Vibrations and Waves – Bullet-Point Study Notes

Objectives

• Master prerequisites and outcomes for Chapter 11 on Vibrations & Waves.
• Become able to:
– State physical conditions that create SHM (linear restoring force toward equilibrium).
– Compute SHM period for mass $m$ on a spring of force constant $k$ using $T = 2\pi\sqrt{m/k}$.
– Find instantaneous velocity, acceleration, kinetic (KE) & potential energy (PE) anywhere in the motion.
– Write (and use) sinusoidal equations for $x(t),\,v(t),\,a(t)$ when amplitude $A$ and angular speed $\omega$ are known.
– Obtain the period of a simple pendulum of length $L$ via $T = 2\pi\sqrt{L/g}$.
– Define resonance, name beneficial vs. destructive scenarios, and explain damping strategies.
– Distinguish transverse vs. longitudinal waves; give everyday examples.
– Calculate wave speeds in liquids, solids, strings/ropes from elastic or tension parameters.
– Evaluate wave energy transmission, power, and intensity through an area $A$.
– Describe reflection, refraction, interference (constructive & destructive) and diffraction.
– Explain standing-wave formation on a string and compute harmonic frequencies.

Key Terms & Phrases

• Simple harmonic motion (SHM) – periodic oscillations obeying two conditions:

1. $F\propto x$ and $|a|\propto |x|$.
2. $\vec F$ and $\vec a$ are opposite in direction to displacement $\vec x$.
   • Amplitude $A$ – max displacement from equilibrium.
   • Period $T$ – time for one full cycle; Frequency $f = 1/T$.
   • Simple pendulum – point mass on mass-less string, small-angle approximation $\theta_{\max}\lesssim15^\circ$.
   • Damping – dissipative losses causing amplitude decay.
   • Forced vibration – oscillation driven by an external periodic force.
   • Resonance – large-amplitude response when driving frequency matches a natural (resonant) frequency.
   • Wave classifications:
   – Transverse: particle motion ⟂ wave direction (e.g.
   guitar string, EM waves).
   – Longitudinal: particle motion ∥ wave direction (e.g.
   sound in air, slinky compression).
   • Crest / trough – extreme displacements in a transverse wave.
   • Compression / rarefaction – high / low-density regions in a longitudinal wave.
   • Wavelength $\lambda$ – spatial period between repeating points.
   • Intensity $I$ – power per unit area normal to propagation.
   • Principle of superposition – total displacement = algebraic sum of overlapping waves.
   • Constructive / destructive interference – superposition giving larger / smaller amplitude.
   • Diffraction – bending around obstacles; amount increases with larger $\lambda$ or smaller opening.
   • Standing wave – stationary pattern from two identical, opposite-direction waves; nodes vs. antinodes.
   • Harmonics – discrete allowed frequencies: fundamental (1st harmonic) and overtones (2nd, 3rd…).

Simple Harmonic Motion Fundamentals

• Hooke’s Law restoring force: $F = -k x$ (negative sign: force toward equilibrium).
• Net force zero at $x=0$ ⇒ acceleration zero; speed maximum.
• Two coupled differential equations (taking right as +):
– $\dfrac{d^2x}{dt^2} + \dfrac{k}{m}x = 0$ with solution $x(t)=A\cos(\omega t+\phi)$.
– $\omega = 2\pi f = 2\pi/T = \sqrt{k/m}$.

Energy in the SH Oscillator

• Potential energy in spring: $PE = \tfrac12 k x^2$.
• Kinetic energy: $KE = \tfrac12 m v^2$.
• Total mechanical energy (conserved if no damping): $E = KE + PE = \tfrac12 k A^2$.
• Velocity as a function of position:
$v(x) = \pm\sqrt{\dfrac{k}{m}(A^2 - x^2)} = v<em>{\max}\sqrt{1 - x^2/A^2}$, where $v</em>{\max}=\omega A$.

