Waves, Sound, and Electricity

Overview

This study guide reviews key concepts from Physics 103, focusing on waves, sound, and electricity. It covers fundamental definitions, mathematical relationships, and physical principles relevant to college-level physics.

Wave Motion

Basic Properties of Waves

Waves are disturbances that transfer energy through a medium or space. The medium's particles oscillate but do not travel with the wave.

• Waveform: A graphical representation of a wave at a given instant. Crests are the highest points; troughs are the lowest.

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

• Frequency (f): Number of wave cycles passing a point per unit time.

• Period (T): Time for one complete cycle; $T = \frac{1}{f}$

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

Example: Water waves on a lake show energy propagation while water molecules oscillate vertically.

Mathematical Representation of a Traveling Wave

The displacement of a point on a wave as a function of position and time is given by:

• $y(x, t) = A \sin(kx - \omega t)$

• Where $A$ is amplitude, $k = \frac{2\pi}{\lambda}$ is the wave number, and $\omega = 2\pi f$ is angular frequency.

Types of Waves

Transverse and Longitudinal Waves

Waves can be classified based on the direction of particle motion relative to wave propagation.

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

• Longitudinal Waves: Particle motion is parallel to wave direction (e.g., sound waves).

Example: Sound waves in air are longitudinal; waves on a rope are transverse.

Speed of a Wave on a String

• For a string under tension $F$ and linear density $\mu$:

• $v = \sqrt{\frac{F}{\mu}}$

Application: Musical instruments use string tension and mass to control pitch.

Standing Waves and Resonance

Standing Waves

Standing waves form when two waves of the same frequency travel in opposite directions and interfere. Nodes are points of zero amplitude; antinodes are points of maximum amplitude.

• For a string fixed at both ends, only certain wavelengths are allowed:

• $\lambda_n = \frac{2L}{n}$, $n = 1, 2, 3, ...$

• Corresponding frequencies: $f_n = \frac{nv}{2L}$

Example: Harmonics in musical instruments.

Sound Waves

Nature of Sound

Sound is a longitudinal wave that propagates through a medium by compressions and rarefactions.

• Speed of Sound in Air: $v = 331 + 0.6T$ (where $T$ is temperature in °C)

• Speed in Liquids: $v = \sqrt{\frac{B}{\rho}}$ (Bulk modulus $B$, density $\rho$)

• Speed in Solids: $v = \sqrt{\frac{E}{\rho}}$ (Young's modulus $E$, density $\rho$)

Example: Sound travels faster in solids than in gases.

Categories of Sound Waves

• Audible: 20 Hz to 20,000 Hz

• Infrasonic: Below 20 Hz

• Ultrasonic: Above 20,000 Hz

Intensity and Decibels

Intensity ($I$) is the power per unit area:

• $I = \frac{P}{A}$

• Sound level in decibels: $\beta = 10 \log_{10} \left( \frac{I}{I_0} \right)$, where $I_0 = 1.0 \times 10^{-12}$ W/m2

Doppler Effect

The observed frequency of a wave changes due to relative motion between source and observer.

• General formula: $f' = f \frac{v + v_o}{v - v_s}$

• $v$ = speed of sound, $v_o$ = observer velocity, $v_s$ = source velocity

Application: Used in radar, medical imaging, and astronomy.

Electricity

Electric Charge

Electric charge is a fundamental property of matter. There are two types: positive and negative. Like charges repel; unlike charges attract.

• Quantization: Charge exists in discrete packets; $q = N e$

• Conservation: Total charge in an isolated system remains constant.

Conductors and Insulators

• Conductors: Allow free movement of charge (e.g., metals).

• Insulators: Do not allow free movement of charge (e.g., glass, rubber).

Coulomb’s Law

The force between two point charges is:

• $F = k \frac{|q_1 q_2|}{r^2}$

• $k = 8.99 \times 10^9$ Nm2/C2

• Force is attractive for opposite charges, repulsive for like charges.

Electric Field

An electric field exists around a charged object and exerts a force on other charges.

• Field at a point: $E = \frac{F}{q}$

• For a point charge: $E = k \frac{|q|}{r^2}$

• Direction: Away from positive, toward negative charges.

Electric Field Lines

• Field lines start on positive charges and end on negative charges.

• Density of lines indicates field strength.

• Lines never cross.

Motion of Charged Particles in Electric Fields

• Force: $F = qE$

• Acceleration: $a = \frac{F}{m}$

• Direction depends on sign of charge.

Electric Potential and Potential Energy

• Potential energy change: $\Delta PE = -qEd$

• Potential difference: $\Delta V = \frac{\Delta PE}{q}$

• Unit: Volt (V), where $1\,\text{V} = 1\,\text{J}/\text{C}$

Relationship Between Electric Field and Potential

• Uniform field: $E = \frac{\Delta V}{d}$

• Work done: $W = qEd$

Summary Table: Key Equations

Additional info: These notes synthesize and expand upon the provided lecture slides and equations, ensuring coverage of all major topics relevant to Physics 103: Waves, Sound, and Electricity.