A string on a musical instrument is held under tension T and extends from the po

Question

A string on a musical instrument is held under tension T and extends from the point x = 0 to the point x = L. The string is overwound with wire in such a way that its mass per unit length ?(x) increases uniformly from ?0at x = 0 to ?L at x = L.

(a) Find an expression for ?(x) as a function of x over the range 0 ? x ? L. (Let ?(0) = ?0 and ?(L) = ?L. Use any variable or symbol stated above along with the following as necessary: x.)

(b) Find an expression for the time interval required for a transverse pulse to travel the length of the string. (Use any variable or symbol stated above as necessary.)

Explanation / Answer

a) u(x) = u0 + (uL -u0) x/L

b) v = dx/dt = sqrt( T/u)

sqrt( u0 + (uL - u0) x/L) = sqrt(T) dt

so

integral of sqrt(u0 + (uL - u0) x/L ) form 0 to L = sqrt(T) t

let y = u0 + (uL - u0)x/L

dy = (uL - u0)/L dx

so becomes

integral of sqrt(y) *L/(uL - u0) from u0 to uL = 2/3 L/(uL - u0) ( uL^(3/2) - u0^(3/2) ) = sqrt(T) t

t = 2/3 L/(uL - u0) ( uL^(3/2) - u0^(3/2) )/sqrt(T)

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