Wave Mechanics and Sound Notes

Types of Vibrations

• Free Vibrations: Vibrations occurring at the body's natural frequency without external periodic forces.
• Forced or Driven Vibrations: Vibrations caused by an external periodic force (e.g., sonometer wire, resonance tube, microphone diaphragm).
• Resonance: A special case of forced vibration where the external driving frequency matches the body's natural frequency ($n_1 = n_2$), resulting in maximum amplitude and sound intensity.
• Undamped Vibrations: Occur when frictional forces are zero, maintaining a constant amplitude ($a$) with no energy dissipation.
• Damped Vibrations: Occur in the presence of friction, causing continuous dissipation of energy and decreasing amplitude over time.

Plane Progressive Waves

• Definition: A wave continuously propagating forward through periodic particle vibrations, transferring energy across space.
• Properties:
  • All particles vibrate with equal amplitude ($a$) and period ($T$).
  • All particles pass equilibrium positions successively at maximum speed ($a\rho$ or $a\frac{2\text{π}}{T}$).
  • No particles remain permanently at rest or cross equilibrium simultaneously.
• General Equation:
  • $y = a \sin((\omega t \pm kx) + \phi_0)$
  • Negative sign between $t$ and $x$ indicates wave propagation along the positive x-axis; positive sign indicates propagation along the negative x-axis.
  • Wave Number / Propagation Constant: $k = \frac{2\pi}{\lambda}$
  • Angular Frequency: $\omega = 2\pi f = \frac{2\pi}{T}$
  • Phase Velocity: $v = \frac{\omega}{k}$
• Particle Velocity ($v_{\text{particle}}$) & Wave Slope:
  • $v_{\text{particle}} = \frac{dy}{dt} = \omega a \cos(\omega t - kx) = \omega \sqrt{a^2 - y^2}$
  • Maximum velocity occurs at mean position ($y = 0$): $v_{\text{max}} = \omega a$
  • Minimum velocity occurs at extreme position ($y = a$): $v_{\text{min}} = 0$
  • Relation to wave slope: $v_{\text{particle}} = -\text{Wave velocity} \times \text{Slope of the wave}$

Velocity of Transverse and Longitudinal Waves

• Transverse Waves on a String:
  • Velocity: $v = \sqrt{\frac{T}{m}}$, where $T$ is tension and $m = \rho A$ is mass per unit length.
  • In solid media (e.g., S-seismic waves): $v = √{\frac{\eta}{\rho}}$, where $\eta$ is the modulus of rigidity.
  • For a heavy string of length $L$ suspended vertically under gravity:
  • Speed at bottom free end A: $V_A = 0$
  • Speed at distance $x$ from free end (point B): $V_B = \sqrt{x \, g}$
  • Speed at top support C: $V_C = \sqrt{L \, g}$
  • Speed at B relative to C: $V_{B \text{ w.r.t } C} = \sqrt{(L - x) g}$
  • Time taken by wave from A to B: $T_{AB} = \sqrt{\frac{x}{g}}$
• Longitudinal Sound Waves:
  • General relation: $v = \sqrt{\frac{E}{\rho}}$, where $E$ is elastic modulus and $\rho$ is density.
  • Elasticity order: $E_s > E_l > E_g \implies v_{\text{solid}} > v_{\text{liquid}} > v_{\text{gas}}$.
  • Sound velocity in solid rods: $v_s = \sqrt{\frac{Y}{\rho}}$ ($Y$ = Young's modulus).
  • Sound velocity in unbounded solids: $v = \sqrt{\frac{B + \frac{4}{3}\eta}{\rho}}$ ($B$ = Bulk modulus).
  • Sound velocity in liquid or gas: $v = \sqrt{\frac{B}{\rho}}$.
• Newton's Formula & Laplace Correction:
  • Newton assumed isothermal propagation ($B_{\text{iso}} = P$): $v = \sqrt{\frac{P}{\rho}} = 280\,\text{m/s}$ at NTP (lower than experimental $332\,\text{m/s}$).
  • Laplace correction assumed adiabatic propagation ($B_{\text{adia}} = \gamma P$): $v = \sqrt{\frac{\gamma P}{\rho}} = \sqrt{\frac{\gamma R T}{M}} = 331.3\,\text{m/s}$ at NTP, where $\gamma = \frac{C_p}{C_v} = 1.4$.

Factors Affecting Velocity of Sound

• Density: Inverse relation to square root of gas density ($v \propto \frac{1}{\sqrt{\rho}}$).
• Temperature: Direct relation to square root of absolute temperature ($v \propto \sqrt{T}$).
  • For small temperature changes near $0\,^\circ\text{C}$: $v' = v_0 + 0.61 \times t\,(\text{m/s})$.
• Pressure: Sound velocity is independent of pressure changes at constant temperature.
• Humidity: Sound travels faster in moist air than in dry air because moist air is less dense than dry air at equal temperatures.
• Wind: $v = v_s + v_w \cos(\theta)$.
  • In direction of wind: $v = v_s + v_w$
  • Opposite to wind: $v = v_s - v_w$
• Invariance: Amplitude, frequency, phase, loudness, pitch, quality, and wave shape do not affect sound velocity.

Reverberation and Sound in Gas Mixtures

• Reverberation: Persistent repetition of sound due to multiple reflections in an enclosed space.
• Sabines Formula: Reverberation time $T = \frac{K V}{A S} = \frac{0.18 V}{\sum S A}$, where $V$ is volume, $A$ is absorption coefficient, $S$ is area, and $K = 0.18$.
• Refraction of Sound: Follows Snell's law: $\frac{\sin(i)}{\sin(r)} = {1}\mu_2 = \frac{\mu_2}{\mu_1} = \frac{v_2}{v_1}$.
• Gas Mixture Properties:
  • Velocity in mixture: $v_{\text{mix}} = \sqrt{\frac{\gamma_{\text{mix}} P}{\rho_{\text{mix}}}} = \sqrt{\frac{\gamma_{\text{mix}} R T}{M_{\text{mix}}}}$
  • Density of mixture: $\rho_{\text{mix}} = \frac{\rho_1 V_1 + \rho_2 V_2}{V_1 + V_2}$
  • Specific heats: $(C_v)_{\text{mix}} = \frac{n_1 C_{v1} + n_2 C_{v2}}{n_1 + n_2}$, $(C_p)_{\text{mix}} = (C_v)_{\text{mix}} + R$
  • Adiabatic index: $\frac{n_1 + n_2}{\gamma_{\text{mix}} - 1} = \frac{n_1}{\gamma_1 - 1} + \frac{n_2}{\gamma_2 - 1}$
  • Molecular weight: $M_{\text{mix}} = \frac{M_1 n_1 + M_2 n_2}{n_1 + n_2}$

Displacement, Pressure Waves, and Wave Energy

• Displacement Wave Equation: $y = a \sin(\omega t - kx)$
• Pressure Wave Equation: $P = -B \frac{dy}{dx} = a B k \cos(\omega t - kx) = P_0 \cos(\omega t - kx)$
  • Pressure Amplitude: $P_0 = a B k$
  • Phase relationship: Pressure is maximum where displacement is minimum, and pressure is minimum where displacement is maximum.
• Energy Transport: Total energy transported equals maximum kinetic energy:
  • $dE = \frac{1}{2} (d M) \times (v_{\text{max}})^2 = \frac{1}{2} \mu \, dx \times (a \omega)^2$