CSD 202 Week 2b Gibbs

Course Introduction

• CSD 202: Normal Aspects of Hearing

• Spring 2025, Week 2b: Simple Harmonic Motion

• Instructor: Prof. Bobby Gibbs

• Contact: bgibbs5@wisc.edu

• Department: Communication Sciences and Disorders, University of Wisconsin–Madison

Review of Previous Class Topics

• Key Concepts Covered:

  • Mass

  • Length

  • Time

  • Displacement

  • Velocity

  • Acceleration

• Force Definitions:

  • Newton's Second Law: F = ma

  • Hooke's Law: F = -kx

  • Work Equation: Work (W) = ΔE

• Types of Energy:

  • Potential Energy

  • Kinetic Energy

  • Power: P = ΔE/Δt

Understanding Force

• Importance of Understanding Formulas:

  • Units of Force: Newtons (N)

  • Relationship: F = ma, where

    • m = mass

    • a = acceleration

  • 1 Newton is defined as the force necessary to accelerate a 1-kg mass at a rate of 1 m/s².

Force Explained

• Types of Force:

  • Newton's Law:

    • To accelerate an object, apply force based on mass and desired acceleration.

  • Hooke's Law:

    • Describes restoring force proportional to displacement in elastic materials: F = -kx. Stiffness affects how an object returns to equilibrium when deformed.

Class Learning Objectives

• Current Objective:

  • Describe sound, its characteristics, and laws of propagation.

• Class Goals:

  • Learn parameters of sound in simple harmonic motion:

    • Amplitude (A)

    • Frequency (f)

    • Phase (Φ)

Motion Types

• Focus on Motion Types:

  • Previous focus: linear, unidirectional motion.

  • New focus: back-and-forth (vibratory) motion, critical for understanding sound waves.

Importance of Air

• Overview of Air Composition and Its Role in Sound Propagation.

Atmospheric Pressure

• Definition and Importance:

  • Atmospheric Pressure = Force/Area (N/m² or Pascals)

  • Decreases with altitude changes.

  • Sound defined as local pressure deviations from static atmospheric pressure.

Atmospheric Pressure Context

• Visual Reference:

  • Less pressure on Everest, relating to sound propagation.

Organizational Context

• Reference to Local Organizations and Universities.

Air Qualities

• Key Aspects of Air Composition Relevant to Sound:

  • Mass

  • Elasticity

Sound as Vibration

• Sound Definition:

  • Sound = Vibration of an object.

  • Vibration in air molecules results in sound transmission.

  • Eardrum responds to these vibrations, contributing to hearing.

Sound Pressure Variations

• Definition of Sound:

  • Caused by small pressure variations in atmospheric pressure due to disturbances (sound sources).

  • Static atmosphere pressure example: 100 kPa.

  • Acoustic Pressure: p0 = 20 μPa at threshold level.

Pressure Fundamentals

• Pressure Dynamics:

  • Increasing force yields larger pressure with constant area.

  • Increasing area while applying constant force yields smaller pressure.

  • Pressure Unit: Pascals (Pa) defines force distributed over an area.

Sound Pressure Level (SPL)

• SPL Definition:

  • Measures sound force in a given point in space.

  • SPL calculated as a ratio against reference sound pressure: 2 x 10⁻⁵ N/m² (0.00002 Pa).

  • Measured in decibels (dB SPL).

Compression and Rarefaction

• Sound Dynamics:

  • Compression and rarefaction create pressure variations.

  • Book terminology: "condensation" and "rarefaction" describe pressure changes in sound waves.

Sound Propagation

• Sound Travels Through Air:

  • Exhibiting three-dimensional propagation characteristics.

Wave Dynamics

• Wave Characteristics:

  • Compression and rarefaction illustrated; sound creates fluctuations in air pressure.

Wave Types

• Two Types of Waves:

  • Transverse (e.g., light)

  • Longitudinal (e.g., sound)

Longitudinal Wave Defined

• Longitudinal Waves Explained:

  • Disturbance transporting energy through a medium (e.g., air) without relocating matter.

  • Local oscillations occur but particles do not travel with the wave.

Wave Types Comparison

• Visual Representation of:

  • Transverse vs. Longitudinal Waves.

Recap on Sound Waves

• Key Recap:

  • Sound characterized by fluctuations in pressure.

  • Propagates as longitudinal waves through a material medium (air).

  • Medium supports vibratory motion due to mass and elasticity.

Simple Harmonic Motion (SHM)

• SHM Definition:

  • Represents simplest form of vibratory motion requiring mass and elasticity/stiffness.

  • Example systems: spring-mass, pendulums, stretched strings.

Spring-Mass System

• Oscillation Dynamics:

  • Key equations governing spring-mass oscillation:

    • F = -kx (Hooke's Law)

    • F = ma (Newton's Second Law)

Hooke’s Law Reiterated

• SHM Condition:

  • Restoring force proportional to displacement ensures SHM occurs.

