>> 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!
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 shift 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.
As you study this section, please keep in mind that fluctuations of different physical nature are described from common mathematical positions. Here it is necessary to clearly understand such concepts as harmonic oscillation, phase, phase difference, amplitude, frequency, oscillation period.
It must be borne in mind that in any real oscillatory system there is resistance of the medium, i.e. the oscillations will be damped. To characterize the damping of oscillations, a damping coefficient and a logarithmic damping decrement are introduced.
If oscillations occur under the influence of an external, periodically changing force, then such oscillations are called forced. They will be undamped. The amplitude of forced oscillations depends on the frequency of the driving force. As the frequency of forced oscillations approaches the frequency of natural oscillations, the amplitude of forced oscillations increases sharply. This phenomenon is called resonance.
When moving on to the study of electromagnetic waves, you need to clearly understand thatelectromagnetic waveis an electromagnetic field propagating in space. The simplest system emitting electromagnetic waves is an electric dipole. If a dipole undergoes harmonic oscillations, then it emits a monochromatic wave.
Formula table: oscillations and waves
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Physical laws, formulas, variables |
Oscillation and wave formulas |
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Harmonic vibration equation: where x is the displacement (deviation) of the fluctuating quantity from the equilibrium position; A - amplitude; ω - circular (cyclic) frequency; α - initial phase; (ωt+α) - phase. |
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Relationship between period and circular frequency: |
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Frequency: |
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Relationship between circular frequency and frequency: |
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Periods of natural oscillations 1) spring pendulum: where k is the spring stiffness; 2) mathematical pendulum: where l is the length of the pendulum, g - free fall acceleration; 3) oscillatory circuit: where L is the inductance of the circuit, C is the capacitance of the capacitor. |
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Natural frequency: |
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Addition of oscillations of the same frequency and direction: 1) amplitude of the resulting oscillation where A 1 and A 2 are the amplitudes of the vibration components, α 1 and α 2 - initial phases of the vibration components; 2) the initial phase of the resulting oscillation |
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Equation of damped oscillations: e = 2.71... - the base of natural logarithms. |
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Amplitude of damped oscillations: where A 0 is the amplitude at the initial moment of time; β - attenuation coefficient; |
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Attenuation coefficient: oscillating body where r is the resistance coefficient of the medium, m - body weight; oscillatory circuit where R is active resistance, L is the inductance of the circuit. |
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Frequency of damped oscillations ω: |
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Period of damped oscillations T: |
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Logarithmic damping decrement: |
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Relationship between the logarithmic decrement χ and the damping coefficient β: |
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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.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] 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
Definition
Initial phase of oscillation is a parameter that, together with the oscillation amplitude, determines the initial state of the oscillatory system. The value of the initial phase is set in the initial conditions, that is, at $t=0$ c.
Let's consider harmonic oscillations of some parameter $\xi $. Harmonic vibrations are described by the equation:
\[\xi =A(\cos ((\omega )_0t+\varphi)\ )\ \left(1\right),\]
where $A=(\xi )_(max)$ is the amplitude of oscillations; $(\omega )_0$ - cyclic (circular) oscillation frequency. The parameter $\xi $ lies within $-A\le \xi \le $+A.
Determination of the oscillation phase
The entire argument of the periodic function (in this case, cosine: $\ ((\omega )_0t+\varphi)$), which describes the oscillatory process, is called the oscillation phase. The magnitude of the oscillation phase at the initial moment of time, that is, at $t=0$, ($\varphi $) is called the initial phase. There is no established phase designation; we have the initial phase designated $\varphi$. Sometimes, to emphasize that the initial phase refers to the moment of time $t=0$, the index 0 is added to the letter denoting the initial phase; for example, $(\varphi )_0.$ is written.
The unit of measurement for the initial phase is the angle unit - radian (rad) or degree.
