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 . Harmonic vibration equation 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 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 using the Thomson formula. Frequency is inversely related to period.
Another characteristic of harmonic oscillations is the phase of oscillations.
As we already know, with a given amplitude of oscillations, at any time we can determine the coordinate of the body. It will be uniquely specified by the argument of the trigonometric function φ = ω0*t. The value of φ, which is under the sign of the trigonometric function, called the oscillation phase.
For phase, the units are radians. The phase uniquely determines not only the coordinate of the ted at any moment of time, but also the speed or acceleration. Therefore, it is believed that the phase of oscillations determines the state of the oscillatory system at any time.
Of course, provided that the amplitude of the oscillations is given. Two oscillations that have the same frequency and period of oscillation may differ from each other in phase.
- φ = ω0*t = 2*pi*t/T.
If we express the time t in the number of periods that have passed since the beginning of the oscillations, then any value of time t corresponds to the value of the phase, expressed in radians. For example, if we take the time t = T/4, then this value will correspond to the value of the phase pi/2.
Thus, we can plot the dependence of the coordinate not on time, but on phase, and we will get exactly the same dependence. The following figure shows such a graph.
Initial phase of oscillation
When describing the coordinate of the oscillatory motion, we used the sine and cosine functions. For cosine, we wrote the following formula:
- x = Xm*cos(ω0*t).
But we can describe the same trajectory of motion with the help of a sine. In this case, we need to shift the argument by pi / 2, that is, the difference between the sine and the cosine is pi / 2 or a quarter of the period.
- x=Xm*sin(ω0*t+pi/2).
The value of pi/2 is called the initial phase of the oscillation. The initial phase of the oscillation is the position of the body at the initial moment of time t = 0. In order to make the pendulum oscillate, we must remove it from the equilibrium position. We can do this in two ways:
- Take him aside and let him go.
- Hit him.
In the first case, we immediately change the coordinate of the body, that is, at the initial moment of time, the coordinate will be equal to the value of the amplitude. To describe such an oscillation, it is more convenient to use the cosine function and the form
- x = Xm*cos(ω0*t),
or the formula
- x = Xm*sin(ω0*t+&phi),
where φ is the initial phase of the oscillation.
If we hit the body, then at the initial moment of time its coordinate is equal to zero, and in this case it is more convenient to use the form:
- x = Xm*sin(ω0*t).
Two oscillations that differ only in the initial phase are said to be out of phase.
For example, for oscillations described by the following formulas:
- x = Xm*sin(ω0*t),
- x = Xm*sin(ω0*t+pi/2),
the phase shift is pi/2.
Phase shift is also sometimes referred to as phase difference.
Functions cos (wt + j), which describes a harmonic oscillatory process (w√ circular frequency, t √ time, j√ initial F. c., i.e. F. c. at the initial moment of time t = 0). The F. c. is determined up to an arbitrary term that is a multiple of 2p. Usually, only the differences between F. to. various harmonic processes are significant. For oscillations of the same frequency, the difference between F. c. is always equal to the difference between the initial F. c. j1 √ j2 and does not depend on the origin of time. For oscillations of different frequencies w1 and w2, the phase relations are characterized by the reduced difference of the F. c. j1 - (w1 / w2) × j2, which is also independent of the origin of time. Auditory perception of the direction of sound arrival is associated with the difference in F. to. waves coming to one and the other ear.
Wikipedia
Oscillation phase
Oscillation phase total - the argument of a periodic function that describes an oscillatory or wave process.
Oscillation phase initial - the value of the oscillation phase at the initial moment of time, i.e. at t= 0 , as well as at the initial moment of time at the origin of the coordinate system, i.e. at t= 0 at point ( x, y, z) = 0 .
Oscillation phase Counted from the zero-crossing point of the value to a positive value.
As a rule, one speaks of phase in relation to harmonic oscillations or monochromatic waves. When describing a quantity experiencing harmonic oscillations, for example, one of the expressions is used:
A cos( ω t + φ ), A sin( ω t + φ ), Ae.Similarly, when describing a wave propagating in one-dimensional space, for example, expressions of the form are used:
A cos( kx − ω t + φ ), A sin( kx − ω t + φ ), Ae,for a wave in space of any dimension:
$A \cos(\mathbf k\cdot \mathbf r - \omega t + \varphi _0)$, $A \sin(\mathbf k\cdot \mathbf r - \omega t + \varphi _0)$, $A e^(i(\mathbf k\cdot \mathbf r - \omega t + \varphi _0))$.
The phase of oscillations in these expressions is argument functions, i.e. an expression written in brackets; oscillation phase initial - magnitude φ , which is one of the terms of the total phase. Speaking of full phase, word complete often omitted.
Since the sin and cos functions coincide with each other when the argument is shifted by π /2, then, in order to avoid confusion, it is better to use only one of these two functions to determine the phase, and not both at the same time. According to the usual convention, the phase is cosine argument, not sine argument.
That is, for the oscillatory process
φ = ω t + φ ,for a wave in one-dimensional space
φ = kx − ω t + φ ,for a wave in three-dimensional space or space of any other dimension:
$\varphi = \mathbf k\mathbf r - \omega t + \varphi _0$,
where ω - angular frequency (a value showing how many radians or degrees the phase will change in 1 s; the higher the value, the faster the phase grows over time); t- time ; φ - the initial phase (that is, the phase at t = 0); k- wave number ; x- coordinate of the point of observation of the wave process in one-dimensional space; k- wave vector ; r- radius-vector of a point in space (a set of coordinates, for example, Cartesian).
In the above expressions, the phase has the dimension of angular units (radians, degrees). The phase of the oscillatory process, by analogy with the mechanical rotational process, is also expressed in cycles, that is, fractions of the period of the repeating process:
1 cycle = 2 π radian = 360 degrees.
In analytical expressions in technology, it is relatively rare.
Sometimes (in the semiclassical approximation, where quasi-monochromatic waves are used, i.e. close to monochromatic, but not strictly monochromatic) and also in the path integral formalism, where the waves can be far from monochromatic, although still similar to monochromatic), the phase is considered, which is a non-linear function of time t and spatial coordinates r, in principle - an arbitrary function:
$\varphi = \varphi(\mathbf r, t).$
>> 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 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!
Please, format it according to the rules for formatting articles.
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 uniformly growing 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 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 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 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, 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 minima 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 - is, in its meaning, 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 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
The argument of the function cos (ωt + φ), which describes the harmonic oscillatory process (ω is the circular frequency, t is time, φ is the initial F. c., i.e. F. c. at the initial moment of time t = 0). F. c. is determined up to an arbitrary term ...
initial phase of oscillation- pradinė virpesių fazė statusas T sritis automatika atitikmenys: engl. initial phase of oscillation vok. Anfangsschwingungsphase, f rus. initial phase of oscillations, fpranc. phase initiale d oscillations, f … Automatikos terminų žodynas
- (from the Greek phasis appearance) period, stage in the development of a phenomenon, stage. The oscillation phase is a function argument describing a harmonic oscillatory process or an argument of a similar imaginary exponent. Sometimes just an argument ... ... Wikipedia
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- (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
- (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; and. [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
