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Illustration of the phase difference between two oscillations of the same frequency
Oscillation phase - physical quantity, used primarily to describe harmonic or close to harmonic oscillations, varying with time (most often uniformly growing with time), at a given amplitude (for damped oscillations - at a given initial amplitude and damping coefficient) defining the state of the oscillatory system at (any) given moment of time. It is equally used to describe waves, mainly monochromatic or close to monochromatic.
Oscillation phase(in telecommunications for 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.
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:
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.).
One of the most fundamental physical quantities on which it is built modern description almost any sufficiently fundamental physical system- action - in its meaning is a phase.
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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.K., A amplitude, w circular frequency, t time, j0 initial (fixed) F.K. (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] Optics topics, optical instruments and measurements General terms of oscillations and waves EN phase of oscillation DE Schwingungsphase FR... ... Technical Translator's Guide
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. k., i.e. F. k. at the initial moment of time t = 0). The 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 oscillation, f pranc. phase initiale d oscillations, f … Automatikos terminų žodynas
- (from the Greek phasis appearance) period, stage in the development of a phenomenon, stage. Oscillation phase is an argument of a function describing a harmonic oscillatory process or an argument of a similar imaginary exponential. Sometimes it’s just an argument... ... Wikipedia
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 the animal and flora. Enter into your new, decisive,... encyclopedic Dictionary
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.
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 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 according to Thomson's formula. Frequency and period are inversely proportional.
Oscillation phase complete - argument of a periodic function describing an oscillatory or wave process.
Oscillation phase initial - the value of the oscillation phase (total) at the initial moment of time, i.e. at t= 0 (for an oscillatory process), 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 (for the wave process).
Oscillation phase(in electrical engineering) - the argument of a sinusoidal function (voltage, current), counted from the point where the value passes through zero to a positive value.
Oscillation phase- harmonic oscillation ( φ ) .
Size φ, standing under the sign of the cosine or sine function is called oscillation phase described by this function.
φ = ω៰ t
As a rule, phase is spoken of 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 0) (\displaystyle A\cos(\omega t+\varphi _(0))), A sin (ω t + φ 0) (\displaystyle A\sin(\omega t+\varphi _(0))), A e i (ω t + φ 0) (\displaystyle Ae^(i(\omega t+\varphi _(0)))).Similarly, when describing a wave propagating in one-dimensional space, for example, expressions of the form are used:
A cos (k x − ω t + φ 0) (\displaystyle A\cos(kx-\omega t+\varphi _(0))), A sin (k x − ω t + φ 0) (\displaystyle A\sin(kx-\omega t+\varphi _(0))), A e i (k x − ω t + φ 0) (\displaystyle Ae^(i(kx-\omega t+\varphi _(0)))),for a wave in space of any dimension (for example, in three-dimensional space):
A cos (k ⋅ r − ω t + φ 0) (\displaystyle A\cos(\mathbf (k) \cdot \mathbf (r) -\omega t+\varphi _(0))), A sin (k ⋅ r − ω t + φ 0) (\displaystyle A\sin(\mathbf (k) \cdot \mathbf (r) -\omega t+\varphi _(0))), A e i (k ⋅ r − ω t + φ 0) (\displaystyle Ae^(i(\mathbf (k) \cdot \mathbf (r) -\omega t+\varphi _(0)))).The oscillation phase (total) in these expressions is argument functions, i.e. expression written in parentheses; initial oscillation phase - value φ 0, which is one of the terms of the total phase. Speaking of the full phase, the word full often omitted.
Oscillations with the same amplitudes and frequencies may differ in phase. Because ω៰ =2π/T, That φ = ω៰t = 2π t/T.
Attitude t/T indicates how many periods have passed since the start of the oscillations. Any time value t , expressed in the number of periods T , corresponds to the phase value φ , expressed in radians. So, as time passes t=T/4 (quarter period) φ=π/2, after half the period φ =π/2, after a whole period φ=2 π etc.
Because the functions sin(...) and cos(...) coincide with each other when the argument (i.e. phase) is shifted by π / 2 , (\displaystyle \pi /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, a phase is considered the argument is cosine, not sine.
That is, for the oscillatory process (see above) the phase (full)
φ = ω t + φ 0 (\displaystyle \varphi =\omega t+\varphi _(0)),for a wave in one-dimensional space
φ = k x − ω t + φ 0 (\displaystyle \varphi =kx-\omega t+\varphi _(0)),for a wave in three-dimensional space or space of any other dimension:
φ = k r − ω t + φ 0 (\displaystyle \varphi =\mathbf (k) \mathbf (r) -\omega t+\varphi _(0)),Where ω (\displaystyle \omega )- angular frequency (a value indicating 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 0 (\displaystyle \varphi _(0))- initial phase (that is, the phase at t = 0); k- wave number; x- coordinate of the observation point 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 (\displaystyle \pi ) radian = 360 degrees.
