Chapter 10 – Oscillations and Waves (JEE Physics)
1. Oscillatory Motion
A motion which repeats itself about a fixed position (mean position) at regular intervals of time
is called oscillatory motion.
Examples:
- Motion of a pendulum
- Vibration of a spring
- AC current
2. Simple Harmonic Motion (SHM)
An oscillatory motion is called Simple Harmonic Motion if the restoring force is directly
proportional to the displacement and always acts towards the mean position.
$F \propto -x$ or $F = -kx$
3. Equation of SHM
$x = A \sin(\omega t + \phi)$ or $x = A \cos(\omega t + \phi)$
Where:
- $A$ = Amplitude
- $\omega$ = Angular frequency
- $\phi$ = Phase constant
4. Velocity and Acceleration in SHM
$v = \frac{dx}{dt} = A\omega \cos(\omega t + \phi)$
$a = -\omega^2 x$
Acceleration is always directed towards the mean position.
5. Time Period of SHM
$T = \frac{2\pi}{\omega}$
Time period is independent of amplitude (for ideal SHM).
6. SHM of a Spring–Mass System
$T = 2\pi \sqrt{\frac{m}{k}}$
Where:
- $m$ = mass
- $k$ = spring constant
7. Simple Pendulum
A simple pendulum consists of a point mass suspended by a light, inextensible string.
$T = 2\pi \sqrt{\frac{L}{g}}$
Valid only for small angular oscillations.
8. Energy in SHM
Total Energy = $\frac{1}{2}kA^2$
Kinetic Energy = $\frac{1}{2}k(A^2 - x^2)$
Potential Energy = $\frac{1}{2}kx^2$
9. Phase and Phase Difference
Phase determines the state of motion of a particle executing SHM.
Phase difference = $\Delta \phi = \omega \Delta t$
10. Wave Motion
Wave motion is the propagation of disturbance through a medium without
the actual transfer of matter.
Types:
- Transverse waves
- Longitudinal waves
11. Wave Equation
$y(x,t) = A \sin(kx - \omega t + \phi)$
Where:
- $k = \frac{2\pi}{\lambda}$
- $\omega = 2\pi f$
12. Speed of Wave
$v = f\lambda$
Wave speed depends on properties of the medium, not on amplitude.
13. Sound Waves
Sound waves are longitudinal mechanical waves.
$v = \sqrt{\frac{\gamma P}{\rho}}$
14. Beats
Beats are produced due to superposition of two waves of slightly different frequencies.
Beat frequency = $|f_1 - f_2|$
15. Doppler Effect
The apparent change in frequency of sound due to relative motion between
source and observer is called Doppler Effect.
$f' = f \left(\frac{v \pm v_o}{v \mp v_s}\right)$
Where:
- $v_o$ = velocity of observer
- $v_s$ = velocity of source
16. Standing Waves
Standing waves are formed by superposition of two identical waves travelling
in opposite directions.
$\lambda_n = \frac{2L}{n}$
17. Organ Pipes
Organ pipes produce sound by standing waves of air columns.
Open pipe: $f_n = \frac{nv}{2L}$
Closed pipe: $f_n = \frac{nv}{4L}$
18. Resonance
Resonance occurs when frequency of external force equals natural frequency of system.
Results in maximum amplitude.
19. Damped Oscillations
When amplitude decreases with time due to resistive forces, oscillation is damped.
20. Forced Oscillations
Oscillations maintained by an external periodic force are called forced oscillations.
21. Important JEE Exam Tips
- Memorize SHM formulas with physical meaning
- Be careful with Doppler sign convention
- Use energy method to solve tough SHM problems
- Draw phase and energy graphs