fluctuations called movements or processes that are characterized by a certain repetition in time. Fluctuations are widespread in the surrounding world and can have a very different nature. These can be mechanical (pendulum), electromagnetic (oscillatory circuit) and other types of oscillations. free, or own oscillations are called oscillations that occur in a system left to itself, after it has been brought out of equilibrium by an external influence. An example is the oscillation of a ball suspended on a thread. Harmonic vibrations such oscillations are called, in which the oscillating value varies with time according to the law sinus or cosine . The equation harmonic vibrations looks like:, where A - oscillation amplitude (the value of the greatest deviation of the system from the equilibrium position); - circular (cyclic) frequency. Periodically changing cosine argument - called oscillation phase . The oscillation phase determines the displacement of the oscillating quantity from the equilibrium position at a given time t. The constant φ is the value of the phase at time t = 0 and is called the initial phase of the oscillation .. This period of time T is called the period of harmonic oscillations. The period of harmonic oscillations is : T = 2π/. Mathematical pendulum- an oscillator, which is a mechanical system consisting of a material point located on a weightless inextensible thread or on a weightless rod in a uniform field of gravitational forces. The period of small natural oscillations of a mathematical pendulum of length L motionless suspended in a uniform gravitational field with free fall acceleration g equals

and does not depend on the amplitude of oscillations and the mass of the pendulum. physical pendulum- An oscillator, which is a rigid body that oscillates in the field of any forces about a point that is not the center of mass of this body, or a fixed axis perpendicular to the direction of the forces and not passing through the center of mass of this body.

24. Electromagnetic oscillations. Oscillatory circuit. Thomson formula.

Electromagnetic vibrations- These are fluctuations in electric and magnetic fields, which are accompanied by a periodic change in charge, current and voltage. The simplest system where free electromagnetic oscillations can arise and exist is an oscillatory circuit. Oscillatory circuit- this is a circuit consisting of an inductor and a capacitor (Fig. 29, a). If the capacitor is charged and closed to the coil, then current will flow through the coil (Fig. 29, b). When the capacitor is discharged, the current in the circuit will not stop due to self-induction in the coil. The induction current, in accordance with the Lenz rule, will have the same direction and recharge the capacitor (Fig. 29, c). The process will be repeated (Fig. 29, d) by analogy with pendulum oscillations. Thus, electromagnetic oscillations will occur in the oscillatory circuit due to the conversion of energy electric field capacitor() into energy magnetic field coils with current (), and vice versa. The period of electromagnetic oscillations in an ideal oscillatory circuit depends on the inductance of the coil and the capacitance of the capacitor and is found by the Thomson formula. Frequency is inversely related to period.

Fluctuations - a process of changing the states of the system around the equilibrium point, repeating to one degree or another in time.

Harmonic oscillation - oscillations in which a physical (or any other) quantity changes over time according to a sinusoidal or cosine law. The kinematic equation of harmonic oscillations has the form

where x is the displacement (deviation) of the oscillating point from the equilibrium position at time t; A - oscillation amplitude, this is the value that determines the maximum deviation of the oscillating point from the equilibrium position; ω - cyclic frequency, a value showing the number of complete oscillations occurring within 2π seconds - complete phase oscillations, 0 - initial phase of oscillations.

Amplitude - the maximum value of the displacement or change of a variable from the average value during oscillatory or wave motion.

The amplitude and initial phase of the oscillations are determined by the initial conditions of motion, i.e. position and speed of a material point at the moment t=0.

Generalized harmonic oscillation in differential form

the amplitude of sound waves and audio signals usually refers to the amplitude of the air pressure in the wave, but is sometimes described as the amplitude of displacement from equilibrium (air or the speaker's diaphragm)

Frequency is a physical quantity, a characteristic of a periodic process, equal to the number of complete cycles of the process completed per unit of time. The frequency of oscillations in sound waves is determined by the oscillation frequency of the source. High frequency vibrations decay faster than low frequency vibrations.

The reciprocal of the oscillation frequency is called the period T.

The oscillation period is the duration of one complete cycle of oscillations.

In the coordinate system from the point 0 we draw the vector А̅, the projection of which on the OX axis is equal to Аcosϕ. If the vector А̅ rotates uniformly with an angular velocity ω˳ counterclockwise, then ϕ=ω˳t + ϕ˳, where ϕ˳ is the initial value of ϕ (oscillation phase), then the oscillation amplitude is the modulus of the uniformly rotating vector А̅, the oscillation phase (ϕ ) is the angle between the vector А̅ and the ОХ axis, the initial phase (ϕ˳) -initial value this angle, the angular frequency of oscillations (ω) - angular velocity rotation of the vector А̅..

