Showing posts with label Waves. Show all posts
Showing posts with label Waves. Show all posts

Intensity of a wave

Definition of Intensity of a wave:
In a medium, the energy per unit area per unit time delivered perpendicuar to the direction of the wave propagation s caled the intensity of the wave. It is denoted by $I$.

If the energy $E$ is delivered in the time $t$ rom area $A$ perpendicular to the wave propagation, then

$I=\frac{E}{At} \qquad{1}$

Unit: $Joule/m^{2}-sec$ or $watt/m^{2}$

Dimensional formula: $[MT^{-3}]$

We know that the total mechanical energy of a vibrating particle is

$E=\frac{1}{2}m \omega^{2} a^{2}$

Where $\omega$ is the angular frequency and $a$ is the amplitude of the wave.

$E=\frac{1}{2}m (2\pi n)^{2} a^{2} \qquad \left( \omega=2\pi n \right)$

$E=2 \pi^{2} m n^{2} a^{2} \qquad(2)$

Where $m$ is the mass of the vibrating particle.

Now substitute the value of $E$ from equation $(2)$ to equation $(1)$. So the intensity of the wave

$I=\frac{2 \pi^{2} m n^{2} a^{2}}{At} \qquad(3)$

If the wave travels the distance $x$ in time $t$ with velocity $v$, Then


Now substitute the value of the above equation in equation $(3)$

$I=\frac{2 \pi^{2} m v n^{2} a^{2}}{Ax}$

$I=\frac{2 \pi^{2} m v n^{2} a^{2}}{V}$

Where $V$ is the volume of the corresponding medium during the wave propagation in time $t$.

$I=2 \pi^{2} \rho v n^{2} a^{2} \qquad \left(\because \rho=\frac{m}{V} \right)$

It is clear that for wave propagation in a medium with a constant velocity, i.e. wave's intensity is directly proportional to the square of amplitude and frequency both.

$I\propto a^{2}$ and $I \propto n^{2}$

Difference between sound waves and light waves

Sound Waves:

  • Sound waves are mechanical waves in nature.
  • They need a medium to propagate and so cannot be produced in a vacuum.
  • They can move in all types of mediums as solid liquid or gas whether they are transparent or not.
  • They propagate in the form of longitudinal waves.
  • The particle of the medium vibrates along the direction of the propagation and so contraction and rarefaction are formed there.
  • These are three-dimensional waves.
  • These waves do not show a polarization effect.
  • For a normal human being the audible frequency range is $20$ to $20000 Hertz$.
  • The speed of the sound waves is more in a dense medium than in a rare medium.

  • Light waves:

  • light waves are electromagnetic waves in nature.
  • They don't need any medium and so can produce and propagate in a vacuum.
  • Their velocity in a vacuum is the maximum of value $3 \times 10^{8} m/s$
  • They can move in a transparent medium only.
  • They propagate in the form of transverse waves.
  • The electric field and magnetic field vibration are perpendicular to the direction of wave propagation.
  • These wave waves are also three-dimensional waves.
  • These waves source the polarization effect.
  • For a normal human being the visible frequency range is $4 \times 10^{18} Hz$ to $8 \times 10^{14} Hz$.
  • The speed of light is more in rare mediums than in dense ones.
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