At a wavelength of a certain magnitude. Determine what the wavelength is - formula. How is the sound wavelength calculated - formula. Calculation of wavelength using photon energy

During the lesson you will be able to independently study the topic “Wavelength. Wave propagation speed." In this lesson you will get to know special characteristics waves First of all, you will learn what wavelength is. We will look at its definition, how it is designated and measured. Then we will also take a closer look at the speed of wave propagation.

To begin with, let us remember that mechanical wave is a vibration that propagates over time in an elastic medium. Since it is an oscillation, the wave will have all the characteristics that correspond to an oscillation: amplitude, oscillation period and frequency.

In addition, the wave has its own special characteristics. One of these characteristics is wavelength. The wavelength is denoted by the Greek letter (lambda, or they say “lambda”) and is measured in meters. Let us list the characteristics of the wave:

What is wavelength?

Wavelength - this is the smallest distance between particles vibrating with the same phase.

Rice. 1. Wavelength, wave amplitude

It is more difficult to talk about wavelength in a longitudinal wave, because there it is much more difficult to observe particles that perform the same vibrations. But there is also a characteristic - wavelength, which determines the distance between two particles performing the same vibration, vibration with the same phase.

Also, the wavelength can be called the distance traveled by the wave during one period of oscillation of the particle (Fig. 2).

Rice. 2. Wavelength

The next characteristic is the speed of wave propagation (or simply wave speed). Wave speed denoted in the same way as any other speed, by a letter and measured in . How to clearly explain what wave speed is? The easiest way to do this is using a transverse wave as an example.

Transverse wave is a wave in which disturbances are oriented perpendicular to the direction of its propagation (Fig. 3).

Rice. 3. Transverse wave

Imagine a seagull flying over the crest of a wave. Its flight speed over the crest will be the speed of the wave itself (Fig. 4).

Rice. 4. To determine the wave speed

Wave speed depends on what the density of the medium is, what the forces of interaction between the particles of this medium are. Let's write down the relationship between wave speed, wave length and wave period: .

Velocity can be defined as the ratio of the wavelength, the distance traveled by the wave in one period, to the period of vibration of the particles of the medium in which the wave propagates. In addition, remember that the period is related to frequency by the following relationship:

Then we get a relationship that connects speed, wavelength and oscillation frequency: .

We know that a wave arises as a result of the action of external forces. It is important to note that when a wave passes from one medium to another, its characteristics change: the speed of the waves, the wavelength. But the oscillation frequency remains the same.

Bibliography

Sokolovich Yu.A., Bogdanova G.S. Physics: a reference book with examples of problem solving. - 2nd edition repartition. - X.: Vesta: publishing house "Ranok", 2005. - 464 p.
Peryshkin A.V., Gutnik E.M., Physics. 9th grade: textbook for general education. institutions / A.V. Peryshkin, E.M. Gutnik. - 14th ed., stereotype. - M.: Bustard, 2009. - 300 p.

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Homework

Absolutely everything in this world happens at some speed. Bodies do not move instantly, it takes time. Waves are no exception, no matter in what medium they propagate.

Wave propagation speed

If you throw a stone into the water of a lake, the resulting waves will not reach the shore immediately. It takes time for waves to travel a certain distance; therefore, we can talk about the speed of wave propagation.

The speed of a wave depends on the properties of the medium in which it propagates. When moving from one medium to another, the speed of the waves changes. For example, if a vibrating iron sheet is inserted with its end into water, the water will be covered with ripples of small waves, but the speed of their propagation will be less than in the iron sheet. This is easy to check even at home. Just don't cut yourself on the vibrating iron sheet...

Wavelength

There is another important characteristic: wavelength. Wavelength is the distance over which a wave propagates during one period of oscillatory motion. It's easier to understand this graphically.

If you sketch a wave in the form of a picture or graph, then the wavelength will be the distance between any nearest crests or troughs of the wave, or between any other nearest points of the wave that are in the same phase.

