Shulou(Shulou.com)11/24 Report--

Wave equations give us a deeper understanding of water waves, sound waves, light waves and elastic vibrations. Seismologists use wave equations to infer the structure of the earth's interior. Oil companies look for oil in a similar way. Physicists use it to predict the existence of electromagnetic waves, leading to the emergence of radio, television, radar and modern communications.

We live in a world of waves. Our ears can detect compression waves in the air: we call it "hearing". Our eyes can detect electromagnetic waves: we call it "vision". When a ship is tossed up and down on the sea, it reacts to the waves in the water. Surfers use waves for entertainment; radios, televisions and mobile phones use electromagnetic waves. Microwave oven... The name speaks for itself.

Mathematicians began to think about "waves" centuries ago. They first studied music, especially the violin. How do violin strings sound?

The string of a violin can be reasonably assumed as an infinitely thin line, and its vibration occurs in a plane. Such assumptions play a role in reducing the dimension, thus making the problem easier. Once we understand the principle of simple waves, we can extend this understanding to real waves. Without some mathematicians thinking about the vocal principle of the violin, there would not be today's electronic world and global communication.

Pythagorean school and music Pythagorean school believe that the world is based on numbers, and they think that numbers refer to integers or the ratio between integers.

Some of Pythagoras's important worldviews come from music. There is a story about Pythagoras passing by a blacksmith shop. He noticed that hammers of different sizes make different tones. Hammers linked with simple numbers (for example, one hammer is twice the size of the other) make a harmonious sound.

Pythagoras used a stretchable string to conduct a series of experiments. Ptolemy mentioned these experiments in the Harmonics around A.D. 150. By moving the support to different positions along the string, Pythagoras found that when the length of two strings of equal tension is in simple proportion (such as 2:1 or 3:2), they can make a harmonious sound, while the complex proportion is disharmonious.

A little bit of music musicians describe a pair of notes according to the intervals between notes. The most basic interval is an octave. On a violin, the way to play an octave note on an open string is to press the string in the middle of the fingerboard. So the octave is related to a simple 2:1 ratio.

Other harmonious notes are also related to simple numerical ratios. The most important things in western music are the 4:3 ratio and the 3:2 ratio. If you consider the scale of the whole note, these names make sense, such as C, D, E, F, G, A, B, C. Based on C, the fourth degree corresponds to F, the fifth degree to G, and the octave to C.

These proportions provide a theoretical basis for musical scales and have developed into the scales used in most western music today. For convenience in the future, I will write down the score of 3:2 as a score of 3. Start with a basic note, rise to a fifth, and get a string of notes of length.

Calculate and get

All these notes, except the first two, are too high to stay within an octave, but we can lower them by one or more octaves and repeatedly divide the score by 2 until the result is between 1 and 2. That's what you get.

Finally, the sort gets:

These notes are quite close to the C, D, E, G, An and B notes on the piano. Notice that F is gone. In fact, the gap between 81amp 64 and 3x2 sounds wider to the ear than the others. To fill this gap, we insert the ratio of 4max 3mer-the fourth degree, which is very close to F on the piano. Now we have a scale based entirely on the fourth, fifth and octave.

We have now explained the white notes on the piano, but there are also black notes. This is because there are two different ratios for consecutive numbers in the scale: 9-stroke 8 (called a tone) and 256-pound 243 (chromatic). For example, the ratio of 81amp 64 to 9x8 is 9x8, while the ratio of 4amp 3 to 81x64 is 256x243. The names "tone" and "semitone" denote an approximate comparison of intervals. The values are 1.125 and 1.05 respectively. The first note is larger, so the pitch change of a note is greater than that of a semitone. The ratio of two semitones is 1.05 ^ 2 (about 1.11), so the two semitones are very close to one tone.

Continuing this vein, we can divide each note into two intervals and get a scale of 12 notes. This can be done in several different ways and produce slightly different results. In any case, when we change the tone of a piece of music, there may be a slight but audible problem: if you move each note up one semitone, the interval will change slightly. If we choose a specific ratio for the semitone and make it equal to 2 to the power of 12, this effect can be avoided. Then the two notes will form an accurate chromatic, and the 12 semitones will form an octave, and you can change the scale by moving all the notes up and down by a fixed amplitude.

Pythagoras' theory of natural harmony is actually based on western music. In order to explain why simple ratios are closely related to musical harmony, we must look at the physical phenomena of string vibration.

Disclaimer: I am not an expert in music theory. I would like to leave a message to point out what is wrong.

