of propagation of vibrations in a medium is called wave.
When a wave propagates, the particles of the medium do not move with the wave, but oscillate around their equilibrium positions. Together with the wave, only the state of vibrational motion and energy are transferred from particle to particle. That's why the main property of all waves,regardless of their nature,is the transfer of energy without the transfer of matter.
There are waves transverse (vibrations occur in a plane perpendicular to the direction of propagation) And longitudinal (condensation and rarefaction of particles of the medium occurs in the direction of propagation).
where υ is the speed of wave propagation, – period, ν – frequency. From here, the speed of wave propagation can be found using the formula:
| . | (5.1.2) |
The geometric location of points oscillating in the same phase is called wave surface. The wave surface can be drawn through any point in space covered by the wave process, i.e. There are an infinite number of wave surfaces. The wave surfaces remain stationary (they pass through the equilibrium position of particles oscillating in the same phase). There is only one wavefront, and it moves all the time.
Wave surfaces can be of any shape. In the simplest cases, wave surfaces have the shape plane or spheres, respectively, the waves are called flat or spherical . In a plane wave, the wave surfaces are a system of planes parallel to each other, in a spherical wave - a system of concentric spheres.
The process of propagation of vibrations in an elastic medium is called a wave. The distance over which a wave propagates in a time equal to the oscillation period is called the wavelength. The wavelength is related to the period of vibration of the particles T and wave propagation speed υ ratio
λ = υT or λ = υ /ν,
where ν = 1/ T– vibration frequency of medium particles.
If two waves of the same frequency and amplitude propagate towards each other, then as a result of their superposition, under certain conditions, a standing wave can arise. In a medium where standing waves are established, particle vibrations occur with different amplitudes. At certain points in the medium the amplitude of the oscillation is zero, these points are called nodes; at other points the amplitude is equal to the sum of the amplitudes of the added oscillations; such points are called antinodes. The distance between two adjacent nodes (or antinodes) is equal to l/2, where l is the length of the traveling wave (Fig. 1).
A standing wave can be formed by the superposition of incident and reflected waves. Moreover, if the reflection occurs from a medium many times denser than the medium in which the wave propagates, then in place
Rice. 1 reflection, the displacement of particles is zero, that is, the image
there is a node. If a wave is reflected from a less dense medium, then due to the weak retarding effect of particles of the second medium, oscillations with double amplitude occur at the boundary, that is, an antinode is formed. In the case when the densities of the media differ little from each other, partial reflection of waves from the interface between the two media is observed.
Let us consider standing waves that are formed in a pipe with air of length l, closed on both sides (Fig. 1, A). Through a small hole at one end of the pipe, using a speaker, we excite vibrations of sound frequency. Then a sound wave will spread in the air inside the pipe, which will be reflected from the other closed end and run back. It would seem that a standing wave should arise at any oscillation frequency. However, in a pipe that is closed on both sides, knots should form at the ends. This condition is satisfied if half the length of the traveling wave fits in the pipe: l= l/2 (Fig. 1, b). Here, the displacement amplitudes of air particles are plotted vertically. The solid line shows a traveling wave, the dotted line shows a reflected wave. In a pipe, such a standing wave is also possible, where there is one more node, and in this case two halves of the wavelength fit: l= 2l/2 (Fig. 1, V). The next standing wave occurs when the traveling wave length satisfies the condition l= 3λ/2 (Fig. 1, G). Thus, in a pipe closed on both sides, a standing wave is formed in those cases when an integer number of half wavelengths fit along the length of the pipe:
Where m= 1, 2, 3. Expressing l from (1) and substituting ν = into the formula υ /λ,
The resulting formula expresses the natural frequencies of oscillations of the air column in a pipe of length l, Where m= 1 corresponds to the fundamental tone, m= 2, 3 – overtones. In general, the oscillation of an air column can be represented as a superposition of its own oscillations.
Loading...