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 $\phi<em>0=0$): $x(t)=A\cos!\left(\omega t\right)$ $v(t)=-\omega A\sin!\left(\omega t\right)\;\;(v</em>{max}=\omega A)$
$a(t)=-\omega^2 A\cos!\left(\omega t\right)\;\;(a<em>{max}=\omega^2A)$ • 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: $x(t)=A\cos!\left(\omega t+\phi</em>0\right)$.

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 $E=\tfrac12 kx<em>0^2+\tfrac12 m v</em>0^2$.
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 $f<em>{ext}$ can replenish energy lost to damping. • Resonance: $f</em>{ext}=f_0$ (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 $T<em>s$. $v=\sqrt{\tfrac{T</em>s}{\mu}}$.
• 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 $v<em>p=\partial D/\partial t=-\omega A\cos(\dots)$ (max $\omega A$). • Wave speed $v$ constant along string; distinct from $v</em>p$.

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
$v<em>{sound}=\sqrt{\tfrac{\gamma k</em>B T}{m}}=\sqrt{\tfrac{\gamma R T}{M}}$
– 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 $\Delta f=-\frac{2v<em>o\cos\theta}{v}f</em>0$.

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 $\lambda<em>{mat}=\lambda</em>{vac}/n$ (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
$\beta(\text{dB})=10\log<em>{10}!\left(\tfrac{I}{I</em>0}\right),\;I_0=1.0\times10^{-12}\,\text{W m}^{-2}$.
– ×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 $v<em>s$, observer at $v</em>o$, positive when moving toward the other):
$f' = f \left(\tfrac{v\,\pm v<em>o}{v\,\mp v</em>s}\right)$
• 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.)