The initial phase of oscillations on the chart. We study oscillations - the phase of oscillations
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§ 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 plotted on the horizontal axis instead of time various meanings phases.
Representation of harmonic oscillations using cosine and sine. You already know that during 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 vibrations are called harmonic!
2. How are acceleration and coordinate related in harmonic oscillations!
3. How connected cyclic frequency oscillation and oscillation period!
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!
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 of a material body relative to any coordinate system, which is conditionally considered immovable.
Among huge amount possible movement options special interest represents an 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 of the oscillatory system. It cannot be otherwise - after all, the phase of oscillations is an argument of the function that describes these oscillations. true value phases for the movement of an oscillatory nature can mean coordinates, speed and other physical parameters that change according to a harmonic law, but the common thing for them is the time dependence.
Demonstrating oscillations is not at all difficult - for this you need the simplest mechanical system- a thread, length r, and a “material point” suspended on it - a small weight. Fasten the thread in the center rectangular system coordinates and make our “pendulum” spin. Let us assume that he willingly does this with an 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, full angle 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 the angular measure of time, which expresses time in units of angle. During this time, the load makes a turn through a certain angle, which can be uniquely determined based on the fact that for a cyclic oscillation 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 the angle as a fraction of the 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 of harmonic ones 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, of 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.
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 phase of oscillation determines the displacement of the oscillating quantity from the equilibrium position in this moment time t. The constant φ is the value of the phase at time t = 0 and is called the initial phase of the oscillation .. This period of time T is called the period of harmonic oscillations. The period of harmonic oscillations is : T = 2π/. Mathematical pendulum- an oscillator, which is a mechanical system consisting of a material point located on a weightless inextensible thread or on a weightless rod in a uniform field of gravitational forces. The period of small natural oscillations of a mathematical pendulum of length L motionless suspended in a uniform gravitational field with free fall acceleration g equals
and does not depend on the amplitude of oscillations and the mass of the pendulum. physical pendulum- An oscillator, which is a rigid body that oscillates in the field of any forces about a point that is not the center of mass of this body, or a fixed axis perpendicular to the direction of the forces and not passing through the center of mass of this body.
24. Electromagnetic oscillations. Oscillatory circuit. Thomson formula.
Electromagnetic vibrations- These are fluctuations in electric and magnetic fields, which are accompanied by a periodic change in charge, current and voltage. The simplest system where free electromagnetic oscillations can arise and exist is an oscillatory circuit. Oscillatory circuit- this is a circuit consisting of an inductor and a capacitor (Fig. 29, a). If the capacitor is charged and closed to the coil, then current will flow through the coil (Fig. 29, b). When the capacitor is discharged, the current in the circuit will not stop due to self-induction in the coil. The induction current, in accordance with the Lenz rule, will have the same direction and recharge the capacitor (Fig. 29, c). The process will be repeated (Fig. 29, d) by analogy with pendulum oscillations. Thus, electromagnetic oscillations will occur in the oscillatory circuit due to the conversion of energy electric field capacitor() into energy magnetic field coils with current (), and vice versa. The period of electromagnetic oscillations in an ideal oscillatory circuit depends on the inductance of the coil and the capacitance of the capacitor and is found using the Thomson formula. Frequency is inversely related to period.
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 (full) at the initial moment of time, i.e. at t= 0 (for an oscillatory process), as well as at the initial 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 positive value.
Oscillation phase- harmonic oscillation ( φ ) .
the value 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, 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 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 (full) in these expressions is argument functions, i.e. an expression written in brackets; oscillation phase initial - magnitude φ 0 , which is one of the terms of the total phase. Speaking of full phase, word complete 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 oscillations. Any value of time t , expressed in the number of periods T , corresponds to the phase value φ , expressed in radians. So, as time goes by t=Т/4 (quarters of the period) φ=π/2, after half a period φ =π/2, after a whole period φ=2 π etc.
Since the sin(…) and cos(…) functions coincide with each other when the argument (that is, the 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, the phase is cosine argument, not sine.
That is, for an oscillatory process (see above), the phase (total)
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 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 0 (\displaystyle \varphi _(0))- the 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 representation of the phase in radians is predominantly (and by default), the representation in degrees is also quite common (apparently, as extremely explicit and not leading to confusion, since it is never customary to omit the degree sign in any oral speech, nor in the records). The indication of the phase in cycles or periods (with the exception of verbal formulations) is relatively rare in technology.
Sometimes (in the semiclassical approximation, where quasimonochromatic 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, is an arbitrary function.
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