Question

A uniform rope having mass m hangs vertically from rigid support. A transverse wave pulse is produced at the lower end. The frequency of wave pulse varies with x from the lower ends as

Answer 1

Let a uniform rope of length L and mass m is given as shown in figure .

we cut an elementary length of dx form the lower end of rope.

so, mass of element = $\bold{\frac{m}{L}dx}$

Tension on elementary part = mg.dx/L

Now, formula of velocity is $\bold{\sqrt{\frac{T}{\mu}}}$

Where T is tension in string e.g., T = mg.dx/L

μ is mass per unit length e.g., mass per unit length ( μ ) =( m.dx/L)/x = m.dx/Lx

∴ velocity of wave at x from the lower point = $\bold{\sqrt{\frac{mg.dx/L}{m.dx/Lx}}}$

= $\bold{\sqrt{gx}}$

but we know, velocity of wave = wavelength × frequency

so, frequency = velocity of wave/wavelength

e.g., frequency is directly proportional to velocity of wave if wavelength remains constant.

so, frequency, $f\propto\sqrt{gx}$
hence, graph between frequency and x will be parabolic as shown in figure.