Sinusoidal Equations (taking $x=A$ at $t=0$)

• Displacement: $x(t)=A\cos\omega t$.
• Velocity: $v(t)=-\omega A\sin\omega t$.
• Acceleration: $a(t)=-\omega^2 A\cos\omega t$ (out of phase with displacement by $180^\circ$, in phase with force).

Reference-Circle Interpretation

• Projecting uniform circular motion onto one axis reproduces SHM.
• Angle $\theta = \omega t$ maps arc to time; projection radius $A$ gives physical displacement.

Simple Pendulum

• Period (small angles): $T = 2\pi\sqrt{L/g}$ – independent of mass & amplitude (isochronous property).
• Sensitivity: lower $g$ (high altitude, lunar surface) ⇒ longer period ⇒ clock loses time.
• Experimental gravity measurement: rearrange $g = 4\pi^2 L/T^2$.

Damped Harmonic Motion

• Real oscillators lose energy to air drag, internal friction; amplitude decays exponentially.
• Equation: $m\ddot x + b\dot x + kx = 0$, where $b$ = damping constant.
• Critical applications: car shock absorbers; suspension bridges to prevent resonance.

Forced Vibrations & Resonance

• Driven system: $m\ddot x + b\dot x + kx = F<em>0\cos\omega</em>d t$.
• Amplitude response peaks at $\omega<em>d\approx\omega</em>0=\sqrt{k/m}$ (sharpness depends on damping).
• Beneficial: pushing a swing, musical sound amplification.
• Destructive: Tacoma Narrows Bridge collapse, wine-glass shatter by voice.
• Engineering fix: add damping or detune natural frequencies.

Mechanical Waves – Generalities

• A wave transports energy & momentum, not (significantly) mass.
• Periodic wave speed relation: $v = f\lambda$.

Transverse Wave on a String

• Speed determined by tension $F<em>T$ and linear mass density $\mu = m/L$: $v = \sqrt{F</em>T/\mu}$.

Longitudinal Waves in Media

• Solids (rod): $v = \sqrt{E/\rho}$, $E$ = Young’s modulus.
• Fluids & gases: $v = \sqrt{B/\rho}$, $B$ = bulk modulus.
• Sound in aluminum example: $v \approx 5.09\times10^3\,\text{m/s}$.

Wave Energy, Power & Intensity

• Energy through distance $vt$ in time $t$: $E = 2\pi^2\rho A v t f^2 x<em>0^2$ (here $A$ is area, $x</em>0$ amplitude).
• Average power: $P = E/t = 2\pi^2\rho A v f^2 x<em>0^2$. • Intensity: $I = P/A = 2\pi^2\rho v f^2 x</em>0^2$.
• Spherical spreading: $I \propto 1/r^2$, amplitude also $\propto 1/r$.

Wave Behaviors

Reflection

• Law: angle of incidence $\theta<em>i$ equals angle of reflection $\theta</em>r$.
• Fixed end reflection on a string inverts phase (crest→trough); free end keeps phase.

Refraction

• Change of speed in new medium bends wave; shallow water ⇒ slower ⇒ wavefront bends toward normal.

Interference

• Superposition principle governs net displacement.
• Constructive: path difference $n\lambda$ ⇒ amplitudes add.
• Destructive: path difference $(n+\tfrac12)\lambda$ ⇒ cancellation.

Diffraction

• Pronounced when obstacle/opening size ≲ $\lambda$; explains why low-frequency sound bends around corners.

Standing Waves in Strings

• Arise from interference of incident & reflected waves of same $f,\,\lambda$.
• Node spacing: $\tfrac12\lambda$; antinode halfway between nodes.
• Condition for a string fixed at both ends (length $L$):
$\tfrac12\lambda = \dfrac{L}{n}$ ⇒ $\lambda<em>n = \dfrac{2L}{n}$. • Harmonic frequencies: $f</em>n = n\dfrac{v}{2L},\;n=1,2,3,\ldots$.
• Fundamental $n=1$; first overtone $n=2$ (2×fundamental), etc.
• Third-harmonic example (given $f=60\,\text{Hz}$, $L=0.250\,\text{m}$) yields $v=10.0\,\text{m/s}$.

