$7\mathrm{.}5$ Designing Stringed Instruments

How are stringed instruments such as pianos, violins, and guitars designed? Stringed instruments are created using strings that are tautly attached at both ends. You can pluck, bow, or strike a string to create vibrations. The frequency of the vibrations determines the sounds produced. The higher the frequency, the higher the note. You can alter the frequency by pressing down on the string, effectively shortening the string length. Marin Mersenne $(15881648)$ determined three laws relating the frequency, f$,$ to string length, L$,$ the tension, T$,$ and the mass per length, W:

where the symbol $$ means $$proportional to$.$

a$.$ Describe in words how changes in each of the following affects the frequency, f:

These laws can be combined to give the formula for the fundamental frequency of a string fixed on both ends:

where $1/2$ is the constant of proportionality.

b$.$ Why are strings for lower notes thicker than those for higher notes?

c$.$ Which string on a violin produces the highest sound: the one with the highest tension or the lowest? Why?

d$.$ What is the result of cello strings being longer than those of a viola?

e$.$ How would doubling the mass per length of a string, without changing anything else, change the frequency?