Initial phase of oscillations and method of excitation of oscillations
Let us assume that at $t=0$ the displacement of the system from the equilibrium position is equal to $(\xi )_0$, and the initial velocity is $(\dot(\xi ))_0$. Then equation (1) takes the form:
\[\xi \left(0\right)=A(\cos \varphi =\ )(\xi )_0\left(2\right);;\] \[\ \frac(d\xi )(dt) =-A(\omega )_0(\sin \varphi =\ )(\dot(\xi ))_0\to -A(\sin \varphi =\frac((\dot(\xi ))_0)(( \omega )_0)\ )\ \left(3\right).\]
Let us square both equations (2) and add them:
\[(\xi )^2_0+(\left(\frac((\dot(\xi ))_0)((\omega )_0)\right))^2=A^2\left(4\right). \]
From expression (4) we have:
Divide equation (3) by (2), we get:
Expressions (5) and (6) show that the initial phase and amplitude depend on the initial conditions of oscillations. This means that the amplitude and initial phase depend on the method of excitation of oscillations. For example, if the weight of a spring pendulum is deflected from the equilibrium position and by a distance $x_0$ and released without a push, then the equation of motion of the pendulum is the equation:
with initial conditions:
With such excitation, the oscillations of a spring pendulum can be described by the expression:
Addition of oscillations and initial phase
A body that vibrates is capable of taking part in several oscillatory processes simultaneously. In this case, it becomes necessary to find out what the resulting fluctuation will be.
Let us assume that two oscillations with equal frequencies occur along one straight line. The equation of the resulting oscillations will be the expression:
\[\xi =(\xi )_1+(\xi )_2=A(\cos \left((\omega )_0t+\varphi \right),\ )\]
then the amplitude of the total oscillation is equal to:
where $A_1$; $A_2$ - amplitudes of folding oscillations; $(\varphi )_2;;(\varphi )_1$ - initial phases of summed oscillations. In this case, the initial phase of the resulting oscillation ($\varphi $) is calculated using the formula:
Equation of the trajectory of a point that takes part in two mutually perpendicular oscillations with amplitudes $A_1$ and $A_2$ and initial phases $(\varphi )_2 and (\varphi )_1$:
\[\frac(x^2)(A^2_1)+\frac(y^2)(A^2_2)-\frac(2xy)(A_1A_2)(\cos \left((\varphi )_2-(\ varphi )_1\right)\ )=(sin)^2\left((\varphi )_2-(\varphi )_1\right)\left(12\right).\]
In the case of equality of the initial phases of the oscillation components, the trajectory equation has the form:
which indicates the movement of a point in a straight line.
If the difference in the initial phases of the added oscillations is $\Delta \varphi =(\varphi )_2-(\varphi )_1=\frac(\pi )(2),$ the trajectory equation becomes the formula:
\[\frac(x^2)(A^2_1)+\frac(y^2)(A^2_2)=1\left(14\right),\]
which means the trajectory of movement is an ellipse.
Examples of problems with solutions
Example 1
Exercise. The oscillations of the spring oscillator are excited by a push from the equilibrium position, while the load is given an instantaneous speed equal to $v_0$. Write down the initial conditions for such an oscillation and the function $x(t)$ that describes these oscillations.
Solution. Giving the bob of a spring pendulum an instantaneous speed equal to $v_0$ means that when describing its oscillations using the equation:
the initial conditions will be:
Substituting $t=0$ into expression (1.1), we have:
Since $A\ne 0$, then $(\cos \left(\varphi \right)\ )=0\to \varphi =\pm \frac(\pi )(2).$
Let's take the first derivative $\frac(dx)(dt)$ and substitute the moment of time $t=0$:
\[\dot(x)\left(0\right)=-A(\omega )_(0\ )(\sin \left(\varphi \right)\ )=v_0\to A=\frac(v_0) ((\omega )_(0\ ))\ \left(1.4\right).\]
From (1.4) it follows that the initial phase is $\varphi =-\frac(\pi )(2).$ Let us substitute the resulting initial phase and amplitude into equation (1.1):
Answer.$x(t)=\frac(v_0)((\omega )_(0\ ))(\sin (\ )(\omega )_0t)$
Example 2
Exercise. Two oscillations in the same direction are added. The equations of these oscillations have the form: $x_1=(\cos \pi (t+\frac(1)(6))\ ;;\ x_2=2(\cos \pi (t+\frac(1)(2))\ )$. What is the initial phase of the resulting oscillation?
Solution. Let's write the equation of harmonic vibrations along the X axis:
Let us transform the equations specified in the problem statement to the same form:
\;;\ x_2=2(\cos \left[\pi t+\frac(\pi )(2)\right](2.2).\ )\]
Comparing equations (2.2) with (2.1) we find that the initial phases of oscillations are equal to:
\[(\varphi )_1=\frac(\pi )(6);;\ (\varphi )_2=\frac(\pi )(2).\]
Let us depict in Fig. 1 a vector diagram of oscillations.
$tg\ \varphi $ of total oscillations can be found from Fig. 1:
\ \[\varphi =arctg\ \left(2.87\right)\approx 70.9()^\circ \]
Answer.$\varphi =70.9()^\circ $