The key to physical phenomena is Newton's second law of motion, which connects acceleration with force. We also need to know how the string changes with the movement, slight stretch or contraction of the string under the action of tension. For this we need Hooke's law: the change in the length of the spring is proportional to the force exerted on it. The string of a violin is actually a spring. But there is still an obstacle: the string of a violin is a continuum, a line made up of infinitely many points. So mathematicians who study the cycle think that strings are a large number of tight particles connected by springs. This makes it possible to write the vibration equation of the violin string.

In 1727, John Bernoulli began to solve the problem. In his mathematical model, there is only one string fixed at both ends, without a violin; the string vibrates up and down on a plane. In this experiment, Bernoulli found that the shape of the string vibrating at any time is a sinusoidal curve; the amplitude of the vibration also follows a sinusoidal curve (in time rather than space). His solution is sinct sinx, where c is a constant.

A continuous vibrating string. Its shape is sinusoidal. The amplitude also varies sinusoidal with time. Sinx tells us the shape of the vibration, at t moment, multiplied by a factor sinct. The period of the oscillation is 2 π / c.

This is the simplest solution Bernoulli can get, but there are other forms.

Modes 1, 2, 3 of the vibrating string. In each case, the string vibrates up and down, and its amplitude varies sinusoidal with time. The more waves there are, the faster they vibrate. Similarly, a sinusoidal curve is the shape of a chord at any time, and its amplitude is multiplied by a time-dependent factor, which is also sinusoidal. The formula is sin2ct sin2x,sin3ct sin3x and so on. The vibration period is 2 π / 2c, 2 π / 3c and so on. So the more waves there are, the faster the string vibrates.

Through the construction of the musical instrument and the assumption of the mathematical model, some points on the strings are always at rest. These "points" are the reason for the simple numerical ratio in the Pythagorean experiment. For example, because vibration modes 2 and 3 (pictured above) occur on the same string, the gap between mode 2 nodes is twice the gap that mode 3 should have. This explains why proportions like 3:2 naturally arise from the dynamics of the vibrating spring, but not why they are harmonious. Before we solve this problem, let's introduce the theme of this article-wave equation.

Wave equation comes from Newton's second law of motion. In 1746, Jean Leronda Lambert regarded vibrating violin strings as a collection of particles. He derived an equation to describe how the shape of the string changes over time. But before I explain what it looks like, we need to understand a concept called partial derivative.

If the function u depends on only one variable x, we write its derivative as

The small change of u divided by the small change of x, but the wave height u depends not only on x, but also on the time t. At any fixed time, we can find the du / dx, which tells us the local slope of the wave. But we can also fix the space and let time change, and it tells us the rate at which a particle jumps up and down.

We can use du / dt to express the time derivative and interpret it as a small change in u divided by a small change in t. But this representation hides a kind of fuzziness: the height of the small change du, in these two cases may be different, and usually different. When we differentiate space, we change the space variable a little bit and see how the height changes; when we take the derivative of time, we change the time variable a little bit to see how the height changes. There is no reason to say that the change with time should be equal to the change with space.

Therefore, mathematicians decided to deal with this fuzziness by changing the symbol d. They chose the symbol ∂.

A symbol that gives a lot of people a headache, and then they write these two derivatives as

As soon as you see ∂, it tells you that you will see the rate of change of several different variables. These rates of change are called partial derivatives because conceptually you only change part of the set of variables and leave the rest unchanged.

When D'Alembert solved the equation of the vibrating string, he was faced with this situation. The shape of the string depends on space and time. Newton's second law of motion tells him that the acceleration of a small string is proportional to the force acting on it. Acceleration is the derivative of velocity to time. But this force is the pull of adjacent segments, and "adjacent" means a small change in space. When he calculated these forces, he got the equation.

Where u (x, t) is the vertical position of x on the string at t time, and c is a constant related to the tension and elasticity of the string.

D'Alembert 's formula is the wave equation, which, like Newton's second law, is a differential equation that involves the second derivative of u. Because these are partial derivatives, it is a partial differential equation. The second spatial derivative represents the resultant force acting on the string, and the second time derivative is acceleration. The wave equation sets a precedent: the key equations of most classical mathematical physics are partial differential equations.

Once the wave equation is written, it can be solved. Because it's a linear equation. Partial differential equations have many solutions, usually infinite, because each initial state has an independent solution. For example, the strings of a violin can be bent into any shape you like. "Linear" means that if u (x, t) and v (x, t) are solutions, then any linear combination au (x, t) + bv (x, t) is also the solution, where an and b are constants.

The linear property of the wave equation stems from the approximation made by Bernoulli and D'Alembert: all disturbances are assumed to be small. Now, the force on the string can be approximately expressed by a linear combination of the displacements of each mass. A better approximation will result in a nonlinear partial differential equation, which is much more complicated.

