Lecture 6 - Standing waves

1
Lecture 6 - Standing waves

• Standing waves on a stretched string.
• Aims
• Semi-infinite string
• Standing waves arising from reflection
• counter-propagating waves.
• String of finite length
• Quantisation
• violins and laser cavities.
• 2-D and 3-D standing waves.
• Rectangular, elastic sheet
• Waves in a box (black-body radiation).

2
Standing waves in 1-D

• General considerations
• Standing waves arise whenever there are two
  counter-propagating waves.
• For example in a clamped stretched-string.
• In last lecture we considered reflection when Z2
  . The reflection coefficient is -1.
• What does the resulting wave look like?
• A standing wave.

Real part
3
Standing waves continued

• Nodes and anti-nodes
• Open ended string
• Here the reflection coefficient r1. The
  argument, however, is similar.
• Again we have a standing wave but the free end of
  the string (x0) is an anti-node.
• String of finite length
• Analysis has many applications
• Violin, guitar etc..
• Laser cavity
• 1-D infinite potential in Quantum mechanics
• etc.
• The basic phenomenon to emerge is quantisation.

4
Quantisation on a finite string

• String of fixed length
• With a string clamped at both ends, each end is a
  node. Between the nodes we have a standing wave.
• Boundary conditions Y0 at x0 and xl.Or
  earlier solutiondescribes the wave provided
• An integral number of half wavelengths must fit
  on the string. Using k2p/l, gives
  lnl/2.
• In acoustics, these are known as harmonics. Since
  wvk we have wo, 2wo, 3wo etc.
• Other names normal modes, eigen-states etc.

Quantisationcondition
5
Other boundary conditions

• N.B. not on handout.
• Anti-nodes at each end
• e.g. wind instruments, open organ pipe
  etc.
• kl np
• One node, one anti-node
• e.g. stopped organ pipe.
• kl (n1/2)p

6
Standing waves in 2- and 3-D

• Examples
• acoustics of drums, soundboards etc.
• electron states in 2-D and 3-D potential wells
• black-body radiation (photons in a cavity)
• phonon modes in solids (will be used in Thermal
  Physics to describe the Debye theory for the
  thermal capacity of a solid)
• 2-D standing waves on a rectangular sheet
• Boundary x0, a and y0, b. z2 outside.
• Reflection at each boundary results in a phase
  change of p. Thus,Superposition gives

7
2-D standing wave solution

• Boundary conditions
• Must have Y0 at x0, a and y0, b.
• Each pair of integers, (nx,ny), specifies a
  normal mode.
• Frequency of vibration follows from

Quantisationconditions
8
Typical 2-D modes

• Rectangular membrane
• Circular membrane

9
3-D stabding waves in a box

• Extend argument to 3-D
• boundary conditions lead to quantisation
  conditions
• Thus, gives the frequency of the normal modes
  as