Oscillations & Waves – Chapters 15–16

Simple Harmonic Motion (SHM)

• Oscillatory motion that is sinusoidal in time and occurs about an equilibrium position.
• Two universal traits of any oscillator
    – Motion is centered on a stable equilibrium.
    – Motion is periodic with a well-defined period T.
• Sinusoidal oscillation (glider on an air track, mass–spring, pendulum for small angles, etc.) is called simple harmonic motion; non-sinusoidal but repetitive motion (bouncing ball) is merely “harmonic.”
• Key vocabulary
    – Amplitude A: maximum displacement from equilibrium.
    – Cycle: one complete back-and-forth swing.
    – Frequency f: number of cycles per second; SI unit hertz (Hz).
    – Angular frequency \omega: \omega=2\pi f=\frac{2\pi}{T} (rad s⁻¹).
    – Period T=\frac{1}{f}=\frac{2\pi}{\omega} (s).
    – Phase constant \phi_0: specifies where in its cycle the oscillator is at t=0.
• Frequency & period do not depend on amplitude for ideal SHM.
• Real-world analogies
    – Cochlea membrane segment ≈ mass-spring; period ∝ membrane mass & stiffness.
    – Leg swing while walking ≈ pendulum; period ∝ \sqrt{L/g}.

Kinematics of SHM

• Starting at maximum displacement (typical convention \phi0=0): x(t)=A\cos!\left(\omega t\right) v(t)=-\omega A\sin!\left(\omega t\right)\;\;(v{max}=\omega A)
a(t)=-\omega^2 A\cos!\left(\omega t\right)\;\;(a{max}=\omega^2A) • Velocity is zero at the turning points x=\pm A; speed is maximum at equilibrium x=0. • Vertical spring: choose y=0 at static equilibrium; equations identical with y replacing x. • General start position: x(t)=A\cos!\left(\omega t+\phi0\right).

Sinusoidal Relationships

• Changing amplitude shifts peaks vertically; changing \omega changes horizontal spacing (period); changing \phi_0 shifts the graph left–right.

Energy in a Spring–Mass Oscillator (15.3)

• Elastic potential energy U=\tfrac12 kx^2.
• Total mechanical energy (conserved for no damping)
E=K+U=\tfrac12 m v^2+\tfrac12 kx^2=\tfrac12 kA^2=\tfrac12 m v_{max}^2
• Derive motion parameters
\omega=\sqrt{\frac{k}{m}},\;f=\frac{1}{2\pi}\sqrt{\frac{k}{m}},\;T=2\pi\sqrt{\frac{m}{k}}
• Speed–position relation (from energy)
v(x)=\pm\omega\sqrt{A^2-x^2}.

Concept checks (poll highlights)

• Doubling amplitude does not change T.
• For f=0.15\,\text{Hz} earthquake, half-cycle (crest → trough) time =\tfrac{T}{2}=\tfrac{1}{2f}=3.3\,\text{s}.

Example Problem Snapshot

• 5 kg block, k=1.0\times10^3\,\text{N m}^{-1}, initial stretch 0.50 m, release w/ 10 m s⁻¹ toward equilibrium.
a) f=\tfrac{1}{2\pi}\sqrt{k/m}=\;2.2\,\text{Hz}.
b) Total energy E=\tfrac12 kx0^2+\tfrac12 m v0^2.
c) Use E=\tfrac12 kA^2 to solve A.
d) Determine \phi_0 from initial conditions.

Pendulums (15.5)

Simple Pendulum

• Small-angle approximation \sin\theta\approx\theta gives SHM with
\omega=\sqrt{\tfrac{g}{L}},\;T=2\pi\sqrt{\tfrac{L}{g}} (independent of mass).

Physical Pendulum

• Rigid body of moment of inertia I about pivot, center-of-mass distance l:
\omega=\sqrt{\tfrac{mgl}{I}},\;T=2\pi\sqrt{\tfrac{I}{mgl}}.
• Moments of inertia: rod, disk, hoop, sphere (table provided).

Sample conceptual polls

• Halving g ⇒ T\to T\sqrt{2} (~2.8 s if original 2 s).
• Edge-pivot hoop vs. disk of same M,R: hoop has larger I ⇒ longer T.

Driven Oscillations & Resonance (15.7)

• External periodic force at driving frequency f{ext} can replenish energy lost to damping. • Resonance: f{ext}=f_0 (natural frequency) ⇒ large amplitude.
• Biological importance: basilar membrane has position-dependent resonances → pitch discrimination.

Wave Fundamentals (Chap 16)

What is a Wave?

• Traveling disturbance that transports energy (not matter) with speed v characteristic of the medium.
• Types

  1. Mechanical (string, sound, water).

  2. Electromagnetic (light, radio) – need no medium.

  3. Matter waves (electrons).
    • Transverse vs. longitudinal depending on particle motion direction relative to propagation.

