How to determine the initial phase of oscillations by the equation. initial phase. Oscillation phase. The quality factor of the circuit determines the sharpness of the resonant

The waves look like

Equations of a plane monochromatic electromagnetic

Instantaneous values at any point are related by the relation

oscillate in the same phases, and their

The plane perpendicular to the propagation velocity vector

The magnetic fields are mutually perpendicular and lie in

Electromagnetic waves are transverse,

The environments are determined by the formula

The phase velocity of electromagnetic waves in various

Wave.

The space process and is an electromagnetic

Point to another. This periodic in time and

Spreading in the surrounding space from one

Mutual transformations of electric and magnetic fields,

electromagnetic field, then a sequence arises

To excite a variable with vibrating charges

Maxwell's equations for electromagnetic field. If a

The existence of electromagnetic waves follows from

Electromagnetic waves

Shimi, will be weak. Thus, for example,

The voltage created on the capacitor by other components

Exceeding the value of this component, while

Ideal stresses, the desired component. Having set up

complex tension, equal to the sum several sine

The phenomenon of resonance is used to isolate from

Equal to the value of the inverse quality factor of the circuit, i.e.

Relative width of the resonance curve

The quality factor of the circuit determines the sharpness of the resonant

Loop resistance.

So the quality factor is inversely proportional to

With cut U

The capacitor can exceed the applied voltage, i.e.

The resonant properties of the circuit are characterized by the quality

A steady current in a circuit with a capacitor cannot flow.

Ires LC

Coincides with the natural frequency of the circuit

Therefore, the resonant frequency for the current strength

Rice. 1.22

R1< R2 < R3

    . (1.96)

At ω →0, I= 0, since at a constant voltage

ness Q, which shows how many times the voltage on

 (1.97)

At low damping ω res≈ ω0 and

Q  1 (1.98)

curves. On fig. 1.23 shows one of the resonance curves

for the current in the circuit. Frequencies ω1 and ω2 correspond to the current

max I I 2 .

  

contour (by changing R and C) to the required frequency

, you can get a voltage on the capacitor in Q once

tune the radio to the desired wavelength.

    1 0 2

mmax I

Rice. 1.7

Fig.1.23

 , (1.100)

 - speed of electromagnetic waves in vacuum.

since the vectors E

and H

electrical and

wave formation, forming a right-handed system (Fig. 1.24). At

this vectors E

and H

0 0   E  N. (1.101)

cos() m Е  Е t  kx  , (1.102)

cos() m H  H t  kx  , (1.103)

where ω is the wave frequency, k = ω/υ = 2π/λ is the wave number, α-

Fig.1.24

Electromagnetic waves carry energy. Volumetric

Oscillatory processes are an important element modern science and technology, so their study has always been given attention as one of the "eternal" problems. The task of any knowledge is not mere curiosity, but its use in Everyday life. And for this there are and daily there are new technical systems and mechanisms. They are in motion, they manifest their essence by performing some kind of work, or, being motionless, they retain the potential opportunity, under certain conditions, to move into a state of motion. What is movement? Without delving into the wilds, we will accept the simplest interpretation: a change in the position material body relative to any coordinate system, which is conventionally considered fixed.

Among huge amount of possible variants of movement, of particular interest is the oscillatory one, which differs in that the system repeats the change in its coordinates (or physical quantities) at certain intervals - cycles. Such oscillations are called periodic or cyclic. Among them, a separate class is distinguished in which characteristics(speed, acceleration, position in space, etc.) change in time according to a harmonic law, i.e. having a sinusoidal shape. A remarkable property of harmonic oscillations is that their combination represents any other options, incl. and inharmonious. A very important concept in physics is the “oscillation phase”, which means fixing the position of an oscillating body at some point in time. The phase is measured in angular units - radians, quite conditionally, just as a convenient technique for explaining periodic processes. In other words, the phase determines the value of the current state oscillatory system. It cannot be otherwise - after all, the phase of oscillations is an argument of the function that describes these oscillations. true value phase for a character can mean coordinates, speed and others physical parameters, changing according to the harmonic law, but the common thing for them is the time dependence.

Demonstrating vibrations is not at all difficult - for this you need the simplest mechanical system - a thread, length r, and suspended on it “ material point”- a weight. We fix the thread in the center of the rectangular coordinate system and make our “pendulum” spin. Let us assume that he willingly does this with angular velocity w. Then, during the time t, the angle of rotation of the load will be φ = wt. Additionally, this expression should take into account the initial phase of the oscillations in the form of the angle φ0 - the position of the system before the start of movement. So, the total angle of rotation, phase, is calculated from the relation φ = wt + φ0. Then the expression for the harmonic function, and this is the projection of the load coordinate on the X axis, can be written:

x \u003d A * cos (wt + φ0), where A is the vibration amplitude, in our case equal to r - the radius of the thread.

Similarly, the same projection on the Y axis will be written as follows:

y \u003d A * sin (wt + φ0).

It should be understood that the phase of oscillations in this case does not mean the measure of rotation “angle”, but angle measure time, which expresses time in units of angle. During this time, the load rotates through a certain angle, which can be uniquely determined based on the fact that for cyclic oscillations w = 2 * π /T, where T is the oscillation period. Therefore, if one period corresponds to a rotation of 2π radians, then part of the period, time, can be proportionally expressed by an angle as a fraction of a full rotation of 2π.

