Lecture 3 - Introduction to Waves

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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.

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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
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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
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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.

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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
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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).

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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
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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 toY2, energy
  flowing across a sphere of radius r is
  proportional to 4pr2 Y2. 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