Sine wave is a signal wave with the most singular frequency component, and is called a sine wave because the waveform of the signal is a mathematical sinusoidal curve.
Any complex signal, such as a spectral signal, can be broken down by the Fourier transform into a superposition of many sinusoidal signals of different frequencies and amplitudes.
Sine waves are also found in **, which are the natural vibration modes of guitar, violin, and piano strings.
This type of partial differential equation can be derived by applying classical mechanics and calculus to an ideal model of a taut string.
In an ideal model, a string is seen as a continuous array of infinitesimal particles stacked side by side, with adjacent particles connected by elastic force.
At any time t, each particle in the string moves according to the forces it is subjected to, and these forces are generated by the tension of the string as the adjacent particles pull against each other.
On the premise that these forces are known, each particle moves according to f=ma, and this process takes place at every point x of the string.
The differential equation thus established is related to t and x at the same time, it is a partial differential equation, called the wave equation, and it shows that the typical motion of the vibrating string is a wave.
In the heat flow problem, some sine waves can regenerate themselves when they vibrate, and if the ends of the string are fixed, these sine waves cannot propagate and can only vibrate in place.
As an ideal building block, the sine wave is based on the premise that the string vibrates in a sine wave mode and maintains such a frequency on the premise of ignoring the air resistance and internal friction that the string vibrates when it vibrates.
Other mode shapes can also be superimposed by an infinite number of sine waves, such as a string that is often pulled into a triangular shape before being released.
Although a triangular wave has a sharp angle, it can also be represented in the form of an infinite series sum of perfectly smooth sine waves. In other words, sharp corners can be created without sharp corners.
Through three approximations that are getting closer and closer to reality, an approximate triangular wave is constructed from the sine wave, as shown by the dotted line in the figure below.
The result of the first approximation is a sine wave with the optimal possible amplitude, and the results of the second and third approximations are the optimal sum of two and three sine waves, respectively.
The amplitude of the optimal sine wave follows a formula discovered by Fourier:
Triangle wave = this infinite series and the Fourier series known as triangle waves, in which only odd frequencies appear. And the corresponding amplitude is the reciprocal of the odd square of the alternating plus and minus signs.
With this formula, it is possible to synthesize triangular waves and other arbitrarily complex waveform curves from a much simpler sine wave.
So far, Fourier thought has provided a theoretical basis for the ** synthesizer.
For example, in order to produce an accurate pitch, a metal tuning fork with a set vibration frequency can be struck.
The tuning fork vibrates to disturb the nearby air, compresses the air when it vibrates outward, and thins the surrounding air when it vibrates inward. In this way, the air molecules sway back and forth to produce sinusoidal pressure disturbances, and the ears hear the monotonous and dull sound of the tuning fork.
However, the same frequency pitch, played on a violin or piano, sounds more vivid and softer than the sound of a tuning fork, because the instrument is paired with different overtones.
Overtones are the corresponding sounds of waves such as sin3x and sin5x in the triangle wave formula, which add color to the note by adding multiple times the fundamental frequency.
In addition to the original sine wave, the synthesized triangle wave also includes an overtone sine wave with a frequency of 3 times, the overtone is not as strong as the basic sine wave mode, the relative amplitude is only 1 9 of the basic mode, and the other odd times sine wave mode is weaker.
From the point of view of **, the amplitude determines the loudness of the overtones, and the richness of the violin's sound is related to the specific combination of its soft overtones and loud overtones.
The power of Fourier's thought lies in the fact that the sound of any musical instrument can be synthesized with an infinite number of tuning forks.
All we have to do is strike the tuning fork at the right time and with the right amount of force, albeit with a single monotonous sine wave, to create an incredible combination of specific sounds for each instrument.
From a theoretical point of view, Fourier's use of calculus ** particles for motion and change on a continuum is undoubtedly another big step forward compared to Newton's study of the motion of discrete particles.
After Fourier, scientists continued to use the Fourier method to move other continuums, such as the flutter of the wings of a Boeing 787 airliner, the appearance of patients after facial surgery, the rumble of the earth behind them, and so on.
Today, these techniques are ubiquitous in science and engineering, and they are used to analyze a variety of wave phenomena, including shock waves from thermonuclear **, radio waves for communication, digestive waves in the intestines, pathological radio waves in the brain associated with epilepsy, Parkinson's tremor, traffic congestion waves on highways, and more.
