Basic formulas in physics - vibrations and waves. Initial phase. Phase shift What is the phase of harmonic oscillations called?

Oscillations movements or processes that are characterized by a certain repeatability over time are called. Oscillations 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 vibrations. 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 are called such oscillations in which the oscillating quantity changes with time according to the law sine or cosine . Harmonic Equation has the form:, where A - vibration amplitude (the magnitude of the greatest deviation of the system from the equilibrium position); - circular (cyclic) frequency. The periodically changing argument of the cosine is 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 represents the phase value at time t = 0 and is called initial phase of oscillation .. This period of time T is called the period of harmonic oscillations. The period of harmonic oscillations is equal to : 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. 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 solid body that oscillates in a field of any forces relative to a point that is not the center of mass of this body, or a fixed axis perpendicular to the direction of action of the forces and not passing through the center of mass of this body.

24. Electromagnetic vibrations. Oscillatory circuit. Thomson's formula.

Electromagnetic vibrations- these are oscillations of electric and magnetic fields, which are accompanied by periodic changes 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 connected 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 induced current, in accordance with Lenz's rule, will have the same direction and will 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 the energy of the electric field of the capacitor () into the energy of the magnetic field of the current coil (), 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 according to Thomson's formula. Frequency and period are inversely proportional.

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Illustration of the phase difference between two oscillations of the same frequency

Oscillation phase- a physical quantity used primarily to describe harmonic or close to harmonic oscillations, varying with time (most often growing uniformly with time), at a given amplitude (for damped oscillations - at a given initial amplitude and damping coefficient) that determines the state of the oscillatory system in ( any) given point in time. It is equally used to describe waves, mainly monochromatic or close to monochromatic.

Oscillation phase(in telecommunications for a periodic signal f(t) with period T) is the fractional part t/T of period T by which t is shifted relative to an arbitrary origin. The origin of coordinates is usually considered to be the moment of the previous transition of the function through zero in the direction from negative to positive values.

In most cases, phase is spoken of in relation to harmonic (sinusoidal or imaginary exponential) oscillations (or monochromatic waves, also sinusoidal or imaginary exponential).

For such fluctuations:

, , ,

or waves

For example, waves propagating in one-dimensional space: , , , or waves propagating in three-dimensional space (or space of any dimension): , , ,

the oscillation phase is defined as the argument of this function(one of the listed ones, in each case it is clear from the context which one), describing a harmonic oscillatory process or a monochromatic wave.

That is, for the oscillation phase

,

for a wave in one-dimensional space

,

for a wave in three-dimensional space or space of any other dimension:

,

where is the angular frequency (the higher the value, the faster the phase grows over time), t- time, - phase at t=0 - initial phase; k- wave number, x- coordinate, k- wave vector, x- a set of (Cartesian) coordinates characterizing a point in space (radius vector).

The phase is expressed in angular units (radians, degrees) or in cycles (fractions of a period):

1 cycle = 2 radians = 360 degrees.

• In physics, especially when writing formulas, the radian representation of the phase is predominantly (and by default) used; its measurement in cycles or periods (except for verbal formulations) is generally quite rare, but measurement in degrees occurs quite often (apparently, as extremely explicit and not leading to confusion, since it is customary to never omit the degree sign either in speech or in writing), especially often in engineering applications (such as electrical engineering).

Sometimes (in the semiclassical approximation, where waves close to monochromatic, but not strictly monochromatic, are used, as well as in the formalism of the path integral, where waves can be far from monochromatic, although still similar to monochromatic) the phase is considered as depending on time and spatial coordinates not as a linear function, but as a basically arbitrary function of coordinates and time:

Related terms

If two waves (two oscillations) completely coincide with each other, they say that the waves are located in phase. If the moments of maximum of one oscillation coincide with the moments of minimum of another oscillation (or the maxima of one wave coincide with the minima of another), they say that the oscillations (waves) are in antiphase. Moreover, if the waves are identical (in amplitude), as a result of addition, their mutual destruction occurs (exactly, completely - only if the waves are monochromatic or at least symmetrical, assuming the propagation medium is linear, etc.).