In analytical expressions (in formulas), the phase representation in radians is predominantly (and by default) used; the representation in degrees is also found quite often (apparently, as extremely obvious and not leading to confusion, since it is never customary to omit the degree sign either in oral speech or in recordings). Indicating the phase in cycles or periods (except for verbal formulations) is relatively rare in technology.
Sometimes (in the quasi-classical approximation, where quasi-monochromatic waves are used, i.e. close to monochromatic, but not strictly monochromatic), as well as in the path integral formalism, where waves can be far from monochromatic, although still similar to monochromatic) the phase is considered, being nonlinear function time t and spatial coordinates r, in principle, an arbitrary function.
Another characteristic of harmonic oscillations is the phase of oscillations.
As we already know, for a given amplitude of oscillations, at any moment in time we can determine the coordinates of the body. It will be uniquely specified by the argument trigonometric functionφ = ω0*t. The quantity φ, which is under the sign of the trigonometric function, called the oscillation phase.
The units for phase are radians. The phase uniquely determines not only the coordinate of the body at any time, but also the speed or acceleration. Therefore, it is believed that the oscillation phase determines the state of the oscillatory system at any time.
Of course, provided that the amplitude of oscillations is specified. Two oscillations that have the same frequency and period of oscillation may differ from each other in phase.
If we express time t in the number of periods that have passed since the beginning of the oscillations, then any value of time t corresponds to a phase value expressed in radians. For example, if we take time t = T/4, then this value will correspond to the phase value 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.
When describing the coordinates of oscillatory motion, we used the sine and cosine functions. For cosine we wrote the following formula:
But we can describe the same trajectory of motion using a sine. In this case, we need to shift the argument by pi/2, that is, the difference between sine and cosine is pi/2 or a quarter of the period.
The value pi/2 is called the initial phase of the oscillation. The initial phase of 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 its equilibrium position. We can do this in two ways:
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 amplitude value. To describe such an oscillation, it is more convenient to use the cosine function and the form
or the formula
where φ is the initial phase of 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:
Two oscillations that differ only in the initial phase are called phase-shifted.
For example, for vibrations described by the following formulas:
the phase shift is pi/2.
Phase shift is also sometimes called phase difference.
The concept of phase, and even more so of phase shift, is difficult for students to grasp. Phase is a physical quantity that characterizes an oscillation at a certain point in time. The state of oscillation in accordance with the formula can be characterized, for example, by the deviation of a point from the equilibrium position. Since for given values the value is uniquely determined by the value of the angle, the phase in the equations of oscillatory motion is usually called the value of the angle
Time can be measured in fractions of a period. Therefore, the phase is proportional to the fraction of the period that has passed since the start of the oscillation. Therefore, the phase of oscillations is also called a quantity measured by the fraction of a period that has passed from the beginning of the oscillations.
Problems involving the addition of harmonic oscillatory motions are solved mainly graphically with a gradual complication of the conditions. First, oscillations that differ only in amplitude are added, then - in amplitude and initial phase, and, finally, oscillations that have different amplitudes, phases and periods of oscillations.
All these tasks are uniform and not complicated in terms of solution methods, but require careful and painstaking execution of drawings. To facilitate the labor-intensive work of compiling tables and drawing sinusoids, it is advisable to prepare their templates in the form of slits in cardboard or tin. Three or four sinusoids can be made on one stencil. This device allows students to focus their attention on the addition of vibrations and relative position sinusoids, and not on their plotting. However, when resorting to such an auxiliary technique, the teacher must be sure that the students already know how to draw graphs of sine and cosine waves. Special attention you need to pay attention to the addition of oscillations with the same period and phases, which will lead students to the concept of resonance.
Using students’ knowledge of mathematics, one should also solve a number of problems involving the addition of harmonic vibrations analytical method. The following cases are of interest:
1) Addition of two oscillations with the same periods and phases:
The amplitudes of oscillations can be either the same or different.
2) The addition of two oscillations with the same periods, but different amplitudes and phases. IN general view the addition of such oscillations gives the resulting displacement:
and the value is determined from the formula
IN high school with all students there is no need to solve this problem in such a general form. It is quite sufficient to consider the special case when the phase difference or
This will make the problem (see No. 771) quite accessible and will not interfere with obtaining from it important conclusions about the oscillations that are obtained by adding two harmonic oscillations that have the same periods, but different phases.
766. Are the wings of a flying bird in the same or different phases? human hands when walking? two chips that fell on the crest and trough of a wave from the ship.
