Acoustics, Vibration, and Waves

Acoustics, Vibration, and Waves is the branch of physics that studies mechanical waves and the vibrations that produce them, with a primary focus on the science of sound. This field investigates the fundamental principles governing the generation, propagation, and perception of these energetic disturbances as they travel through various media. Its applications are highly interdisciplinary, ranging from noise control engineering and architectural design for concert halls to medical imaging with ultrasound, the creation of musical instruments, and the study of seismic activity.

1. Fundamentals of Vibration and Oscillation
   1. Introduction to Oscillatory Motion
      1. Definition of Oscillation
      2. Equilibrium Position
      3. Restoring Force Concept
      4. Types of Oscillatory Systems
         1. Linear Oscillators
         2. Nonlinear Oscillators
         3. Conservative Systems
         4. Non-conservative Systems
      5. Periodic vs. Aperiodic Motion
      6. Degrees of Freedom
   2. Simple Harmonic Motion (SHM)
      1. Mathematical Formulation of SHM
         1. Differential Equation of SHM
         2. General Solution and Initial Conditions
         3. Complex Exponential Representation
      2. The Ideal Mass-Spring System
         1. Hooke's Law
         2. Equation of Motion
         3. Physical Parameters
            1. Mass
            2. Spring Constant
         4. Force Analysis
      3. The Simple Pendulum
         1. Small Angle Approximation
         2. Equation of Motion
         3. Physical Parameters
            1. Length
            2. Gravity
         4. Large Amplitude Corrections
      4. Kinematics of SHM
         1. Displacement as a Function of Time
         2. Velocity as a Function of Time
         3. Acceleration as a Function of Time
         4. Amplitude
         5. Period
         6. Frequency
         7. Angular Frequency
         8. Phase and Phase Constant
         9. Initial Conditions and Phase Determination
      5. Energy in SHM
         1. Kinetic Energy in SHM
         2. Potential Energy in SHM
         3. Total Mechanical Energy
         4. Conservation of Energy in SHM
         5. Energy Exchange During Oscillation
         6. Energy Diagrams
   3. Damped Oscillations
      1. Damping Forces
         1. Viscous Damping
         2. Coulomb Damping
         3. Structural Damping
         4. Air Resistance
      2. The Damped Harmonic Oscillator Equation
         1. Equation of Motion with Damping
         2. Characteristic Equation
         3. Solution Methods
      3. Types of Damping
         1. Underdamped Systems
            1. Oscillatory Decay
            2. Logarithmic Decrement
            3. Envelope Function
         2. Critically Damped Systems
            1. Non-oscillatory Return to Equilibrium
            2. Fastest Return Condition
         3. Overdamped Systems
            1. Slow Return to Equilibrium
            2. Exponential Decay
      4. Damping Parameters
         1. Damping Ratio
         2. Damping Coefficient
         3. Natural Frequency vs. Damped Frequency
      5. Quality Factor
         1. Definition and Physical Meaning
         2. Relation to Energy Loss
         3. Calculation Methods
         4. Bandwidth Relationship
   4. Forced Oscillations and Resonance
      1. Driving Force and Frequency
         1. Equation of Motion with Forcing Term
         2. Harmonic Driving Forces
         3. Non-harmonic Driving Forces
      2. Solution Methods
         1. Steady-State Solutions
         2. Transient Solutions
         3. Complete Solution
      3. Transient and Steady-State Behavior
         1. Response to Initial Conditions
         2. Long-Term Behavior
         3. Time Constants
      4. Resonance Phenomenon
         1. Resonant Frequency
         2. Amplitude Response at Resonance
         3. Phase Shift at Resonance
         4. Power Absorption
      5. Frequency Response
         1. Amplitude Response Curves
         2. Phase Response Curves
         3. Transfer Functions
      6. Resonance Characteristics
         1. Resonance Curve and Bandwidth
         2. Full Width at Half Maximum
         3. Relation to Q Factor
         4. Sharpness of Resonance
   5. Multi-Degree-of-Freedom Systems
      1. Coupled Oscillators
         1. Two Coupled Pendulums
         2. Coupled Mass-Spring Systems
         3. Coupling Mechanisms
      2. Normal Modes
         1. Mode Shapes
         2. Modal Frequencies
         3. Orthogonality of Modes
      3. Matrix Methods
         1. Mass Matrix
         2. Stiffness Matrix
         3. Eigenvalue Problems