8.7. Representation of Vibrations with the Help of Fourier Series

Any complex periodic vibration x = f(t) can be represented as the sum of the simple harmonic vibrations with the cyclic frequencies multiple to the basic cyclic frequency  = 2/T:

where

Such representation of the periodic function f(t) is called the decomposition of this function into Fourier series, or the harmonic analysis of the complex periodic vibration. The terms of Fourier series which correspond to the harmonic vibrations with the cyclic frequency , 2, 3 etc. are called the first (or basic), second, third etc. harmonics of the complex periodic vibration x = f(t). The aggregate of these harmonics forms the spectrum of the vibration x = f(t).

8.8. Waves. Equation of Mechanical Harmonic Wave

We have studied several types of motion that can be performed by “particles” of various kinds before. Associated with each particle there is an instantaneous position in space as well as mass, kinetic energy etc. A fundamental property of classical particles is that no two particles can occupy the same point in space at the same time. Now we turn to the study of classical waves, which have properties very different from those of classical particles.

There are many types of wave motion and in each type a disturbance of some sort is propagated, when the mechanical waves pass through the medium, and individual particle oscillates up and down or back and forth but does not progress with the wave. Energy is transferred through the medium although the particles themselves are not transmitted. In a typical wave motion vibrating center produces motion in particles of the medium immediately in contact with it; these particles in turn impart motion to their neighbors. In many cases, but by no means always, the particles describe the simple harmonic motion about their equilibrium positions, with the phases varying along the wave.

If in any place of medium a vibration is created this vibration will spread from particle to particle as a result of the interaction of particles of the medium.

The process of spreading vibrations in the space is called the wave.

Types of waves:

1. The wave impulse is the single perturbation which is restricted in time and spreads in the space (Fig. 8.14)

Examples: spark, discharge.

2. The pack of waves is a sequential raw of perturbations which is restricted in time and spread in the space (Fig. 8.15).

Examples: radar irradiation.

3. The harmonic wave is a process when laws of sinus or cosines describe all oscillating in the wave values. This type of waves does not really exist in the nature.

Figure 8.14 Figure 8.15

Let us obtain the equation of the running wave.

Main simplifications:

1. Only the mediums with linear properties are considered, namely the properties of the medium do not change during wave spreads.

2. The shape of the perturbation does not change.

3. The velocity of the wave during the spreading is constant.

L et us suppose that the source of harmonic vibration is located at the point O (Fig. 8.16): y = B sint.