Vibrations do not exist by themselves - sounds, light, vibration are always a superposition, an overlay, a large number fluctuations from different sources. Of course, the result of the superposition of two or more oscillations is influenced by their parameters, incl. and phase of oscillation. The formula of the total oscillation, as a rule, is non-harmonic, while it can have a very complex view but that only makes it more interesting. As mentioned above, any non-harmonic oscillation can be represented as a large number harmonic with different amplitude, frequency and phase. In mathematics, such an operation is called “expansion of a function in a series” and is widely used in calculations, for example, the strength of structures and structures. The basis of such calculations is the study of harmonic oscillations, taking into account all parameters, including the phase.

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

Oscillation phase- a physical quantity used primarily to describe harmonic or close to harmonic oscillations, changing with time (most often growing uniformly with time), at a given amplitude (for damped oscillations - at a given initial amplitude and damping coefficient) determining the state of the oscillatory system in ( any) at a given point in time. It is also 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 from an arbitrary origin. The origin of coordinates is usually considered the moment of the previous transition of the function through zero in the direction from negative values to the positive.

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

For such fluctuations:

, , ,

or the 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 an argument of this function(one of the listed, in each case it is clear from the context which one), which describes a harmonic oscillatory process or a monochromatic wave.

That is, for phase oscillation

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), measuring it in cycles or periods (with the exception of verbal formulations) is generally quite rare, but measuring in degrees is quite common (apparently, as 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 are used that are close to monochromatic, but not strictly monochromatic, and also in the path integral formalism, where waves can be far from monochromatic, although still similar to monochromatic), the phase is considered as depending on time and space coordinates not like linear function, but as, in principle, an arbitrary function of coordinates and time:

Related terms

If two waves (two oscillations) completely coincide with each other, the waves are said to be in phase. In the event that the moments of the maximum of one oscillation coincide with the moments of the minimum of another oscillation (or the maxima of one wave coincide with the minimums of the other), they say that the oscillations (waves) are in antiphase. In this case, if the waves are the same (in amplitude), as a result of the addition, their mutual annihilation 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 the "Phase of Oscillations" is in other dictionaries:

The periodically changing argument of the function describing the oscillations. or waves. process. In harmonic. oscillation u(х,t)=Acos(wt+j0), where wt+j0=j F. c., А 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 value that varies according to the law harmonic oscillation. [GOST 7601 78] Topics optics, optical instruments and measurements Generalizing terms oscillations and waves EN phase of oscillation DE Schwingungsphase FR… … Technical Translator's Handbook 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 course of the development of any process (social, geological, physical, etc.). In physics and technology, the phase of oscillations of the state is especially important. oscillatory process in a certain ... ... Modern Encyclopedia

- (from the Greek phasis appearance) ..1) a certain moment in the course of the development of any process (social, geological, physical, etc.). In physics and technology, the phase of oscillations is especially important, the state of an oscillatory process in 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

s; well. [from Greek. phasis appearance] 1. A separate stage, period, stage of development of what l. phenomena, processes, etc. The main phases of the development of society. The phases of the process of interaction between the animal and flora. Enter your new, decisive, ... ... encyclopedic Dictionary

>> Oscillation phase

§ 23 PHASE OF OSCILLATIONS

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

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

The value under the sign of the cosine or sine function is called the phase of the oscillations 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, such as velocity and acceleration, which also change according to the harmonic law. Therefore, we can say 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.

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, expressed in radians. So, after the lapse of time t \u003d (quarter of the period), after the lapse of half of the period = , after the lapse of the whole period = 2, 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 the horizontal axis plots different phase values 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:

But in this case, the initial phase, i.e., the value of the phase at the 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 shift from the hypoposition of equilibrium 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 = x m sin 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 , then the oscillation equation can be written as

x = xm 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 for oscillations shifted in phase by . Graph 1 corresponds to oscillations that occur according to the sinusoidal law: x \u003d x m sin 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.

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 frequency of oscillation of a body attached to a spring depend on its mass, and the frequency of oscillation 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!

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Please, format it according to the rules for formatting articles.

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

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 from 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.

For such fluctuations:

, , ,

or the waves

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

That is, for phase oscillation

for a wave in one-dimensional space

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

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), measuring it in cycles or periods (with the exception of verbal formulations) is generally quite rare, but measuring in degrees is quite common (apparently, as 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 are used that are close to monochromatic, but not strictly monochromatic, and also in the path integral formalism, where waves can be far from monochromatic, although still similar to monochromatic), the phase is considered as depending on time and space coordinates not as a linear function, but as a basically arbitrary function of coordinates and time:

Related terms

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

Wikimedia Foundation. 2010 .

See what the "Phase of Oscillations" is in other dictionaries:

- (φ) Argument of a function describing a value that changes according to the law of harmonic oscillation. [GOST 7601 78] Topics optics, optical instruments and measurements Generalizing terms oscillations and waves EN phase of oscillation DE Schwingungsphase FR… … Technical Translator's Handbook 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 course of the development of any process (social, geological, physical, etc.). In physics and technology, the phase of oscillations is especially important, the state of an oscillatory process in a certain ... ... Modern Encyclopedia

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

s; well. [from Greek. phasis appearance] 1. A separate stage, period, stage of development of what l. phenomena, processes, etc. The main phases of the development of society. Phases of the process of interaction between the animal and plant world. Enter your new, decisive, ... ... encyclopedic Dictionary