How to Calculate Wavelength

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Wavelength is the distance between two adjacent maximum points in a wave’s oscillation and is often referred to when referring to electromagnetic waves. ^{[1] X Research Source} Depending on the available information, the wavelength can be determined in one way or another. If we know the speed and frequency, we can apply the basic wavelength formula. If we want to calculate the wavelength of light when the characteristic energy of a photon is known, we can apply the energy formula. The correct application of the formula will make it easy to calculate the wavelength.

Table of Contents

Calculate wavelength based on speed and frequency

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Calculate the wavelength according to the formula. To calculate the wavelength, we just need to divide the propagation speed by the frequency. The formula is as follows:

Wavelength

=

Spreading speed

Frequency

{displaystyle {text{Wavelength}}={frac {text{Planning rate}}{text{Frequency}}}}

{{text{Wavelength}}}={frac {{text{Planning rate}}}{{text{Frequency}}}}

. ^{[2] X Research Source}

Wavelength is usually represented by the letter lamda $λ$ ${displaystyle lambda }$ $lambda$ in Greek.
Speed is usually denoted by . $v$ ${displaystyle v}$ $v$ .
Frequency is usually denoted as . $f$ ${displaystyle f}$ $f$ .
$λ$ $=$ $v$ $f$ ${displaystyle lambda ={frac {v}{f}}}$ $lambda ={frac {v}{f}}$

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Use the correct units. Speed can be recorded as standard or British units of measurement such as miles per hour (mph), kilometers per hour (kph), meters per second (m/s) etc. However, wavelengths are usually are written in standard units of measurement such as nanometers, meters, millimeters, etc. Frequency is generally in Hertz (Hz). ^{[3] X Research Sources}

Always use the same unit form when applying formulas. Often in physics, mathematical operations will be calculated using standard units of measurement.
If the frequency is given in kilohertz (kHz) or the propagation speed of the wave is given in km/s, we need to convert these values to Hertz and m/s by multiplying the given value by 1000 ( 10 kHz = 10 000 Hz).

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Substitute the known values into the formula. To calculate the wavelength according to the formula, once we have determined the propagation speed and frequency of the wave, we just need to plug these values into the formula and do the speed division by the frequency. ^{[4] X Research Sources}

Example: Find the wavelength of a wave traveling with a speed of 20 m/s and with a frequency of 5 Hz.
- $Wavelength$ $=$ $Spreading speed$ $Frequency$ ${displaystyle {text{Wavelength}}={frac {text{Planning rate}}{text{Frequency}}}}$ ${{text{Wavelength}}}={frac {{text{Planning rate}}}{{text{Frequency}}}}$
  $λ$ $=$ $v$ $f$ ${displaystyle lambda ={frac {v}{f}}}$ $lambda ={frac {v}{f}}$
  $λ$ $=$ $20$ $m$ $/$ $S$ $5$ $H$ $z$ ${displaystyle lambda ={frac {20m/s}{5Hz}}}$ $lambda ={frac {20m/s}{5Hz}}$
  $λ$ $=$ $4$ $m$ ${displaystyle lambda =4m}$ $lambda =4m$

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Use this formula to calculate propagation speed or frequency. By transforming the above formula, we can calculate the propagation speed or frequency if the wavelength is known in advance. To calculate the speed when the frequency and wavelength are known, we use the formula

v

=

λ

f

{displaystyle v={frac {lambda }{f}}}

v={frac {lambda }{f}}

. To calculate the frequency when the propagation speed and wavelength are known, we use the formula

f

=

v

λ

{displaystyle f={frac {v}{lambda }}}

f={frac {v}{lambda }}

. ^{[5] X Research Sources}

Example: Find the propagation speed of a wave with a wavelength of 450 nm with a frequency of 45 Hz. $v$ $=$ $λ$ $f$ $=$ $450$ $n$ $square meter$ $45$ $H$ $z$ $=$ $ten$ $n$ $m$ $/$ $S$ ${displaystyle v={frac {lambda }{f}}={frac {450nm}{45Hz}}=10nm/s}$ $v={frac {lambda }{f}}={frac {450nm}{45Hz}}=10nm/s$ .
Example: Find the frequency of a wave with wavelength 2.5 m and propagating with speed 50 m/s. $f$ $=$ $v$ $λ$ $=$ $50$ $square meter$ $/$ $S$ $2.5$ $square meter$ $=$ $20$ $H$ $z$ ${displaystyle f={frac {v}{lambda }}={frac {50m/s}{2.5m}}=20Hz}$ $f={frac {v}{lambda }}={frac {50m/s}{2.5m}}=20Hz$ .

Calculate the wavelength knowing the photon energy

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Calculate the wavelength using the energy formula. The wavelength-dependent energy formula is

E

=

H

c

λ

{displaystyle E={frac {hc}{lambda }}}

E={frac {hc}{lambda }}

with

E

{displaystyle E}

E

is the energy of the system in Joule (J),

H

{displaystyle h}

H

is the Planck constant: 6.626 x 10 ^-34 Joule seconds (J s),

c

{displaystyle c}

c

is the speed of light traveling in a vacuum: 3.0 x 10 ⁸ meters per second (m/s), and

λ

{displaystyle lambda }

lambda

is the wavelength in meters (m). ^{[6] X Research Source}

In this type of problem, the energy of a photon is usually given.

