Wave CT 2 Compact

Forced Vibration is the vibration of a system under the influence of a continuous, external, periodically applied force of a frequency ( $p$ ) same as the oscillator.
Governing Equation: $\frac{d ^{2} y}{d t ^{2}} + 2 λ \frac{d y}{d t} + ω^{2} y = f_{0} sin pt$ .
Amplitude ( $A$ ): $A = \frac{f _{0}}{( ω ^{2} - p ^{2} ) ^{2} + 4 λ ^{2} p ^{2}}$ .
Resonance Condition ( $P_{R}$ ): Maximum displacement occurs when $p = P_{R} = ω^{2} - 2 λ^{2}$ .
Maximum Amplitude ( $A_{ma x}$ ): $A_{ma x} = \frac{f _{0}}{2 λ ω ^{2} - λ ^{2}}$ .

Wave Motion transports energy and momentum without transporting matter.
Types: Mechanical (needs media; Ex: Sound/Water waves); Non-mechanical (no media; Ex: Light/Radio waves).
Progressive Wave Equation: $y = a sin (ω t - k x)$ OR $y = a sin \frac{2 π}{λ} (v t - x)$ .
Wave Velocity ( $V$ ): $V = \frac{ω}{K} = f λ = \frac{λ}{T}$ .
Differential Equation: $\frac{d ^{2} y}{d t ^{2}} = V^{2} \frac{d ^{2} y}{d x ^{2}}$ .
Energy Density ( $E$ ): $E = 2 π^{2} f^{2} a^{2} ρ$ .
Intensity ( $I$ ): $I = 2 π^{2} f^{2} ρ a^{2} V$ (where $I \propto a^{2}$ ).

Sound Wave: Mechanical, longitudinal wave requiring a medium (velocity $v = f λ$ ).
Audible Frequency: $20 Hz \leq f \leq 20, 000 Hz$ .
Doppler Effect: Apparent change in frequency ( $f^{'}$ ) due to relative motion between source and observer.
- $f^{'} = f \times \frac{V \pm V _{o}}{V \mp V _{s}}$ .
Beats: Formed by interference of two wave trains.
- Beat Frequency: Always equal to the difference of the two nearly equal frequencies, $n_{1} - n_{2}$ beats per second.
- Maximum Sound Time: $t = \frac{k}{n _{1} - n _{2}}$ (where $k = 0, 1, 2...$ ).
Intensity Level ( $L$ ): $L = k lo g_{10} (\frac{I}{I _{0}})$ .
Reverberation: Persistence of audible sound after the source has stopped.

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