Action

One of the most fundamental physical quantities on which the modern description of almost any sufficiently fundamental physical system is built - action - in its meaning is a phase.

Notes

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See what “Oscillation phase” is in other dictionaries:

A periodically changing argument of the function describing the oscillation. or waves. process. In harmonious oscillations u(x,t)=Acos(wt+j0), where wt+j0=j F.c., A amplitude, w circular frequency, t time, j0 initial (fixed) F.c. (at time t =0,… … Physical encyclopedia

oscillation phase- (φ) Argument of a function describing a quantity that changes according to the law of harmonic oscillation. [GOST 7601 78] Topics: optics, optical instruments and measurements General terms of oscillations and waves EN phase of oscillation DE Schwingungsphase FR… … Technical Translator's Guide Phase - Phase. Oscillations of pendulums in the same phase (a) and antiphase (b); f is the angle of deviation of the pendulum from the equilibrium position. PHASE (from the Greek phasis appearance), 1) a certain moment in the development of any process (social, ... ... Illustrated Encyclopedic Dictionary

- (from the Greek phasis appearance), 1) a certain moment in the development of any process (social, geological, physical, etc.). In physics and technology, the oscillation phase is the state of the oscillatory process at a certain... ... Modern encyclopedia

- (from the Greek phasis appearance) ..1) a certain moment in the development of any process (social, geological, physical, etc.). In physics and technology, the oscillation phase is the state of the oscillatory process at a certain... ... Big Encyclopedic Dictionary

Phase (from the Greek phasis √ appearance), period, stage in the development of a phenomenon; see also Phase, Oscillation phase... Great Soviet Encyclopedia

Y; and. [from Greek phasis appearance] 1. A separate stage, period, stage of development of which l. phenomenon, process, etc. The main phases of the development of society. Phases of the process of interaction between flora and fauna. Enter into your new, decisive,... encyclopedic Dictionary

Oscillations are a process of changing the states of a system around the equilibrium point that is repeated to varying degrees over 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 is the amplitude of oscillations, this is the value that determines the maximum deviation of the oscillating point from the equilibrium position; ω - cyclic frequency, a value indicating the number of complete oscillations occurring within 2π seconds - the full phase of oscillations, 0 - the initial phase of oscillations.

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

The amplitude and initial phase of oscillations are determined by the initial conditions of movement, i.e. position and speed of the 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 the displacement relative to equilibrium (the 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 vibration in sound waves is determined by the frequency of vibration of the source. High frequency oscillations decay faster than low frequency ones.

The reciprocal of the oscillation frequency is called period T.

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

In the coordinate system, from point 0 we draw a vector A̅, the projection of which onto the OX axis is equal to Аcosϕ. If the vector A̅ 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 amplitude of the oscillations is the modulus of the uniformly rotating vector A̅, the oscillation phase (ϕ ) is the angle between the vector A̅ and the OX axis, the initial phase (ϕ˳) is the initial value of this angle, the angular frequency of oscillations (ω) is the angular velocity of rotation of the vector A̅..

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

The surface separating at a given moment in 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 wave front leaves, 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. They are reflected and refracted at the interface between 2 media.

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

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

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

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

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

Free (natural) vibrations are called those that occur without external influences due to the energy initially obtained by heat. Characteristic models of such mechanical oscillations 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 initial energy (deviation of a material point from the position of equilibrium and motion without initial speed), or due to kinetic (the body is imparted speed in the initial equilibrium position), or due to both energy (imparting speed 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 you pull the spring a distance x, then a large elastic force will act on the material point. 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 is such a small angle α that the trajectory of a material point can be considered a straight line coinciding with the OX axis. In this case, the approximate equality is satisfied: α ≈sin α≈ tanα ≈x/L

Undamped oscillations. Let us consider a model in which the resistance force is neglected.
The amplitude and initial phase of oscillations are determined by the initial conditions of movement, i.e. position and speed of the material point moment t=0.
Among the various types of vibrations, harmonic vibration is the simplest form.

