Vibrations & Waves: Key Formulas and Concepts

Vibrations & Waves: Formula Table

1. Simple Harmonic Motion (SHM)

Simple Harmonic Motion describes oscillatory motion where the restoring force is proportional to displacement. It is fundamental in physics for modeling systems like springs and pendulums.

• Displacement, Velocity, Acceleration: $x(t) = A \cos(\omega t + \phi)$ $v(t) = -A \omega \sin(\omega t + \phi)$ $a(t) = -\omega^2 x(t)$

• Angular Frequency: $\omega = 2\pi f = \frac{2\pi}{T}$

• Period of a Mass-Spring System: $T = 2\pi \sqrt{\frac{m}{k}}$

• Angular Frequency of Mass-Spring: $\omega = \sqrt{\frac{k}{m}}$

• Period of a Simple Pendulum (small angles): $T = 2\pi \sqrt{\frac{l}{g}}$

Example: A 0.5 kg mass attached to a spring with $k = 200$ N/m has $T = 2\pi \sqrt{0.5/200} \approx 0.314$ s.

Energy in SHM

• Total Mechanical Energy: $E = \frac{1}{2} k A^2$

• Potential Energy: $U = \frac{1}{2} k x^2$

• Kinetic Energy: $K = E - U = \frac{1}{2} k (A^2 - x^2)$

2. Damped & Driven Oscillations

Damped oscillations occur when energy is lost from the system, typically due to friction or resistance. Driven oscillations involve an external periodic force.

• Damped Motion (light damping): $x(t) = A_0 e^{-bt/2m} \cos(\omega' t)$

• Damped Angular Frequency: $\omega' = \sqrt{\frac{k}{m} - \left(\frac{b}{2m}\right)^2}$

• Driven Oscillator (resonance peak): $\omega_{max} = \omega_0 = \sqrt{\frac{k}{m}}$

Example: Car shock absorbers use damping to reduce oscillations after hitting a bump.

3. Mechanical Waves

Mechanical waves transfer energy through a medium via oscillations. Examples include sound waves and waves on strings.

• Wave Speed: $v = f \lambda$

• Wave Function (Traveling Wave): Right-moving: $y(x, t) = A \sin(kx - \omega t)$ Left-moving: $y(x, t) = A \sin(kx + \omega t)$

• Wave Number & Angular Frequency: $k = \frac{2\pi}{\lambda}$ $\omega = 2\pi f$

• Speed of a Wave on a String: $v = \sqrt{\frac{T}{\mu}}$ where $\mu = \frac{m}{L}$ is the linear mass density.

Example: Guitar strings produce sound by vibrating at specific frequencies determined by tension and mass per unit length.

4. Interference, Standing Waves, Harmonics

Interference occurs when waves overlap, producing patterns of constructive and destructive interference. Standing waves form when two waves of the same frequency travel in opposite directions.

• Standing Wave Pattern: $y(x, t) = 2A \sin(kx) \cos(\omega t)$

• Node Spacing: Node separation: $\frac{\lambda}{2}$

• Antinode Spacing: Antinode separation: $\frac{\lambda}{2}$

• Strings Fixed at Both Ends: $\lambda_n = \frac{2L}{n}$ $f_n = \frac{nv}{2L}$ $n = 1, 2, 3, \ldots$

• Open Pipe: $\lambda_n = \frac{2L}{n}$ $f_n = \frac{nv}{2L}$

• Closed Pipe (one end closed): $\lambda_n = \frac{4L}{n}$ $f_n = \frac{nv}{4L}$ Only odd harmonics: $n = 1, 3, 5, \ldots$

Example: Organ pipes and wind instruments use standing waves to produce musical notes.

5. Sound Waves

Sound waves are longitudinal mechanical waves that propagate through air and other media. Their properties depend on the medium and temperature.

• Speed of Sound in Air: $v = 331 + 0.6T_c$ (m/s), where $T_c$ is temperature in Celsius.

• Intensity: $I = \frac{P}{A}$

• Intensity Level (decibels): $\beta = 10 \log \left(\frac{I}{I_0}\right)$ where $I_0 = 10^{-12}$ W/m$^2$

• Doppler Effect (general formula): $f' = f \left(\frac{v \pm v_{obs}}{v \mp v_{src}}\right)$ Top sign: observer moving toward source / source moving away, etc.

Example: The pitch of a siren changes as an ambulance passes due to the Doppler effect.

6. Energy & Power in Waves

Waves carry energy, and the rate at which energy is transmitted is called power. For waves on a string, power depends on amplitude, frequency, and string properties.

• Power Transmitted by a Wave on a String: $p = \frac{1}{2} \mu \omega^2 A^2 v$

Example: The loudness of a sound from a speaker depends on the power transmitted by the sound waves.

Summary Table: Key Formulas

Topic	Formula	Description
SHM Displacement	$x(t) = A \cos(\omega t + \phi)$	Position as a function of time
Wave Speed	$v = f \lambda$	Speed of a wave
Standing Wave Node Spacing	$\frac{\lambda}{2}$	Distance between nodes
Speed of Sound in Air	$v = 331 + 0.6T_c$	Depends on temperature
Doppler Effect	$f' = f \left(\frac{v \pm v_{obs}}{v \mp v_{src}}\right)$	Frequency shift due to motion
Power in Wave	$p = \frac{1}{2} \mu \omega^2 A^2 v$	Power transmitted by a wave