Speed of propagation of vibrations. Wavelength

Important physical parameter, necessary for solving many problems in acoustics and radio electronics. It can be calculated in several ways, depending on what parameters are specified. It is most convenient to do this if you know the frequency or period and speed of propagation.

Formulas

The basic formula that answers the question of how to find wavelength through frequency is presented below:

Here l is the wavelength in meters, v is the speed of its propagation in m/s, u is the linear frequency in hertz.

Since frequency is related to period in an inverse relationship, the previous expression can be written differently:

T is the oscillation period in seconds.

This parameter can be expressed in terms of cyclic frequency and phase speed:

l = 2 pi*v/w

In this expression, w is the cyclic frequency expressed in radians per second.

The frequency of the wave through the length, as can be seen from the previous expression, is found as follows:

Let's consider an electromagnetic wave that propagates in a substance with n. Then the frequency of the wave in terms of length is expressed by the following relation:

If it propagates in a vacuum, then n = 1, and the expression takes on the following form:

In the last formula, the wave frequency in terms of length is expressed using the constant c - the speed of light in vacuum, c = 300,000 km/s.

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.

Encyclopedic YouTube

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Amplitude, period, frequency and wavelength of periodic waves

Sound vibrations - Wavelength

5.7 Wavelength. Wave speed

Lesson 370. Wave phase speed. Shear wave speed in a string

Lesson 369. Mechanical waves. Mathematical description of a traveling wave

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. This disturbance might look something like this if I jerk the rope up and down once. The disturbance will be transmitted along the rope in approximately this way. Let's paint it black. Immediately after the first cycle - jerking up and down - the rope will look something like this. But if you wait a little, it will look something like this, considering that we pulled once. The impulse is transmitted further along the rope. In the last video, we defined this disturbance as being transmitted along a rope or in a given environment, although the environment is not a necessary condition. We called it a wave. And, in particular, this wave is an impulse. This is an impulse wave because there was essentially only one disturbance in the rope. But if we continue to periodically pull the rope up and down at regular intervals, it will look something like this. I will try to depict it as accurately as possible. It will look like this, and the vibrations, or disturbances, will be transmitted to the right. They will be transmitted to the right at a certain speed. And in this video I want to look at waves of this type. Imagine that I periodically jerk the left end of the rope up and down, up and down, creating periodic vibrations. We will call them periodic waves. This is a periodic wave. The movement is repeated again and again. Now I would like to discuss some properties of a periodic wave. Firstly, you can notice that when moving, the rope rises and falls a certain distance from its original position, here it is. How far are the highest and lowest points from the starting position? This is called the amplitude of the wave. This distance (I'll highlight it in purple) - this distance is called amplitude. Sailors sometimes talk about wave height. Height usually refers to the distance from the base of a wave to its crest. We are talking about amplitude, or the distance from the initial, equilibrium position to the maximum. Let's denote the maximum. This is the highest point. The highest point of a wave, or its top. And this is the sole. If you were sitting in a boat, you would be interested in the height of the wave, the entire distance from your boat to the highest point of the wave. Okay, let's not go off topic. That's what's interesting. Not all waves are created by me pulling the left end of the rope. But I think you understand that this diagram can demonstrate many different types waves And this is essentially a deviation from the average, or zero, position, amplitude. A question arises. It is clear how far the rope deviates from the average position, but how often does this happen? How long does it take for the rope to rise, fall and return? How long does each cycle last? A cycle is a movement up, down and back to the starting point. How long does each cycle last? Can you say how long each period is? We said that this is a periodic wave. A period is a repetition of a wave. The duration of one complete cycle is called a period. And the period is measured by time. Maybe I pull the rope every two seconds. It takes two seconds for it to rise, fall and return to the middle. The period is two seconds. And another related characteristic is how many cycles per second do I do? In other words, how many seconds are there in each cycle? Let's write this down. How many cycles per second do I make? That is, how many seconds are there in each cycle? How many seconds are there in each cycle? So the period, for example, could be 5 seconds per cycle. Or maybe 2 seconds. But how many cycles occur per second? Let's ask the opposite question. It takes a few seconds to go up, go down and return to the middle. How many cycles of descent, ascent and return fit into each second? How many cycles occur per second? This is the opposite of period. A period is usually denoted by a capital T. It is a frequency. Let's write it down. Frequency. It is usually denoted with a lowercase f. It characterizes the number of vibrations per second. So if a full cycle takes 5 seconds, that means we'll have 1/5 of the cycle happening per second. I just reversed this ratio. This is quite logical. Because period and frequency are opposite characteristics to each other. How many seconds is this in a cycle? How long does it take to ascend, descend and return? And this is how many descents, ascents and returns in one second? So they are the inverse of each other. We can say that frequency is equal to the ratio of one to the period. Or the period is equal to the ratio of unity to frequency. So, if the rope vibrates at a frequency of, say, 10 cycles per second... And by the way, the unit for frequency is the hertz, so let's write it as 10 hertz. You've probably already heard something similar. 