

§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
Total energy is equal
Quasielastic 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 W_{K} and W_{P} have a frequency 2ω_{0}, 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
(1) Laid from point A on the vector length, making an angle φ_{0} with Ox. If this vector to begin rotating with angular velocity ω_{0}, 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 xaxis angle of the initial phase φ_{0}. 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=X_{1} + X_{2}. 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 φ_{10} = φ_{20}, А= А_{1} + А_{2}are called inphase. Fluctuations that have φ_{10}  φ_{20} = π, А= А_{2} – А_{1}called antiphase. If А_{1} = А_{2}, when φ_{10} = φ_{20} А = 2А_{1}, at φ_{10}  φ_{20} = π, А= А_{2} – А_{1} = 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= А_{1} = А_{2}, φ_{10} = φ_{20} = 0. Then
(2) 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 ω_{1} and ω_{2}. The second factor has in virtue of proximity ω_{1} and ω_{2} low frequency, ie large period. This allows us to consider the resulting motion as nearly harmonic oscillation with an average angular frequency and slowly varying amplitude .
1,2  graph the slowly varying amplitude. 3  graph of the resulting oscillation. When φ_{1} ≈ φ_{2}, А_{рез} ≈ 2А. After a interval , one of the vibrations behind the other in phase by π and А_{res} → 0. This gradual increase and decrease the amplitude of the resulting oscillation is called a beat. If ω_{1} and ω_{2} are comparable, ie can be found two numbers n_{1} and n_{2}, that then after that interval of timethe 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 nonperiodic. 4. Addition perpendicular vibrations. Consider the result of the addition of two harmonic oscillations of the same frequency ω_{1} = ω_{2} = ω, occurring in mutually perpendicular directions along the x and y axes.(1) а) Let φ_{10} = φ_{20}.
b) Let φ_{10} = φ_{20 }+π. Then
c) Let φ_{10} = φ_{20 }+π/2
If А = В –circle. d) φ_{10} = φ_{20 } π/2 – ellipse, but changes the direction of the circuit e) Arbitrary φ_{10} and φ_{20}  also an ellipse with equationIn the general case
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 ω_{1} and ω_{2}.
