Off-tune tuning fork

The natural vibrations of the tuning forks are standing waves with sound velocity $c = \lambda f$ . For the fundamental vibration, the wavelength is $\lambda = 4 l$ , since the top corresponds to anti-node at and the bottom corresponds to a node, so that the oscillation frequency is $f = \frac{1}{4} c / l$ .

The superimposition of two sine waves with the frequencies $f_1$ and $f_2$ results $\begin{aligned} \sin(2 \pi f_1 t) + \sin (2 \pi f_2 t) &= \sin(\pi (f_+ + f_-) t) + \sin(\pi (f_+ - f_-) t)\\ &= [\sin(\pi f_+ t)\cos(\pi f_- t) + \cos(\pi f_+ t)\sin(\pi f_- t)] \\ &\quad + \,[\sin(\pi f_+ t)\cos(\pi f_- t) - \cos(\pi f_+ t)\sin(\pi f_- t)] \\ &= 2 \sin(\pi f_+ t)\cos(\pi f_- t) \end{aligned}$ with $f_+ = f_1 + f_2$ and $f_- = f_1 - f_2$ . The result is a sine wave with the mean frequency $f_+/ 2$ whose amplitude is modulated by a cosine oscillation with the frequency $f_- / 2$ . This phenomenon is also called (acoustic) beating.

The amplitude is maximized for the times $t = n / f_- = n \Delta t$ , $n = 0,1,2, \dots$ , so that $\Delta t = 0.5 \, \text{s }$ yields a difference frequency $f_- = 2 \, \text{Hz}$ . It follows $\begin{aligned} f_- &= \frac{c}{4} \left(\frac{1}{l_1} - \frac{1}{l_2} \right) \\ \Rightarrow \quad c &= \frac{4 f_-}{\frac{1}{l_1} - \frac{1}{l_2}} \\ \Rightarrow \quad f_1 &= \frac{c}{4 l_1} = \frac{f_-}{1 - \frac{l_1}{l_2}} = 402\,\text{Hz} \end{aligned}$