Standing Waves in a Chain Vertically Hung In the lowest order, the velocity of transverse waves in a chain freely hanging maybe given by where x is the distance from the lower, free end of the chain. The wave velocity is stronglydependent on the location and the chain can be regarded as a typical nonuniform wave medium. The differential equation to describe transverse waves in the chain is given by (See Note No. 8 for detailed derivation.) This is not in the form of standard wave equation. For a narrow pulse, the term of first order derivative can be ignored, which yields the approximated propagation velocity given in the first equation. satisfies This can be satisfied by the zero-th order Bessel function where L is the length of he chain. At the upper end x = L, the displacement should vanish. The Bessel function J0(y) becomes zero when y = .4048, 5.5201, 8.6537, ?. Therefore, the resonance frequencies are and so on. Animation shows standing wave patterns with those frequencies.Standing waves in a chain vertically hung (typical nonuniform wave medium). Lowest order mode. > with(plots): animate([.1*BesselJ(0,2.4048*sqrt(x))*sin(t),x,x=0..1],t=0..2*Pi,frames=60,color=red); Second mode. Third mode. |
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