Solved Problems V (Wave Phenomenon)

1.      A transverse force F_tr(t) applies on starting point O of  a long rope, a harmonic transverse wave is formed with angular frequency ω. The amplitude of the periodic force is y_o. ρ is the mass per meter of the rope and F_s is the tension force applied on the rope.

a.       Give the general expression of the disturbance y (x,t) on the rope as a function of place and time. Give also the complex amplitude Y (x), hint: use the complex expression, the complex value of the disturbance at x = 0 is A₀. Then find the expression for complex value V_tr (x) of the transverse velocity v_tr (x,t).

b.      Define the relationship between F_tr and F_S.

c.       The maximum disturbance of the wave is y₀. What is the relationship between A₀ and y₀?

d.      Calculate the absolute value of the impedance Z on the rope at x = 0 with |Z| = |F_tr/V_tr (0)| given that f₀ = 100 N, y₀ = 0,01 m and ω = 2π.10² s^-1. F_tr is the complex value of F_tr (t). Show that Z = F_S/c given that $c=\frac{\omega }{k}=\sqrt{\frac{{{F}_{S}}}{\rho }}$.

e.       Calculate the tension force F_S on the rope, given that ρ = 0,1 kg/m. Calculate also c.

f.       Calculate the energy per second in every point on the rope.

g.       A part of the rope with length L is now fixed with a tension force F_S = 1000 N. By ‘ticking’ the rope, a wave pattern of 4 nodes appears (2 nodes at the ends of the pattern and 2 nodes in between). Give a sketch of the wave pattern.

h.      What is the frequency of the wave, if the length L of the fixed part is equal to 1 m?

Solution:

2.      End of a rope with density ρ₁ is fixed at x = 0 with a second rope with density ρ₂. Neglect all dissipated forces in this case, the tension force works on both ropes is F.

a.       What is the wave equation of wave propagates on the rope with density ρ and tension force F? Give also the one-dimensional Helmholtz equation of a wave propagates on a rope.

b.      What is the boundary conditions at the ‘transition’ area x = 0 of both ropes?

Assume now that the incoming harmonic wave from negative x-direction with angular frequency ω is given by: $y(x,t)=$ Re $\left[ Y(x){{e}^{-i\omega t}} \right]$. The complex amplitude of the incoming wave is given by: $Y(x)=A{{e}^{i{{k}_{1}}x}}$ (with A real). The complex amplitudes of reflected and transmitted waves are respectively: $B{{e}^{-i{{k}_{1}}x}}$ and $C{{e}^{i{{k}_{2}}x}}$.

c.       Calculate the coefficient of reflection r = B/A and the transmission coefficient t = C/A.

d.      If ρ₁ > ρ₂, how’s the phase changed by reflection?

e.       If ${{\rho }_{2}}=\infty $, i.e. for the case of a very rigid/fixed ends of rope, what is then r and t?          

Solution: