Oscillations

OSCILLATIONS, processes (in the broadest sense) that periodically change their direction over time. These processes can be very diverse. If, for example, a heavy ball is suspended from a steel coil spring, pulled down, and then left to itself, it will perform oscillations under the action of elastic forces. Such are also the elastic oscillations of strings, rods, etc. However, the oscillating quantity can also be the strength of an electric current (alternating current), the voltage of an electric field (electrical oscillations), and so on. The prototype of any oscillation can be considered the oscillatory motion of a point, i.e., its back-and-forth movement along a certain segment of a straight line between two extreme positions. Let us consider the simplest and at the same time most important oscillatory motion—the so-called simple harmonic oscillation. Let us imagine a point moving uniformly along a circle of radius a with a period T. This means that if the point

at point y

at I^m it will return again to point \ Y

q onto the vertical dia-

Figure 1. meter PQ, i.e., the motion of the base of the perpendicular dropped from the point onto this diameter. It is easy to see that while the point moving along the circle counterclockwise, starting from M, makes one full revolution, the projection makes one full oscillation along the diameter, which consists of movement from O to P, then back from P to O, further from O to Q, and back to O. Thus, the period of revolution T of the point around the circle will at the same time be the period of the complete oscillation of the projection of this point along the diameter. The quantity reciprocal to T obviously shows how many complete oscillations are performed in a unit of time; it is called the frequency of oscillation and is usually denoted by the Greek letter v = 1/T. The distance of the oscillating point from the initial position O is called the elongation; the greatest removal of the oscillating point from the initial position (O in Fig. 1) characterizes the span of the oscillation and is called the amplitude. The state of oscillation of a point, i.e., its position on the diameter PQ and the direction of its motion, is completely determined by the position of the auxiliary point moving along the circle. The latter, in turn, is known if the angle formed by the radius drawn to the given point and any diameter taken as initial, for example with the horizontal diameter MN, is given. This angle, which completely determines the state of the oscillatory motion of the point, is called the phase of oscillation. For example, the angle MOA will be the phase of oscillation of point A; the obtuse angle MOB is the phase of oscillation of point B. If, as in this case, the phases differ by 180° (or, what is the same, by ̀̄), it is said that the points are in opposite phases: in this case, they are located symmetrically relative to point O and move in opposite directions. During the period T, the point completes a full oscillation, as a result of which its phase changes by a full circle, or by 2̀̄. It is clear that at any moment t the phase of oscillation ́ͩ will be related to the period T by the following relation: ́ͩ/2̀̄ = t/T, or ́ͩ = 2̀̄t/T = 2̀̄vt.

(1) From the consideration of triangle AA₁O on the basis of the rules of trigonometry, the following formula is directly obtained for the elongation s of the oscillating point: s = OA sin ́ͩ = a " sin (2̀̄t/T) = a " sin 2̀̄vt, (2) where OA = a is the amplitude of the oscillation. Formula (2) represents the basic law of harmonic oscillation. Since the value of the sine oscillates between the values +1 and -1, the value of the elongation oscillates between +a and -a. If we graphically represent the relationship between elongation and phase, we obtain the curve shown in Fig. 2. This

Figure 2.

curve is called a sinusoid, and therefore the oscillation itself is called sinusoidal. According to what was said at the beginning, this applies not only to the oscillatory motion of a point, but can also apply, for example, to an alternating electric current. This is the origin of such terms as "sinusoidal currents of electrotherapy." The velocity of a point performing harmonic oscillation does not remain constant during the period. Indeed, at M and N, the point on the circle and its projection move parallel to each other, and near P and Q almost perpendicularly. From this it follows that a harmonically oscillating point possesses the greatest velocity when passing through the middle O of the segment along which it moves, and the smallest at the ends of this segment; at P and Q it stops for a moment and subsequently changes the direction of its motion. Mathematically, the law of velocity change over time is expressed by the formula u = (2̀̄a/T) cos (2̀̄t/T) or, if instead of the period T we introduce the frequency v = 1/T, u = 2̀̄va cos 2̀̄vt. Since with a continuous change of the angle the cosine oscillates, like the sine, within the limits from +1 to -1, the velocity will also periodically change, oscillating between the values +2̀̄av and -2̀̄av. Acceleration (w) can also be easily found mathematically, and it is equal to w = -(4̀̄"a/T") sin (2̀̄t/T); if we compare this expression with formula (2), we find w = -(4̀̄"/T")s or, denoting the constant quantity (4̀̄"/T") by k, we obtain w = -ks. We thus see that if a point oscillates harmonically, its acceleration is directed toward the center (the minus sign in the previous formula) and is proportional to the elongation, the distance from the center. This property holds generally and usually serves as a criterion that a given oscillatory motion is harmonic. Finally, the energy of the oscillation turns out to be proportional to the square of the amplitude. Harmonic oscillation is the most important because, according to Fourier's theorem, any periodic motion can be represented as a combination of harmonic oscillations with various amplitudes, but with frequencies related as 1:2:3:4 (fundamental tone, octave, fourth, fifth, etc.). Such a decomposition of periodic motions is important in many practical applications. There are devices that allow this decomposition to be carried out mechanically (harmonic analyzers).

E. Shpolsky