10 Hz simply means 10 cycles per second. If the frequency is 10 cycles per second, then the period is equal to its ratio to unity. We divide 1 by 10 seconds, which is quite logical. 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. We are also interested in how quickly the wave propagates to the right in this case? If I pull on the left end of the rope, how fast does it move to the right? This is speed. To find out, we need to calculate how far the wave travels in one cycle. Or in one period. After I pull once, how far does the wave go? What is the distance from this point at the neutral level to this point? This is called wavelength. Wavelength. It can be defined in many ways. We can say that the wavelength is the distance that the initial pulse travels in one cycle. Or that it is the distance from one highest point to another. This is also a wavelength. Or the distance from one sole to another sole. This is also a wavelength. But in general, wavelength is the distance between two identical points on a wave. From this point to this. This is also a wavelength. This is the distance between the beginning of one complete cycle and its completion at exactly the same point. At the same time, when I talk about identical points, this point does not count. Because at a given point, although it is in the same position, the wave descends. And we need a point where the wave is in the same phase. Look, there is an upward movement here. So we need a rise phase. This distance is not the wavelength. To walk the same length, you need to walk the same phase. It is necessary that the movement be in the same direction. This is also the wavelength. So, if we know how far the wave travels in one period... Let's write: the wavelength is equal to the distance the wave travels in one period. The wavelength is equal to the distance that the wave travels in one period. Or, you could say, in one cycle. It's the same thing. Because a period is the time during which a wave completes one cycle. One ascent, descent and return to the zero point. So, if we know the distance and the time it takes the wave to travel, that is, the period, how can we calculate the speed? Speed is equal to the ratio of distance to time of movement. Speed is the ratio of distance to time of movement. And for a wave, the speed could be designated as a vector, but this, I think, is already clear. So, speed reflects how far the wave travels in a period? And the distance itself is the wavelength. The wave impulse will travel exactly that long. This will be the wavelength. So we go this distance, and how long does it take? This distance is covered in a period. That is, it is the wavelength divided by the period. Wavelength divided by period. But we already know that the ratio of unit to period is the same as frequency. So we can write this as wavelength... And by the way, an important point. Wavelength is usually denoted by the Greek letter lambda. So, we can say that the speed is equal to the wavelength divided by the period. Which is equal to the wavelength times one divided by the period. We just learned that the ratio of unit to period is the same as frequency. So speed is equal to the product of wavelength and frequency. This way, you will solve all the main problems that you may encounter in the topic of waves. For example, if we are given that the speed is 100 meters per second and directed to the right... Let's make this assumption. Speed is a vector, and you need to indicate its direction. Let the frequency be, say, 20 cycles per second, this is the same as 20 Hz. So, again, the frequency will be 20 cycles per second or 20 Hz. Imagine looking out a small window and seeing only this part of the wave, only this part of my rope. If you know about 20 Hz, then you know that in 1 second you will see 20 descents and ascents. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13... In 1 second you will see the wave rise and fall 20 times. This is what a frequency of 20 Hz, or 20 cycles per second, means. So, we are given speed, we are given frequency. What will be the wavelength? In this case, it will be equal... Let's return to speed: speed is equal to the product of wavelength and frequency, right? Let's divide both sides by 20. By the way, let's check the units of measurement: these are meters per second. It turns out: λ multiplied by 20 cycles per second. λ multiplied by 20 cycles per second. If we divide both sides by 20 cycles per second, we get 100 meters per second times 1/20 of a second per cycle. Here remains 5. Here 1. We get 5, the seconds are reduced. And we get 5 meters per cycle. 5 meters per cycle in this case will be the wavelength. 5 meters per cycle. Amazing. One could say it is 5 meters per cycle, but the wavelength assumes that it means distance traveled per cycle. In this case, if the wave travels to the right at a speed of 100 meters per second, and this is the frequency (we see that the wave oscillates up and down 20 times per second), then this distance must be 5 meters. The period can be calculated in the same way. The period is equal to the ratio of unity to frequency. It is equal to 1/20 of a second per cycle. 1/20 second per cycle. I don't want you to memorize the formulas, I want you to understand their logic. I hope this video helped you. Using formulas you can answer almost any question, if there are 2 variables and you need to calculate the third. I hope you find this helpful. Subtitles by the Amara.org community

