§4 The energy of the harmonic oscillations

By definition, the kinetic energy of a body of mass m, moving with speed  equal


The potential energy is equal to

W p


Total energy is equal









Quasi-elastic force is conservative, so the total energy of the harmonic motion is constant. In the process of oscillations is turning kinetic energy into potential energy and vice versa. Fluctuations WK and WP have a frequency 2ω0, ie twice the frequency of harmonic oscillations.



§ 5 The addition of harmonic oscillations

Image fluctuations in vector diagram

1. Let oscillations are described by the equation



Laid from point A on the vector length, making an angle φ0 with Ox. If this vector to begin rotating with angular velocity ω0, then the projection of the end of the vector will change with time as the cosine (1), ie, harmonic motion can be described by a vector whose length is equal to the amplitude of the oscillations A, and the direction of the vector form the x-axis angle of the initial phase φ0.

2. Addition of two harmonic oscillations of the same direction and the same frequency.


 The resulting vector  is equal to

And find on the parallelogram, its projection on the X axis equal to

X=X1 + X2.

Length of the resulting vector, or the amplitude of the resulting oscillation is on the law of cosines and equal


The initial phase of the resulting oscillation is determined by the condition

The addition of two harmonic oscillations with the same frequency and the same direction, the resulting motion is also a harmonic oscillation with the same period and an amplitude A, which lies within

Fluctuations that have φ10 = φ20, А= А1 + А2are called in-phase.

Fluctuations that have φ10 - φ20 = π, А=| А2 – А1called antiphase.

If А1 = А2, when φ10 = φ20  А = 2А1, at φ10 - φ20 = π, А=| А2 – А1| = 0.


3. Beats

Beats - Addition of oscillations with close frequencies ω1 ≈ ω2.

With the addition of harmonic oscillations differ slightly in frequency resulting motion is a harmonic oscillation with pulsing amplitude. Such vibrations are called beats.
For simplicity, assume

А= А1 = А2, φ10 = φ20 = 0.


,  where 



The resulting expression is the product of two oscillations.

Factor  has a frequency an average of two terms of vibrations. ie close to their frequencies ω1 and ω2. The second factor has in virtue of proximity ω1 and ω2 low frequency, ie large period. This allows us to consider the resulting motion as nearly harmonic oscillation with an average angular frequency wand slowly varying amplitude .

1,2 - graph the slowly varying amplitude.

3 - graph of the resulting oscillation.

When   φ1 ≈ φ2, Арез ≈ 2А. After a interval ,    

one of the vibrations behind the other in phase by π and Аres → 0. This gradual increase and decrease the amplitude of the resulting oscillation is called a beat.

If ω1 and ω2 are comparable, ie can be found two numbers n1 and n2, that  then after that interval of time

the arguments of both factors in (2) to change the whole (though different) number of times 2π, their product will take the same value as in the beginning of period τ. The value of τ is the time period of the resulting oscillation.

If the frequency is not comparable, the resulting oscillation will non-periodic.

4. Addition perpendicular vibrations.

Consider the result of the addition of two harmonic oscillations of the same frequency ω1 = ω2 = ω, occurring in mutually perpendicular directions along the x and y axes.


а) Let φ10 = φ20.

Then, т.е. - тtrajectory - the diagonal of a rectangle with sides 2A (x-axis) and 2B (y-axis)

b) Let φ10 = φ20 +π.



c) Let φ10 = φ20 +π/2

 - ellipse.

If А = Вcircle.

d) φ10 = φ20 - π/2 – ellipse, but changes the direction of the circuit

e) Arbitrary φ10 and φ20 - also an ellipse with equation

In the general case

  1. φ20 - φ10 = 2kπ;


2. Δφ = (2k + 1)π;


 3. Δφ =  ±π/2k;


f) Lissajous figures.

In the case where the frequency of oscillation perpendicular, in which both involved the point under consideration are as integers, the trajectory is a complex curves, known as Lissajous figures. The shape of the curve depends on the ratio of amplitude, frequency and phase difference summed vibrations.
Ratio of frequency foldable vibrations is the ratio of the number of intersections of the Lissajous figures with lines parallel to the axes. By type of Lissajous figures can be determined from the known unknown frequency, or to determine the frequency ratio
ω1 and ω2.



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