Sound: Numerical Problems Set (5)

Question

A tuning fork of frequency $480\text{ Hz}$ produces a sound that travels at $340\text{ m/s}$ in air at room temperature. (a) Find the wavelength. (b) If the same fork is sounded under water (where sound travels at $1500\text{ m/s}$), find the new wavelength. (c) An observer moves towards the stationary fork at $20\text{ m/s}$ in air — find the apparent frequency.

Solution — Step by Step

Why This Works

The source vibrates at a fixed rate — that fixes frequency. Wavelength then adjusts to match the speed of the medium: $\lambda = v/f$. This is why sound from underwater speakers sounds odd to us when we surface.

The Doppler formula simply counts: when we walk towards the source, we cross more wavefronts per second than a stationary observer. The fraction $(v + v_o)/v$ captures this rate-of-crossing increase.

Alternative Method

For the Doppler part, think of it physically: if the observer approaches at $20\text{ m/s}$, the relative speed of sound and observer is $360\text{ m/s}$. Number of wavefronts crossed per second = relative speed / wavelength = $360/0.708 \approx 508\text{ Hz}$. Same answer, no formula memorisation.

Common Mistake