Medium Requirements

• Elasticity provides restoring force.
• Random thermal motion ≠ organized wave disturbance.

Wave Speed on a String

• Linear density \mu=\frac{m}{L}.
• Tension Ts. v=\sqrt{\tfrac{Ts}{\mu}}.
• Need 4× tension to double speed (Poll 21).

Visualization Tools

• Snapshot graph: displacement vs. position at fixed time.
• History graph: displacement vs. time at fixed position.
• Convert between the two using known speed & direction.

Sinusoidal Waves (16.3)

• Generated by SHM source; same f as oscillator.
• Wavelength \lambda: spatial period.
• Wave speed relation v=\lambda f=\frac{\omega}{k} (independent of f for given medium).
• Entering new medium: v, \lambda change; f constant.

Mathematical Form

Right-moving wave
D(x,t)=A\sin!\left(kx-\omega t+\phi_0\right)
Left-moving wave: replace kx\to -kx.
Wave number k=\tfrac{2\pi}{\lambda}.

Particle Motion vs. Wave Motion

• Particle speed vp=\partial D/\partial t=-\omega A\cos(\dots) (max \omega A). • Wave speed v constant along string; distinct from vp.

Phase

• Phase \phi=kx-\omega t+\phi_0.
• Phase difference between points separated by \Delta x:
\Delta\phi=\frac{2\pi\,\Delta x}{\lambda}.

Sound Waves & Speed in Gases

• Longitudinal density variations (compressions & rarefactions).
• Ideal-gas speed
v{sound}=\sqrt{\tfrac{\gamma kB T}{m}}=\sqrt{\tfrac{\gamma R T}{M}}
– Monatomic \gamma=1.67; diatomic \gamma=1.40.
• Air: v\approx331+0.6T_C\;\text{m s}^{-1} (343 m s⁻¹ at 20 °C).
• Faster in liquids, fastest in solids.

Ultrasound Imaging

• Uses f>20\,\text{kHz}; pulse reflects at tissue boundaries; reflection strength ∝ difference in sound speed.
• Doppler ultrasound: color-coded velocity info using frequency shift \Delta f=-\frac{2vo\cos\theta}{v}f0.

Electromagnetic Waves

• Self-sustaining E & B field oscillations; no medium required.
• Speed in vacuum c=3.00\times10^8\,\text{m s}^{-1}.
• Index of refraction n=c/v.
• Wavelength in material \lambda{mat}=\lambda{vac}/n (frequency unchanged).

Power, Intensity, and Decibels

• Power P (W); Intensity I=P/A (W m⁻²).
• Spherical source: I=\tfrac{P}{4\pi r^{2}}\;(I\propto1/r^{2}).
• Sound intensity level
\beta(\text{dB})=10\log{10}!\left(\tfrac{I}{I0}\right),\;I_0=1.0\times10^{-12}\,\text{W m}^{-2}.
– ×10 intensity → +10 dB; ÷10 → –10 dB; ×2 → +3 dB.
• Example: 10 flutes in unison raise level from 70 dB to 80 dB (Poll 32).
• Practical hearing range: 0 dB threshold → 130 dB pain. Aging reduces high-frequency sensitivity.

Doppler Effect for Sound (16.7)

• Relative motion source–observer changes perceived frequency.
• Convenient sign rule
– Motion toward ⇒ frequency increases.
– Motion away ⇒ frequency decreases.
• General formula (source at speed vs, observer at vo, positive when moving toward the other):
f' = f \left(\tfrac{v\,\pm vo}{v\,\mp vs}\right)
• Applications: ambulance siren pitch shift, weather radar, medical blood-flow meters, bats tracking insects.

Example Numbers

• Ambulance 33.5 m s⁻¹, siren 400 Hz; car 24.6 m s⁻¹ opposite: approaching f'\approx\;530\,\text{Hz}, receding f'\approx\;305\,\text{Hz}.

2-D & 3-D Waves; Circular & Spherical Fronts

• Wavefront = locus of points in same phase (crests).
• Circular fronts on water surface; spherical fronts from bulb or speaker.
• Far from source, curvature negligible → plane wave approximation.

Quick Reference Summary of Period–Frequency Formulas

| System | \omega | T | Key Parameters |
| Spring-mass | \sqrt{\tfrac{k}{m}} | 2\pi\sqrt{\tfrac{m}{k}} | m,k |
| Simple pendulum | \sqrt{\tfrac{g}{L}} | 2\pi\sqrt{\tfrac{L}{g}} | L,g |
| Physical pendulum | \sqrt{\tfrac{mgl}{I}} | 2\pi\sqrt{\tfrac{I}{mgl}} | I,l |

(Poll, example, and video references in slides serve as checkpoints for each concept above.)