Wavefronts and rays

∙Consider the transverse waves shown here.
∙The wavefronts are by convention located at the crests of the waves.
∙A top view simplifies the drawing:
∙Wavefronts don’t have to be straight.

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FYI
∙Wavefronts are perpendicular to the wave velocity.

Wavefronts can also travel in 3D, such as the “cross-section” of the spherical wavefront shown here:

∙Such a wave can come from a point source of sound or light, or even from an explosion.
∙Even if a wavefront is curved to begin with, as you get farther from the point source the wavefronts become more “flat” or planar.

 
∙Wavefronts can also be bent by obstacles in the environment, or by the properties of the medium through which the wave passes:

∙The animation shows flat wavefronts being focused by a lens.
∙Oftentimes rather than drawing wavefronts we draw rays, which are perpendicular to the wavefronts.
FYI
∙Rays are parallel to the wave velocity

Longitudinal waves also have wavefronts and rays.
Instead of crests and troughs, longitudinal waves have compressions and rarefactions.

Just as crests were arbitrarily chosen to be wavefronts in transverse waves, compressions are usually chosen as wavefronts in longitudinal waves.Amplitude and intensity

∙Intensity is the rate energy is being transmitted per unit area and is measured in (W m^-2).

Since $A = 4\pi x^2$ for a spherical wave, we can rewrite our intensity formula.

$$I = \frac{power}{area}$$

$$I \propto x^{-2}$$   Intensity vs. distance x

Recall that the total energy $E_T$ of a particle in SHM was

$$E_T = \frac{1}{2}kx_{MAX}^2$$

where $x_{MAX}$ was the amplitude A of the oscillation.

3. Relationship between Power and Amplitude

Since $P = \frac{E_T}{time}$, clearly $P \propto E_T$ so that $P \propto A^2$.

But $I = \frac{power}{area}$ so that the following is true:

Interpreting diagrams involving reflection

∙Wave reflection occurs when a wave meets a boundary, such as a solid object, or a change in the medium, and is at least partially diverted backwards.

∙The angles of the rays are always measured with respect to the normal which is perpendicular to the surface.
∙The relationship between the angle of incidence θ_incident and the angle of reflection θ_reflect is simple:

 
∙We can also look at the wave fronts: Observe…

∙During reflection the frequency and the wavelength of the wave do not change.

Wave refraction occurs when a wave meets a boundary, such as a solid object, or a change in the medium ,and is at least partially allowed through the boundary.

∙It may help to imagine the ranks of a marching band.

∙Obviously, the cadence does not change.
∙Thus the period and the frequency do not change.
∙But the speed and the wavelength do change.

∙During refraction the frequency and the period do not change.

∙It should be clear that the incident wave is faster than the refracted wave in this example.

Snell’s law

∙Snell’s law relates the wave’s velocities in two mediums to their angles:

∙A refractive index n_m is often used when dealing with light waves. It is defined to be the ratio of the speed of light in vacuum c to the speed of light in the medium v_m.

Snell’s law – total internal reflection

∙When a wave moving in an optically dense region arrives at a boundary with a less dense medium it is possible to have all of the light reflected – trapping it inside. This is the principle behind fiber optics.

 
∙The critical angle θc is that incident angle at which the angle of refraction θrefr is 90°:

∙Snell’s law gives us a relationship between the indices of refraction and the critical angle:
n₁sinθ_c = n₂sin90° = n₂(1).