Oscillations, Traveling Waves, and Sound: Study Notes

Oscillations

Equilibrium and Restoring Forces

Oscillatory systems are characterized by a restoring force that acts to return the system to its equilibrium position, where the net force is zero.

• Equilibrium: The state where the sum of all forces acting on a system is zero.

• Restoring Force: A force that acts in the direction opposite to displacement, tending to bring the system back to equilibrium.

• Linear Restoring Force: For many systems (e.g., springs), the restoring force is proportional to displacement: $F = -kx$

Simple Harmonic Motion (SHM)

Simple harmonic motion occurs when the restoring force is directly proportional to displacement and directed toward equilibrium.

• Displacement as a function of time: $x(t) = A \cos(\omega t)$

• Velocity: $v(t) = -A\omega \sin(\omega t)$

• Acceleration: $a(t) = -\omega^2 x$

• Amplitude (A): Maximum displacement from equilibrium.

• Angular Frequency (\(\omega\)): Determines how rapidly the system oscillates.

Angular Frequency and Period

The angular frequency, period, and frequency are fundamental quantities describing oscillatory motion.

• Angular Frequency: $\omega = 2\pi f = 2\pi / T$

• Frequency: $f = 1 / T$

• Period (T): Time for one complete oscillation.

Mass–Spring System

A mass attached to a spring exhibits SHM, with frequency determined by the spring constant and mass.

• Frequency: $f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$

• Increasing the spring constant (k) increases the frequency.

• Increasing the mass (m) decreases the frequency.

Energy in SHM

The total mechanical energy in SHM is conserved and alternates between kinetic and potential forms.

• Total Energy: $E = \frac{1}{2}kA^2$

• At equilibrium: Kinetic energy is maximum, potential energy is zero.

• At maximum displacement: Potential energy is maximum, kinetic energy is zero.

Pendulum (Small Angle Approximation)

A simple pendulum exhibits SHM for small angles (typically less than 15°).

• Period: $T = 2\pi \sqrt{\frac{L}{g}}$

• The period is independent of mass and depends only on the pendulum length (L) and gravitational acceleration (g).

Damped Oscillations

Real oscillatory systems experience damping, causing amplitude to decrease over time.

• Amplitude decay: $A(t) = A_0 e^{-t/\tau}$

• Damping constant (\(\tau\)): Characterizes the rate of amplitude decrease.

Resonance

Resonance occurs when a system is driven at its natural frequency, resulting in large amplitude oscillations.

• Driving frequency equals natural frequency.

• Can lead to dramatic effects (e.g., bridge collapse, glass shattering).

Traveling Waves and Sound

Wave Model

Waves are disturbances that transfer energy through space or a medium, without transferring matter.

• Mechanical Waves: Require a medium (e.g., sound, water waves).

• Electromagnetic Waves: Do not require a medium (e.g., light, radio waves).

Types of Waves

Waves are classified by the direction of particle motion relative to wave propagation.

• Transverse Waves: Particle motion is perpendicular to wave direction (e.g., waves on a string).

• Longitudinal Waves: Particle motion is parallel to wave direction (e.g., sound waves).

• Sound Waves: Are longitudinal mechanical waves.

Wave Speed

The speed of a wave depends on the properties of the medium.

• For a stretched string: $v = \sqrt{\frac{T}{\mu}}$ where T is tension and \mu is mass per unit length.

• Wave speed does not depend on amplitude.

Fundamental Wave Relationship

The relationship between wave speed, frequency, and wavelength is fundamental to all wave phenomena.

• $v = f\lambda$

• If frequency increases in the same medium, wavelength decreases.

Sinusoidal Wave Equation

A sinusoidal wave traveling to the right can be described mathematically as:

• $y(x,t) = A \cos(kx - \omega t)$

• Wave number: $k = \frac{2\pi}{\lambda}$

Sound Waves

Sound waves consist of alternating compressions and rarefactions in a medium.

• Human hearing range: 20 Hz to 20,000 Hz.

Intensity

Intensity measures the power transmitted by a wave per unit area.

• $I = \frac{P}{A}$

• For spherical waves: $I = \frac{P}{4\pi r^2}$

• Intensity decreases with the square of the distance from the source.

Decibel Scale

The decibel (dB) scale is a logarithmic measure of sound intensity relative to a reference level.

• $\beta = 10 \log_{10}\left(\frac{I}{I_0}\right)$

• Reference intensity: $I_0 = 1.0 \times 10^{-12}$ W/m2

• Each 10 dB increase corresponds to a tenfold increase in intensity.

Doppler Effect

The Doppler effect describes the change in observed frequency due to relative motion between source and observer.

• Approaching source: Observed frequency increases.

• Receding source: Observed frequency decreases.

Shock Waves

Shock waves occur when a source moves faster than the speed of sound in the medium, producing a sonic boom.

• Associated with supersonic aircraft and explosions.