Oscillations and Waves Cheatsheet

Cheat Codes & Shortcuts

• Simple Harmonic Motion (SHM): \( a = -\omega^2 x \), where \( \omega \) is angular frequency.
• Displacement: \( x(t) = A \cos(\omega t + \phi) \), \( A \) is amplitude, \( \phi \) is phase.
• Angular Frequency: \( \omega = \sqrt{\frac{k}{m}} \) for a mass-spring, \( \omega = \sqrt{\frac{g}{l}} \) for a pendulum.
• Period: \( T = \frac{2\pi}{\omega} \).
• Wave Equation: \( \frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2} \), where \( v \) is wave speed.
• Wave Speed: \( v = f \lambda \), where \( f \) is frequency, \( \lambda \) is wavelength.
• Superposition: Waves add linearly when they overlap.
• Standing Waves: \( \lambda_n = \frac{2L}{n} \), where \( L \) is length, \( n \) is mode number.
• Doppler Effect: \( f' = f \frac{v \pm v_o}{v \mp v_s} \), observer/source moving relative to wave speed \( v \).
• Energy in SHM: \( E = \frac{1}{2} k A^2 \), total energy conserved.

Quick Reference Table

Advice

Oscillations and Waves Quick Tips

• SHM Equations: Use \( x = A \cos(\omega t + \phi) \) or \( x = A \sin(\omega t + \phi) \) based on initial conditions.
• Frequency and Period: Relate via \( f = \frac{1}{T} \), and \( \omega = 2\pi f \).
• Wave Equation: Solve \( \frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2} \) for traveling waves.
• Standing Waves: Nodes are at \( x = \frac{n \lambda}{2} \), anti-nodes at \( x = \frac{(2n-1) \lambda}{4} \).
• Doppler Effect: Adjust signs based on relative motion direction of source and observer.