Answer
a. 253 overtones
b. 253 overtones
Work Step by Step
a. The harmonics for an open pipe are given by $f_n=\frac{nv}{2 \mathcal{l}}$. The human range of hearing extends up to a frequency of 20 kHz, so that puts an upper limit on n.
$$\frac{nv}{2 \mathcal{l}} \lt 2\times10^4\;Hz$$
$$n \lt \frac{2 \mathcal{l}(2\times10^4\;Hz)}{v}=\frac{2 (2.18\;m)( 2\times10^4\;Hz)}{343\;m/s}=254.2$$
There are 254 harmonics: one fundamental and 253 overtones.
b. The harmonics for a closed pipe are given by $f_n=\frac{nv}{4 \mathcal{l}}$, with n being an odd number. The human range of hearing extends up to a frequency of 20 kHz, so that puts an upper limit on n.
$$\frac{nv}{4 \mathcal{l}} \lt 2\times10^4\;Hz$$
$$n \lt \frac{4 \mathcal{l}(2\times10^4\;Hz)}{v}=\frac{4 (2.18\;m)( 2\times10^4\;Hz)}{343\;m/s}=508.5$$
However, n must be odd, so that means n = 1, 3, 5, …507.
There are 254 harmonics: one fundamental and 253 overtones.