2. Characteristics of wave processes: wave front, beam, wave speed, wavelength. Longitudinal and transverse waves; examples.

The surface separating at a given moment of time the medium already covered and not yet covered by oscillations is called the wave front. At all points of such a surface, after the departure of the wave front, oscillations are established that are identical in phase.

The beam is perpendicular to the wave front. Acoustic rays, like light rays, are rectilinear in a homogeneous medium. Reflected and refracted at the interface between two media.

Wavelength - the distance between two points closest to each other, oscillating in the same phases, usually the wavelength is indicated by the Greek letter. By analogy with the waves that arise in water from a thrown stone, the wavelength is the distance between two adjacent wave crests. One of the main characteristics of vibrations. Measured in units of distance (meters, centimeters, etc.)

• longitudinal waves (compression waves, P-waves) - the particles of the medium oscillate parallel(along) the direction of wave propagation (as, for example, in the case of sound propagation);
• transverse waves (shear waves, S-waves) - the particles of the medium oscillate perpendicular wave propagation direction ( electromagnetic waves, waves on media separation surfaces);

The angular frequency of oscillations (ω) is the angular velocity of rotation of the vector А̅(V), the displacement x of the oscillating point is the projection of the vector А̅ onto the OX axis.

V=dx/dt=-Aω˳sin(ω˳t+ϕ˳)=-Vmsin(ω˳t+ϕ˳), where Vm=Aω˳ is maximum speed(speed amplitude)

3. Free and forced vibrations. Natural frequency of oscillations of the system. Resonance phenomenon. Examples .

Free (natural) vibrations they are called those that are performed without external influences due to the energy initially received by heat. Typical models of such mechanical vibrations are a material point on a spring (spring pendulum) and a material point on an inextensible thread (mathematical pendulum).

In these examples, oscillations arise either due to the initial energy (deviation of the material point from the equilibrium position and movement without initial velocity), or due to kinetic energy (the body is given speed in the initial equilibrium position), or due to both of these energies (velocity is communicated to the body deviated from the equilibrium position).

Consider a spring pendulum. In the equilibrium position, the elastic force F1

balances the force of gravity mg. If the spring is pulled a distance x, then material point there will be a large elastic force. The change in the value of the elastic force (F), according to Hooke's law, is proportional to the change in the length of the spring or the displacement x of the point: F= - rx

Another example. The mathematical pendulum of deviation from the equilibrium position ha is such a small angle α that it is possible to consider the trajectory of the motion of a material point as a straight line coinciding with the axis OX. In this case, the approximate equality is fulfilled: α ≈sin α≈ tgα ≈x/L

Undamped vibrations. Consider a model in which the drag force is neglected.
The amplitude and initial phase of oscillations are determined by the initial conditions of motion, i.e. position and speed of the material point moment t=0.
Among the various modes of oscillation, harmonic oscillation is the simplest form.

Thus, a material point suspended on a spring or thread performs harmonic oscillations, if the resistance forces are not taken into account.

The oscillation period can be found from the formula: T=1/v=2P/ω0

damped vibrations. AT real case resistance (friction) forces act on the oscillating body, the nature of the motion changes, and the oscillation becomes damped.

With regard to one-dimensional motion, we give the last formula the following form: Fс= - r * dx/dt

The rate of decrease in the oscillation amplitude is determined by the damping coefficient: the stronger the retarding effect of the medium, the greater ß and the faster the amplitude decreases. In practice, however, the degree of damping is often characterized by a logarithmic damping decrement, understanding by this value equal to natural logarithm the ratio of two successive amplitudes separated by a time interval equal to the oscillation period, therefore, the damping coefficient and the logarithmic damping decrement are related by a fairly simple relationship: λ=ßT

With strong damping, it can be seen from the formula that the oscillation period is an imaginary quantity. The motion in this case will no longer be periodic and is called aperiodic.

Forced vibrations. Forced oscillations are called oscillations that occur in the system with the participation of an external force that changes according to a periodic law.

Let us assume that, in addition to the elastic force and the friction force, an external driving force acts on the material point F=F0 cos ωt

The amplitude of the forced oscillation is directly proportional to the amplitude of the driving force and has a complex dependence on the attenuation coefficient of the medium and the circular frequencies of natural and forced oscillations. If ω0 and ß are given for the system, then the amplitude of forced oscillations has a maximum value at a certain specific frequency of the driving force, called resonant The phenomenon itself - the achievement of the maximum amplitude of forced oscillations for given ω0 and ß - is called resonance.

The resonant circular frequency can be found from the condition of the minimum denominator in: ωres=√ωₒ- 2ß

Mechanical resonance can be both beneficial and detrimental. The harmful effect is mainly related to the destruction it can cause. So, in technology, taking into account different vibrations, it is necessary to provide for the possible occurrence of resonant conditions, otherwise there may be destruction and catastrophes. Bodies usually have several natural vibration frequencies and, accordingly, several resonant frequencies.

