Waves & Sound

Introduction to Wave Types and Wave Speed

Waves are disturbances that transfer energy through a medium (such as a string, water, or air) without transporting matter. They are fundamental to understanding sound, light, and many other physical phenomena.

• Wave Types:

  • Transverse Waves: Displacement is perpendicular to the direction of wave motion (e.g., waves on a string).

  • Longitudinal Waves: Displacement is parallel to the direction of wave motion (e.g., sound waves in air).

• Key Properties:

  • Wavelength (\( \lambda \)): Distance between consecutive crests (transverse) or compressions (longitudinal).

  • Amplitude (A): Maximum displacement from equilibrium. For transverse waves, it's half the vertical distance from crest to trough.

  • Period (T): Time to complete one cycle.

  • Frequency (f): Number of cycles per second. \( f = \frac{1}{T} \)

  • Wave Speed (v): Relationship for all waves:

• Example: Sound is a longitudinal wave traveling through air at 343 m/s. For a frequency of 260 Hz, the wavelength is:

Wave Speed on Strings

The speed of a wave on a string depends on the string's tension, mass, and length.

• Wave Speed Formula for Strings:

  • Where \( F_T \) is the tension in the string (N), \( \mu = \frac{m}{L} \) is the mass per unit length (kg/m).

• Frequency and Wavelength:

• Example: A string with tension 100 N, mass 0.5 kg, length 1.2 m, and wavelength 0.15 m:

Wave Speed in Fluids and Solids

Longitudinal wave speed depends on the medium's properties:

• In Fluids: Where \( \beta \) is the bulk modulus, \( \rho \) is density.

• In Solids: Where \( Y \) is Young's modulus.

• Example: For a liquid with \( \rho = 1200\,\text{kg/m}^3 \), \( f = 400\,\text{Hz} \), \( \lambda = 8\,\text{m} \):

Wave Functions

A wave function mathematically describes the displacement of a wave at any position and time.

• General Form: or

  • \( A \): Amplitude

  • \( k = \frac{2\pi}{\lambda} \): Wavenumber

  • \( \omega = 2\pi f \): Angular frequency

  • \( \phi \): Phase constant (accounts for initial displacement)

• Direction: Use minus sign for rightward motion, plus for leftward.

• Example: (for a wave with \( \lambda = 0.32\,\text{m} \), )

Transverse Velocity of Waves

The transverse velocity refers to the velocity of a particle on the string, not the wave itself.

• Transverse Velocity: (for cosine form)

• Maximum Transverse Velocity:

Average Power of Waves on Strings

Waves carry energy, and the rate of energy transfer is power.

• Average Power:

• Example: For \( \mu = 0.05\,\text{kg/m} \), , , :

Wave Intensity and the Inverse-Square Law

Intensity is the power per unit area carried by a wave. For spherical waves, intensity decreases with the square of the distance from the source.

• Intensity: For a point source:

• Inverse-Square Law:

• Example: If intensity at 3.4 m is 0.3 W/m2, at 2.5 m it is:

Sound Intensity Level (Decibels)

Because the human ear perceives sound logarithmically, sound intensity is measured in decibels (dB).

• Decibel Scale: Where (threshold of hearing).

• Example: For :

• Doubling Distance: Each time distance from the source doubles, intensity level decreases by approximately 6 dB.

Wave Interference & Superposition

When two or more waves meet, their displacements add according to the principle of superposition.

• Constructive Interference: Displacements have the same sign; amplitudes add.

• Destructive Interference: Displacements have opposite signs; amplitudes subtract.

• Superposition Principle: (at each point and time)

Standing Waves

Standing waves are formed when two waves of the same frequency and amplitude travel in opposite directions and interfere.

• Standing Wave Equation:

• Harmonics:

  • n = 1: Fundamental frequency

  • n = 2, 3, ...: Higher harmonics (overtones)

• Frequencies and Wavelengths:

  • For a string fixed at both ends (length L):

    • (fundamental)

  • For open and closed pipes:

    • Open at both ends: ,

    • Closed at one end: ,

Beats

Beats occur when two waves of slightly different frequencies interfere, causing periodic variations in amplitude (loudness).

• Beat Frequency:

• Resulting Sound Frequency:

• Example: Two notes at 527 Hz and 527.5 Hz produce a beat frequency of 0.5 Hz.

The Doppler Effect

The Doppler Effect is the change in observed frequency due to relative motion between a sound source and a listener.

• Doppler Effect Formula: Where is the frequency heard by the listener, is the source frequency, is the speed of sound, is the velocity of the listener (positive if moving toward the source), and is the velocity of the source (positive if moving away from the listener).

• Key Points:

  • If source and listener move toward each other, .

  • If they move apart, .

  • No relative motion: .

• Example: A stationary car alarm emits 550 Hz. A listener moving toward it hears 600 Hz. The listener's speed can be found using the Doppler formula.

Summary Table: Key Wave Equations

Additional info: Some context and explanations have been expanded for clarity and completeness, including explicit formulas, definitions, and examples for each major concept. This guide covers all foundational aspects of waves and sound relevant to introductory college physics.