Since the wavelength is the distance traveled by it, this value can be found, like any other distance, by multiplying the speed of passage per unit of time. Thus, the wavelength is directly proportional to the speed of wave propagation. Find The wavelength can be used by the formula:

where λ is the wavelength, v is the wave speed, and T is the oscillation period.

And taking into account that the period of oscillations is inversely proportional to the frequency of the same oscillations: T=1⁄υ, we can deduce relationship between wave propagation speed and oscillation frequency:

v=λυ .

Oscillation frequency in different environments

The oscillation frequency of waves does not change when moving from one medium to another. For example, the frequency of forced oscillations coincides with the oscillation frequency of the source. The oscillation frequency does not depend on the properties of the propagation medium. When moving from one medium to another, only the wavelength and the speed of its propagation change.

These formulas are valid for both transverse and longitudinal waves. When longitudinal waves propagate, the wavelength will be the distance between the two closest points with the same stretching or compression. It will also coincide with the distance traveled by the wave in one period of oscillation, so the formulas will be fully suitable in this case.

A wave is a disturbance of matter, which, propagating in space, transfers energy without transferring matter itself. Each wave has certain characteristics. One of the important characteristics of disturbance processes is the wavelength, the formula for calculating it is given in the article.

Types of waves

All waves are classified according to their physical nature, according to the type of movement of matter particles, according to their periodicity and according to the method of propagation in space.

According to the type of movement of matter particles when waves propagate in it, the following types are distinguished:

Transverse waves are a type of disturbance in which particles of matter vibrate in a direction that is perpendicular to the direction of propagation of the wave. An example of a transverse wave is light.
Longitudinal waves are waves in which particles of matter vibrate in the direction of propagation of the wave. The sound is good example longitudinal wave.

According to their physical nature, the following types of waves are distinguished:

Mechanical. This type of wave requires a substance to occur, that is, a solid, liquid or gaseous medium. An example of mechanical waves is sea waves.
Electromagnetic. This type of wave does not require matter to propagate, but can propagate in a vacuum. A striking example of electromagnetic waves are radio waves.
Gravitational. These waves lead to a disturbance in space-time. Such waves are generated by large space objects, for example, a double star, which rotates around a common center of gravity.

Depending on the wave size, they can be:

One-dimensional, that is, those that propagate in one dimension, for example, the vibration of a rope.
Two-dimensional or surface. These waves travel in two dimensions, such as waves on the surface of water.
Three-dimensional or spherical. These waves travel in three dimensions, such as light or sound.

According to the periodicity of the wave, we can say that there are:

Periodic disturbances that have strictly repeating characteristics after a certain period of time, for example, sound waves.
Not periodic, such waves do not repeat their characteristics at certain time intervals, for example, electrocardiogram waves.

Physical characteristics of the wave

The wave is characterized by 6 parameters, of which only 3 are independent, the rest are derived from these three using the appropriate formulas:

Wavelength L is the distance between two wave maxima.
Height H is the vertical distance between the maximum and minimum of the wave.
Amplitude is a value equal to half the height.
Period T is the time during which two maxima or two minima of a wave will pass through the same point in space.
Frequency is the reciprocal of the wave period, that is, it describes the number of maxima or minima that pass through a specific point in space per unit time.
Velocity is a quantity characterizing the propagation of a wave. It is calculated by the formula: wavelength divided by period, that is, v = L/T.

Independent characteristics are, for example, wavelength, period and amplitude.

Wavelength

This characteristic contains information about the wave, which largely describes its properties. In physics, a wavelength is defined as the distance between its two maxima (minima), or more generally as the distance between two points that oscillate in the same phase. The wave phase refers to the instantaneous state of each point of the wave. The concept of "phase" only makes sense for periodic wavelengths, usually denoted by the Greek letter λ (lambda).

In physics, the formula for wavelength depends on the initial information that is available about a given vibration. For example, in case electromagnetic vibrations you can know the frequency and speed of propagation of the wave, and then apply the usual calculation formula to calculate the wavelength, or you can know the energy of an individual photon, then you should apply a specific formula specifically for energy.

