Worked example: Why can't we transmit 20 kHz audio directly? Antenna size calculation. Step 1: Wavelength of a 20 kHz audio wave: λ = c/f = (3×10⁸ m/s)/(20×10³ Hz) = 15,000 m = 15 km. Step 2: A practical antenna needs length ≈ λ/4 = 3.75 km. A 3.75 km tall antenna is impossible. Step 3: Now modulate onto a 1 MHz carrier: λ = (3×10⁸)/(10⁶) = 300 m, so λ/4 = 75 m — a feasible tower. Step 4: Bonus — at 100 MHz (FM band): λ = 3 m, λ/4 = 0.75 m, which is why an FM radio whip antenna is under a metre. Conclusion: modulation onto a high-frequency carrier shrinks the antenna from kilometres to metres — the single most practical reason modulation exists.