Pj Problems - Overview
Celestial Stars
The Number Line
Geometries
7 Spaces Of Interest - Overview
Triadic Unit Mesh
Creation
The Atom
Survival
Energy
Light
Heat
Sound
Music
Language
Stories
Work
States Of Matter
Buoyancy
Nuclear Reactions
Molecular Shapes
Electron Configurations
Chemical Bonds
Energy Conversion
Chemical Reactions
Electromagnetism
Continuity
Growth
Human-cells
Proteins
Nucleic Acids
COHN - Natures Engineering Of The Human Body
The Human-Body Systems
Vision
Walking
Behaviors
Sensors Sensings
Beauty
Faith, Love, Charity
Photosynthesis
Weather
Systems
Algorithms
Tools
Networks
Search
Differential Calculus
Antiderivative
Integral Calculus
Economies
Inflation
Markets
Money Supply
Painting
(a) The sound produced by a tuning fork (figure 113.4) is considered a simple sound. In general, the sound wave of a simple sound can be represented by the following simple sinusoid:
y = asin2πft --------------(1)
Where a is the amplitude of the sound wave, f is the frequency and t is time.
What is the amplitude and frequency of the simple sound represented by 10sin(π/2)16t?
(b) Sounds from musical instruments and the human voice are complex sounds so they are not representable by only the simple sinusoid of equation (1). However, intelligible sounds (simple or complex) are periodic even if they are not sinusoids. Consider the following complex sound:
y = 0.07sin480πt + 0.05sin760πt + ... -------------(2)
(i) What are the frequencies of the fundamental, first harmonic and first partial?
(ii) What is the frequency of the second harmonic?
(iii) Why is the frequency of a complex sound always that of the first harmonic?
(iv) What is the difference between natural frequency (resonant frequency) and fundamental frequency?
(v) Can humans hear an infrasonic or an ultrasonic sound?
The strings:
S7P4A44 (Motion - Oscillatory).
The math:
Pj Problem of Interest is of type motion (oscillatory).
(a) General sinusoid of simple sound: y = asin2πft
Given sinusoid:10sin(π/2)16t?
Comparing both sinusoids:
2πf = (π/2)16
So, f = 8/2 = 4 cycles per second.
Amplitude = 10.
(b)i Fundamental, first harmonic and first partial mean the same thing. Fourier's theorem says that any periodic function is a sum of simple sine functions representable by equation (1) and that the frequencies of the simple sine functions are all integral multiple of one frequency (the lowest frequency in the series) called the fundamental frequency.
The simple sine function with the fundamental frequency is called the fundamental or first harmonic. The second simple sine function is called the second harmonic and so on.
So, 0.07sin480πt contains the fundamental frequency
So, 480 = 2f; f = fundamental frequency = 240 cycles per sec.
(ii) So, the frequency of the second harmonic is 2(240) = 480 cycles per sec.
So, the frequency of the third harmonic is 3(240) = 720 cycles per sec
Frequencies of subsequent harmonics are similarly calculated. The harmonics beyond the first harmonic are also called overtones.
(iii) The cycles of the overtones are contained within the cycle of the fundamental tone.
(iv) All objects have a natural frequency (resonant frequency) at which they vibrate. They create standing waves (waves that do not appear to be moving) when vibrating at their natural frequencies or multiples of their natural frequency. The lowest frequencies in the series of these multiples is the fundamental frequency.
(v) Sound waves with frequencies below 20 hertz are called infrasonic. Sound waves with frequencies above 20,000 hertz are called ultrasonic. Infrasonic and ultrasonic sounds are naturally inaudible to the human ears.
Range of artificially unaided human hearing is 20 hertz - 20,000 hertz.
The point . is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.
Derivation Of The Area Of A Circle, A Sector Of A Circle And A Circular Ring
Derivation Of The Area Of A Trapezoid, A Rectangle And A Triangle
Derivation Of The Area Of An Ellipse
Derivation Of Volume Of A Cylinder
Derivation Of Volume Of A Sphere
Derivation Of Volume Of A Cone
Derivation Of Volume Of A Torus
Derivation Of Volume Of A Paraboloid
Volume Obtained By Revolving The Curve y = x2 About The X Axis
Single Variable Functions
Absolute Value Functions
Conics
Real Numbers
Vector Spaces
Equation Of The Ascent Path Of An Airplane
Calculating Capacity Of A Video Adapter Board Memory
Probability Density Functions
Boolean Algebra - Logic Functions
Ordinary Differential Equations (ODEs)
Infinite Sequences And Series
Introduction To Group Theory
Advanced Calculus - Partial Derivatives
Advanced Calculus - General Charateristics Of Partial Differential Equations
Advanced Calculus - Jacobians
Advanced Calculus - Solving PDEs By The Method Of Separation Of Variables
Advanced Calculus - Fourier Series
Advanced Calculus - Multiple Integrals
Production Schedule That Maximizes Profit Given Constraint Equation
Separation Of Variables As Solution Method For Homogeneous Heat Flow Equation
Newton And Fourier Cooling Laws Applied To Heat Flow Boundary Conditions
Fourier Series
Derivation Of Heat Equation For A One-Dimensional Heat Flow
Homogenizing-Non-Homogeneous-Time-Varying-IBVP-Boundary-Condition
The Universe is composed of matter and radiant energy. Matter is any kind of mass-energy that moves with velocities less than the velocity of light. Radiant energy is any kind of mass-energy that moves with the velocity of light.
Periodic Table
Composition And Structure Of Matter
How Matter Gets Composed
How Matter Gets Composed (2)
Molecular Structure Of Matter
Molecular Shapes: Bond Length, Bond Angle
Molecular Shapes: Valence Shell Electron Pair Repulsion
Molecular Shapes: Orbital Hybridization
Molecular Shapes: Sigma Bonds Pi Bonds
Molecular Shapes: Non ABn Molecules
Molecular Orbital Theory
More Pj Problem Strings