Fourier Expansion and Harmonics

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Questions and Answers

What is the minimum number of parameters required to characterize a harmonic wave?

Three (amplitude, frequency, and period)

Two (amplitude and frequency) (correct)

Four (amplitude, frequency, period, and time)

One (amplitude)

What is the result of the superposition of a set of harmonic behaviours?

A non-harmonic but periodic wave (correct)

A harmonic wave with a single frequency

A non-periodic wave

A silent sound

What is the relationship between the fundamental frequency (w1) and the period (T) of a wave?

w1 = T

w1 = T / 2π

w1 = π / T

w1 = 2π / T (correct)

What is the term used to describe the combination of multiple harmonic waves?

Fourier expansion Signup and view all the answers

What is the term used to describe the unique 'tone colour' of a sound?

Timbre behaviour Signup and view all the answers

How long did it take to demonstrate that a non-harmonic sound can be composed of harmonic waves?

90 years Signup and view all the answers

What is the primary tool used to analyze the presence of different frequencies in a sound?

Fourier expansion Signup and view all the answers

What is the main reason why we can differentiate between the same pitch played on a guitar and a piano?

Different timbre characteristics Signup and view all the answers

Where do high frequencies produce vibration of the basilar membrane?

Close to the oval window Signup and view all the answers

What is the consequence of the non-linear response of the cochlea to different frequencies?

Differentiation of timbre between instruments Signup and view all the answers

What is the characteristic of sound that is perceived in a logarithmic way?

Sound level Signup and view all the answers

Why do different instruments produce different timbres when playing the same pitch?

Due to the unique combination of frequencies present in each instrument Signup and view all the answers

What is the Fourier expansion primarily used to describe?

The decomposition of a sound wave into its component frequencies Signup and view all the answers

What is the primary factor that determines the timbre of a sound?

The number of harmonics present in the sound wave Signup and view all the answers

Why do two sounds with the same fundamental frequency have different timbre?

Because they have different numbers of harmonics Signup and view all the answers

What is the consequence of the superposition of multiple harmonics in a sound wave?

The creation of a sound wave with multiple frequencies Signup and view all the answers

What is the relationship between the fundamental frequency and its harmonic frequencies?

The harmonic frequencies are derived from the fundamental frequency Signup and view all the answers

What is the visual representation of a sound wave's frequency components?

A diagram with frequency on one axis and amplitude on the other Signup and view all the answers

What is the fundamental frequency that determines the pitch of a sound?

The lowest frequency in a set of frequencies Signup and view all the answers

What happens when a string is played on a guitar?

A whole set of frequencies, including the fundamental frequency and its harmonics, are triggered Signup and view all the answers

What is the sequence of events when playing a guitar?

The guitar string is plucked, then the frequencies are triggered, and finally the cochlea is excited Signup and view all the answers

Why is it impossible to generate only one frequency when playing a guitar?

Because the harmonics of the fundamental frequency are always triggered Signup and view all the answers

What is the mathematical concept that describes the behaviour of a set of frequencies triggered by a vibrating object?

Fourier Expansion Signup and view all the answers

What is unique about the sound produced by a flute?

It produces a single frequency Signup and view all the answers

What is the primary reason for the difference in sound between sound n1 and sound n2?

Difference in amplitude ratios of harmonics Signup and view all the answers

Which of the following is a consequence of a person having an increased hearing threshold?

Perceiving a distorted sound Signup and view all the answers

What is the main characteristic of the harmonics produced when a person is talking or singing?

They have a specific amplitude Signup and view all the answers

What is the result of speaking louder to a person with an increased hearing threshold?

The distortion remains Signup and view all the answers

Why are consonants particularly affected by a person's increased hearing threshold?

They are more sensitive to high pitches Signup and view all the answers

What is the fundamental principle underlying the Fourier expansion of sound?

Sound waves can be decomposed into harmonics Signup and view all the answers

Study Notes

Fundamental Concepts of Waves

Waves are categorized into fundamental waves (w1) and higher harmonics (wn), with every non-zero amplitude indicating active harmonic modes in a system.
The theory explaining wave behavior is called Fourier Expansion, represented compactly as P(t).
The fundamental frequency determines all derived frequencies, with amplitude being the only variable that can be adjusted.

Sound and Timbre

Timbre refers to the distinct quality of sound produced by different instruments or voices, determined by the number of activated harmonics.
Sounds with identical pitches can exhibit different timbres based on their harmonic content.
An instrument like the flute has approximately 10 harmonics, whereas others can produce more, indicating varied harmonic complexity.

Cochlea Response to Sound

Different frequencies excite unique responses within the cochlea, with high frequencies affecting areas near the oval window and low frequencies impacting the apex.
This selective response is crucial for pitch differentiation, allowing the perception of distinct sound sources despite shared fundamental frequencies.

Harmonic and Non-Harmonic Behavior

Harmonic waves oscillate sinusoidally and are characterized by amplitude and frequency.
In contrast, non-harmonic behaviors, such as those found in real-world sound waves, are produced by the superposition of multiple harmonic behaviors.
Instruments, like guitars, produce complex sounds resulting from the simultaneous excitation of several harmonic frequencies.

Frequencies and Amplitudes

The fundamental frequency (f1) is central to sound production, with higher harmonics (w2, w3, ...) being integral multiples of f1.
Each sound produced contains a set of harmonics associated with its fundamental frequency, distinguished by different amplitudes.

Hearing and Perception Challenges

Sound levels are perceived logarithmically, measured in decibels, impacting how different frequencies are heard.
Individuals with hearing impairments may struggle to perceive certain harmonics, leading to distorted sound perception.
For example, a person with an increased hearing threshold may struggle to distinguish consonants in speech, altering their audible experience significantly.

Key Ideas of Sound Interaction

When sounds are produced, they generate a harmonic series in the ear, leading to a complex auditory experience, where differentiating pitches relies on the cochlea's response to varying frequencies.
Every sound source evokes multiple frequencies, echoing the behavior of musical instruments, contributing to overall sound complexity through Fourier Expansion.

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Description

Test your understanding of Fourier Expansion, a mathematical concept that decomposes a function into its constituent frequencies. Learn about fundamental waves, higher harmonics, and modes in a system.