## The Fourier Analysis of Sound Waves

Question:

Discuss about the Fourier Series and Waves Analysis.

This research paper is about the use of Fourier series and analysis to differentiate the sound waves produced by a piano and keyboard. Fourier analysis was discovered by Joseph Fourier by finding out that any periodic wave with a pattern that is repeating may be broken down into waves that are simpler. This means that a periodic wave that is complicated may be divided into waves that are simpler. The simplicity of the sine and cosine waves have made many mathematicians to use these simpler waves when expressing waves that are more complicated as a summation of cosines and sines with diverse amplitudes.

Joseph Fourier came up with an analysis method which is commonly known as Fourier series to be used in the determination of the waves that are simpler and their amplitudes from the periodic function that is more complicated. Sound waves are those categories of waves which can be analysed by the use of Fourier series and analysis by enabling diverse instruments of music to be analysed by the use of this method. The instruments of music produce sound due to the vibration of the physical objects like keyboard, piano, guitar, and violin management. The vibration is caused by a variation of air pressure that is periodic which is heard as sound (Dorfler 147).

Fourier series and analysis is a way of representing the function as a summation of the simple sine wave through decomposing any function that is periodic or even signal that is periodic into the summation of a set of simple functions that are oscillating such as cosines and sines. A Fourier transform that has a discrete time is a periodic function which is normally explained in terms of Fourier series. The branch of Fourier analysis if the Fourier series. For the mathematical expression of the Fourier series, s(x) signifies a function of the actual variable x, whereas s can be integrated at an interval of [x_{0}, x_{0}+P], given that both P and x_{0 }are real numbers.

The function s can be represented in the intervals above as a series or infinite sum of sinusoidal functions related to harmonics. Separate to the interval, the series is said to be periodic with the frequency of 1/P or period of P. Therefore, if s has the characteristic above, then the estimate is effective on the whole of the real line (Hammond 214). The partial sum or finite summation can be expressed as:

## Fourier Series and Sound Waves From Piano

The function, S_{N}(x) can be seen to be approximating s(x) at the interval [x_{0}, x_{0}+P], and the approximated values increases as N approaches infinity, s ∞. This infinite summation of s(x) is what is known as Fourier series (Harry 178).

Music is a mixture of sound waves of different frequencies. The component music’s frequencies are separate or discrete and their ratios form fractions that are simple with a discernible frequency that is dominant. The sound is an example of a wave that is longitudinal which means that the components of the medium vibrate parallel to the wave's direction of propagation. A wave of sound coming out of an instrument of music such as keyboard or piano pushes the air backwards or forward during the outward propagation of sound. This has the impact of pulling or squeezing the air around the instrument by varying the pressure of air slightly (Joe 174).

These variations in pressure may be detected by the eardrum located in the middle ear, then changed into pulses of the neural system located in the inner ear, and then sent for the purposes of processing in the brain. Music is a composition of pure tones and can be categorized as either monotonic or polytonic. A monotonic music is a simple tone composed of particularly pure tones such as watch alarm while polytonic music is composed of a mixture of pure tones that are being played together in a way that sound is pleasant-sounding. A sound possessing numerous frequencies such as that produced by piano or keyboard would still be periodic despite being more complex than a simple sine curve (Joseph 168).

The difference between the sounds produced by the piano and keyboard is the vibration produced by each of these instruments. The piano can be categorized as chordophone and the part involved in the vibration is the stretched string. The keyboard can be categorized as an electrophone and the part involved in the vibration is the electric circuit. Both piano and keyboard naturally vibrate at numerous related frequencies which are commonly known as harmonics. The lowest vibration frequency is known as the fundamental and is normally the loudest. When the sound waves meet they do not collide with the physical objects, rather they pass one another like spectres (Lenssen 189). The sound waves interfering join each other by the linear superposition principle. With the correct combination of cosine and sine functions, functions of different shapes can be made as shown in the figure below:

## Fourier Series and Sound Waves From Keyboard

The process of combination of tones together to produce waveforms that are complex is known as additive synthesis. Music instrument such as piano can naturally perform this process of additive synthesis. Electric instruments such as the keyboard are designed with the mind-set of additive synthesis (Lionheart 149).

A wave of sound coming out of an instrument of music such as keyboard or piano pushes the air backwards or forward during the outward propagation of sound. This has the impact of pulling or squeezing the air around the instrument by varying the pressure of air slightly. The piano can be categorized as chordophone and the part involved in the vibration is the stretched string. Since the sound waves from music signals are time-varying signals, the typical Fourier transform is not enough for analysing such signals. The tool that should be used in this analysis is the time-frequency analysis and can be done using Wigner function of distribution, Gabor transforms, and short-time Fourier transform (William 217).

A piano sound is produced by striking of strings. The fundamental frequency is the frequency that is lowest in the series of harmonics. In the signal that is periodic, the fundamental frequency is the inverse of the length of the period. By decomposing the complex periodic waveform into series of sinusoids that are simple, the following equations can be derived:

The complex waves produced by the piano are a combination of the square waves, sawtooth waves, and triangular waves. These waves are explained below:

Square wave: By considering the square wave below which is produced by the key in a piano:

This type of wave has a period of T= 2 and ω = π, the integration can be done on the sections of t = 0, to t=1and t=1 to t=2. To find the Fourier series of this square wave, there is need of calculating the coefficients b_{n}, a_{n}, and a_{0}.