D'Alembert knew he was right because he found a solution with a fixed shape moving along a string, just like a wave. The velocity of the wave is a constant c in the equation. Waves can travel to the left or right, and the superposition principle plays a role here. D'Alembert proved that each solution is the superposition of two waves, one propagating to the left and the other to the right. In addition, each individual wave can have any shape. The standing wave found in a violin string with a fixed end is a combination of two waves of the same shape, one moving to the left and the other to the right. At both ends, the two waves cancel each other out: the peaks of one wave coincide with the troughs of the other. So they conform to the physical boundary conditions.

There are two ways to solve the wave equation: the Bernoulli equation can get the sine and cosine, and the D'Alembert equation can get waves of arbitrary shape. At first, D'Alembert 's solution seemed more general: sine and cosine were functions. But most functions are not sine and cosine. However, the wave equation is linear, so Bernoulli solutions can be combined. For simplicity, you only need to consider fixed-time waves and get rid of time dependence. The following figure takes 5sinx + 4sin2x − 2cos6x as an example. Its shape is quite irregular and its swing is large, but it is still smooth and wavy.

A typical combination of sine and cosine with different amplitudes and frequencies. What bothers mathematicians is that some functions are so rough or jagged that we cannot write them as a linear combination of sine and cosine. But if you use a limited number of sinusoids and cosines, this is not the case-- this suggests a solution. The sine and cosine of a convergent infinite series also satisfy the wave equation. Can it represent a sawtooth function? Mathematicians debated this question endlessly, and when the same problem appeared in the thermal theory, the debate finally reached the best part. The problem with heat flow naturally involves discontinuous functions with sudden jumps, which is worse than jagged functions. But as it turns out, most "reasonable" waveforms can be represented by infinite series of sine and cosine, so they can be represented approximately by a limited combination of sine and cosine.

Sine and cosine explain the harmony ratio advocated by the Pythagorean school. These specially shaped waves are important in sound theory because they represent "pure" tones. Any real instrument can produce a mixture of pure tones. If you pluck the strings of the violin, the main note you hear is the sinx wave, but it is also superimposed with a little bit of sin2x, maybe sin3x and so on. The tonic is called the pitch, and the rest is its harmony. The number before x is called wavenumber. In particular, the frequency of sin2x is twice that of sinx, and it is an octave higher than before. This note is most harmonious when played with the pitch.

Mathematicians first derive the wave equation in the simplest way: a vibration line (an one-dimensional system). However, in practical application, a more general theory is needed to simulate two-dimensional and three-dimensional waves. Even in music, two dimensions are needed to simulate the mode of drum skin vibration. Many other areas of physics involve two-dimensional or three-dimensional models. It is easy to extend the wave equation to a higher dimension, and all you have to do is repeat the methods for calculating violin strings.

For example, in three-dimensional space, we use three spatial coordinates (x, y, z) and a time t. Waves are described by a function u that depends on these four coordinates. For example, it can describe the pressure in the air as sound waves pass through the air. Using the same assumptions as D'Alembert, an equally beautiful equation can be obtained by the same method:

The formula in parentheses is called Laplace formula. This expression often appears in mathematical physics, so it has its own special symbol:

Another symbol that gives a lot of people a headache. In order to get a two-dimensional Laplace equation, you just need to remove the term z.

The main problem with high-dimensional is that the shape produced by waves (called the domain of equations) can be very complex. In one-dimensional space, the only connected shape is an interval, a line segment. However, in two-dimensional space, it can be any shape on a plane, while in three-dimensional space, it can be any shape in space.

The wave equation has achieved amazing success, and in some areas of physics, it is very close to describing reality. However, its derivation requires several hypotheses. When these assumptions are unrealistic, the same physical ideas can be modified to meet practical needs, resulting in different versions of the wave equation.

Earthquake is a typical example. The main problem here is not D'Alembert 's assumption that the amplitude of the wave is very small, but the change in the physical properties of the domain. These characteristics will have a strong impact on seismic waves, which are vibrations that pass through the earth. By understanding these effects, we can go deep into the interior of the earth and understand its composition.

The biggest goal in the field of seismology is to find a reliable way to predict earthquakes and volcanic eruptions. This has proved difficult because the conditions that trigger these events are a complex combination of many factors in many locations. However, the study of wave equations by seismologists provides a theoretical basis for many projects being studied.

Wave equations also have some commercial applications. Oil companies explore "liquid gold" a few kilometers underground by exploding on the surface and using seismic echoes generated by the explosion to map the underground geology. The main mathematical problem here is to reconstruct geology based on the received signals, which is the reverse use of wave equations.

This article comes from the official account of Wechat: Lao Hu Shuo Science (ID:LaohuSci). Author: I am Lao Hu.

Welcome to subscribe "Shulou Technology Information " to get latest news, interesting things and hot topics in the IT industry, and controls the hottest and latest Internet news, technology news and IT industry trends.