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=√(T/μ)
  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
  Z=T/v=√(μT)=μv
  
• 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
  v=√(Y/ρ)
  

(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
  ε(∇∙E⃗)=ρ
  ∇∙H =0
  
• Here in above equations E ⃗ is electric field , H ⃗ is the magnetic field and ρ is charge density