Sawtooth wave: This type of waveform is also produced by the piano when specific keys are pressed. The Fourier series for this type of waveform is as shown in the equation below:

The diagram of the sawtooth waves is as shown in the figure below:

Triangular wave: This is another example of the waveform produced by the sound waves of the piano (Lenssen 217). The Fourier series of the waveform of this triangular wave is as shown in the equation below:

The diagram of the triangular wave produced by the sound waves from the piano as shown in the figure below:

## Conclusion

The piano can be categorized as chordophone and the part involved in the vibration is the stretched string. When the string of the piano is plucked as shown in the figure below, the amplitude of every mode of harmonic and overtone can be predicted:

By the use of the method of measuring the Fourier coefficients, it is possible to determine the amplitude of every harmonic tone. There is need to know that every mode has its own frequency movement and also its own shape before being plucked (Ravetz 167). The shape of the mode after being released should also be determined as shown below:

Every harmonic of the sound waves from the piano has its own oscillation frequency, the m-th harmonic propagates m times that of the fundamental mode or at a frequency f_{m} = mf_{0}.

Fourier series and Analysis of Sound waves from Keyboard

Just like in the sound waves from a piano, the sound waves from the keyboard are made up of complex waves composed of square waves, triangular waves, and sawtooth waves. A wave of sound coming out of an instrument of music such as keyboard or piano pushes the air backwards or forward during the outward propagation of sound. This has the impact of pulling or squeezing the air around the instrument by varying the pressure of air slightly. The difference between the sounds produced by the piano and keyboard is the vibration produced by each of these instruments (Joseph 168).

The piano can be categorized as chordophone and the part involved in the vibration is the stretched string. The keyboard can be categorized as an electrophone and the part involved in the vibration is the electric circuit. Both piano and keyboard naturally vibrate at numerous related frequencies which are commonly known as harmonics. In the determination of the frequency of the component that is present in the musical note from the keyboard, there is need of computing the Fourier series of the musical note that has been the sample (Lenssen 275). When a single key of the keyboard (B) was pressed, the following sound wave was produced an electric circuit as measured by the oscilloscope:

The Fourier transform of the above signal of the sound waves from the keyboard with the major focus being on the amplitude and the frequency of the signal is as shown in the figure below:

For the analysis of the sound waves produced by the keyboard, there is need to re-synthesize the same sound wave by incorporating the component of frequency as shown in the Fourier analysis. Hence the convergence summation of all the periods of the square waves, triangle waves, and sawtooth waves produced by the keyboard can be represented by the Fourier series. The coefficients of the series above are samples of the function of the related continuous time (Dorfler 289).

Hence a convergent summation of all the periods of different types sound waves in the in the sound of the keyboard can be represented by the equation below, whose coefficients are samples of a function of related continuous time as shown the equation below:

The fundamental frequency is a characteristic shared of all the keys in the keyboard. The lowest frequency in the series of harmonics is the fundamental frequency. In the signal that is periodic, the fundamental frequency is the inverse of the length of the period (Ravetz 149).

The Fourier series an analysis is applied in many areas in the field of mathematics and also music. Some of the areas in which the Fourier series and analysis is applied include thin-walled shell theory, econometrics, quantum mechanics, image processing, signal processing, optics, acoustics, and vibration analysis. In music, this analysis is used in the development of different musical equipment to produce different tones required (Harry 248).

Conclusion

This research paper is about the use of Fourier series and analysis to differentiate the sound waves produced by a piano and keyboard. Fourier analysis was discovered by Joseph Fourier by finding out that any periodic wave with a pattern that is repeating may be broken down into waves that are simpler. The difference between the sounds produced by the piano and keyboard is the vibration produced by each of these instruments. The piano can be categorized as chordophone and the part involved in the vibration is the stretched string. The keyboard can be categorized as an electrophone and the part involved in the vibration is the electric circuit. Both piano and keyboard naturally vibrate at numerous related frequencies which are commonly known as harmonics.

Dorfler, Monika. "What Time–Frequency Analysis Can Do To Music Signals,. London: William Publication, 2013.

Hammond, Nicola. Mathematics of Music. Colorado: UW-L Journal of Undergraduate Research XIV, 2012.

Harry. Music, Physics and Engineering. London: Dover Publications, 2014.

Joe, Wolfe. "What is a Sound Spectrum?". New York: ww.phys.unsw.edu.au/~jw/sound.spectrum.html. , 2011.

Joseph, George. The Crest of the Peacock. Perth: Princeton University Press, 2014.

Lenssen, Neone. An Introduction to Fourier Analysis with Applications to Music. Paris: Journal of Humanistic Mathematics, 2013.

Lionheart, Bavon. Fourier Series. 2014: from https://www.maths.manchester.ac.uk/~bl/teaching/2m2/fseries, 2016.

Ravetz, Joseph Fourier 1768-1830. London: Institute of Technology, 2011.

Robertson, Kay. Jean Baptiste Joseph Fourier. Michigan: School of Mathematics and Statistics University of St. Andrews, 2013.

Tymoczko, David. A Geometry of Music. Colorado: Harmony and Counterpoint in the Extended Common Practice, 2013.

William, John. Time–frequency Analysis of Musical Signals. Moscow: IEEE, 2012.

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