Solution. Having agreed on the starting point, as well as the positive and negative (for example, left and down) direction of movement, we conclude that the wings of a flying bird move equally and in the same direction, they are in the same phase; the person’s hands, as well as the wood chips, have deviated from the equilibrium position by the same distance, but are moving in opposite directions - they are in different, as they say, “opposite” phases.
767(e). Hang two identical pendulums and set them in oscillation, deflecting them different sides at the same distance. What is the phase difference between these oscillations? Does it decrease over time?
Solution. The movements of pendulums are described by the equations:
or in general where is an integer. Phase difference for given movements
does not change over time.
768(e). Perform an experiment similar to the previous one, taking pendulums of different lengths. Could there come a time when the pendulums
will they move in the same direction? Calculate when this will happen for the pendulums you took.
Solution. Movements differ in phase and period of oscillations
Pendulums will move in the same direction when their phases become the same: from where
769. Figure 239 shows graphs of four oscillatory movements. Determine the initial phase of each oscillatory movement and the phase shift for oscillations I and II, I and III, I and IV; II and III, II and IV; III and IV.
Solution 1. Let us imagine that the graphs show the oscillation of four pendulums at the moment when pendulum I began to oscillate, pendulum II has already deviated to its extreme position, pendulum III has returned to the equilibrium position, and pendulum IV has deviated completely in the opposite direction. From these considerations it follows that the phase difference
Solution 2. All vibrations are harmonic, and therefore they can be described by the equation
Let's consider all the oscillations at any particular moment in time, for example. Let us take into account that the sign of x is determined by the sign of the trigonometric function. The value of A is taken in absolute value, i.e. positive.
I. ; since at subsequent times therefore, therefore
III. ; since at subsequent moments of time, therefore,
Having made the appropriate calculations, we get the same result as in the first solution:
Despite the somewhat cumbersome nature of the second solution, it should be used to develop students’ skills in applying the equation of harmonic oscillatory motion.
770. Add two oscillatory movements with the same periods and phases, if the amplitude of one oscillation is cm, and the second is cm. What amplitude will the resulting oscillatory movement have?
Solution 1. Draw sinusoids of oscillations I and II (Fig. 240).
When constructing sinusoids using tables, it is enough to take 9 characteristic values phases: 0°, 45°, 90°, etc. The amplitude of the resulting oscillation is found for the same phases as the sum of the amplitudes of the first and second oscillations (graph III).
Solution 2.
Consequently, the amplitude of the resulting oscillation is cm, and the oscillation occurs according to the law. Using trigonometric tables, a sinusoid of the resulting oscillation is constructed using this formula.
771. Add two oscillations with the same periods and amplitudes if they: do not differ in phase; have a phase difference differ in phase by
Solution 1.
The first case is quite similar to the one considered in the previous problem and does not require any special explanation.
For the second case, the addition of vibrations is shown in Figure 241, a.
The addition of oscillations that differ in phase is shown in Figure 241, b.
Solution 2. For each case, we derive the equation of the resulting oscillation.
The resulting vibration has the same frequency and twice the amplitude.
For the second and third cases, we can write the following equation:
where is the phase difference between the two oscillations.
When the equation takes the form
As can be seen from this formula, when adding two harmonic oscillations of the same period that differ in phase, a harmonic oscillation of the same period is obtained, but with a different amplitude and initial phase than those of the oscillation components.
When Therefore, the result of addition also depends significantly on the phase difference. With a phase difference and equal amplitudes, one oscillation completely “quenches” the other.
When analyzing solutions, you should also pay attention to the fact that the resulting oscillation will have the greatest amplitude in the case when the phase difference between the added oscillations is zero (resonance).
772. How does the rocking of a ship depend on the period of wave oscillation?
Answer. The motion will be greatest when the period of wave oscillation coincides with the period of the ship's own oscillations.
773. Why do periodically recurring depressions (dents) form over time on the road along which dump trucks transport stone, sand, etc. from the quarry?
Answer. It is enough for the slightest irregularity to form, and the body, which has a certain period of oscillation, will begin to move, as a result of which, when the dump truck moves,
periodic increased and decreased loads on the ground will be created, leading to the formation of depressions (dents) on the road.
774. Using the solution to problem 760, determine at what speed the greatest vertical oscillations of the car will occur if the length of the rail is
Solution. The period of oscillation of the car is sec.
If the impacts of the wheels at the joints coincide with this oscillation frequency, resonance will occur.
775. Is it correct to say that forced vibrations reach significant sizes only when the natural frequency of the oscillating body is equal to the frequency of the driving force? Give examples to explain your statement.
Answer. Resonance can also occur when a periodically changing force, but not according to a harmonic law, has a period that is an integer number of times less than the body’s own period.
An example would be periodic shocks that act on a swing not every time it swings. In this regard, the answer to the previous problem should be clarified. Resonance can occur not only at the speed of the train, but also at a speed several times greater, where is an integer.