Resonant phenomena under the action of external mechanical vibrations occur in the internal organs. This, apparently, is one of the reasons for the negative impact of infrasonic oscillations and vibrations on the human body.

6. Sound research methods in medicine: percussion, auscultation. Phonocardiography.

Sound can be a source of information about the state of a person’s internal organs; therefore, such methods for studying a patient’s condition as auscultation, percussion, and phonocardiography are well-spread in medicine.

Auscultation

For auscultation, a stethoscope or phonendoscope is used. The phonendoscope consists of a hollow capsule with a sound-transmitting membrane applied to the patient's body, rubber tubes go from it to the doctor's ear. The resonance of the air column occurs in the capsule, as a result of which the sound is amplified and auscultation improves. During auscultation of the lungs, breath sounds, various wheezing, characteristic of diseases, are heard. You can also listen to the heart, intestines and stomach.

Percussion

In this method, the sound of individual parts of the body is listened to when they are tapped. Imagine a closed cavity inside some body, filled with air. If called in this body sound vibrations, then at a certain frequency of sound, the air in the cavity will begin to resonate, highlighting and amplifying a tone corresponding to the size and position of the cavity. The human body can be represented as a combination of gas-filled (lungs), liquid (internal organs) and solid (bones) volumes. When hitting the surface of the body, oscillations occur, the frequencies of which have a wide range. From this range, some oscillations will die out rather quickly, while others, coinciding with the natural oscillations of the voids, will intensify and, due to resonance, will be audible.

Phonocardiography

It is used to diagnose the state of cardiac activity. The method consists in graphic recording of heart sounds and murmurs and their diagnostic interpretation. The phonocardiograph consists of a microphone, an amplifier, a system of frequency filters and a recording device.

9. Ultrasonic research methods (ultrasound) in medical diagnostics.

1) Methods of diagnostics and research

They include location methods using mainly impulsive radiation. This is echoencephalography - the definition of tumors and swelling of the brain. Ultrasound cardiography - measuring the size of the heart in dynamics; in ophthalmology - ultrasonic location for determining the size of the eye media.

2) Methods of influence

Ultrasonic physiotherapy - mechanical and thermal effects on the tissue.

11. Shock wave. Production and use of shock waves in medicine.
shock wave – discontinuity surface, which moves relative to the gas and at the intersection of which the pressure, density, temperature and velocity experience a jump.
With large disturbances (explosion, supersonic motion of bodies, powerful electrical discharge etc.) the speed of oscillating particles of the medium can become comparable to the speed of sound , a shock wave occurs.

The shock wave can have significant energy, so, at nuclear explosion to the formation of a shock wave in environment about 50% of the energy of the explosion is expended. Therefore, the shock wave, reaching biological and technical objects, is capable of causing death, injury and destruction.

Shock waves are used in medical technology, which are extremely short, powerful impulse pressures with high pressure amplitudes and a small stretching component. They are generated outside the patient's body and transmitted deep into the body, producing a therapeutic effect, provided by the specialization of the equipment model: crushing of urinary stones, treatment of pain zones and consequences of injuries of the musculoskeletal system, stimulation of the recovery of the heart muscle after myocardial infarction, smoothing of cellulite formations, etc.

When reading this section, keep in mind that fluctuations of different physical nature are described from a unified mathematical standpoint. Here it is necessary to clearly understand such concepts as harmonic oscillation, phase, phase difference, amplitude, frequency, oscillation period.

It must be borne in mind that in any real oscillatory system there are resistances of the medium, i.e. oscillations will be damped. To characterize the damping of oscillations, the damping coefficient and the logarithmic damping decrement are introduced.

If vibrations are made under the action of an external, periodically changing force, then such vibrations are called forced. They will be unstoppable. The amplitude of forced oscillations depends on the frequency of the driving force. When the frequency of forced oscillations approaches the frequency of natural oscillations, the amplitude of forced oscillations increases sharply. This phenomenon is called resonance.

Turning to the study of electromagnetic waves, you need to clearly understand thatelectromagnetic waveis an electromagnetic field propagating in space. The simplest system that emits electromagnetic waves is an electric dipole. If the dipole performs harmonic oscillations, then it radiates a monochromatic wave.