Wavelength - spatial period of the wave process

Wavelength in the medium

In an optically denser medium (the layer is highlighted dark color) length electromagnetic wave is shrinking. 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.

During the lesson you will be able to independently study the topic “Wavelength. Wave propagation speed." In this lesson you will learn about the special characteristics of 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

Talk about wavelength in longitudinal wave more difficult, 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.

References

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

Each wave travels at a certain speed. Under wave speed understand the speed of propagation of the disturbance. For example, a blow to the end of a steel rod causes local compression in it, which then propagates along the rod at a speed of about 5 km/s.

The speed of the wave is determined by the properties of the medium in which the wave propagates. When a wave passes from one medium to another, its speed changes.

In addition to speed, important characteristic wave is the wavelength. Wavelength is the distance over which a wave propagates in a time equal to the period of oscillation in it.

Since the speed of a wave is a constant value (for a given medium), the distance traveled by the wave is equal to the product of the speed and the time of its propagation. Thus, to find the wavelength, you need to multiply the speed of the wave by the period of oscillation in it:

v - wave speed; T is the period of oscillation in the wave; λ ( greek letter"lambda") - wavelength.

By choosing the direction of wave propagation as the direction of the x axis and denoting by y the coordinate of the particles oscillating in the wave, we can construct wave chart. A graph of a sine wave (at a fixed time t) is shown in Figure 45. The distance between adjacent crests (or troughs) in this graph coincides with the wavelength λ.

Formula (22.1) expresses the relationship between wavelength and its speed and period. Considering that the period of oscillation in a wave is inversely proportional to the frequency, i.e. T = 1/ν, we can obtain a formula expressing the relationship between the wavelength and its speed and frequency:

The resulting formula shows that the speed of the wave is equal to the product of the wavelength and the frequency of oscillations in it.

The frequency of oscillations in the wave coincides with the frequency of oscillations of the source (since the oscillations of the particles of the medium are forced) and does not depend on the properties of the medium in which the wave propagates. When a wave passes from one medium to another, its frequency does not change, only the speed and wavelength change.

1. What is meant by wave speed? 2. What is wavelength? 3. How is wavelength related to the speed and period of oscillation in the wave? 4. How is wavelength related to the speed and frequency of oscillations in the wave? 5. Which of the following wave characteristics change when the wave passes from one medium to another: a) frequency; b) period; c) speed; d) wavelength?

Experimental task. Pour water into the bath and, by rhythmically touching the water with your finger (or ruler), create waves on its surface. Using different oscillation frequencies (for example, touching the water once and twice per second), pay attention to the distance between adjacent wave crests. At what oscillation frequency is the wavelength longer?