Formula for calculating wave speed. How to calculate wavelength

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

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

1. 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.
2. 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.

1. Internet portal "eduspb" ()
2. Internet portal "eduspb" ()
3. Internet portal “class-fizika.narod.ru” ()

Homework

Oscillations T point with a constant will travel a certain distance. This distance can be wavelength. Wavelength in letter? and equal? = vT, where v is its phase velocity. The phase speed of a wave can also be expressed in terms of its wave number k: v = w/k. Wavelength is expressed in terms of wave number as? = 2*pi/k.

The period of a wave can be written in terms of its frequency as T = 1/f. Then? = v/f. The wavelength can also be expressed in terms of the circular frequency. By definition, the circular frequency is f = w/(2*pi). From here, ? = 2*pi*v/w.

According to wave-particle duality, any microparticle is also associated with a wave called a de Broglie wave. De Broglie waves are inherent in electrons, protons, neutrons and other microparticles. This wave has a certain length. It has been established that the de Broglie wavelength is inversely proportional to the momentum of the particle and is equal to? = h/p, where h is Planck's constant. The wave frequency is directly proportional to the particle energy: ? = E/h. The phase speed of the de Broglie wave will be equal to E/p

In dispersive media, the concept of group velocity is introduced. For one-dimensional waves it is equal to Vgr = dw/dk, where w is the circular frequency and k is the wave number.

Video on the topic

Waves are different. Sometimes you need to measure the amplitude and wavelength of the surf on the coast, and sometimes you need to measure the frequency and voltage of an electrical signal wave. For each case there are different ways to obtain wave parameters.

You will need

• foot rod, stopwatch, electronic pressure gauge, standard signal generator, oscilloscope, frequency meter.

Instructions

To determine the height of the wave near the shore in shallow water, insert a foot rod into the bottom. Notice the divisions on the foot rod that coincide with the upper and lower (crest and ) levels of the wave passing by it. Subtract the smaller value from the larger value to get the wave height. For a more accurate measurement, use an electronic pressure gauge. Place its sensor at the place where you want to measure the wave height. Observe the readings of the device as a crest and wave pass over the sensor. Subtract the smaller value from the larger value and get the pressure drop corresponding to the wave height.

For wave movement, use a stopwatch to measure the time between the passage of two adjacent wave crests over the sensor or footpole. Using two foot rods, determine. To do this, position them in such a way that the tops of two adjacent waves pass by the foot poles at the same time. Then measure the distance between the foot rods (in meters). It will be equal to the wavelength. Divide 60 by the time measured with a stopwatch and multiply by the wavelength. Get the speed of the wave (in meters per minute). Example: the wave travel time is 2 seconds and the length is 3.5 meters. In this case, the speed of the wave will be (60/2) × 3.5 = 105 meters per minute.

To convert to meters per second, divide this result by 60 (105/60 = 1.75 meters per second), and to convert to kilometers per hour, multiply by 60 and then divide by a thousand (105 × 60 = 6300 meters per hour, 6300 /1000=6.3 kilometers per hour).

To determine the parameters of the electrical signal, use special instruments. Connect the standard signal generator to the oscilloscope. Set the output signal amplitude in the generator to 1 Volt. Turn on the oscilloscope and adjust its sensitivity so that the upper signal level coincides with the first wide vertical bar on the screen grid. Disconnect the generator and connect the source of the signal being studied. Calculate the amplitude of the input signal using vertical wide stripes.

Connect the source of the signal under study to the input of the frequency meter. Take frequency readings from the frequency meter indicator. To obtain the wavelength, divide the speed of light by the frequency of the signal being studied. Example: the measured frequency is 100 MHz, the wavelength is 299792458/100000000=2.99 meters.

A mechanical wave is the process of propagation of vibrations in an elastic medium, accompanied by the transfer of energy of a vibrating body from one point in the elastic medium to another. Important wave characteristics: length and phase speed.

You will need

• - calculator.

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

The speed of a 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.

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:

Where v— wave speed, T- period of oscillations in the wave, λ (Greek letter lambda) - wavelength.