Formula Table: Oscillations and Waves

Physical laws, formulas, variables	Oscillation and wave formulas
Harmonic vibration equation: where x is the displacement (deviation) of the oscillating value from the equilibrium position; A - amplitude; ω - circular (cyclic) frequency; α - initial phase; (ωt+α) - phase.
Relationship between period and circular frequency:
Frequency:
Relation of circular frequency to frequency:
Periods of natural oscillations 1) spring pendulum: where k is the stiffness of the spring; 2) mathematical pendulum: where l is the length of the pendulum, g - free fall acceleration; 3) oscillatory circuit: where L is the inductance of the circuit, C is the capacitance of the capacitor.
Frequency of natural vibrations:
Addition of oscillations of the same frequency and direction: 1) the amplitude of the resulting oscillation where A 1 and A 2 are the amplitudes of the component oscillations, α 1 and α 2 - the initial phase of the components of the oscillations; 2) the initial phase of the resulting oscillation
Damped oscillation equation: e \u003d 2.71 ... - the base of natural logarithms.
Amplitude of damped oscillations: where A 0 - amplitude at the initial time; β - damping factor;
Attenuation factor: oscillating body where r is the coefficient of resistance of the medium, m - body weight; oscillatory circuit where R is active resistance, L is the inductance of the circuit.
Frequency of damped oscillations ω:
Period of damped oscillations T:
Logarithmic damping decrement:
Relationship between logarithmic decrement χ and damping factor β:

We introduce one more quantity characterizing harmonic oscillations, - oscillation phase.

For a given oscillation amplitude, the coordinate of an oscillating body at any time is uniquely determined by the cosine or sine argument: φ = ω 0 t.

The value φ, which is under the sign of the cosine or sine function, is called oscillation phase described by this function. The phase is expressed in angular units - radians.

The phase determines not only the value of the coordinate, but also the value of other physical quantities, for example, speed and acceleration, which also change according to a harmonic law. Therefore, it can be said that the phase determines the state of the oscillatory system at a given amplitude at any time. This is the meaning of the concept of phase.

Oscillations with the same amplitudes and frequencies may differ in phase.

Since then

The ratio indicates how many periods have passed since the start of oscillations. Any value of time t, expressed in the number of periods T, corresponds to the value of the phase φ, expressed in radians. So, after the passage of time (a quarter of the period), after the passage of half of the period φ = π, after the passage of a whole period φ = 2π, etc.

It is possible to depict on a graph the dependence of the coordinate of an oscillating point not on time, but on phase. Figure 3.7 shows the same cosine wave as in Figure 3.6, but on the horizontal axis, different values ​​of the phase φ are plotted instead of time.

Representation of harmonic oscillations using cosine and sine. You already know that with harmonic oscillations, the coordinate of the body changes with time according to the law of cosine or sine. After introducing the concept of a phase, we will dwell on this in more detail.

The sine differs from the cosine by the shift of the argument by , which corresponds, as can be seen from equation (3.21), to a time interval equal to a quarter of the period:

Therefore, instead of the formula x \u003d x m cos ω 0 t, you can use the formula to describe harmonic oscillations

But at the same time initial phase, i.e. the value of the phase at time t = 0, is not equal to zero, but .

Usually, we excite the oscillations of a body attached to a spring, or the oscillations of a pendulum, by removing the pendulum body from its equilibrium position and then releasing it. The displacement from the equilibrium position is maximum at the initial moment. Therefore, to describe oscillations, it is more convenient to use formula (3.14) using the cosine than formula (3.23) using the sine.

But if we excited oscillations of a body at rest with a short-term push, then the coordinate of the body at the initial moment would be equal to zero, and it would be more convenient to describe changes in the coordinate with time using a sine, i.e., by the formula

x \u003d x m sin ω 0 t, (3.24)

since in this case the initial phase is equal to zero.

If at the initial moment of time (at t - 0) the oscillation phase is equal to φ, then the oscillation equation can be written as

x \u003d x m sin (ω 0 t + φ).

The oscillations described by formulas (3.23) and (3.24) differ from each other only in phases. The phase difference, or, as is often said, the phase shift of these oscillations is . Figure 3.8 shows plots of coordinates versus time for two harmonics shifted in phase by . Graph 1 corresponds to oscillations that occur according to the sinusoidal law: x \u003d x m sin ω 0 t, and graph 2 corresponds to oscillations that occur according to the cosine law:

To determine the phase difference of two oscillations, it is necessary in both cases to express the oscillating value through the same trigonometric function- cosine or sine.

Questions for the paragraph

1. What oscillations are called harmonic?

2. How are acceleration and coordinate related in harmonic oscillations?

3. How are the cyclic frequency of oscillations and the period of oscillations related?

4. Why does the oscillation frequency of a body attached to a spring depend on its mass, while the oscillation frequency of a mathematical pendulum does not depend on the mass?

5. What are the amplitudes and periods of three different harmonic oscillations, the graphs of which are presented in figures 3.8, 3.9?