Parametric resonance. Standing waves.

In order to determine vibrator’s frequency we use standing waves that appear in tense thread connected to vibrator. Oscillations of any oscillatory system can be made forced by driving any of its parameters. The resonance that will appear in this case is called the parametric resonance.

Let’s consider a horizontally tense thread fixed at one end. Another end of the thread connected to vibrator that is able to oscillate along the thread. In such system will appear transverse oscillations caused by periodically varying tension of thread. Let at initial moment of time the vibrator was shifted left and the thread is freely sags in gravity field (Fig. 14, a). When vibrator-rod moves right the tension increases until thread wouldn’t come in its horizontal position. In this case all points of thread will move upwards (Fig. 14, b) (velocities of some points shown as arrows). At reverse (right to left) motion of vibrator-rod thread’s tension decreases but its points will move upwards by inertia (Fig. 14, c). Over next complete oscillation of vibrator-rod the thread will sag again. Thus, by the time of two complete oscillations of vibrator the thread oscillates once. Hence, vibrator frequency is double thread’s frequency. Obviously, that considered above method of excitation of oscillations is parametric – external affection (vibrator’s oscillations) causes periodical alternations of one of the parameters of a thread (thread’s tension).

Any system which is able to oscillate will perform natural oscillations under the action of random external affection. If the frequency of this parametric affection is double the system’s natural frequency the oscillations become forced (holds the conditions of parametric resonance). Same phenomena can be achieved in electric circuit by external alternation of capacitance or inductance. Work of parametric generators and amplifiers is based on parametric resonance phenomenon.

Oscillations of vibrator produce elastic wave, propagating along the thread with frequency ω and amplitude A. All points of thread will perform undamped oscillations if thread is not quite long and dissipations of energy are small.

Let’s set O (point of pulley and thread connection) as origin of coordinates. A wave, which has reached this point, will be reflected back. Thus we’ll have two waves travelling in opposite directions:

All points of a thread will be behind vibrator’s oscillations as much, as smaller will be x in chosen coordinate system (Fig. 15). That is why equation (68) is equation of incident plane travelling wave, and (69) – equation of reflected wave. Minus sign before kx in reflected wave equation means that phase of oscillations of each thread’s point in reflected wave as smaller as far this point is from origin of coordinates. Besides, as point O immovable, phase of reflected wave will be deferent at that point from phase of incident wave on π.

The result of superposition of incident and reflected waves will be complex oscillations of all points of thread under the law:

ξ = ξ₁ + ξ₂ = 2A× cos × cos ,

or ξ = 2A×cos(kx – )×cos(ωt + )

(70)

Thus, we’ve obtain equation of steady state standing wave in a thread. A time independent expression A_SW(x)=|2A cos(kx– )| represents the amplitude of standing wave. As it seen, the amplitude varies under harmonic law with respect to distance x from origin of coordinates. For points at which

kx – = nπ, (n = 0, 1, 2, …)

(71)

amplitude gets maximal value equal to A_SW(x_AN)=2A. These points called antinodes or wave crests. From (71) we can obtaine coordinates of antinodes:

xAN = (2n + 1) .

(72)

Points, at which

kx - = (n - )π, (n = 0, 1, 2, …)

(73)

do not oscillate and A_SW(x_N)= 0. These points called nodes of standing wave. The nodes appear in points at which incident and reflected waves are in antiphase. As it follows from (73), the coordinates of nodes are

xN = 2n .

(74)

Distance between two nodes or antinodes called standing wavelength. That is why

λ_ST = λ.

Since there are nodes at both fixed ends of a thread, so, thread’s length l must contain integer number of standing wavelengths

l = n λ_ST = n .

On a Fig. 16 shown appearance of standing wave by reflection, with phase, changed on π. Solid line represents displacement of thread’s points produced by incident wave, dashed line – by reflected wave. The resultant displacement represented by bold line. The figure shows moment at which incident wave come to point O with non-zero phase.

The main feature of standing wave is that it doesn’t transport energy. That follows from the fact that position of nodes and antinodes doesn’t change with time. Such particularity is the result of the fact that the superposing wave transfer equal quantities of energy in the opposite directions.

Date: 2015-12-24; view: 705