Lecture 3 - Introduction to Waves

1. Lecture 3 - Introduction to Waves Waves and the wave equation • Aims: • Review of wave motion: • “Snapshots” and “waveforms”; • Wave equation. • Harmonic waves: • Phase velocity. • Representation of waves. • Plane waves: • 1-, 2-, and 3-D waves. • Spherical waves.

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

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

4. Wave equation: superposition • Important features: • Generality: no specification of the type of wave. • Linearity: all terms in Y are raised to first power only. Superposition is, thus, an implicit property. • Superposition: consider Y = Y1 + Y2. • If Y1 and Y2 are solutions so is Y. • Any 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. Harmonic waves • 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) ~ eiwu. • k is the wave-number, given by k = 2p/l. • v = w/k - Phase velocity of the wave • In -ve x-direction • Nomenclature and convention: • The following are equally valid for a wave in the +ve x-direction: • There is no “agreed convention”. In general we use the i(wt-kx) form. The i(kx-wt) form is common in optics and quantum theory. • The following are equivalent: Use this form

6. Plane waves (sect 2.2) • Plane waves: • In a 3-D medium Y = Y(x,y,z,t). For a wave propagating in the +ve x-directionY(x,y,z,t) = A(y,z)ei(wt-kx). • If A(y,z) = constant, we have a plane wave. • Surfaces of constant phase (i.e. wt-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. 3-D plane-wave • Wavevector • Phase at Q = phase at P • Vector is the wavevector. It is in the direction of propagation and has magnitude 2p/l. • General 3-D plane-wave • it is the solution of the 3-D wave equation Wavevector Plane wave Wave equation

8. Spherical waves • Waves expanding symmetrically from a point source. • Conservation of energy demands the amplitude decreases with distance from the source. • Since energy is proportional to|Y|2, energy flowing across a sphere of radius r is proportional to 4pr2|Y|2. Thus Yµ1/r. • Spherical wave: • At large r, it approximates a plane wave. • Summary • General wave: • 1-D wave equation • 3-D plane wave • 3-D wave equation • Spherical wave Spherical wave