Energy Transfer in SHM

• Energy Dynamics:

  • Potential energy in a spring converts to kinetic energy during oscillation towards equilibrium.

  • Momentum gained causes position to overshoot equilibrium, leading to repeat energy conversion.

Example of SHM

• Air in Cavities:

  • Like Helmholtz resonators behaves as spring-mass systems, undergoing SHM oscillations.

Resonance in the Ear Canal

• Ear Canal Function:

  • Air column in the ear canal acts analogously to a resonator, supporting oscillations similar to spring-mass systems.

SHM Recap

• SHM Overview:

  • Mass and stiffness characteristics of medium (e.g., air in tubes) dictate SHM production and propagation.

  • Importance of SHM in understanding air vocal cavities.

Quantifying SHM

• Description of SHM:

  • SHM modeled using sine functions.

  • Sine waves capable of representing both simple and complex harmonic motion (i.e., Fourier analysis).

Sine Function Application

• Sine as a Periodic Function:

  • Visualizes SHM as a function of time.

  • Relates to equilibrium and displacement shifts over time.

Further Sine Function Analysis

• Displacement Over Time in SHM Captured by Sine Equation:

  • x(t) = A sin(ωt + ϕ)

  • Indicates alternating positive/negative displacements through time.

SHM and Circular Motion

• Relating SHM to Uniform Circular Motion:

  • Visual framework depicting oscillation dynamics in circular forms.

Motion Dynamics in SHM

• Front and Side View Dynamics Illustrated in the Study of Oscillation.

Motion and Time Correlation

• Examples of SHM as Uniform Circular Motion Leading to Periodic Oscillation Pattern Over Time.

Circular Motion Analysis

• Further Detailing How SHM Correlates with Uniform Circular Motion From Various Perspectives.

Trigonometry in SHM

• Application of High School Trigonometry to Understand Relationships in SHM:

  • Sine and cosine definitions in terms of triangle dimensions.

SHM Visual Representation

• Unit Circle Representation in SHM Depicting Sine and Cosine Wave Patterns as They Relate to Circular Motion.

Recap of Motion Dynamics

• SHM Recap:

  • Mass and stiffness as fundamental components influencing vibratory motion.

  • Relation to quantifying SHM explored.

Components of SHM

• Key Ingredients in SHM:

  • Amplitude

  • Period

  • Frequency

  • Phase

  • Wavelength

  • Pressure

  • Distance/Time

Amplitude Explained

• Amplitude (A):

  • Measurement indicating magnitude of oscillation, vital in sound wave characterization.

Measuring Sound Amplitude

• Methods of Measuring Amplitude:

  • Absolute amplitude determined by peak measurements (baseline to peak).

  • RMS (Root Mean Square) as a statistical measure useful for varying signals.

RMS Measurement in Sound

• RMS Calculation for Sound Waves Explained:

  • Arms = √(x₁² + x₂² + ... + xₙ²/n)

  • Provides a measure of average sound amplitude over time.

Period Definition

• Definition of Period (T):

  • Time required to complete one full cycle of sound waves (alternating condensation & rarefaction).

Frequency Definition

• Frequency (f) Overview:

  • Rate of oscillation measured in cycles per second (Hz).

  • Inversely related to period (f = 1/T).

Frequency Examples

• Examples Demonstrating Frequency Calculations for Various Cycles and Their Durations.

Wavelength Defined

• Wavelength (λ):

  • Distance between identical points in adjacent cycles of a sound wave.

  • Formula: λ = c/f, where c = speed of sound in air (~340 m/s).

Phase Definition

• Understanding Phase (ϕ):

  • Initial point of oscillation and its importance in wave dynamics.

  • Description of relative displacement among waves.

Mathematical Representation of Sound

• Sound Mathematically Represented as Periodic Sine Functions:

  • x(t) = A sin(ωt + ϕ), illustrating how air pressure changes over time.

Complex Nature of Sound

• Real-life Sound Characterized as a Combination of Infinite Mass and Spring Components, Creating Complex Waveforms.

Tuning Fork Functionality

• Tuning Fork Dynamics:

  • Tuning fork vibrates through elasticity and mass-induced resonance when force is applied.

  • Generates sinusoidal waves in the surrounding air.

Visual Representation of Tuning Fork Action

• Tuning Fork Illustration Depicting Air Vibrations Produced by Tine Movements.

Pure Tones Explained

• Pure Tone Characteristics:

  • Defined as a sinusoidal waveform at a unique frequency.

  • Generated by tuning forks of differing lengths and masses.

Final Recap

• Summary of SHM:

  • Sound embodies SHM and requires a medium to propagate (dictated by mass/stiffness).

  • Quantified through amplitude, frequency, phase, and wavelength.

  • Visualized on a 2D plane via waveforms.

Next Class Focus

• Upcoming Topics:

  • Exploration of sound measurement methods, emphasizing intensity and decibel scale.

  • Essential textbook readings and tutorials on sound pressure and intensity relationships.