Problem-Solving Strategies

• SHM spring:

1. Tabulate given $m,\,\Delta x,\,F,\,A$.
2. Use Hooke to find $k$; then $T$, $E<em>{total}$ via energy conservation. • Simple pendulum: direct use of $T(L),\,g$; check small-angle assumption. • Wave speed: identify medium (string, solid, fluid) ⇒ correct formula. • Energy / intensity: compute $v, f, x</em>0, A$; apply formulas; watch for spherical spreading.
   • Standing waves: sketch harmonic, find $\lambda$ from node spacing; use $v=f\lambda$ and string tension relation when necessary.

Numerical Examples & Insights

• Spring stretches $0.300\,\text{m}$ with $1.00\,\text{kg}$ mass ⇒ $k=32.7\,\text{N/m}$; period $1.10\,\text{s}$; total energy $0.235\,\text{J}$.
• Sinusoidal equation $x=2.00\cos(\pi t)$ ⇒ $A=2.00\,\text{m},\,f=0.500\,\text{Hz},\,T=2.00\,\text{s}$; at $t=1/2\,\text{s}$, $x=1.00\,\text{m}, v=-5.44\,\text{m/s}, a=-9.86\,\text{m/s}^2$.
• Aluminum rod (sound) speed $\approx5090\,\text{m/s}$; 440 Hz note has $\lambda=11.6\,\text{m}$.
• Earthquake intensity falls off with $r^2$; 48 km-to-1 km drop gives gain factor $\sim2300$ ⇒ huge increase in $I$ near epicenter.
• Guitar string (0.62 m speaking length, 520 N tension, 3.6 g mass): fundamental $290\,\text{Hz}$, overtones $580$ & $870\,\text{Hz}$.

Connections & Applications

• Clocks: pendulum timekeeping limited by local $g$ variations → need compensation for altitude.
• Seismology: intensity formulas underpin Richter-like magnitude scales.
• Engineering: designing bridges/buildings with tuned mass dampers to suppress resonance.
• Musical instruments: fretting a guitar changes $L$, altering $\lambda$ and $f$; bowing a violin supplies continual forced vibration, while body shape enhances resonance.
• Medical ultrasound: intensity control critical for safety; high-frequency (short-wavelength) waves provide fine imaging but attenuate quickly.
• Ethical note: Ensure structural designs include damping to protect lives; monitor industrial vibration to prevent harmful noise exposure.

Quick-Reference Formula List

• Spring SHM period: $T = 2\pi\sqrt{m/k}$.
• Pendulum (small-angle): $T = 2\pi\sqrt{L/g}$.
• Sinusoidal kinematics: $x=A\cos\omega t, v=-\omega A\sin\omega t, a=-\omega^2 A\cos\omega t$.
• Wave speed: $v=f\lambda$.
• String wave: $v = \sqrt{F<em>T/\mu}$. • Longitudinal solid: $v = \sqrt{E/\rho}$; fluid: $v = \sqrt{B/\rho}$. • Wave energy: $E = 2\pi^2\rho A v t f^2 x</em>0^2$.
• Power: $P = 2\pi^2\rho A v f^2 x<em>0^2$. • Intensity: $I = 2\pi^2\rho v f^2 x</em>0^2$.
• Standing-wave wavelengths: $\lambda<em>n = 2L/n$; frequencies: $f</em>n = n v/(2L)$.

These bullet-point notes condense every major & minor concept, definitions, formulas, worked-example results, problem-solving tactics, and real-world or ethical insights from the full Chapter 11 transcript, allowing for standalone study without the original text.