Tru Physics

Transverse Waves

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Tru Physics
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Introduction

Transverse waves are waves in which the direction of displacement of the medium’s particles is perpendicular to the direction of the wave propagation. Examples of transverse waves include light waves, radio waves, and waves on a stretched string. This is in contrast to longitudinal waves for which the particle displacement is parallel to the direction of propagation.

Mathematical Representation

The mathematical representation of a transverse wave moving in the positive x-direction is given by:

y(x, t) = A \sin(kx - \omega t + \phi)

where:

  • y(x, t) is the displacement of the particle from its equilibrium position,
  • A is the amplitude of the wave,
  • k is the wave number,
  • \omega is the angular frequency,
  • t is time,
  • \phi is the phase constant.

Wave Speed

The speed v of a transverse wave is given by:

v = \lambda f = \dfrac{\omega}{k}

where:

  • \lambda is the wavelength,
  • f is the frequency.

Wave Number and Angular Frequency

The wave number k and the angular frequency \omega are related to the wavelength \lambda and the frequency f respectively, as follows:

k = \dfrac{2\pi}{\lambda}

\omega = 2\pi f

Polarization

One of the unique characteristics of transverse waves (which is not found in longitudinal waves) is the ability to be polarized. Polarization refers to the direction of oscillation in the plane perpendicular to the direction of propagation. For example, light and other electromagnetic waves can be polarized. However, light only comes in two polarizations as it cannot be polarized in its direction of motion (since nothing can go faster than the speed of light). Generally, waves will come in three polarizations due to the three orthogonal directions in real space.

Energy and Intensity

The energy carried by a transverse wave is proportional to the square of its amplitude. The intensity I, which is the energy transported per unit time per unit area, is given by:

I = \dfrac{1}{2} \rho v \omega^2 A^2

where:

  • \rho is the density of the medium,
  • v is the wave speed.

Conclusion

Transverse waves play a crucial role in many areas of physics and engineering, including optics, electromagnetism, and wave mechanics. Understanding their properties is essential for a wide range of applications, from telecommunications to medical imaging.

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