Wave Motion and Sound: Study Notes (Chapters 15 & 16)

Wave Motion and Sound

15.1 Characteristics of Wave Motion

Waves are vibrational disturbances that propagate through a medium or space, transferring energy without transporting matter. They are fundamental to many physical phenomena, including sound, light, and water waves.

• Wave: A vibrational disturbance from equilibrium that travels through a medium or space.

• Medium: The substance that carries the wave (e.g., water, air, solids).

• Examples: Sound waves, light waves, radio waves, microwaves, water waves, earthquake waves.

Types of Waves

• Mechanical Waves: Oscillations of matter that require a medium to propagate (e.g., sound, water waves).

• Electromagnetic Waves: Created by oscillating electric and magnetic fields; do not require a medium (e.g., light, radio waves).

• Matter Waves: Quantum mechanical waves associated with particles (e.g., electrons).

Classifications of Waves

• By Orientation:

  • Transverse Waves: Oscillations are perpendicular to the direction of wave propagation (e.g., light, waves on a string).

  • Longitudinal Waves: Oscillations are parallel to the direction of wave propagation (e.g., sound waves).

• By Appearance:

  • Traveling Waves: Move through the medium.

  • Standing Waves: Stationary patterns formed by interference.

• By Dimension:

  • 1D: Waves on a string.

  • 2D: Water surface waves.

  • 3D: Spherical sound waves.

• By Duration:

  • Periodic Waves: Continuous, repeating oscillations.

  • Pulse Waves: Single, non-repeating disturbances.

Wave Parameters

• Amplitude (A): Maximum displacement from equilibrium.

• Wavelength (λ): Distance between successive crests or troughs.

• Frequency (f): Number of cycles per second (Hz).

• Period (T): Time for one cycle; $T = 1/f$.

• Wave Velocity (v): Speed at which the wave propagates; $v = \\lambda f$.

Wave Relationships

• For a given medium, $v = \\lambda f$ is constant.

• Example: In vacuum, light propagates at $c = 3 \\times 10^8$ m/s for any frequency.

15.2 Types of Waves: Transverse and Longitudinal

The motion of particles in a wave can be perpendicular (transverse) or parallel (longitudinal) to the direction of wave propagation.

• Transverse Wave: Particles move up and down while the wave moves horizontally.

• Longitudinal Wave: Particles oscillate back and forth in the same direction as the wave.

Velocity of Waves

• Transverse Wave (on a cord): $v = \\sqrt{F_T/\\mu}$ Where $F_T$ is tension, $\\mu$ is linear mass density.

• Longitudinal Wave (in solids): $v = \\sqrt{E/\\rho}$ Where $E$ is Young's modulus, $\\rho$ is mass density.

• Longitudinal Wave (in liquids/gases): $v = \\sqrt{B/\\rho}$ Where $B$ is bulk modulus.

15.4 Mathematical Representation of a Traveling Wave

Traveling waves are described by the wave equation, a second-order partial differential equation.

• Wave Equation: $\\frac{\\partial^2 D}{\\partial x^2} = \\frac{1}{v^2} \\frac{\\partial^2 D}{\\partial t^2}$

• General Solution: $D(x,t) = A \\sin(kx \\pm \\omega t + \\phi)$ or $D(x,t) = A \\cos(kx \\pm \\omega t + \\phi)$

• $\\omega = 2\\pi f$ (angular frequency), $k = 2\\pi/\\lambda$ (wave number), $\\phi$ (phase constant)

• Direction: $kx - \\omega t$ (right), $kx + \\omega t$ (left)

• Particle Velocity: $v_p = \\frac{dD(x,t)}{dt}$

15.3 Energy Transported by Waves

Waves transport energy through a medium. The energy and power associated with a wave depend on its amplitude and frequency.

• Energy: $E = 2\\pi^2 \\rho S v t f^2 A^2$

• Average Power: $P = E/t = 2\\pi^2 \\rho S v f^2 A^2$

• Intensity: $I = P/S = 2\\pi^2 \\rho v f^2 A^2$

• For spherical waves: $I = P/(4\\pi r^2)$

16.3 Intensity of Sound: Decibels

Sound intensity is a measure of the energy transported by sound waves per unit area. The perceived loudness is measured in decibels (dB), which is a logarithmic scale.

• Sound: Mechanical, longitudinal wave; requires a medium.

• Speed of Sound in Air: About 340 m/s at 20°C.

• Intensity Level (dB): $\\beta = 10 \\log_{10} \\left(\\frac{I}{I_0}\\right)$

• Threshold of Hearing: $I_0 = 1.0 \\times 10^{-12}$ W/m2

Table: Intensity of Various Sounds

Source	Sound Level (dB)	Intensity (W/m2)
Jet plane at 30 m	140	100
Loud rock concert	120	1
Busy street traffic	80	10-4
Quiet radio	40	10-8
Threshold of hearing	0	10-12

Additional info:

• Wave graphs can be plotted as a function of time (at a fixed position) or as a function of position (at a fixed time).

• Sound waves are pressure waves, with regions of compression and rarefaction.

• Sound waves can be converted to electrical signals using a diaphragm and coil (microphone principle).