STANDING WAVES

STANDING WAVES ON A STRING

• standing waves on a string:
  reflection of wave at end of string, interference of outgoing with reflected wave ``standing wave''
• nodes
  string fixed at ends displacement at end must be = 0
  ``(displacement) nodes'' at ends of string not all wavelengths possible
• length must be an integer multiple of half-wavelengths:

  

• possible wavelengths are:

  

• possible frequencies:

  

  called ``characteristic'' or ``natural'' frequencies of the string

  

  is the ``fundamental frequency'', or ``first harmonic''; the others are called ``harmonics'' or ``overtones''

RESONANCE:
when a system is excited by a periodic disturbance whose frequency equals one of its characteristic frequencies, a standing wave develops in the system, with large amplitudes
at resonance, energy transfer to the system is maximal
examples:

• pushing a swing,
  
• shape of throat and nasal cavity overtones sound of voice
  
• musical instruments
  
• Tacoma Narrows Bridge
  
• oscillator circuits in radio and TV

THE DUAL NATURE OF LIGHT

• wave theory of light explains most phenomena involving light:
  • propagation in straight line
  • reflection
  • refraction
  • superposition, interference, diffraction
  • polarization
  • Doppler effect

• wave theory does not explain:
  • frequency dependence of thermal radiation
  • photoelectric effect

THERMAL RADIATION

• experimental observations:
  • atoms of a hot solid emit radiation;
  • increase in temperature more radiation, and component of maximum intensity shifted towards higher frequency (shorter wavelength)
• ``classical'' explanation: the hotter the solid, the more vibrational energy
  higher frequency of vibration of atoms/electrons
  higher frequency of radiation
• but frequency spectrum of this radiation (``black body radiation'' calculated within framework of electromagnetism and thermodynamics did not agree with measured spectrum;
  predicted ``ultraviolet catastrophe''
  

  

• Max Planck's hypothesis (1900):
  energy is quantized;
  ``oscillators'' (oscillating atoms) can only have certain amounts of energy
  relation between energy and frequency of oscillator:

  

  h = ``Planck's constant''

• calculation of black body spectrum using Planck's hypothesis gives formula (``Planck formula'') which describes measured spectra.
• = first evidence that energy is quantized

PHOTOELECTRIC EFFECT:

• (first observed by Heinrich Hertz in 1887)
• Electrons are emitted when certain metallic materials exposed to light
  (now used in photocells in cameras, and solar energy cells)
• some aspects of photoelectric effect could not be explained by classical theory:
  • classical theory: if light continuos flow of e.m. energy takes some (calculable) time for wave to supply sufficient energy for electron to be emitted
  • find experimentally: current flows almost immediately upon exposure to light

  • classical theory: light of any frequency could cause photoelectric effect - need only sufficient intensity
  • find experimentally: only light with frequency above certain minimum frequency causes electrons to be emitted

  • classical theory: energy of electrons depends on light intensity
  • find experimentally: energy of electrons depends on frequency

• Albert Einstein's explanation:
  assume that not only energy in atoms is quantized, but also energy carried by light
  light comes in ``packets of energy'' called light quanta or photons
  energy of one photon = h f, where f = frequency of the light.
  with this assumption, all aspects of photoelectric effect could be explained
• photon energy vs color of light:

  

  blue light has more energy than red light
  

IS LIGHT A WAVE OR A PARTICLE?
answer:
it is both, depending on what question you ask, it has a ``wave'' aspect and a ``particle'' aspect

Note:
according to quantum theory, ``particles'' (e.g. electrons, protons,..) have also a ``wave'' aspect!
(depends on what question you ask)