Waves and Sound – Comprehensive Study Notes

Waves vs. Particles: Fundamental Idea

• Wave = traveling disturbance

  • Requires a medium or can be self-sustaining (e.g., light)

  • Transports only energy, not matter

  • Medium’s individual particles oscillate about equilibrium positions to create the disturbance

  • Energy moves from source → receiver without bulk migration of the material

Categorizing Waves by Particle Motion Relative to Wave Propagation

• ## 1. Transverse Waves

  • Particle displacement ⟂ wave velocity direction

  • Classic visualization: shake a stretched spring up–down while pulse travels left→right

  • Examples

  • Waves on strings (guitar, violin, piano)

  • Electromagnetic waves (light, radio, X-rays) – to be elaborated next semester

• ## 2. Longitudinal Waves

  • Particle displacement ∥ wave velocity direction

  • Visualization: push–pull compression pulses along a Slinky

  • Examples: sound in air, ultrasound in tissues, seismic P-waves

• ## 3. Mixed or Surface Waves

  • Water waves: particle paths are nearly circular → combination of transverse & longitudinal components

  • Any surface or interface wave whose restoring forces include gravity + surface tension often displays this hybrid motion

Everyday Analogies for Concept Reinforcement

• Football-stadium “wave”

  • People stand/sit (vertical motion) while the disturbance races around stands → behaves like a transverse wave

• Fishing float on a lake

  • As ripples pass underneath, float executes simultaneous up–down & back–forth → circular path like water particles → mixed wave

Describing Periodic (Sinusoidal) Waves

• Periodic wave: pattern repeats via Simple Harmonic Motion (SHM) of source

• Key measurable quantities

  1. Amplitude, A – maximum displacement from equilibrium (energy ∝ $A^{2}$)

  2. Wavelength, (\lambda) – distance between successive like points (crest→crest, compression→compression)

  3. Period, T – time for one complete oscillation (360° phase)

  4. Frequency, f – oscillations per second; $f = 1/T$ (unit: Hz = $\text{s}^{-1}$)

• Concept check: Longitudinal waves do possess amplitude (magnitude of compression/rarefaction displacement or pressure variation)

Universal Wave Speed Relation

• Travel speed connects space & time aspects:
  $v = \frac{\lambda}{T} = \lambda f$

• Holds for any periodic wave provided medium properties remain uniform

Worked Numerical Example — Radio Wave

• AM broadcast: $f = 1230\,\text{kHz} = 1.230\times10^{6}\,\text{Hz}$

• EM wave speed: $v = 3.00\times10^{8}\,\text{m/s}$

• Wavelength: $\lambda = \frac{v}{f} = \frac{3.00\times10^{8}}{1.230\times10^{6}} \approx 244\,\text{m}$

• Physical insight: lower-frequency AM waves possess long wavelengths → good diffraction & long-range coverage

Conceptual & Quantitative Practice Questions (Slide references)

• Coil in Slinky (Q6): Although the pulse advances 1 m in 1 s, an individual coil merely oscillates about its original position → answer B: No

• Sound entering water (Q7)

  • $f$ fixed by source

  • $v<em>{\text{air}} = 343\,\text{m/s},\; v</em>{\text{water}} = 1482\,\text{m/s}$

  • $\lambda = v/f$ ∴ larger speed ⇒ increase in $\lambda$ → answer A: Increase

• Boat crest→trough (Q8)

  • Given $\lambda = 20\,\text{m},\; v = 5\,\text{m/s}$

  • $T = \lambda / v = 4\,\text{s}$

  • Crest→trough is half a cycle ⇒ $T/2 = 2\,\text{s}$ → answer B: 2 s

• Graph pairs (Q9)

  • Determine $\lambda$ by x-axis spacing; given speed $v = 12\,\text{m/s}$, compute $f = v/\lambda$ for each curve

Wave Speed on a String – Microscopic View

• Neighbor interactions transmit the pulse

• According to Newton II: $F_{\text{net}} = ma$; higher pulling force ⇒ larger acceleration of particle ⇒ faster information transfer

• Two controlling material parameters

  1. Tension, F_T (N)

  • Greater tension → stronger restoring forces between adjacent segments ⇒ larger $v$

  1. Linear mass density, (\mu = m/L) (kg / m)

  • Lower inertia (smaller $\mu$) → higher acceleration ⇒ larger $v$

• Ideal string wave speed formula
  $v = \sqrt{\frac{F_T}{\mu}}$

Musical Connections

• Stringed instruments exploit transverse standing waves; tuning performed by changing tension or effective vibrating length to manipulate $v$ and hence resonance frequencies $f_n = n\,v/(2L)$

• Example (Q10): Electric-guitar E strings

  • Data: $L = 0.628\,\text{m},\;F_T = 226\,\text{N}$

  • $\mu_{\text{high}} = 0.208\,\text{g}/0.628\,\text{m} = 3.31\times10^{-4}\,\text{kg/m}$

  • $v_{\text{high}} = \sqrt{\frac{226}{3.31\times10^{-4}}} \approx 830\,\text{m/s}$

  • $\mu_{\text{low}} = 3.32\,\text{g}/0.628 = 5.29\times10^{-3}\,\text{kg/m}$

  • $v_{\text{low}} = \sqrt{\frac{226}{5.29\times10^{-3}}} \approx 207\,\text{m/s}$

  • Insight: thicker (larger $\mu$) bass string vibrates more slowly, yielding lower pitch

Strategic Tension Question (Q11)

• Two identical strings with equal tension; to make pulse on B overtake A, increase tension on B → raises its wave speed (Answer A)

Linear Density Problem (Q12)

• Piano middle-C string

  • Given $T = 944\,\text{N},\;T_{\text{period}} = 3.82\,\text{ms}=3.82\times10^{-3}\,\text{s},\;\lambda =1.26\,\text{m}$

  • Wave speed: $v=\lambda/T = 1.26/3.82\times10^{-3}\approx 330\,\text{m/s}$

  • Linear density: $\mu = F_T / v^{2} = 944/330^{2} \approx 8.67\times10^{-3}\,\text{kg/m}$

Ethical & Practical Implications

• Acoustic engineering: adjusting tension & material density influences sound quality and sustainability of musical instruments

• Communication tech: precise control of EM wave frequency & wavelength underpins radio allocation and broadcasting regulations

• Oceanography: mixed-mode water waves affect ship stability, coastal erosion, and renewable-energy devices (wave power)

Recap & Key Takeaways

• Waves convey energy; medium’s particles oscillate about equilibrium

• Direction of particle motion defines transverse, longitudinal, or surface waves

• Periodic waves characterized by $A,\;\lambda,\;T,\;f$ with universal relation $v = \lambda f$

• Medium properties dictate speed; on strings, $v = \sqrt{F_T/\mu}$

• Real-world examples (stadium, fishing float, guitars, radios) cement conceptual understanding and show practical relevance