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

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

Damped oscillations. In a real case, resistance (friction) forces act on an oscillating body, the nature of the movement changes, and the oscillation becomes damped.

In relation to one-dimensional motion, we give the last formula the following form: Fc = - r * dx/dt

The rate at which the oscillation amplitude decreases is determined by the damping coefficient: the stronger the braking 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, meaning by this a value equal to the natural logarithm of 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 is clear from the formula that the period of oscillation is an imaginary quantity. The movement in this case will no longer be periodic and is called aperiodic.

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

Let us assume that the material point, in addition to the elastic force and the friction force, is acted upon by an external driving force 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 damping 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 some 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 harmful. The harmful effects are mainly due to the destruction it can cause. Thus, in technology, taking into account various vibrations, it is necessary to provide for the possible occurrence of resonant conditions, otherwise there may be destruction and disasters. Bodies usually have several natural vibration frequencies and, accordingly, several resonant frequencies.

Resonance phenomena under the action of external mechanical vibrations occur in internal organs. This is apparently one of the reasons for the negative impact of infrasonic vibrations 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, which is why methods for studying the patient’s condition such as auscultation, percussion and phonocardiography are widely used in medicine.

Auscultation

For auscultation, a stethoscope or phonendoscope is used. A phonendoscope consists of a hollow capsule with a sound-transmitting membrane that is applied to the patient’s body, from which rubber tubes go to the doctor’s ear. A resonance of the air column occurs in the capsule, resulting in increased sound and improved auscultation. When auscultating the lungs, breathing sounds and 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 by tapping them. Let's imagine a closed cavity inside some body, filled with air. If you induce sound vibrations in this body, then at a certain frequency of sound, the air in the cavity will begin to resonate, releasing and amplifying a tone corresponding to the size and position of the cavity. The human body can be represented as a collection of gas-filled (lungs), liquid (internal organs) and solid (bones) volumes. When hitting the surface of a body, vibrations occur, the frequencies of which have a wide range. From this range, some vibrations will fade out quite quickly, while others, coinciding with the natural vibrations of the voids, will intensify and, due to resonance, will be audible.

Phonocardiography

Used to diagnose cardiac conditions. The method consists of graphically recording heart sounds and murmurs and their diagnostic interpretation. A phonocardiograph consists of a microphone, an amplifier, a system of frequency filters and a recording device.

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

1) Diagnostic and research methods

These include location methods using mainly pulsed radiation. This is echoencephalography - detection of tumors and edema of the brain. Ultrasound cardiography – measurement of heart size in dynamics; in ophthalmology - ultrasonic location to determine the size of the ocular media.

2)Methods of influence

Ultrasound physiotherapy – mechanical and thermal effects on tissue.

11. Shock wave. Production and use of shock waves in medicine.
Shock wave – a discontinuity surface that moves relative to the gas and upon crossing which the pressure, density, temperature and speed experience a jump.
Under large disturbances (explosion, supersonic movement of bodies, powerful electric 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 Thus, during a nuclear explosion, about 50% of the explosion energy is spent on the formation of a shock wave in the environment. Therefore, a shock wave, reaching biological and technical objects, can cause death, injury and destruction.

Shock waves are used in medical technology, representing an extremely short, powerful pressure pulse with high pressure amplitudes and a small stretch component. They are generated outside the patient’s body and transmitted deep into the body, producing a therapeutic effect provided for by the specialization of the equipment model: crushing urinary stones, treating pain areas and the consequences of injuries to the musculoskeletal system, stimulating the recovery of the heart muscle after myocardial infarction, smoothing cellulite formations, etc.

>> Oscillation phase

§ 23 PHASE OF OSCILLATIONS

Let us introduce another quantity characterizing harmonic oscillations - the phase of oscillations.

For a given amplitude of oscillations, the coordinate of the oscillating body at any time is uniquely determined by the cosine or sine argument:

The quantity under the sign of the cosine or sine function is called the phase of oscillation described by this function. The phase is expressed in angular units of radians.

