Electromagnetic waves: which relationship correctly links wavelength (λ), frequency (f), and propagation speed (C) of an electromagnetic wave in a given medium?

Introduction / Context:
In remote sensing, optics, and communications, every electromagnetic wave is described by three tightly coupled quantities: its wavelength (λ), its frequency (f), and the speed at which it propagates (C). Recognizing the correct relationship among these parameters is essential for converting between spectral bands, designing antennas, and interpreting sensor data.

Given Data / Assumptions:

• We consider a uniform medium where electromagnetic waves travel with speed C (for example, in vacuum C is approximately 3 * 10^8 m/s).
• Wavelength λ is the distance between successive crests of the wave, measured in metres.
• Frequency f is the number of cycles per second, measured in hertz (s^-1).

Concept / Approach:
By definition, during one period T (seconds) the wave advances one wavelength λ (metres). The frequency is f = 1 / T. The distance covered per second is therefore λ per T seconds, which yields the wave speed C. This fundamental kinematic reasoning gives the core relation that links the three quantities.

Step-by-Step Solution:
Start from period and frequency: f = 1 / T.Relate distance advanced in one period: distance = λ in time T.Compute speed as distance / time: C = λ / T.Substitute T = 1 / f into C = λ / T to obtain: C = λ * f.Hence the correct relationship among speed, wavelength, and frequency is C = λ * f.

Verification / Alternative check:
Rearrange the same identity to useful forms: λ = C / f and f = C / λ. These equivalent expressions are widely used to convert between frequency (GHz, MHz) and wavelength (m, cm, mm) for radar, lidar, and optical systems.

Why Other Options Are Wrong:

• C = λ / f: has incorrect units (m / s^-1 = m*s), not speed.
• C = f / λ: also dimensionally inconsistent (s^-1 / m = 1 / (m*s)).
• λ = C * f: implies wavelength grows with both speed and frequency; dimensionally wrong (m = m/s * s^-1).
• f = λ / C: would give units s^-1 = m / (m/s) = s, which is incorrect.

Common Pitfalls:

• Forgetting that in media other than vacuum, C equals the phase speed in that medium (C = c / n), so wavelength changes with refractive index while frequency remains constant across boundaries.
• Mixing units (e.g., using MHz with metres without converting), which leads to numerical errors even if the formula is right.

Final Answer:
C = λ * f