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foreword
The previous chapter recorded how sound is produced, as well as some basic characteristics and terms of sound. We know that sound is a wave produced by the vibration of objects. For example, the following are the waveforms of violin, trumpet, flute, and oboe:
Then why do their waveforms look like this? The content of this chapter will deeply analyze where the waveform of the sound comes from. Before entering the content, the following passage will appear many times in this article: The
important thing is said three times:
Fourier's principle shows that any repeated waveform can be decomposed is a sine wave component containing the fundamental frequency and a series of harmonics that are multiples of the fundamental.
Fourier's principle shows that any repeating waveform can be decomposed into sine wave components containing the fundamental frequency and a series of harmonics that are multiples of the fundamental.
Fourier's principle shows that any repeating waveform can be decomposed into sine wave components containing the fundamental frequency and a series of harmonics that are multiples of the fundamental.
First use a picture to explain the meaning of the above sentence. The following is the synthesis of several waves, which become the waves we often see: In fact, if you just
want to understand, then you can actually read the preface; If you have a certain understanding of what a sine wave is, it is recommended to read the third paragraph directly. If you don’t understand it at all, the author will introduce it in detail here.
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1. The basic waveform of sound
Typical basic waveforms of sound are as follows:
1. Sine wave
A sine wave is a waveform with the most single frequency component. This waveform is mathematically a sine curve. It sounds clear, includes only the first harmony, and is a fundamental tone
2. Triangle wave
The shape of the triangular wave and the horizontal axis together form two triangles, often considered a sine wave, because the roll-off speed is too fast, so it sounds softer
3. Sawtooth wave
The waveform shape of the sawtooth wave is similar to that of the triangle wave, but there are abrupt points in the waveform of the sawtooth wave. So it sounds clearer and brighter.
4. Square wave
A square wave is a non-sinusoidal waveform, ideally a square wave has only two values, "high" and "low". So it sounds relatively empty.
Two, sine wave
The reason why the sine wave is singled out is that it is a relatively important waveform:
Fourier's principle shows that any repetitive waveform can be decomposed into a sine wave component containing the fundamental frequency and a series of harmonics that are multiples of the fundamental.
1. What is a sine
Review junior high school mathematics: the sine function is a kind of trigonometric function, and the trigonometric function is divided into sine, cosine, and tangent. In a right triangle, the ratio of the side opposite to the hypotenuse of any acute angle ∠A is called the sine of ∠A.
θ is the required angle, the opposite side of the angle is the opposite side, and the longest side of the triangle is the hypotenuse, and the other side is the adjacent side.
The definition of the trigonometric function sin cos tan is:
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sinθ=opposite side/hypotenuse=b/c
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cosθ = adjacent side / hypotenuse = a / c
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tanθ=opposite side/adjacent side=b/a
2. What are sine waves and sinusoidal curves
Explanation on Baidu Encyclopedia:
Sine wave is a signal with the most single frequency component. It is named because the waveform of this signal is a mathematical sine curve.
Now suppose a function is y=sin X, when X takes 0, 30, 60, 90, 120, 150, 180 respectively (unit: degree), the corresponding values of Y are 0, 0.5, 0.8660, 1, 0.8660, 0.5, 0. Draw the corresponding point in the coordinate system to get the image of the sine wave. One feature of this image is that it changes periodically, for example, when X = 0, Y = 0, and when X = 180, Y = 0; if X takes the value [180~360], we can see that the image is exactly the same as the original the opposite of (in the fourth quadrant). This is the image of a sine wave.
Intuitively feel what a sine curve looks like:
x is 0 angle:
when x is 30 degrees:
when x is 60 degrees
when x is 90 degrees:
the overall look is as follows
----------- Pictures from Zhihu Meet Mathematics -----------
We can intuitively see that the coordinates of the y-axis of the curve are directly related to the angle, speed, and radius of the center of the circle.
3. Sine wave and sound
Fourier's principle shows that any repeating waveform can be decomposed into sine wave components containing the fundamental frequency and a series of harmonics that are multiples of the fundamental.
1. Angular frequency
The angular frequency represents the radian value of the phase angle changing per unit time
Learn more about the sinusoidal curve in the second section of the second section.
If it is like in the picture: the radius of the known circle is x and the frequency is f, then at t, the value of y is
Then there is the formula: y = x * sin(f * t * 360°)
Knowing that one revolution is a period (f * t = 1), f = 1 / t; and knowing that one revolution, the angular arc turns 2π, so the angular frequency: w = 2π / t = 2πf
And because each cycle is 360°, it is also 2π, so there is another formula: y = x * sin(2π f t)
2. Fundamental and Harmonic
Fundamental wave:
Also called the fundamental frequency, the lowest frequency component of a composite wave (synthesis of multiple waves) , that is to say, the one with the largest period (smallest frequency) and the largest amplitude is the fundamental wave. The fundamental wave determines the pitch of the sound.
harmonic:
It refers to the components greater than the integer multiple of the fundamental frequency obtained by Fourier series decomposition of the periodic non-sinusoidal alternating current , and its frequency is higher than the main frequency of the signal.
For example, the fundamental frequency is 50Mhz, and the harmonics appear at 100Mhz, 200Mhz... (integer multiples of the fundamental frequency). In audio, the function of harmonics is to beautify the sound and give color.
diagram
The picture below is the picture of the fundamental wave, the second harmonic, and the third harmonic
3. Waveform synthesis
A formula has been obtained above: y = x * sin(2π f t) , and the synthesis of waves is the addition of y when t is the same
The figure below is the result of adding waves with a radius of 1, a frequency of 100, and a frequency of 200:
the formation of a triangular wave:
----------- The picture comes from the Dezeming blog ------- ---
Square wave formation:
----------- Picture from Dezeming Blog -----------
When there are infinitely many sub-items, the superimposed effect is a square wave:
The waveform addition is actually drawn as follows:
Summarize
The above is a detailed introduction to how the waveform of the sound is formed. It involves a lot of mathematical knowledge. It seems to be a bit confusing, but I personally think it is quite interesting.
If it is helpful to you, please help to like it!