Normal modes

(a)

Do YF problem 19-15 (p. 616).
Note: In (c), they want you to find the combinations of $A$ and $\lambda$ which make the particle velocity achieve the conditions listed in the problem.

(b)

Supposing that the apparatus in YF problem 19-15 is such that only waves with nodes at each end are allowed (closed boundary conditions), what is the length of the string if the situation with the 1.50 kg mass corresponds to the fundamental (longest wavelength) mode?

(c)

For a string of the length you find in (b), what values of the mass are needed to then give the first, second and third next modes?

(d)

A simple model of a clarinet (a single-reed woodwind instrument) is a pipe of length 66.5 cm with one closed end (the reed end, a displacement node) and one open end (a displacement maximum). The speed of sound is 350 m/s. What are the frequencies of the lowest two modes within this simple model?
Just for fun: ``Middle A'' is 440 Hz and each ``octave'' represents a factor of two in frequency, so 110 Hz would be an A two octaves below middle A.