1 Waves 3 Lecture 3 - Introduction to Waves Waves and the wave equation D Aims: ëReview of wave motion: > “Snapshots” and “waveforms”; > Wave equation. ëHarmonic waves: > Phase velocity. ëRepresentation of waves. > Plane waves: > 1-, 2-, and 3-D waves. > Spherical waves.

2 Waves 3 Waveforms and Snapshots D Waves as travelling disturbances.  A scalar wave is specified by a single-valued function of space and time. In 1-D:  =  ( x,t ). ëConsider 3 snapshots of wave moving from left to right (in the +ve x-direction): ë No change in shape, so D Wave ëTravelling in +ve x-direction: ëTravelling in -ve x-direction: Wave velocity

3 Waves 3 Wave equation D Disturbance in time (at fixed x) D Wave equation:  Relates the two, second-order, partial derivatives of  (x,t) = f(x - vt). The chain rule gives, with u = x - vt. (also works for waves in -ve x-direction) Snapshot at t o Waveform at x o 1-D wave equation

4 Waves 3 Wave equation: superposition D Important features: ëGenerality: no specification of the type of wave.  Linearity: all terms in  are raised to first power only. Superposition is, thus, an implicit property.  Superposition:  Superposition: consider  =  1 +  2.  If  1 and  2 are solutions so is . ëAny linear combination of solutions is also a solution. > Basis of justification for Fourier analysis to describe wave properties. i.e. superposition of harmonic waves with different frequencies.

5 Waves 3 Harmonic waves  Wave varies sinusoidally  Wave varies sinusoidally (with both x and t ).  Require f(x-vt) {or equivalently f(t-x/v) } for a wave in the +ve x-direction.  For harmonic dependence we need f(u) ~ e i  u.  k is the wave-number, given by k = 2  /.  v =  /k - Phase velocity of the wave ëIn -ve x-direction D Nomenclature and convention: ëThe following are equally valid for a wave in the +ve x-direction:  There is no “agreed convention”. In general we use the i(  t-kx) form. The i(kx-  t) form is common in optics and quantum theory. ëThe following are equivalent: Use this form

6 Waves 3 Plane waves (sect 2.2) D Plane waves:  In a 3-D medium  =  (x,y,z,t). For a wave propagating in the +ve x-direction  (x,y,z,t) = A(y,z)e i(  t-kx).  If A(y,z) = constant, we have a plane wave.  Surfaces of constant phase (i.e.  t-kx = const) are called wavefronts. They are locally perpendicular to the direction of propagation.  Propagation in an arbitrary direction (defined by a unit vector n ).

7 Waves 3 3-D plane-wave D Wavevector ëPhase at Q = phase at P  Vector is the wavevector. It is in the direction of propagation and has magnitude 2  /. D General 3-D plane-wave ë it is the solution of the 3-D wave equation Wavevector Wave equation Plane wave

8 Waves 3 Spherical waves D Waves expanding symmetrically from a point source. ëConservation of energy demands the amplitude decreases with distance from the source.  Since energy is proportional to |  | 2, energy flowing across a sphere of radius r is proportional to 4  r 2 |  | 2. Thus  1/r. ëSpherical wave: ëAt large r, it approximates a plane wave. D Summary ëGeneral wave: ë1-D wave equation ë3-D plane wave ë3-D wave equation ëSpherical wave Spherical wave