Resonance in a driven string

The simulation computes the steady state solution and can be used to check how resonance happens when the driving frequancy is (nearly) one of the normal frequencies, i.e. , one of the frequencies of the standing waves that may appear in the string when both ends are fixed.

• The unit length is L and the unit time L/v, where v = √(TL/m) is the phase velocity of transverse waves propagating along the string, whete the tension is T.
• You can select below the dimensionless frequency of the driving force, ω ≡ Ω L/π v, and its amplitude A, as well as the number of discretization points N and the time interval between animation frames Δt.
• To get information on an element, put over it the mouse pointer to see the corresponding tooltip.

Activities

• Try in the wave equation of the vibrating string a solution in the form y(t,x) = f(x) cos Ωt and show that in order to satisfy the boundary conditions (which now read f(0) = A, f(L) = 0) one has to choose
  f(x) = A sin k(L-x)/sin kL,
  where k = Ω/v.
• Discuss in which cases resonance will appear.
• Why is unbounded this steady state solution when resonance appear?
• Use the simulation to check the discussion above.