Waves and Sound II: Study Notes for PHYS 111

Waves and Sound II

Introduction to Waves

Waves are a fundamental concept in physics, describing the transfer of energy through a medium. The study of waves includes understanding their types, properties, and behaviors such as reflection, interference, and superposition.

• Wave: A disturbance that transfers energy from one place to another through a medium.

• Periodic Motion: Elements of the medium move in a regular, repeating pattern, but neighboring elements are out of phase.

• Types of Waves:

  • Transverse Waves: Particles move perpendicular to the direction of wave propagation (e.g., light waves).

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

• Wave Speed: The speed of a wave depends on the properties of the medium and is given by: $v = \lambda f = \omega / k$ where $v$ is speed, $\lambda$ is wavelength, $f$ is frequency, $\omega$ is angular frequency, and $k$ is wave number.

Traveling Waves

A traveling wave is described mathematically and shows how the wave propagates through space and time.

• General Equation: $y(x, t) = A \cos\left(\frac{2\pi}{\lambda}x - \frac{2\pi}{T}t\right)$ where $A$ is amplitude, $\lambda$ is wavelength, $T$ is period.

• Phase: To maintain a constant phase, as time increases, position $x$ must also increase.

Reflection of Waves

Reflection occurs when a wave encounters a boundary. The nature of the boundary determines how the wave is reflected.

• Fixed Boundary: The reflected wave is inverted.

• Free Boundary: The reflected wave is not inverted.

• Change in Medium:

  • More dense to less dense: boundary acts as a free end.

  • Less dense to more dense: boundary acts as a fixed end.

Linear Superposition

The principle of superposition states that when two or more waves overlap, the resulting wave is the sum of the individual waves.

• Additive Property: $y = y_1 + y_2$ at all locations.

• Constructive Interference: When waves enhance each other (amplitudes add).

• Destructive Interference: When waves cancel each other (amplitudes subtract).

Interference

Interference occurs when two waves meet and combine, resulting in a new wave pattern.

• Constructive Interference: Occurs when the path difference is $n\lambda$ ($n = 0, 1, 2, ...$).

• Destructive Interference: Occurs when the path difference is $n\lambda/2$ ($n = 1, 3, 5, ...$).

• Example: Two sources emitting waves can create regions of constructive and destructive interference depending on the path difference.

Standing Waves

Standing waves are formed when a wave reflects and interferes with itself, creating nodes and antinodes.

• Nodes: Points of zero amplitude.

• Antinodes: Points of maximum amplitude.

• Fixed Ends: Each harmonic adds a half-wavelength loop.

• Free End: Envelope ends with a quarter-wavelength.

Open End, Fixed End Harmonics

The boundary conditions of a string or pipe affect the harmonics and musical scales produced.

• Each additional harmonic: Includes one extra half-wavelength loop.

• Applications: Predicting musical scales for guitar strings, church pipes, etc.

Example Problem: Harmonics in Pipes

Given the speed of sound $v = 343$ m/s, find the first three harmonics for a church pipe of length $L = 25$ cm:

• (a) Both ends closed

• (b) One end closed

• (c) Both ends open

Additional info: For closed ends, only odd harmonics are present; for open ends, all harmonics are present.

Beats

Beats occur when two waves of slightly different frequencies interfere, resulting in a fluctuating amplitude.

• Beat Frequency: $f_{\text{beat}} = |f_1 - f_2|$

• Average Frequency: $f_{\text{avg}} = \frac{f_1 + f_2}{2}$

• Application: Used in tuning musical instruments.

Physics of Sound

Sound intensity and loudness are important concepts in acoustics, measured using the decibel scale.

• Inverse Square Law: $\text{Intensity} = \frac{\text{Power}}{\text{Area}} = \frac{\text{Power}}{4\pi R^2}$

• Decibel Scale: $B = 10 \log \left( \frac{I}{I_0} \right)$, where $I_0 = 10^{-12}$ W/m2

• Table: Common Sound Levels

Sound Source	Decibels (dB)
Ear drum ruptures	160
Jet taking off	140
Pain threshold	120
Loud rock band	110
Thunder	100
Subway	90
Heavy traffic	80
Conversation	60
Classroom	50
Whisper	20
Threshold of hearing	0

Doppler Effect

The Doppler effect describes the change in frequency or wavelength of a wave in relation to an observer moving relative to the source.

• Moving Observer (Source Fixed): $f' = (1 \pm \frac{u}{v})f$ where $u$ is observer speed, $v$ is wave speed.

• Moving Source (Observer Stationary): $f' = \left( \frac{1}{1 \mp \frac{u}{v}} \right)f$ where $u$ is source speed, $v$ is wave speed.

• Pitch Change: Approaching increases pitch; receding decreases pitch.

Applications of Doppler Effect

The Doppler effect has many practical applications in astronomy, meteorology, and medicine.

• Redshift: Galaxies moving away appear redder.

• Blueshift: Planets rotating toward us appear bluer.

• Doppler Radar: Used to detect tornadoes and measure fluid flow in medical devices.