Fourier thought, and its branches, can help us to understand all these wave phenomena mathematically, to explain them, and in some cases to control or eliminate them.
Theoretically, other types of curves can also be used as building units, and sometimes the effect is better than sine waves, such as wavelets, which perform better than sine waves in image and signal processing tasks in the fields of ** analysis, art restoration and identification, facial recognition, etc.
Sine waves, on the other hand, are well suited to the wave equation, heat conduction equation, and other partial differential equations because it fits perfectly with the derivative. The sine function itself is a property of one sine wave whose derivative is another sine wave, and there is a 1 4 period shift between the two.
This is a remarkable feature that other types of waves do not have. In general, when finding the derivative of an arbitrary curve, it becomes distorted by differentiation, and the shape before and after the derivative is found is very different.
The shape of the curve before and after the derivation of the sine wave will not change, and will only stagger 1 4 cycles to reach the peak.
1 The displacement of 4 periods is combined with the derivative of the sinusoidal function, and the results become extremely ideal and concise. Specifically, if the derivative of the sine wave is sought twice, it will have two 1 4-period displacements, i.e., a total of 1 2 periods earlier.
This means that the previous peak is now a trough and vice versa. The mathematical formula is expressed as:
The formula shows that finding the derivative of sinx twice in a row is equivalent to multiplying it by -1. Replacing two derivatives with one simple multiplication is a very magical simplification. And this is the fundamental reason why sine waves are so well suited for many fields.
If a curve can be formed from a sine wave, then it will inherit the advantages of a sine wave.
The remarkable thing about sine waves in physics is that they form standing waves. Standing waves, although they vibrate up and down, never propagate and vibrate at a single frequency, making them pure waves, which is very rare in the world of waves.
The reason why a musical instrument can produce a melodious and beautiful sound is related to the vibration of the face and body of the instrument, and the sound waves will vibrate and resonate in the wood and cavities of these parts, and these vibration patterns determine the quality and timbre of the instrument.
In 1787, the German physicist and instrument maker Ernst Kladney published an article showing an ingenious way to visualize these vibrational patterns.
He made use of a sheet of metal and played by pulling the bow against the edge of the sheet, allowing the sheet to resonate and make a sound, while sprinkling a thin layer of sand on top of the sheet.
When the bow is pulled, the sand bounces off where the vibrations are strongest and finally falls on the places where there are no vibrations at all, resulting in a visual vibrational mode curve called a Kladney graph.
As the frequency of the sound source is adjusted, the metal plate is excited to resonate in different patterns, and the sand is rearranged into different patterns. The dynamic process can be referred to the following figure:
The Cladney graph enables the visualization of standing waves on a two-dimensional plane. In everyday life, the microwave oven can be described as the three-dimensional counterpart of the Kladney phenomenon.
The inside of the microwave oven is a three-dimensional space, and when it starts working, the inside of the oven will be filled with microwaves in standing wave mode. Although these electromagnetic vibrations are not visible to the naked eye, they can also mimic the two-dimensional experiments described above, and their forms can be indirectly represented.
Towards the end of World War II, the American company Raytheon tried to find a new use for its magnetron (a high-power vacuum tube used in radar).
By chance, an engineer noticed that a peanut chocolate bar in his pocket had melted while using a magnetron, and he realized that the microwaves emitted by the magnetron could heat food efficiently.
To take this idea a step further, the engineers tried to point the magnetron at an egg, which became hot and happened, and the same operation found that popcorn could be made.
It is precisely because of this connection between radar and microwaves that the earliest microwave ovens are called radar stoves.
The original microwave ovens were large in size and expensive as well. However, it was soon improved, and the size became small enough and low enough, so the microwave oven entered thousands of households in industrialized countries.
The story of radar and the microwave oven proves that science is inline, with physics, electrical engineering, materials science, chemistry, and the contingencies of the pre-scientific era nested within each other.
Calculus naturally plays an important role in this, providing the language for describing waves and the tools for analyzing waves.
The wave equation was first discovered as a correlation with the vibrating string, but was eventually used by Maxwell for the existence of electromagnetic waves.
Since then, vacuum tubes, transistors, computers, radars, and microwave ovens have appeared.
Of course, the Fourier method played an integral role in the whole process.