Question

Needing a solution to this question.

5) A rope of mass m and length ℓ hangs from a ceiling. Show that the wave speed on the rope a distance y above the lower end is v= Square root of gy (10). Would such a hanging rope support a “wave”, as per our definition in Q1? Explain in words?

******NOTE this is question one (Q1). It has been solved already. Just for reference. Thanks

Q1) A good definition of a “wave” is a disturbance of a continuous medium that propagates with a fixed shape at constant velocity. (Griffiths, 1999, Introduction to Electrodynamics). Another way of saying this is the medium (e.g. a string, water, or air) will only propagate a wave as long as its shape satisfies the wave equation: d^2 f/dx^2 =1/v^2 . d^2 f/dt^2 a) Show explicitly whether the displacement function f(x,t) = A sin(kx) cos(ωt) will be propagated in a medium as a wave or not. (5) b) Do the same for the function f(x,t) = A exp(−b(bx2 + vt)) (5) c) Could a string support a disturbance of the form f(x,t)=Asin(kx)cos(x−ωt) as a wave? Your professor will show you what this looks like in class and you will immediately

Answer 1

5)

From the concept of Transverse wave on strings (rope) , the speed of the wave is

   v = sqrt(T/mue) ---------(1)

where T is the Tension in the string

   mue is Linear mass density = m/L

here the tension in the string is due to its weight T = mg

and the linear mass density is mue = m/y

substituting in the above equation (1) we get

   v = sqrt((mg)/(m/y)) m/s

   v = sqrt(g*y) m/s

10)
yes it does support a wave

because from the solution of a wave equation it look like

   d^2y/dx^2 = 1/(F/mue) d^2y/dt^2 nothing but the equation of a wave

d^2 f/dx^2 =1/v^2 . d^2 f/dt^2, the term 1/v^2 = 1/(F/mue)

and we can write the speed as V = sqrt(F/mue)
   Where F plays the role of the restoring force , which tends to bring the rope back to its undistrubed or equilibrium configuration

and the linear mass density provides the inertia that [revents the rope from returning instantaneouslu to equilibrium, means the wave will propagate in the rope