Sine waves

According to Fourier's theorem, any periodic wave can be represented by a sum of sine waves of various lengths. This theorem allows us to study each periodic wave by studying its sinusoidal components.

For a sine wave with frequency f, period T and propagation speed v, the wavelength formula is: λ = v/f = v*T.

The speed of wave propagation depends on the type of medium in which the wave process occurs, as well as on the frequency of oscillations. The speed of propagation of an electromagnetic wave in a vacuum is a constant value and is approximately equal to 3*10 8 m/s.

Sound waves

This type of mechanical waves is generated due to a local change in pressure in a substance that occurs during oscillatory processes. For example, in the air we are talking about rarefied and compressed areas that propagate in the form of a spherical wave from the source generating them. This type of wave is periodic, so the formula for sound wavelength is the same as for sine wave.

Note that only longitudinal waves can propagate in liquids and gases, since in these media no elastic force arises when layers of matter are shifted relative to each other, while in a solid body, in addition to longitudinal waves, transverse waves can also exist.

Speed of sound waves in various media

The speed of propagation of such waves is determined by the characteristics of the oscillatory medium: its pressure, temperature and density of the substance. Since the elementary particles that make up solids are closer to each other than these particles in liquids, this structure of a solid allows vibrational energy to be transferred through it faster than through a liquid, so the speed of wave propagation in them is greater. For the same reason, the speed of sound in liquids is higher than in gases.

Data on the speed of sound in some environments:

In the case of air, we note that Newton derived a formula for the speed of sound in this medium depending on temperature, which was subsequently modified by Laplace. This formula looks like: v = 331+0.6*t ºC.

Thus, the formula for the length of a sound wave with frequency f in air at 25 ºC will take the form: λ = v/f = 346/f.

Electromagnetic waves

Unlike mechanical waves, the nature of which is to disturb the substance in which they propagate, electromagnetic waves do not require matter for their propagation. They arise due to two effects: firstly, an alternating magnetic field creates an electric field, and secondly, an alternating electric field creates a magnetic field. The oscillating magnetic and electric fields are directed perpendicular to each other and perpendicular to the direction of motion of the wave, therefore, by their nature, electromagnetic waves are transverse.

In a vacuum, these waves move at a speed of 3*10 8 m/s and can have different frequencies, so the length of the electromagnetic wave is expressed as: λ = v/f = 3*10 8 /f, where f is the oscillation frequency.

Spectrum of electromagnetic radiation

The electromagnetic radiation spectrum is the sum of all electromagnetic wavelengths. The following parts of the spectrum are distinguished:

Radioelectric radiation. The wavelength of the spectrum for this radiation ranges from a few centimeters to thousands of kilometers. These waves are used in television and various types of communications.
Infrared radiation. This thermal radiation has wavelengths of the order of several micrometers.
Visible light. This is the part of the spectrum that the human eye can distinguish. Its wavelength ranges from 400 nm (blue) to 700 nm (red).
Ultraviolet spectrum. Its wavelengths range from 15-400 nm.
X-ray radiation. Used mainly in medicine. Their wavelength lies in the region of 10 nm - 10 pm. The source of their radiation is vibrations of electrons in atoms.
Gamma rays. This is the highest frequency part of the spectrum, with a wavelength less than 10 pc. Gamma rays have enormous penetrating power through any substance. They are generated as a result of processes occurring in the nucleus of an atom.

Calculation of wavelength using photon energy

Very often in physics problems arise that pose the question of what the wavelength is for a photon having energy E. To solve this kind of problem, you should use the following formula: E=h*c/λ, where c is the speed of the photon, h is a constant Plank, which is equal to 6.626 * 10 -34 J * s.

From the above formula we obtain the photon wavelength: λ = h*c/E. For example, let the photon energy E = 2.88*10 -19 J, and the photon moves in vacuum, that is, c = 3*10 8 m/s. Then we get: λ = h*c/E = 6.626*10 -34 *3*10 8 /2.88*10 -19 = 6.90*10 -7 m = 690 nm. Thus, this photon has a wavelength that lies near the upper limit of the visible spectrum, and will be perceived by a person as a red beam of light.

