The D$-$string on a guitar has a linear mass density of $2\mathrm{.}110-3$ kg$/$m and an oscillating length of $0\mathrm{.}64$ m$.$ It is stretched at a tension of $80\mathrm{.}2$ N$.$

a$)$ The guitarist plucks the D$-$string. At what speed, in meters per second, does the pulse travel along the string?

b$)$ The plucked string produces a tone. What is the fundamental frequency, in hertz, of the tone under the given conditions?

c$)$ When a guitar$$s D$-$string is properly tuned, it should produce a fundamental frequency of $146\mathrm{.}8$ Hz$.$ The D$-$string on this guitar might not be properly tuned. To tune it properly, at what tension should it be stretched, in newtons?