The Memory Trick
💡 v = fλ
The fundamental wave equation connects three quantities: speed (v, in m/s), frequency (f, in Hz), and wavelength (λ, in meters): v = fλ. Because wave speed in a given medium is typically fixed (set by the medium's properties, not the wave's source), frequency and wavelength are locked in an inverse relationship — raise one, and the other must fall proportionally.
Why It Works
Speed is literally 'how far the wave travels per unit time,' and each complete wave cycle covers a distance of exactly one wavelength. If more cycles happen per second (higher frequency), each individual cycle must cover proportionally less distance (shorter wavelength) for the total distance-per-second (speed) to stay the same.
Step by Step
Applying v = fλ
1
Know your reference speeds
Speed of sound in air is roughly 343 m/s at 20°C (and increases with temperature). Speed of light in vacuum is c = 3×10⁸ m/s.
These reference values let you quickly estimate wavelength or frequency for common everyday wave problems.
2
In a medium, speed changes via the refractive index
For light entering a medium, v = c/n, where n is the medium's refractive index (always ≥ 1) — meaning light always slows down when entering a medium from vacuum.
Glass has a refractive index of about 1.5, so light travels at roughly c/1.5 ≈ 2×10⁸ m/s inside glass — noticeably slower than in vacuum.
3
Crossing into a new medium: frequency stays fixed
When a wave crosses into a new medium, its frequency stays exactly constant (it's set by the source), while both speed and wavelength change together to satisfy v = fλ in the new medium.
Related concepts: Period T = 1/f (time for one complete cycle), and angular frequency ω = 2πf (used in rotational/oscillation contexts).
🏥 Worked Example
A sound wave in air (v = 343 m/s) has a frequency of 440 Hz (concert A). What is its wavelength?
1
Rearrange v = fλ to solve for λ: λ = v/f.
2
Plug in values: λ = 343/440 ≈ 0.78 m.
3
Sanity check: this wavelength (about 78 cm) is reasonable for an audible sound wave — comparable to the length of a guitar or a person's forearm-to-shoulder, matching real-world intuition about sound wavelengths.
📌 Exam Application
Exams test correctly rearranging v = fλ to solve for any of the three variables, and correctly reasoning about what happens to frequency vs. wavelength when a wave crosses between media with different speeds.
⚠️ Most Common Wave Speed Equation Mistakes
The most common trap is assuming frequency changes when a wave enters a new medium — it doesn't. Only speed and wavelength change together; frequency is fixed by the source and stays constant across any medium the wave travels through.
✓ Quick Self-Test
1) Write the fundamental wave equation connecting speed, frequency, and wavelength. v = fλ. 2) If wave speed is fixed (constant medium) and frequency increases, what happens to wavelength? It decreases proportionally. 3) What is the approximate speed of sound in air at 20°C? About 343 m/s. 4) When a wave crosses from one medium into another, which quantity stays constant — frequency, speed, or wavelength? Frequency. 5) What is the relationship between period (T) and frequency (f)? T = 1/f — period is the reciprocal of frequency.
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