The ranges smoothly transition into each other, there is no clear boundary between them. Therefore, the limiting values of wavelengths are sometimes very arbitrary.

1. Radio waves (L > 1 mm). Sources of radio waves are charge fluctuations in wires, antennas, and oscillatory circuits. Radio waves are also emitted during thunderstorms.

Ultra long waves(L > 10 km). They spread well in water, so they are used for communication with submarines.

Long waves(1 km< Л < 10 км). Используются в радиосвязи, радиовещании, радионавигации.

Medium waves(100 m< Л < 1 км). Радиовещание. Радиосвязь на расстоянии не более 1500 км.

Short waves(10 m< Л < 100 м). Радиовещание. Хорошо отражаются от ионо-сферы; в результате многократных отражений от ионосферы и от поверхности Земли могут распространяться вокруг globe. Therefore, on short waves you can catch radio stations from other countries.

Meter waves(1m< Л < 10 м). Местное радивещание в УКВ-диапазоне. Напри-мер, длина волны радиостанции «Эхо Москвы» составляет 4 м. Используются также в телевидении (федеральные каналы); так, длина волны телеканала «Россия 1» равна примерно 5 м.

Decimeter waves(10 cm< Л < 1м). Телевидение (дециметровые каналы). На-пример, длина волны телеканала «Animal Planet» приблизительно равна 42 см. Это также диапазон mobile communications; Thus, the GSM 1800 standard uses radio waves with a frequency of approximately 1800 MHz, i.e., with a wavelength of about 17 cm. There is another application of decimeter waves that is well known to you - microwave ovens. The standard frequency of a microwave oven is 2450 MHz (this is the frequency at which resonant absorption of electromagnetic radiation by water molecules occurs). It corresponds to a wavelength of approximately 12 cm. Finally, in technology wireless communication Wi-Fi and Bluetooth use the same wavelength - 12 cm (frequency 2400 MHz).

Microwave(1 cm< Л < 10 см). Это — область радиолокации и спутни-ковых телеканалов. Например, канал НТВ+ ведёт своё телевещание на длинах волн около 2 см.

Infrared radiation(780 nm< Л < 1 мм). Испускается молекулами и атомами нагретых тел. Инфракрасное излучение называется ещё тепловым — когда оно попадает на наше тело, мы чувствуем тепло. Человеческим глазом инфракрасное излучение не воспринимается Мощнейшим источником инфракрасного излучения служит Солнце. Лампы накаливания излучают greatest number energy (up to 80%) in just the infrared region of the spectrum. Infrared radiation has a wide range of applications: infrared heaters, remote controls remote control, night vision devices, paint drying and much more. As body temperature increases, the wavelength of infrared radiation decreases, shifting towards visible light. Having inserted a nail into the flame of a burner, we can observe this with our own eyes: at some point the nail “glows red hot”, starting to emit radiation in the visible range.

Visible light(380 nm< Л < 780 нм). Излучение в этом промежутке длин волн воспринимается человеческим глазом. Диапазон видимого света можно разделить на семь интервалов — так называемые спек-тральные цвета.

Red: 625 nm - 780 nm;

Orange: 590 nm - 625 nm;

Yellow: 565 nm - 590 nm;

Green: 500 nm - 565 nm;

Blue: 485 nm - 500 nm;

Blue: 440 nm - 485 nm;

Violet: 380 nm - 440 nm.

The eye has maximum sensitivity to light in the green part of the spectrum.

Ultraviolet radiation(10 nm< Л < 380 нм). Главным источником ультрафиолетового излучения является Солнце. Именно ультрафи-олетовое излучение приводит к появлению загара. Человеческим глазом оно уже не вос-принимается. В небольших дозах ультрафиолетовое излучение полезно для человека: оно повышает иммунитет, улучшает обмен веществ, имеет целый ряд других целебных воздействий и потому применяется в физиотерапии. Ультрафиолетовое излучение обладает бактерицидными свойствами. Например, в боль-ницах для дезинфекции операционных в них включаются специальные ультрафиолетовые лампы. Очень опасным является воздействие УФ излучения на сетчатку глаза — при больших дозах ультрафиолета можно получить ожог сетчатки. Поэтому для защиты глаз (высоко в горах, например) нужно надевать очки, стёкла которых поглощают ультрафиолет.

