Wavelength is commonly represented by the Greek letter lambda, λ{\displaystyle \lambda }. Speed is commonly represented by the letter v{\displaystyle v}. Frequency is commonly represented by the letter f{\displaystyle f}. λ=vf{\displaystyle \lambda ={\frac {v}{f}}}
Always keep units consistent across the equation. Most calculations are done using strictly metric units. If the frequency is in kilohertz (kHz) or the wave speed is in km/s you will need to convert these numbers to Hertz and m/s by multiplying by 1000 (10 kHz = 10,000 Hz).
For example: Find the wavelength of a wave traveling at 20 m/s at a frequency of 5 Hz. Wavelength=WavespeedFrequency{\displaystyle Wavelength={\frac {Wavespeed}{Frequency}}}λ=vf{\displaystyle \lambda ={\frac {v}{f}}}λ=20m/s5Hz{\displaystyle \lambda ={\frac {20m/s}{5Hz}}}λ=4m{\displaystyle \lambda =4m}
For example: Find the speed of a wave with wavelength 450 nm and frequency 45 Hz. v=λ∗f=450nm∗45Hz=20,250nm/s<=20. 25um/s{\displaystyle v={\lambda }{f}={450nm}{45Hz}=20,250nm/s<=20. 25um/s}. For example: Find the frequency of a wave with wavelength 2. 5 m and speed 50 m/s. f=vλ=50m/s2. 5m=20Hz{\displaystyle f={\frac {v}{\lambda }}={\frac {50m/s}{2. 5m}}=20Hz}.
The energy of a photon is usually given to solve these types of problems.
For example: Find the wavelength of a photon with an energy of 2. 88 x 10-19 J. λ=hcE{\displaystyle \lambda ={\frac {hc}{E}}}= (6. 626∗10−34)(3. 0∗108)(2. 88∗10−19){\displaystyle {\frac {(6. 62610^{-34})(3. 010^{8})}{(2. 8810^{-19})}}}=(19. 878∗10−26)(2. 88∗10−19){\displaystyle ={\frac {(19. 87810^{-26})}{(2. 8810^{-19})}}}=6. 90∗10−7meters{\displaystyle =6. 9010^{-7}meters}. Convert to nanometers by multiplying by 109. The wavelength equals 690 nm.
For example: What is the wavelength of a 70 Hertz sound wave traveling at 343 meters per second? You follow the instructions above and get an answer of 4. 9 meters. Check your work by calculating 4. 9 meters x 70 Hz = 343 meters/second. This is the wave speed you started with, so your answer is correct.
For example: Light travels through water at about 225,000,000 meters per second. If the wave’s frequency is 4 x 1014 Hz, what is its wavelength? The wave speed in scientific notation = 2. 25 x 108. The frequency is already written in scientific notation. Wavelength=wavespeedfrequency{\displaystyle Wavelength={\frac {wavespeed}{frequency}}}=2. 25∗1084∗1014=2. 254∗106{\displaystyle ={\frac {2. 2510^{8}}{410^{14}}}={\frac {2. 25}{410^{6}}}}=0. 563∗10−6meters{\displaystyle =0. 56310^{-6}meters}=5. 63∗10−7meters{\displaystyle =5. 63*10^{-7}meters}.
For example: A light wave with frequency f, speed v, and wavelength λ crosses from air to a medium with refractive index 1. 5. How do these three values change? The new speed is equal to v1. 5{\displaystyle {\frac {v}{1. 5}}}. The frequency remains constant at f. The new wavelength is equal to newspeednewfrequency=v1. 5f=v1. 5f{\displaystyle {\frac {newspeed}{newfrequency}}={\frac {\frac {v}{1. 5}}{f}}={\frac {v}{1. 5f}}}.
For example, perhaps you used Joules when you should have used Hertz, so you ended up with the incorrect answer.
