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Lecture 6 - Standing waves

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;

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Lecture 6 - Standing waves

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  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 r=1. The argument, however, is similar. • Again we have a standing wave but the free end of the string (x=0) 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: Y=0 at x=0 andx=l.Or earlier solutiondescribes the wave provided • An integral number of half wavelengths must fit on the string. Using k=2p/l, gives l=nl/2. • In acoustics, these are known as harmonics. Since w=vk 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 = (n+1/2)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 x=0, a; and y=0, 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: Y=0 at x=0, a; and y=0, 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:

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