Sine waves are also found in **, which are the natural vibration modes of guitar, violin, and piano strings. By applying Newtonian mechanics and Leibniz's calculus to an ideal model of a taut string, we can derive the partial differential equation for this vibration. In this model, strings are 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. Under the premise that these forces are known, each particle will move according to Newton's law f=ma, and this process takes place at every point x of the string. The resulting differential equation depends on both x and t, and is another example of a partial differential equation. This equation is known as the wave equation, and sure enough, it shows that the typical motion of a vibrating string is a wave.
Just like in the heat flow problem, some sine waves prove to be effective because they are self-regenerating when they vibrate. If the ends of the string are fixed, these sine waves cannot propagate and simply stay in place and vibrate. If an ideal string experiences negligible air resistance and internal friction and starts vibrating in a sine wave pattern, then it will vibrate like this forever and the vibration frequency will never change. For all these reasons, sine waves are the ideal building blocks to solve this problem.
Other mode shapes can likewise be summed by an infinite number of sine waves. For example, in the harpsichord, which was used in the 18th century, a string was often used as a plucked feather pipe to form a triangle before it was released.
Although a triangular wave has a sharp angle, it can also be represented as an infinite series and form of a perfectly smooth sine wave. In other words, we can create sharp corners without using them. In the figure below, an approximate triangular wave is constructed from a sine wave through three increasingly faithful approximations, as shown by the dotted line at the bottom of the figure.
The result of the first approximation is a sine wave with the optimal possible amplitude ("optimal" means that it minimizes the total squared error of the triangular wave). The result of the second approximation is the optimal sum of the two sine waves. The result of the third approximation is the optimal sum of the three sine waves. The amplitude of the optimal sine wave follows a formula discovered by Fourier:
This infinite series and the Fourier series known as triangular waves. Note the unique numerical pattern: only odd frequencies appear in the sine wave, such as 1, 3, 5, 7..., and their corresponding amplitudes are the reciprocal of the odd squares of the alternating plus and minus signs. Unfortunately, I can't explain in a few words why this formula works; We had to study a lot of specific calculus to figure out where those magical amplitudes in the formulas came from. But the point is that Fourier knows how to calculate them. With this formula, he was able to synthesize a triangular wave or any other arbitrarily complex curve from a much simpler sine wave.
Fourier's great idea is the basis of the synthesizer, and we use a note (such as the A above the c) as an example to illustrate why. In order to produce an accurate pitch, we can strike a tuning fork with a vibrational frequency set to 440 cycles. The tuning fork consists of a handle and two metal forks, which vibrate back and forth 440 times per second when the tuning fork is struck with a rubber hammer.
The vibrations of the metal forks disturb the nearby air: when they vibrate outward, they compress the air; Whereas, when vibrating inward, they thin the surrounding air. The back and forth of the air molecules creates a sinusoidal pressure perturbation, which our ears treat as a pure tone—a monotonous and dull A note that lacks what the ** family calls a timbre. However, we can play the same A note on the violin or piano, and it sounds vivid and warm. Although the violin or piano is also vibrating at a fundamental frequency of 440 cycles and seconds, they produce a different sound than a tuning fork (and other instruments) due to the different overtones. Overtones are the first term for waves like sin3x and sin5x in the triangle wave formula, which adds color to a note by adding multiple times the fundamental frequency.
In addition to the sine wave with a frequency of 440 cycles, the resulting triangle wave also includes a sine wave overtone with a frequency of three times that of the sine wave (3 440 = 1 320 cycles of seconds). This overtone is not as strong as the basic sinx mode, its relative amplitude is only 1 9 of the basic mode, and the other odd modes are weaker. From the point of view, 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 unifying power of Fourier's thought lies in the fact that the sound of any instrument can be synthesized with an infinite number of tuning forks. All we had to do was strike the tuning fork at the right time and with the right amount of force, and even though we were using a monotonous sine wave, we played the sound of a violin, a piano or even a trumpet or oboe uncannificently. This is how the first synthesizers basically worked: by combining a large number of sine waves, they could reproduce the sound of any instrument.
When I was in high school, I took an electronics class and felt the action of sine waves. That was in the stagflationary 70s of the 20th century, when the electronic ** was produced in a big box that looked like an old-fashioned switchboard. My classmates and I plugged the cables into the various sockets, and then turned the knobs back and forth, and we would hear sine, square, and triangle waves. I still remember that the sound of the sine wave was clean and wide, like a flute; Fang Bo's voice sounded sharp and harsh, like a fire alarm; The sound of the triangle wave sounds noisy and noisy. By turning one of the knobs, we can change the frequency of the wave, raising and lowering its pitch.
Turning another knob, we can change the amplitude of the wave to make it sound louder or softer. By plugging in several cables at the same time, we can combine the waves and their overtones in different forms, which is our sensory experience of the abstract Fourier theory. We can hear the shape of the waves as we see them on the oscilloscope. Nowadays you can try it all on the internet, search for something like "triangle sounds" and you'll find an interactive demo program.
More importantly, Fourier took the first step in exploiting the way in which the calculus particle continuum moves and changes. In addition to Newton's study of the motion of discrete particles, this is another great advance. In the centuries that followed, scientists continued to use the Fourier method to address the behavior of other continuums, such as the flutter of the wings of a Boeing 787, the appearance of patients after facial surgery, the flow of blood through arteries, or the rumble of the earth after that.
Today, these techniques are ubiquitous in science and engineering, and they are used to analyze a variety of wave phenomena, including: shock waves generated by thermonuclear **, radio waves for communication, digestive waves that promote nutrient absorption in the intestines and push waste in the right direction; Pathological radio waves in the brain associated with epilepsy and Parkinson's tremor, waves of traffic congestion on highways (like the irritating phenomenon of ghost jams, traffic slows down as a whole for no apparent reason). Fourier thought and its branches can help us to understand all these wave phenomena mathematically, to explain and eliminate them (sometimes with the help of formulas, sometimes through large-scale computer simulations) and, in some cases, to control or eliminate them.