Traveling Waves and Sound: Physics with Calculus Study Notes

Traveling Waves and Sound

Introduction to Waves

Traveling waves are fundamental phenomena in physics, describing the propagation of disturbances through a medium or field. This chapter explores the properties, types, and mathematical descriptions of waves, with a focus on mechanical and sound waves.

The Wave Model

• Wave Model: Describes the basic properties of waves, emphasizing behaviors common to all waves.

• Traveling Wave: An organized disturbance that moves with a well-defined speed through a medium.

• Medium: The substance through which a mechanical wave travels.

• Energy Transfer: Waves transfer energy, not material, from the source outward.

Mechanical and Electromagnetic Waves

• Mechanical Waves: Require a material medium (e.g., water, air, string).

• Electromagnetic Waves: Waves of electromagnetic fields (e.g., light, radio waves) that do not require a medium and can travel through a vacuum.

Types of Waves

Transverse and Longitudinal Waves

Waves are classified based on the direction of particle motion relative to wave propagation:

• Transverse Waves: Particles move perpendicular to the direction of wave travel (e.g., waves on a string).

• Longitudinal Waves: Particles move parallel to the direction of wave travel (e.g., sound waves, spring compression).

Earthquake Waves

• S Waves (Transverse): Move ground up and down, perpendicular to wave direction.

• P Waves (Longitudinal): Compress and expand ground, parallel to wave direction.

Wave Propagation and Medium Response

Motion of Wave vs. Motion of Medium

• Wave Motion: The disturbance moves through the medium.

• Medium Motion: Particles oscillate but do not travel with the wave.

• Newton's Laws: Apply to the medium's particles, not the wave itself.

Waves on a String

• Transverse Pulse: Created by an external force, continues due to internal dynamics.

• Each Point: Moves perpendicular to wave direction.

Sound Waves

• Sound: Longitudinal wave caused by compression and rarefaction of air molecules.

• Loudspeaker Example: Cone compresses air, creating a traveling disturbance.

Wave Speed and Properties of the Medium

Wave Speed on a String

• Depends on: Tension and linear density (mass per unit length).

• Higher Tension: Increases wave speed.

• Higher Linear Density: Decreases wave speed.

Wave Speed Formula (String)

• For a string, the speed $v$ is given by: $v = \sqrt{\frac{T}{\mu}}$ where $T$ is tension and $\mu$ is linear density.

Wave Speed in Gases (Sound)

• Depends on temperature and molecular mass, not pressure or density.

• Speed increases with temperature and decreases with molecular mass.

• Speed of sound in air at 20°C: 343 m/s.

Wave Speed of Light

• Electromagnetic waves travel at speed $c$ in vacuum: $c = 3.00 \times 10^8 \text{ m/s}$

Mathematical and Graphical Descriptions of Waves

Snapshot and History Graphs

• Snapshot Graph: Shows displacement as a function of position at a single instant.

• History Graph: Shows displacement of a single point over time.

Sinusoidal Waves

• Produced by: Sources oscillating with simple harmonic motion.

• Amplitude (A): Maximum displacement.

• Wavelength (\lambda): Distance spanned in one cycle.

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

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

Mathematical Description

• Displacement as a function of position and time: $y(x, t) = A \sin\left(2\pi \left(\frac{x}{\lambda} - \frac{t}{T}\right)\right)$ For a wave traveling to the left: $y(x, t) = A \sin\left(2\pi \left(\frac{x}{\lambda} + \frac{t}{T}\right)\right)$

Wave Velocity Relationship

• During one period $T$, a crest travels one wavelength $\lambda$.

• Wave speed $v$ is: $v = \frac{\lambda}{T} = f \lambda$

Examples and Applications

Spider Sensing Vibrations

• Wave pulse travels along silk thread; speed determined by tension and linear density.

• Time to sense prey: $t = \frac{d}{v}$

Lightning and Thunder

• Light travels almost instantaneously; sound travels slower.

• Distance to lightning: $d = v_{sound} \times t_{delay}$

Boat on Ocean Waves

• Boat's vertical displacement determined by wave equation and boat's position over time.

• Amplitude, wavelength, and period used to calculate displacement.

Summary Table: Types of Waves

Type	Medium Required?	Particle Motion	Examples
Transverse	Yes	Perpendicular	String, S waves
Longitudinal	Yes	Parallel	Sound, P waves
Electromagnetic	No	N/A	Light, radio

Key Equations

• Wave speed on string: $v = \sqrt{\frac{T}{\mu}}$

• Wave speed (general): $v = f \lambda$

• Frequency: $f = \frac{1}{T}$

• Sinusoidal wave: $y(x, t) = A \sin\left(2\pi \left(\frac{x}{\lambda} - \frac{t}{T}\right)\right)$

Additional info:

• Wave speed in solids & liquids is higher than in gases due to stronger intermolecular forces.

• Snapshot and history graphs are essential for visualizing both wave propagation and particle motion.