Final Exam Study Guide

Chapter 16: Traveling Waves

• Introduction to properties of traveling waves.

Transverse and Longitudinal Waves

• Transverse Wave: Moves up and down with a speed V; the displacement occurs perpendicular to the direction of wave propagation.

• Longitudinal Wave: Moves in push/pull motion with a speed V; the displacement occurs parallel to the direction of wave propagation.

Wave Speed

• String Wave Speed:

  • Formula: v_string = ( \sqrt{\frac{T_s}{\mu}} ) where (T_s) is the tension in the string, and (\mu) is the linear mass density.

• Speed of Sound:

  • Formula: v_sound = ( \sqrt{\frac{\gamma k T}{m}} ) where (\gamma) is the specific heat ratio, k is Boltzmann's constant, T is temperature, and m is the mass of the gas molecules.

One-Dimensional Waves: Snapshot Graphs

• Wave moves horizontally; string particle moves vertically.

• At times t1 and t2, the particle at X1 displaces a distance A_y due to wave motion while maintaining shape.

Wavelength and Period of Sinusoidal Waves

• History Graph: Shows displacement over time.

  • Amplitude: A, Crest: 2A, Period: T.

• Snapshot Graph: Shows wave properties at an instant.

• Fundamental relationship for sinusoidal waves: ( v = \lambda f ) where (\lambda = v/f = vT) (T = 1/f).

The Mathematics of Sinusoidal Waves

• Sinusoidal wave equation in the x-direction: ( D(x, t) = A \sin(kx - \omega t + \phi_0) )

  • Where: ( k = \frac{2\pi}{\lambda} ), ( \omega = \frac{2\pi}{T} ).

• Reverse the sign for waves traveling in the -x direction.

Intensity and Decibels

• Sound intensity level in decibels (dB):

  • Formula: ( \beta = (10 \text{ dB}) \log_{10}(\frac{I}{I_0}) ) where ( I_0 = 1 \times 10^{-12} \text{ W/m}^2 ).

The Doppler Effect

• Stationary Observer: Frequencies heard when the sound source moves.

  • Approaching source: ( f^+ = f_0 \left(1 - \frac{v_s}{v}\right) ).

  • Receding source: ( f^- = f_0 \left(1 + \frac{v_s}{v}\right) ).

• Moving Observers:

  • Approaching source: ( f^+ = (1 + \frac{v_o}{v}) f_0 ).

  • Receding source: ( f^- = (1 - \frac{v_o}{v}) f_0 ).

Chapter 17: Superposition

• Understanding and using superposition principles in wave mechanics.

The Principle of Superposition

• Displacement at a point with multiple waves is the sum of individual wave displacements.

Standing Waves on a String

• The modes of a string correspond to unique wavelengths and frequencies.

• Mode number (m): ( \lambda = \frac{2L}{m} ), frequency: ( f_m = \frac{mv}{2L} ).

Standing Sound Waves

• Pressure variation in air columns with different conditions (open-open, closed-closed, open-closed).

Interference in One Dimension

• Phase contributions based on path-length differences leading to constructive/destructive interference
  - Constructive interference: ( \Delta\phi = 0 ) or integral multiples of ( 2\pi ).

  • Destructive interference: half-integer multiples of ( 2\pi ).

Interference in Two and Three Dimensions

• Circular/spherical waves with in-phase sources lead to patterns of constructive/destructive interference.

• General interference condition: ( \Delta\phi = \Delta\phi_0 + \frac{2\pi\Delta r}{\lambda} ).

  • General case for constructive and destructive: based on path length differences ( \Delta r ).

Chapter 18: A Macroscopic Description of Matter

• Characteristics of macroscopic systems to be explored.

Definitions

• Density: ( \rho = \frac{M}{V} )

• Pressure: ( P = \frac{F}{A} )

• Atomic mass number and Avogadro’s number definitions included.

Absolute Temperature and the Kelvin Scale

• Kelvin scale defined; 0 K = -273.15 °C, absolute temperature relationship.

Thermal Expansion

• Linear expansion proportionality: ( \frac{\Delta L}{L} = \alpha \Delta T )

• Volume expansion defined by: ( \frac{\Delta V}{V} = \beta \Delta T )

The Ideal Gas Law

• Written as ( pV = nRT ) or ( pV = Nk_BT )

  • Where R is the universal gas constant and ( k_B ) is Boltzmann’s constant.

p-V Diagrams and Ideal-Gas Processes

• Each point on the p-V diagram represents a unique gas state.

Constant-Volume and Constant-Pressure Processes

• Constant-Volume = isochoric; Constant-Pressure = isobaric process equations detailed.

Constant-Temperature Processes

• Defined as isothermal processes, where ( p \propto T )

Chapter 19: Work, Heat, and the First Law of Thermodynamics

• Energy exchanges in thermal systems focused; conservation lead to first law details.

Work in Ideal-Gas Processes

• Integral representation for work done in varying state conditions.

Three Special Ideal-Gas Processes

• Ideal processes with one term in the first law as zero identified.

Temperature Change and Specific Heat

• Personalization of specific heat definitions and calculations per unit mass or mole.

Phase Changes

• Distinctions between melting, freezing, and vaporization processes.

Calorimetry

• Conservation of energy principle in calorimetry problems defined.

The Specific Heat of Gases

• Two types of specific heat defined: at constant volume and pressure, with notable relationships.

Conduction

• Rate of heat transfer in materials shown by defined heat transfer equations.

Chapter 21: Heat Engines and Refrigerators

• Study topics of thermodynamics expanded to heat engines working principles.

Heat Engines Overview

• Extraction of heat from reservoirs to perform work and exhaust waste heat.

• Efficiency formula identification and its implications for efficiency.

Refrigerators Overview

• Contrast with heat engines; their efficiency measured by the coefficient of performance.

The Carnot Cycle

• Steps of the cycle including isothermal and adiabatic processes.

• Efficiency of Carnot cycle illustrated with key formulas for both engines and refrigerators.