The formula 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 v, i.e. T= 1/ v, we can obtain a formula expressing the relationship between wavelength and its speed and frequency:

,

where

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

Wavelength is the spatial period of the wave. In the wave graph (fig. above), the wavelength is defined as the distance between the two nearest harmonic points traveling wave, being in the same oscillation phase. These are like instant photographs of waves in an oscillating elastic medium at moments in time t And t + Δt. Axis X coincides with the direction of wave propagation, displacements are plotted on the ordinate axis s vibrating particles of the medium.

The frequency of oscillations in the wave coincides with the frequency of oscillations of the source, since the oscillations of particles in the medium are forced and do 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.

Wavelength is the distance between two adjacent points that oscillate in the same phase; Typically, the concept of "wavelength" is associated with the electromagnetic spectrum. The method for calculating the wavelength depends on this information. Use the basic formula if the speed and frequency of the wave are known. If you need to calculate the wavelength of light from a known photon energy, use the appropriate formula.

Steps

Part 1

Use the formula to calculate the wavelength. To find the wavelength, divide the speed of the wave by the frequency. Formula:

• In this formula λ (\displaystyle \lambda)(lambda, letter of the Greek alphabet) – wavelength.
• v (\displaystyle v)– wave speed.
• f (\displaystyle f)– wave frequency.

1. Use appropriate units of measurement. Speed ​​is measured in metric units, such as kilometers per hour (km/h), meters per second (m/s), and so on (in some countries, speed is measured in the imperial system, such as miles per hour). Wavelength is measured in nanometers, meters, millimeters and so on. Frequency is usually measured in hertz (Hz).

   • The units of measurement of the final result must correspond to the units of measurement of the source data.
   • If the frequency is given in kilohertz (kHz), or the wave speed is in kilometers per second (km/s), convert the given values ​​to hertz (10 kHz = 10000 Hz) and to meters per second (m/s).

2. Plug the known values ​​into the formula and find the wavelength. Substitute the values ​​of wave speed and frequency into the given formula. Dividing speed by frequency gives you wavelength.

   • For example. Find the length of a wave traveling at a speed of 20 m/s at an oscillation frequency of 5 Hz.
     • Wavelength = Wave Speed ​​/ Wave Frequency
       λ = v f (\displaystyle \lambda =(\frac (v)(f)))
       λ = 20 5 (\displaystyle \lambda =(\frac (20)(5)))
       λ = 4 (\displaystyle \lambda =4) m.

3. Use the formula provided to calculate the speed or frequency. The formula can be rewritten in another form and calculate the speed or frequency if the wavelength is given. To find the speed from a known frequency and wavelength, use the formula: v = λ f (\displaystyle v=(\frac (\lambda )(f))). To find the frequency from a known speed and wavelength, use the formula: f = v λ (\displaystyle f=(\frac (v)(\lambda ))).

   • For example. Find the speed of wave propagation at an oscillation frequency of 45 Hz if the wavelength is 450 nm. v = λ f = 450 45 = 10 (\displaystyle v=(\frac (\lambda )(f))=(\frac (450)(45))=10) nm/s.
   • For example. Find the oscillation frequency of a wave whose length is 2.5 m and whose propagation speed is 50 m/s. f = v λ = 50 2 , 5 = 20 (\displaystyle f=(\frac (v)(\lambda ))=(\frac (50)(2.5))=20) Hz

   Part 2

   Calculating wavelength from known photon energy

   1. Calculate the wavelength using the formula for calculating photon energy. Formula for calculating photon energy: E = h c λ (\displaystyle E=(\frac (hc)(\lambda ))), Where E (\displaystyle E)– photon energy, measured in joules (J), h (\displaystyle h)– Planck’s constant equal to 6.626 x 10 -34 J∙s, c (\displaystyle c)– speed of light in vacuum, equal to 3 x 10 8 m/s, λ (\displaystyle \lambda)– wavelength, measured in meters.

      • In the problem, the photon energy will be given.

   2. Rewrite the given formula to find the wavelength. To do this, perform a series of mathematical operations. Multiply both sides of the formula by the wavelength, and then divide both sides by the energy; you will get the formula: . If the photon energy is known, the wavelength of the light can be calculated.

   3. Substitute the known values ​​into the resulting formula and calculate the wavelength. Substitute only the energy value into the formula, because the two constants are constant quantities, that is, they do not change. To find the wavelength, multiply the constants and then divide the result by the energy.