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

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

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

You can depict on a graph the dependence of the coordinates of an oscillating point not on time, but on phase. Figure 3.7 shows the same cosine wave as in Figure 3.6, but different phase values ​​are plotted on the horizontal axis instead of time.

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

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

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

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

x = x m sin t (3.24)

since in this case the initial phase is zero.

If at the initial moment of time (at t = 0) the phase of oscillations is equal to , then the equation of oscillations can be written in the form

x = x m sin(t + )

Phase shift. 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 graphs of coordinates versus time of oscillations shifted in phase by . Graph 1 corresponds to oscillations that occur according to the sinusoidal law: x = x m sin t and graph 2 corresponds to oscillations that occur according to the cosine law:

To determine the phase difference between two oscillations, in both cases the oscillating quantity must be expressed through the same trigonometric function - cosine or sine.

1. What vibrations are called harmonic!
2. How are acceleration and coordinate related during harmonic oscillations!

3. How are the cyclic frequency of oscillations and the period of oscillation related?
4. Why does the frequency of oscillation of a body attached to a spring depend on its mass, but the frequency of oscillation of a mathematical pendulum does not depend on 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!

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But because the turns are shifted in space, then the EMF induced in them will not reach amplitude and zero values ​​at the same time.

At the initial moment of time, the EMF of the turn will be:

In these expressions the angles are called phase , or phase . The angles are called initial phase . The phase angle determines the value of the emf at any time, and the initial phase determines the value of the emf at the initial time.

The difference in the initial phases of two sinusoidal quantities of the same frequency and amplitude is called phase angle

Dividing the phase angle by the angular frequency, we obtain the time elapsed since the beginning of the period:

Graphic representation of sinusoidal quantities

U = (U 2 a + (U L - U c) 2)

Thus, due to the presence of a phase angle, the voltage U is always less than the algebraic sum U a + U L + U C. The difference U L - U C = U p is called reactive voltage component.

Let's consider how current and voltage change in a series alternating current circuit.

Impedance and phase angle. If we substitute the values ​​U a = IR into formula (71); U L = lL and U C =I/(C), then we will have: U = ((IR) 2 + 2), from which we obtain the formula for Ohm’s law for a series alternating current circuit:

I = U / ((R 2 + 2)) = U / Z (72)

Where Z = (R 2 + 2) = (R 2 + (X L - X c) 2)

The Z value is called circuit impedance, it is measured in ohms. The difference L - l/(C) is called circuit reactance and is denoted by the letter X. Therefore, the total resistance of the circuit

Z = (R 2 + X 2)

The relationship between active, reactive and impedance of an alternating current circuit can also be obtained using the Pythagorean theorem from the resistance triangle (Fig. 193). The resistance triangle A'B'C' can be obtained from the voltage triangle ABC (see Fig. 192,b) if we divide all its sides by the current I.

The phase shift angle is determined by the relationship between the individual resistances included in a given circuit. From triangle A’B’C (see Fig. 193) we have:

sin? = X/Z; cos? = R/Z; tg? = X/R

For example, if the active resistance R is significantly greater than the reactance X, the angle is relatively small. If the circuit has a large inductive or large capacitive reactance, then the phase shift angle increases and approaches 90°. Wherein, if the inductive reactance is greater than the capacitive reactance, the voltage and leads the current i by an angle; if the capacitive reactance is greater than the inductive reactance, then the voltage lags behind the current i by an angle.

An ideal inductor, a real coil and a capacitor in an alternating current circuit.

A real coil, unlike an ideal one, has not only inductance, but also active resistance, therefore, when alternating current flows in it, it is accompanied not only by a change in energy in the magnetic field, but also by the conversion of electrical energy into another form. Specifically, in the coil wire, electrical energy is converted into heat in accordance with the Lenz-Joule law.

It was previously found that in an alternating current circuit the process of converting electrical energy into another form is characterized by active power of the circuit P , and the change in energy in the magnetic field is reactive power Q .

In a real coil, both processes take place, i.e. its active and reactive powers are different from zero. Therefore, one real coil in the equivalent circuit must be represented by active and reactive elements.