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Transform the formula to calculate wavelength. We can reverse the positions of the variables in the above formula according to arithmetic rules to calculate the wavelength. If we multiply both sides of the formula by the wavelength, and then divide both sides by the energy, we get

λ

=

H

c

E

{displaystyle lambda ={frac {hc}{E}}}

lambda ={frac {hc}{E}}

. If you know the energy of the photon, you can calculate the wavelength using this formula. ^{[7] X Research Sources}

The above formula can also be used to calculate the maximum wavelength of light required to ionize the metal. We just need to substitute the ionization energy of the metal and calculate according to the formula to get the wavelength. ^{[8] X Research Sources}

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Substitute the known variables into the formula. Once we have transformed the formula, we can solve the problem of calculating the wavelength by substituting the value of energy into the formula. Since the other two variables are constants, their values are constant. To solve this math problem, we multiply the two constants together, then divide the result by the energy. ^{[9] X Research Source}

Example: Find the wavelength of a photon with an energy of 2.88 x 10 ^-19 J.
- $λ$ $=$ $H$ $c$ $E$ ${displaystyle lambda ={frac {hc}{E}}}$ $lambda ={frac {hc}{E}}$
  = $($ $6$ $,$ $626$ $*$ $ten$ $-$ $34$ $)$ $($ $3$ $,$ $0$ $*$ $ten$ $8$ $)$ $($ $2$ $,$ $88$ $*$ $ten$ $-$ $19$ $)$ ${displaystyle {frac {(6,626*10^{-34})(3,0*10^{8})}{(2.88*10^{-19})}}}$ ${frac {(6,626*10^{{-34}})(3,0*10^{{8}})}{(2.88*10^{{-19}})}}$
  $=$ $($ $19$ $,$ $878$ $*$ $ten$ $-$ $26$ $)$ $($ $2$ $,$ $88$ $*$ $ten$ $-$ $19$ $)$ ${displaystyle ={frac {(19,878*10^{-26})}{(2.88*10^{-19})}}}$ $={frac {(19,878*10^{{-26}})}{(2.88*10^{{-19}})}}$
  $=$ $6$ $,$ $90$ $*$ $ten$ $-$ $7$ $meters$ ${displaystyle =6,90*10^{-7}{text{meters}}}$ $=6,90*10^{{-7}}{{text{meters}}}$ .
- Convert back to nanometers by multiplying the above result by 10 ⁹ . The wavelength in this example in nanometers is 690 nm.

Proofreading

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Check the results again after calculating by multiplying the wavelength by the frequency. When multiplying the wavelength by the frequency gives the result equal to the known propagation speed, the calculated wavelength result is correct. If the multiplication of wavelength by frequency gives a different result, review the calculation steps. When using the calculator, double check that you have entered the values correctly.

Example: Calculate the wavelength of a 70 Hz sound wave traveling at 343 meters per second.
- After following the instructions above, you have a result of 4.9 meters.
- Double-check the calculation by calculating 4.9 meters x 70 Hz = 343 meters per second. This is also the speed of wave propagation that the problem gives, so the answer of 4.9 meters is correct.

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Use exponentiation to limit rounding errors. The problem of calculating wavelength often involves dealing with large numbers, especially the speed of light. It is such large numbers that can cause your results to be skewed when rounding. You can avoid rounding errors by using exponentiation. ^{[10] X Research Source}

Example: Light travels through water at a speed of 225 000 000 meters per second. Given that the frequency of the light wave is 4 x 10 ¹⁴ Hz, what is the wavelength?
- The propagation rate is written to a power of 2.25 x 10 ⁸ . The frequency has been correctly written to the exponentiation.
- $Wavelength$ $=$ $Spreading speed$ $Frequency$ ${displaystyle {text{Wavelength}}={frac {text{Planning rate}}{text{Frequency}}}}$ ${{text{Wavelength}}}={frac {{text{Planning rate}}}{{text{Frequency}}}}$
  $=$ $2$ $,$ $25$ $*$ $ten$ $8$ $4$ $*$ $ten$ $14$ $=$ $2$ $,$ $25$ $4$ $*$ $ten$ $6$ ${displaystyle ={frac {2,25*10^{8}}{4*10^{14}}}={frac {2,25}{4*10^{6}}}}$ $={frac {2,25*10^{8}}{4*10^{{14}}}}={frac {2,25}{4*10^{6}}}$
  $=$ $0$ $,$ $563$ $*$ $ten$ $-$ $6$ $meters$ ${displaystyle =0.563*10^{-6}{text{meters}}}$ $=0.563*10^{{-6}}{{text{meters}}}$
  $=$ $5$ $,$ $63$ $*$ $ten$ $-$ $7$ $meters$ ${displaystyle =5.63*10^{-7}{text{meters}}}$ $=5.63*10^{{-7}}{{text{meters}}}$ .

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When the wave travels to another medium, the frequency of the wave remains the same. Problems in which waves propagate from one medium to another are quite common. A common mistake in problems of this kind is finding new ways to calculate frequencies. In fact, the frequency of the wave does not change when it crosses the boundary between the two media, only the propagation speed and wavelength change. ^{[11] X Research Source}

Example: A light wave of frequency f , propagation speed v , and wavelength λ propagates from air into another medium with a refractive index of 1.5. How do the three values above change?
- The propagation speed in the new medium is equal to $v$ $first$ $,$ $5$ ${displaystyle {frac {v}{1,5}}}$ ${frac {v}{1.5}}$ .
- The frequency does not change and remains the same value as f .
- The wavelength in the new medium is $New speed$ $Frequency$ $=$ $v$ $first$ $,$ $5$ $f$ $=$ $v$ $first$ $,$ $5$ $f$ ${displaystyle {frac {text{New Speed}}{text{Frequency}}}={frac {frac {v}{1.5}}{f}}={frac {v}{1.5f}} }$ ${frac {{text{New Speed}}}{{text{Frequency}}}}={frac {{frac {v}{1,5}}}{f}}={frac {v}{1 ,5f}}$ .

Steps

Calculate wavelength based on speed and frequency

Calculate the wavelength knowing the photon energy

Proofreading

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