X-ray radiation(5 pm< Л < 10 нм). Возникает в результате торможения быстрых электронов у анода и стенок газоразряд-ных трубок (тормозное излучение), а также при некоторых переходах электронов внутри атомов с одного уровня на другой (характеристическое излучение).

X-ray radiation easily penetrates the soft tissues of the human body, but is absorbed by calcium, which is part of the bones. This makes it possible to take the X-ray images you are familiar with. At airports, you have probably seen the operation of X-ray television introscopes - these devices illuminate hand luggage and baggage with X-rays. The wavelength of X-ray radiation is comparable to the sizes of atoms and interatomic distances in crystals; Therefore, crystals are natural diffraction gratings for X-rays. By observing diffraction patterns obtained when X-rays pass through various crystals, one can study the order of atoms in crystal lattices and complex molecules. Thus, it was with the help of X-ray, genost, manual, and urn analysis that the structure of a number of complex organic molecules was determined - for example, DNA and hemoglobin. In large doses, X-ray radiation is dangerous to humans - it can cause cancer and radiation sickness.

Gamma radiation (L< 5 пм). This is the highest energy radiation. Its penetrating power is much higher than that of X-rays. Gamma radiation occurs during transitions of atomic nuclei from one state to another, as well as during some nuclear reactions. Some insects and birds can see in ultraviolet light. For example, bees use their ultraviolet vision to find nectar on flowers. Sources of gamma rays can be charged particles moving at speeds close to the speed of light - if the trajectories of such particles are curved magnetic field(so-called synchrotron radiation). In large doses, gamma radiation is very dangerous for humans: it causes radiation sickness and cancer. But in small doses it can suppress the growth of cancerous tumors and is therefore used in radiation therapy. The bactericidal effect of gamma radiation is used in agriculture (gamma sterilization of agricultural products before long-term storage), in the food industry (food preservation), and also in medicine (sterilization of materials).

Wavelength can also be determined:

as the distance, measured in the direction of wave propagation, between two points in space at which the phase of the oscillatory process differs by 2π;
as the path that the wave front travels in a time interval equal to the period of the oscillatory process;
How spatial period wave process.

Let's imagine waves arising in water from a uniformly oscillating float, and mentally stop time. Then the wavelength is the distance between two adjacent wave crests, measured in the radial direction. Wavelength is one of the main characteristics of a wave, along with frequency, amplitude, initial phase, direction of propagation and polarization. The Greek letter is used to denote wavelength λ (\displaystyle \lambda), the wavelength dimension is meter.

Typically, wavelength is used in relation to a harmonic or quasi-harmonic (e.g., damped or narrowband modulated) wave process in a homogeneous, quasi-homogeneous, or locally homogeneous medium. However, formally, the wavelength can be determined by analogy for a wave process with a non-harmonic, but periodic space-time dependence, containing a set of harmonics in the spectrum. Then the wavelength will coincide with the wavelength of the main (lowest frequency, fundamental) harmonic of the spectrum.

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Subtitles

In the last video, we discussed what will happen if you take, say, a rope, pull the left end - this, of course, could be the right end, but let it be the left - so, pull up, and then down and then back to the original position. We convey a certain disturbance to the rope. But I think you understand that this diagram can demonstrate many different types waves If a rope can rise, fall, and return to neutral 10 times in a second, then in 1/10 of a second it will do this once. This distance is covered in a period. It is equal to 1/20 of a second per cycle.

Wavelength - spatial period of the wave process

Wavelength in the medium

In an optically denser medium (the layer is highlighted in dark color), the electromagnetic wavelength is reduced. Blue line - distribution of instantaneous ( t= const) values of the wave field strength along the direction of propagation. The change in the amplitude of the field strength due to reflection from the interfaces and interference of the incident and reflected waves is not shown in the figure.