A sound source with an adjustable frequency generates 2 beats per...

Let the frequency of the tuning fork be $f$.

The beat frequency is the absolute difference between the frequencies of the two sound sources.

Given that the beat frequency is $2 \text{ Hz}$ when the source frequency is $476 \text{ Hz}$:

$|f - 476| = 2 \Rightarrow f = 474 \text{ Hz} \text{ or } 478 \text{ Hz}$

Also, the beat frequency is $2 \text{ Hz}$ when the source frequency is $480 \text{ Hz}$:

$|f - 480| = 2 \Rightarrow f = 478 \text{ Hz} \text{ or } 482 \text{ Hz}$

The common value satisfying both conditions is $f = 478 \text{ Hz}$.