Beat Frequency Is the Difference; Loading and Tension Tell You Which Way

The beat frequency is |f1 - f2|, and the sound heard has the average frequency (f1 + f2)/2 with loudness that rises and falls at the beat frequency. Two sources of 499 Hz and 501 Hz give a 500 Hz tone whose intensity pulses 2 times per second. For waves y1 = 4 sin 500πt and y2 = 5 sin 508πt, the frequencies are 250 and 254 Hz, so 4 beats per second are heard, and the ratio of maximum to minimum intensity is ((A1 + A2)/(A1 - A2))² = (9/1)² = 81 : 1.

Loading a fork with wax lowers its frequency, filing it raises it, and increasing the tension of a string raises its frequency. Combine this with whether the beats increase or decrease.

A 512 Hz fork beats 4 times per second with a piano string; when the tension is raised (raising the string frequency) the beats fall to 2, so the string was below the fork, at 508 Hz. If an unknown source gives 4 beats with 250 Hz, it is 254 or 246 Hz;

A trickier case: fork A of 256 Hz gives 4 beats with fork B, so B is 252 or 260 Hz. Waxing B drops it to 250 Hz (6 beats), and when only a little wax is removed the beats fall back to 4, which means B has risen to 252 Hz while still carrying some wax. Since removing only part of the wax gets it to 252 Hz, the unwaxed frequency must be higher than 252 Hz, so B is 260 Hz.

Waves of wavelengths 2.00 m and 2.02 m with the same speed v have frequencies v/2.00 and v/2.02, so 2 beats per second requires v (1/2.00 - 1/2.02) = 2, giving v = 404 m/s.