Concept Notes: Longitudinal Waves, Density/Pressure Variations & Speed in Media
1. Mechanical Wave Nature of Sound
Sound is a mechanical disturbance propagated through an elastic material medium.
- Medium Requirement: In 1660, Robert Boyle proved via the Bell Jar Experiment that as air is pumped out of an enclosed glass jar containing a ringing electric bell, the sound intensity attenuates to zero. In vacuum, sound cannot propagate because there are no material particles to execute oscillations.
Figure 10.1: Acoustic Density & Pressure Fluctuations in Air — Alternating Compressions (High Density) and Rarefactions (Low Density)
2. Anatomy of Longitudinal Waves
- Compression ($C$): Region where particles are crowded together. Represents maximum particle density and maximum pressure ($P_{\text{max}}$).
- Rarefaction ($R$): Region where particles are spaced farthest apart. Represents minimum particle density and minimum pressure ($P_{\text{min}}$).
- Longitudinal vs Transverse Waves:
- In Longitudinal Waves (sound in air/fluids), the medium particles vibrate back and forth parallel to the direction of wave propagation.
- In Transverse Waves (ripples on water surface, light), particles vibrate up and down perpendicular to the wave propagation direction, forming crests and troughs.
3. Speed of Sound Across Different Media ($v$)
Sound speed depends on the medium's elasticity, density, and temperature:
$$v_{\text{solids}} > v_{\text{liquids}} > v_{\text{gases}}$$
| Medium | State | Approximate Speed at $25^\circ\text{C}$ ($m/s$) |
|---|---|---|
| Aluminium | Solid | $6,420\text{ m/s}$ |
| Steel / Iron | Solid | $5,960\text{ m/s}$ / $5,130\text{ m/s}$ |
| Glass (Crown) | Solid | $5,170\text{ m/s}$ |
| Sea Water | Liquid | $1,531\text{ m/s}$ |
| Pure Water ($20^\circ\text{C}$) | Liquid | $1,498\text{ m/s}$ |
| Air ($0^\circ\text{C}$) | Gas | $331\text{ m/s}$ |
| Air ($22^\circ\text{C}$) | Gas | $344\text{ m/s}$ |
Temperature Dependence: As temperature increases, the kinetic energy of particles increases, causing sound speed in air to rise: $v \approx 331 + 0.61 \times T(^\circ\text{C})\text{ m/s}$.