Question

NoN-UNIFORM STRING This problem explores the wave-speed of for a horizontal string with non-uniform linear mass density μ ax, where a is a positive constant and is position within the string, relative to a zero point at one end of the string. The string has a length l and total mass m a) Obtain an expression for the mass of the string, m, in terms of I and a. b) Write the formula that gives the wavespeed as a function of position, r. c) Obtain an expression for the time required for a wave to traverse the entire string. d) As a wave travels from 0 to I does the wavelength, frequency, both, or netiher change? Defend your position using a physical argument.

Answer 1

a)

m = $\mu_{avg}$ x L

$\mu_{avg}$ = aL/2

m = aL^2/2

b)

v = (T/$\mu_{avg}$)^0.5

= (2T/ax)^0.5

where T is tension in string

c)

t = L/v = L/(2T/aL)^0.5

= (aL^3 / 2T)^0.5

d)

f x 2x = (2T/ax)^0.5

as f is dependent on x, it will change. As wavelength = 2x, it will also change