Formula for Wave Speed (c = fλ) Explained Simply

📚 Understanding Wave Speed

The formula $c = f\lambda$ describes the relationship between wave speed ($c$), frequency ($f$), and wavelength ($\lambda$). It's a fundamental concept in physics, applicable to all types of waves, from sound waves to light waves. Understanding this formula allows us to calculate how fast a wave is traveling if we know its frequency and wavelength, or vice-versa.

📜 Historical Context

The development of the wave speed formula is intertwined with the history of understanding wave phenomena. Early scientists, like Christiaan Huygens, contributed to the wave theory of light. Later, physicists such as James Clerk Maxwell formalized the understanding of electromagnetic waves and their speed. The formula $c = f\lambda$ became a cornerstone in wave mechanics as our understanding of waves evolved.

🔑 Key Principles

• 🌊Wave Speed ($c$): The distance a wave travels per unit of time, typically measured in meters per second (m/s).
• 📡Frequency ($f$): The number of complete wave cycles that pass a point in a unit of time, usually measured in Hertz (Hz). 1 Hz means one cycle per second.
• 📏Wavelength ($\lambda$): The distance between two identical points on adjacent waves, such as crest to crest or trough to trough, commonly measured in meters (m).

➗ The Formula Explained

The formula $c = f\lambda$ states that the speed of a wave ($c$) is equal to the product of its frequency ($f$) and its wavelength ($\lambda$). Let's break it down:

• 💡 If you increase the frequency ($f$) while keeping the wavelength ($\lambda$) constant, the wave speed ($c$) increases.
• 📉 If you increase the wavelength ($\lambda$) while keeping the frequency ($f$) constant, the wave speed ($c$) also increases.
• 🧮 If you know any two of the variables, you can solve for the third. For example, $f = \frac{c}{\lambda}$ or $\lambda = \frac{c}{f}$.

🌍 Real-World Examples

Example 1: Sound Waves

A sound wave has a frequency of 440 Hz (that's an A note) and a wavelength of 0.773 meters. What is its speed?

$c = f\lambda = 440 \text{ Hz} \times 0.773 \text{ m} = 340.12 \text{ m/s}$

The speed of the sound wave is approximately 340.12 m/s. This is close to the speed of sound in air at room temperature!

Example 2: Light Waves

A beam of red light has a wavelength of 700 nm (nanometers) and travels through a vacuum. What is its frequency?

First, remember that the speed of light in a vacuum ($c$) is approximately $3 \times 10^8 \text{ m/s}$. Also, convert nm to meters: $700 \text{ nm} = 700 \times 10^{-9} \text{ m} = 7 \times 10^{-7} \text{ m}$.

$f = \frac{c}{\lambda} = \frac{3 \times 10^8 \text{ m/s}}{7 \times 10^{-7} \text{ m}} \approx 4.29 \times 10^{14} \text{ Hz}$

The frequency of the red light is approximately $4.29 \times 10^{14}$ Hz.

📝 Practice Quiz

1. A wave has a frequency of 5 Hz and a wavelength of 2 meters. What is its speed?
2. A water wave travels at 4 m/s and has a wavelength of 0.5 meters. What is its frequency?
3. A radio wave has a frequency of 100 MHz ($100 \times 10^6$ Hz) and travels at the speed of light ($3 \times 10^8$ m/s). What is its wavelength?
4. A sound wave in steel travels at 5000 m/s and has a frequency of 2500 Hz. What is its wavelength?
5. A microwave has a wavelength of 0.03 meters and travels at the speed of light. What is its frequency?

Answers: 1) 10 m/s, 2) 8 Hz, 3) 3 meters, 4) 2 meters, 5) $1 \times 10^{10}$ Hz

⭐ Conclusion

The formula $c = f\lambda$ is a fundamental tool for understanding and calculating wave speed. Whether you're dealing with sound, light, or any other type of wave, this equation provides a powerful way to relate wave speed, frequency, and wavelength. Keep practicing, and you'll master wave mechanics in no time!