Waves in continous medium

  • There are essentially two ways of transporting energy from one place to another (a) Actual transport of matter for example a fired bullet and (b) Waves carry energy but there is no transport of matter for example sound waves carry energy so thay can move diphagram of the ear.
  • Here we will consider the oscillations of open or unbounded systems i.e., systems having no outer boundaries.
  • If such system is disturbed , waves travel in the system with a speed determined by the properties of the system.
  • Waves are not reflected back in such a system.
  • The waves generated by driving force are called travelling waves ; these waves travel from the point where the driving force produces the disturbance.
  • If the driving force produces a harmonic disturbance the travelling wave it produces are called harmonic travelling waves.
  • In the steady state, all moving parts of the system oscillates with simple harmonic motion at the driving frequency.
  • Waves where the displacements or oscillations are transverse (i.e., perpandicular) to the direction of wave propagation is called transverse wave.
  • The wavelength (denoted by λ) of the wave is defined as the distance, measured along the direction of the propagation of the wave, between two nearest points which are in the same state of viberation.
  • Wavelength λ is just the distance travelled by the wave during one time period T of particle oscillation. Thus wave velocity
    v=λ/T=λν
    where ν=1/T - is the frequency of the wave.
  • This relation between wave velocity, frequency and wavelength also holds for longitudinal waves in which the displacements or oscillation in the medium are parallel to the direction of wave propagation.
  • Waves in spring and sound waves are longitudinal waves.
  • Wavelength for longitudinal waves is the distance between two successive compressions or rarefactions.
  • Sound waves are also compressional.
  • Assumptions that are made while obtaining wave equation are:-
    1. Amplitude A of particle oscillations does not change in course of the propagation of wave.
    2. The medium is isotopic and homogeneous so that velocity of wave does not chance from place to  place
  • Displacement of particle at x at any time t is
    Ψ(x,t) = A sin{2π(t-x/v)/T)}
  • The function Ψ(x,t) repeats itself in a distance λ . Wavelength of a wave is also known as spatial periodicity of the wave.
  • The wave is thus doubly periodic. It has temporal periodicity T and spatial periodicity λ.
  • Let us define quantities
    k=2π/λ and
    ω=2π/T
    then wave function can be written as
    Ψ(x,t) = A sin{ωt-kx}
    where quantity k is known as wave number of the wave and ω is called angular frequency of particle oscillations in wave.
  • Harmonic wave travelling in
    + x direction    :     Ψ(x,t) = A sin{ωt-kx}
    - x direction    :     Ψ(x,t) = A sin{ωt+kx}
    above equations can also be equally well described by cosine function.
  • Classicsl wave equation is



  • Important inferences from above wave equation
    1. Whenever second order time derivative of any physical quantity is related to second order space derivative as in above equation , a wave of some sort must travell in the medium.
    2. Velocity of that wave is given by the square root of the coefficent of second order space derivative.
  • Individual derivatives which makes up the medium do not propagate through the medium with the wave; they merely oscillates ( transversly or longitudinally) about there equilibrium positions.
  • It is their phase relationship which we observe as wave.
  • Wave velocity is also called phase velocity with which crest or troughs in case of transverse wave and compressions or rarefactions in case of longitudinal waves travell through the medium.
  • The phase velocity is given by
    v=λ/T=λν
    or,
    v=ω/k
  • Ψ(x,t)=f(vt-x) is the solution of the above given wave equation.

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