Resonant Frequency

(a) Define:
(i) Waves (ii) Amplitude of a wave (iii) Wavelength (iv) Frequency of a wave (v) Speed of a wave
(vi) Transverse waves (vii) Longitudinal waves (viii) Stationary waves (ix) Resonant frequency
(x) Resonance (xi) Wave Reflection (xii) Wave refraction (xiii) Wave diffraction
(xiv) Wave interference (xv) Combination of waves.
(b) Calculate the resonant frequency of the circuit illustrated in figure 7.7.

The strings: S7P4A44 (motion - oscillatory).

The math:
Pj Problem of Interest is of type motion (motion - oscillatory).

(ai) Waves: moving disturbance that transport energy from one place to another. Some waves (mechanical waves) need a medium in which they propagate, some (electromagnetic waves) do not. For example, ocean waves need water as medium. While light waves can propagate with or without a medium. (ii) Amplitude of a wave: maximum vertical distance from rest to crest (figure 7.6)
(iii) Wavelength: distance between two consecutive crests or troughs
(iv)Frequency: number of complete wave or number of cycles per unit time (usually per sec)
(v) Speed: frequency (f) x wavelength (λ)
(vi) Transverse wave: motion of medium is perpendicular to the direction of wave propagation.
(vii) Longitudinal wave: motion of medium is parallel to the direction of wave propagation.
(viii) Stationary waves (Standing waves): waves that appear motionless.
(ix) Resonant frequency (Natural frequency): frequency at which standing waves occur.
(x) Resonance: ability of an object to vibrate by absorbing the energy of its own frequency. Consequently, an object will vibrate if it is in close proximity to another vibrating object having a natural frequency equal to its natural frequency.
(xi) Reflection: bouncing back of unabsorbed wave energy when a wave strikes matter.
(xii) Refraction: bending of wave due to a change in speed which is a function of the medium of propagation.
(xiii) Diffraction: bending of wave around the boundary of a barrier.
(xiv) Interference: simultaneous arrival of two or more waves at a place. Same points on interfering waves combine (Constructive interference). Opposite points on interfering waves combine (Destructive interference). Interfering waves combine at different points, some points add and some points subtract (Varying interference).
(xv) Combination waves: waves are neither transverse nor longitudinal but a hybrid of traverse and longitudinal waves. The hybrid of transverse and longitudinal mediums movements result in circular movement.

(b) figure 7.7 is a parallel LC tank circuit. It is used in frequency applications as a tuning orfrequency selection device.
For a generic parallel LC circuit of figure 7.7, let the resistance = R, the inductance = L, and the capacitance = C
So, for resistor R in series with inductor L, the impedance Z1 is:
Z1 = R + jωL
For the capacitor C, impedance Z2 = 1/jωC
So, admittance Y(ω) = 1/Ztotal = 1/Z1 + 1/Z2 = 1/(R + jωL) + jωC-------(1)
So, Y(ω) = R/(R2 + (ωL)2) + j[ωC - ωL/(R2 + (ωL)2)] -------(2)
Equation (2) is the result of multiplying equation (1) with the complex conjugate (R - jωL).
At resonance, ωoC = ωoL/(R2 + (ωL)2). Where ωo is the resonant frequency in rad/s.
So, ωo = [(1 - (R2C)/L)1/2]/(LC)1/2-------(3)
Where R is in ohms, C is in Farad, L is in Henry.
Now substituting the values given in figure 7.7 into equation (3)
We have resonant frequency, ωo = 1,000 rad/s = 1,000/2π Hz = 159.2 Hz.

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