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Wednesday, October 27, 2010

Wave velocity in a continuous system

  • Any system whose particle motion are governed by classical wave equation is a system in which harmonic waves of any wavelength can travel with the speed v

  • The value of v depends on the elastic and inertial properties of the system under consideration.

(1) Transverse wave on a stretched string

  • Displacement of the string is governed by the equation

    Where T is the tension and µ is the linear density (mass per unit length of the string)
  • Velocity of wave on the string is
    v is the velocity of the wave.
  • Medium through which waves travel will offer impedance to these waves.
  • If the medium is loss less i.e., it does not have any resistive or dissipative components, the impedance is solely determined by its inertia and elasticity.
  • Characteristic impedance of string is determined by
  • Since v is determined by the inertia and elasticity this shows that impedance is also governed by these two properties of the medium.
  • For loss-less medium impedance is real quantity and it is complex if the medium is dissipative.

(2) Longitudinal waves in uniform rod

  • Equation for longitudinal vibrations of a uniform rod is

  • where ξ (x,t )→displacement
    Y is young’s modulus of the rod
    ρ is the density
  • Velocity of longitudinal wave in rod is

(3) Electromagnetic waves in space

  • When electric and magnetic field vary in time they produce EM waves.
  • An oscillating charge has an oscillating electric and magnetic fields around it and hence produces EM waves.
  • Example: - (1) Electrons falling from higher to lower energy orbit radiates EM waves of particular wavelength and frequency. (2) The motion of electrons in an antenna radiates EM waves by a process called Bramstrhlung.
  • Propagation of EM waves in a medium is also due to inertial and elastic properties of the medium.
  • Every medium (including vacuum) has inductive properties described by magnetic permeability µ of the medium.
  • This property provides magnetic inertia of the medium.
  • Elasticity of the medium is provided by the capacitive property called electrical permittivity ε of the medium.
  • Permeability µ stores magnetic energy and the permittivity ε stores the electric field energy.
  • This EM energy propagates in the medium in the form of EM waves.
  • Electric and magnetic fields are connected by Maxwell’s Equations (dielectric medium)
    ∇×H =ε (∂E )/∂t
    ∇×E = - μ (∂H⃗)/∂t
    ∇∙H =0
  • Here in above equations E ⃗ is electric field , H ⃗ is the magnetic field and ρ is charge density

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