Waves, Sound, and Light: Comprehensive Study Guide

Wave Basics

Introduction to Waves

Waves are fundamental phenomena in physics, responsible for the transfer of energy through space and matter without the permanent displacement of the medium. Understanding wave properties is essential for analyzing sound, light, and other physical systems.

• Wave: A disturbance that transfers energy, not matter, through a medium or space.

• Traveling Wave: A wave that moves through space with a specific speed, v.

• Sinusoidal Wave Equation:

$y(x, t) = A \sin(kx - \omega t)$

• Wave Speed: $v = f \lambda$

• Frequency: $f = \frac{1}{T}$

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

• Period (T): The time for one complete cycle.

Example: For a wave with $f = 5\ \mathrm{Hz}$ and $\lambda = 2\ \mathrm{m}$, the speed is $v = f\lambda = 10\ \mathrm{m/s}$.

Simple Harmonic Motion (SHM)

Key Equations and Concepts

Simple harmonic motion describes oscillatory systems where the restoring force is proportional to displacement. It is foundational for understanding waves and vibrations.

• Period and Frequency: $T = \frac{1}{f}$

• Spring-Mass System: $T_{\text{spring}} = 2\pi \sqrt{\frac{m}{k}}$

• Pendulum: $T_{\text{pendulum}} = 2\pi \sqrt{\frac{L}{g}}$

• Energy in SHM: Total mechanical energy remains constant.

• At maximum displacement, potential energy is maximum; at equilibrium, kinetic energy is maximum.

Example: If the mass in a spring-mass system is doubled, the period increases by $\sqrt{2}$, since $T \propto \sqrt{m}$.

Standing Waves & Resonance

Conditions and Applications

Standing waves form when two waves of the same frequency and amplitude travel in opposite directions, leading to resonance under specific conditions.

• Resonance: Occurs when the driving frequency matches the natural frequency ($f_{\text{driving}} = f_{\text{natural}}$).

• Strings/Pipes (Open-Open or Closed-Closed): $\lambda_n = \frac{2L}{n}$, $n = 1, 2, 3, \ldots$

• Open-Closed Pipes: $\lambda_n = \frac{4L}{n}$, $n = 1, 3, 5, \ldots$

Example: For an open-closed pipe of $L = 1\ \mathrm{m}$, the fundamental wavelength is $\lambda = 4L = 4\ \mathrm{m}$.

Wave Speed on Strings

Dependence on Physical Properties

The speed of a wave on a string depends on the tension and the mass per unit length of the string.

• Wave Speed: $v = \sqrt{\frac{T}{\mu}}$

• Tension (T): The force stretching the string.

• Mass per Length (\(\mu\)): $\mu = \frac{m}{L}$

• Wave velocity depends on the medium; frequency is determined by the source.

Sound & Doppler Effect

Frequency Changes Due to Motion

The Doppler effect describes the change in observed frequency due to relative motion between the source and observer.

• Doppler Formula:

$f' = f \frac{v \pm v_o}{v \pm v_s}$

• Source moving toward observer: higher frequency observed.

• Source moving away: lower frequency observed.

Example: A siren emits $400\ \mathrm{Hz}$; you hear $450\ \mathrm{Hz}$, indicating the source is approaching.

Intensity vs Distance

Inverse Square Law

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

• Intensity: $I = \frac{P}{A} \propto \frac{1}{r^2}$

• As the radius increases, intensity decreases for the same power source.

Intensity & Decibels

Measuring Sound Levels

The decibel scale is a logarithmic measure of sound intensity, referencing the threshold of hearing.

• Decibel Level: $\beta = 10 \log \left( \frac{I}{I_0} \right)$

• Reference Intensity: $I_0 = 10^{-12}\ \mathrm{W/m^2}$

• Doubling intensity increases dB by less than double; $+10$ dB corresponds to $10\times$ intensity.

Beats

Interference of Close Frequencies

Beats occur when two waves of similar frequencies interfere, producing a periodic variation in amplitude.

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

• The "envelope" frequency equals the difference between the two frequencies.

Interference & Superposition

Constructive and Destructive Interference

When waves overlap, their displacements add according to the principle of superposition, leading to interference patterns.

• Constructive Interference: $\Delta L = n\lambda$

• Destructive Interference: $\Delta L = (n + \frac{1}{2})\lambda$

Diffraction & Slits

Wave Behavior Through Openings

Diffraction and interference patterns arise when waves pass through slits or around obstacles.

• Single Slit: The width of the central maximum is inversely proportional to slit width.

• Double Slit (Fringe Spacing):

$y = \frac{m\lambda L}{d}$

• Increasing slit spacing ($d$) decreases fringe spacing.

Light & Refraction

Change of Direction and Speed

Refraction occurs when light passes from one medium to another, changing speed and direction due to the difference in optical density.

• Index of Refraction: $n = \frac{c}{v}$

• Wavelength in Material: $\lambda_{\text{mat}} = \frac{\lambda_{\text{vac}}}{n}$

• Inside a material with a different index:

  • Speed and wavelength change

  • Frequency remains constant

Thin Film Interference

Phase Shifts and Path Differences

Thin film interference arises from the superposition of light reflected from the top and bottom surfaces of a thin layer, with phase shifts depending on the indices of refraction.

• If $n_2 > n_1$, add half a wavelength to the path length difference.

Condition	Equation
Constructive (0 phase shift)	$2t = m\frac{\lambda}{n}$
Constructive (1 phase shift)	$2t = (m + \frac{1}{2})\frac{\lambda}{n}$
Destructive (0 phase shift)	$2t = (m + \frac{1}{2})\frac{\lambda}{n}$
Destructive (1 phase shift)	$2t = m\frac{\lambda}{n}$

• t: Film thickness

• m: Integer (order of interference)