1. 18 Waves in more than one dimension

1. 1. 18 Waves in more than one dimension • Often come across situations where a wave evolves in more than one direction • Ripples on a pond - wave propagates in two dimensions • Sunlight emitted by the sun - wave propagates in three dimensions • Bang from an exploding firework - wave propagates in three dimensions • In all these cases we have to come up with a suitable description for evolution of the wave.

2. Circular waves moving across a water table. The waves are generated by placing a vibrating probe in contact with the water’s surface.

3. 1. 18 Waves in more than one dimension • The most important multidimensional wave is the spherical wave • Here the wave evolves in three dimensions.

4. 1. 18 Waves in more than one dimension • At any point in space the surface of the wave looks like a football.

5. 1. 18 Waves in more than one dimension • The wave function for a spherical wave is The term A/|r| is needed to take into account the fact that the energy carried by the wave is distributed over a larger area as the wave evolves.

6. 1. 18 Waves in more than one dimension • If we take a cross section through the wave as r changes from 0 to infinity the wave shape changes from being curved to being planar in nature. To generate a spherical wave we use a point source

7. 1. 19 Wavefronts and Phasefronts • The wave function for a spherical wave is . • The phase of the wave is k.r - wt and any point a distance r from the origin of the wave at a time t will have the same phase.

8. 1. 19 Wave fronts and Phase fronts • As a consequence we will have surfaces of constant phase. • These surfaces are referred to as phase fronts or wave fronts. • A wave front is defined as follows • A wave front is a locus of points at which the wave function has the same phase • The wave front propagates with at the wave speed v and often the wave velocity/speed is called the phase velocity/speed.

9. 1. 19 Wave fronts and Phase fronts • The definition of a wave front is “A wave front is a locus of points at which the wave function has the same phase.” • This means that the particle displacement is the same at all points on a given wave front.

10. T l 1. 19 Wave fronts and Phase fronts • For a harmonic wave the displacement y(x,t) is the same if there is a phase difference of 2π between the waves. • So successive phase fronts are spaced by the wavelength or separated in time by the period.

11. 1. 20 Huygens’ Principle How does a wave front propagate from one point to another? To answer this we have to use Huygens’ Principle? Huygens thought of a wave front as being the source of the next wave front. His Principle states Every point on a primary wave front acts as a source of secondary wavelets. The new wave front is the envelop of these secondary wavelets. Each of the secondary wavelets propagates with the same speed and frequency as the primary wave front.

12. New wave front Secondary sources Secondary wavelet Each secondary source produces its own wavelet that moves forward with a speed v. The leading edges of all the wavelets forms the new wave front.