Chapter 16: Sound and Hearing – Longitudinal Sound Waves, Sound Pressure, and Wave Equation

Longitudinal Sound Waves

Introduction to Sound Waves

Sound waves are mechanical waves that propagate through a medium (such as air, water, or solids) via the oscillation of particles parallel to the direction of wave travel. These are known as longitudinal waves, in contrast to transverse waves where oscillations are perpendicular to the direction of propagation.

• Condensation: Regions where particles are compressed, resulting in higher pressure.

• Rarefaction: Regions where particles are spread apart, resulting in lower pressure.

• Normal air pressure: The equilibrium pressure in the absence of sound.

• Frequency ($f$): Number of oscillations per second, measured in Hertz (Hz).

• Speed of sound: In air, $v_{sound} hickapprox 330$ m/s; in steel, $v_{sound} hickapprox 5000$ m/s.

• Audible range: Human hearing typically spans 20 Hz to 20 kHz. Frequencies above 20 kHz are termed ultrasound.

Examples and Applications

• Slinky demonstration: Used to visualize longitudinal wave propagation.

• Speaker diaphragm: Creates alternating regions of condensation and rarefaction in air, producing sound.

Sound Pressure and Particle Displacement

Relationship Between Pressure and Displacement

Sound pressure is the local deviation from the ambient atmospheric pressure caused by a sound wave. It is directly related to the displacement of particles in the medium.

• Volume change ($rac{dV}{V}$): For a fluid element, the change in volume due to particle displacement is given by: $rac{dV}{V} = rac{ rac{ ext{S}(y(x+ ext{d}x, t) - y(x, t))}{S ext{d}x}}{1} = rac{ ext{d}y}{ ext{d}x}$

• Sound pressure formula: $p(x, t) = -B rac{ ext{d}y}{ ext{d}x}$ where $B$ is the bulk modulus of the medium.

• Pressure deviation: $ext{Sound pressure at } (x, t): ext{ } p(x, t) = P(x, t) - P_0(x, t)$ $P_0(x, t)$ is the pressure in the absence of sound.

Example: Sinusoidal Sound Wave

• Displacement wave: $y(x, t) = A ext{cos}(kx - ext{wt})$

• Pressure wave: $p(x, t) = BkA ext{sin}(kx - ext{wt})$

• Amplitude of pressure fluctuation: $p_{max} = BkA$

Wave Equation and Speed of Sound in Fluids

Derivation of the Wave Equation

The propagation of sound in fluids can be modeled using the wave equation, derived from Newton's laws and the properties of the medium.

• 1-D lattice analogy: The displacement of particles connected by springs is analogous to fluid elements connected by pressure forces.

• Wave equation: $rac{ ext{d}^2 y}{ ext{d} t^2} - v^2 rac{ ext{d}^2 y}{ ext{d} x^2} = 0$

• Wave speed in fluids: $v = ext{sqrt} rac{B}{ ho}$ where $B$ is the bulk modulus and $ho$ is the mass density of the medium.

Physical Interpretation

• Restoring force: Proportional to the bulk modulus $B$.

• Inertia: Proportional to the mass density $ho$.

• Speed of sound in different media:

  • Air (gas): $v hickapprox 330$ m/s

  • Water (liquid): $v hickapprox 1400$ m/s

  • Steel (solid): $v hickapprox 5000$ m/s

Example Calculation

• Given: Bulk modulus of water $B = 2.2 imes 10^9$ N/m2, density $ho = 1000$ kg/m3

• Speed of sound: $v = ext{sqrt} rac{2.2 imes 10^9}{1000} = 1483$ m/s

Summary Table: Speed of Sound in Various Media

Medium	Speed of Sound (m/s)
Air (20°C)	330
Water	1400
Steel	5000

Additional info: The notes above are expanded and clarified for academic completeness, including definitions, formulas, and physical interpretations relevant to college-level physics.