Problem — Ranking wavelengths in different media

Problem

Three waves propagate in different media:

• A: $f = 100\;\text{Hz}$, $v = 340\;\text{m/s}$ (air);
• B: $f = 100\;\text{Hz}$, $v = 1500\;\text{m/s}$ (water);
• C: $f = 50\;\text{Hz}$, $v = 340\;\text{m/s}$ (air, an octave lower).

Rank the wavelengths from smallest to largest. (Careful: the required ordering may surprise you!)

Solution

Setup. $\lambda = v/f$. Let’s calculate the three wavelengths:

$\lambda_A = \frac{340}{100} = 3{,}4\;\mathrm{m}$

$\lambda_B = \frac{1500}{100} = 15\;\mathrm{m}$

$\lambda_C = \frac{340}{50} = 6{,}8\;\mathrm{m}$

Ranking from smallest to largest:

$\ev{\lambda_A < \lambda_C < \lambda_B}$

The surprise: the wave in water (B), despite having the same frequency as A, has the largest $\lambda$ because it travels much faster there.

Links

Topics: Onde Concepts: Periodo e frequenza Skills: Ragionamento di ranking