      • For example. Find the wavelength of light if the photon energy is 2.88 x 10 -19 J.
        • λ = h c E (\displaystyle \lambda =(\frac (hc)(E)))
          = (6 , 626 ∗ 10 − 34) (3 ∗ 10 8) (2 , 88 ∗ 10 − 19) (\displaystyle (\frac ((6,626*10^(-34))(3*10^(8)) )((2.88*10^(-19)))))
          = (19 , 878 ∗ 10 − 26) (2 , 88 ∗ 10 − 19) (\displaystyle =(\frac ((19.878*10^(-26)))((2.88*10^(-19) ))))
          = 6.90 ∗ 10 − 7 (\displaystyle =6.90*10^(-7)) m.
        • Convert the resulting value to nanometers by multiplying it by 10 -9. The wavelength is 690 nm.

What do you need to know and be able to do?

1. Determination of wavelength.
Wavelength is the distance between nearby points oscillating in the same phases.

THIS IS INTERESTING

Seismic waves.

Seismic waves are waves propagating in the Earth from the sources of earthquakes or some powerful explosions. Since the Earth is mostly solid, two types of waves can simultaneously arise in it - longitudinal and transverse. The speed of these waves is different: longitudinal ones travel faster than transverse ones. For example, at a depth of 500 km, the speed of transverse seismic waves is 5 km/s, and the speed of longitudinal waves is 10 km/s.

Registration and recording of vibrations of the earth's surface caused by seismic waves is carried out using instruments - seismographs. Propagating from the source of an earthquake, longitudinal waves arrive first at the seismic station, and after some time - transverse waves. Knowing the speed of propagation of seismic waves in the earth's crust and the delay time of the transverse wave, it is possible to determine the distance to the center of the earthquake. To find out more precisely where it is located, they use data from several seismic stations.

Hundreds of thousands of earthquakes are recorded around the globe every year. The vast majority of them are weak, but some are observed from time to time. which violate the integrity of the soil, destroy buildings and lead to casualties.

The intensity of earthquakes is assessed on a 12-point scale.

1948 - Ashgabat - earthquake 9-12 points
1966 - Tashkent - 8 points
1988 - Spitak - several tens of thousands of people died
1976 - China - hundreds of thousands of victims

It is possible to counteract the destructive consequences of earthquakes only by constructing earthquake-resistant buildings. But in which areas of the Earth will the next earthquake occur?

Predicting earthquakes is a daunting task. Many research institutes in many countries around the world are engaged in solving this problem. The study of seismic waves inside our Earth allows us to study the deep structure of the planet. In addition, seismic exploration helps to detect areas favorable for the accumulation of oil and gas. Seismic research is carried out not only on Earth, but also on other celestial bodies.

In 1969, American astronauts placed seismic stations on the Moon. Every year they recorded from 600 to 3000 weak moonquakes. In 1976, with the help of the Viking spacecraft (USA), a seismograph was installed on Mars.

DO IT YOURSELF

Waves on paper.

You can perform many experiments using a sounding tube.
If, for example, you put a sheet of thick light paper on a soft substrate lying on a table, sprinkle a layer of potassium permanganate crystals on top, place a glass tube vertically in the middle of the sheet and excite vibrations in it by friction, then when sound appears, the potassium permanganate crystals will begin to move and form beautiful lines . The tube should only lightly touch the surface of the sheet. The pattern that appears on the sheet will depend on the length of the tube.

The tube excites vibrations in the paper sheet. A standing wave is formed in a sheet of paper, which is the result of the interference of two traveling waves. A circular wave arises from the end of the oscillating tube, which is reflected from the edge of the paper without changing phase. These waves are coherent and interfere, distributing potassium permanganate crystals on paper into bizarre patterns.

ABOUT THE SHOCK WAVE

In his lecture "On Ship Waves" Lord Kelvin said:
"...one discovery was actually made by a horse that daily pulled a boat along a rope between Glasgow
and Ardrossan. One day the horse rushed, and the driver, being an observant person, noticed that when the horse reached a certain speed, it became clearly easier to pull the boat
and there was no wave trace left behind her.”

The explanation for this phenomenon is that the speed of the boat and the speed of the wave that the boat excites in the river coincided.
If the horse ran even faster (the speed of the boat would become greater than the speed of the wave),
then a shock wave would appear behind the boat.
The shock wave from a supersonic aircraft occurs in exactly the same way.