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Refractive medium: Calculating change of wavelength

Discussion in 'Electronic Design' started by [email protected], Apr 4, 2013.

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  1. Guest

    Does anyone know the procedure (formula) for calculating wavelength of
    radio waves, expressed in MHz, according the refractive index of the
    transmission medium?

    For example, water has a refractive idex of about 1.3. The normal
    wavelength of a 900MHz signal is 15.7cm. What does the latter change
    to when propagating through a sealed container of water?

    Martin Crawford
  2. Guest

    Mark L. Fergerson
  3. Guest

  4. The dielectric properties of water are complex at radio frequencies.
    See for example:

  5. whit3rd

    whit3rd Guest

    F (cycles/second) * Lambda (meters/cycle) = Speed(meters/second)

    There's complications, of course, if the material has more than one
    kind of internal wave propogation (Karo corn syrup has two indexes
    of refraction for visible light).
  6. Artemus

    Artemus Guest

    The speed of light is less than in a vacuum.
  7. Martin Brown

    Martin Brown Guest

    The very simplest definition of refractive index is

    n = <speed of light in vacuum>/<speed of light in material>

    Basically you only have to consider continuity of the wave E field at
    the surface boundary to see how the only possible solution when its
    speed slows down is for the wavelength to shorten proportionately.

    So wavelength in refractive media = lambda_vacuum/n

    It is left as an exercise for the reader to prove this.

    There is actually a tiny refractive index of air vs vacuum, but I cannot
    be bothered to look up what it is at 900MHz.

    So your concrete number example is 15.7/1.3 = 12.077cm

    There is a subtle distinction between phase velocity of the wave crests
    and group velocity of a signal travelling in a dispersive medium like a
    waveguide or edge of a sharp spectral line which is usually the
    fundamental misunderstanding of physics at the root of any claims for
    FTL signalling by electronics engineers.

    Interesting fact:

    Minor corrections for imperfect vacuum in historic experiments used to
    measure the speed of light in a vacuum have been responsible for
    systematic errors that at one point far exceeded the error bars.

    Plotting contemporaneous speed of light with error bars against time is
    very informative - basically every subsequent experimenter that refined
    this method reproduced the original (very famous) experimenters error
    exactly. It was only when a new even more accurate technique came along
    that the discrepancy was observed and the systematic error corrected.

    One of the Relativity texts of the 1970's had this plot in as a salutary
    lesson about assuming that famous experimentalists were infallible and
    measurement errors free from systematic effects.
    (actually if anyone knows of an online retailer with the original book I
    would love to obtain a copy - sadly I can't recall the title)
  8. Guest

    Empirical fact. See Martin Brown's post for expansion.
    Plug in the numbers and do the math. Easy peasy.

    Mark L. Fergerson
  9. Guest

    On Fri, 05 Apr 2013 07:57:27 +0100, Martin Brown
    Thanks to everyone for their helpful answers.

    Martin Crawford
  10. Wimpie

    Wimpie Guest

    El 04-04-13 4:35, escribió:
    Hello Martin,

    Water at 900 MHz doesn't behave as a lossless dielectric like PE, PTFE
    or PP, especially when it is conducting due to contamination.

    The phase velocity in a medium:

    Vph = 1/Re{(sqrt(u*e) }

    Re{..} = Real part of ...
    u = complex permeability (4*pi*10e-7 H/m for water),
    e = complex permittivity for the medium (er*8.854e-12 F/m)

    If you convert u*e to polar notation, taking square root from a
    complex number is just halving the argument and taking the root from
    the modulus.

    Re(Z) = |Z|*cos(arg(Z))

    Though the real part of relative permittivity (er) is around 78 over a
    realtive wide range of frequencies, above 1 GHz, er drops with
    increasing frequency. The imaginary part (the loss part) changes with
    frequency, and depends on the conductivity (due to contaminations).

    For 900 MHz and pure water, er = 78-j8
    for 900 MHz and seawater, er = 78-j60

    When discussing pure water you may use er = 78 + j0 as a first
    approximation, this saves you from complex calculus. This will result
    in Vph = 0.34e8 m/s (that is 0.11*c0)

    Wavelength follows from lambda = c/f = vph/f
    c = propagation velocity (m/s), f = frequency (Hz)

    lambda (900 MHz) = 0.34e8/900M = 38 mm.

    Best regards,
  11. Wimpie

    Wimpie Guest

    El 05-04-13 13:24, escribió:
    Hello Martin,

    If a good answer is important for you, I would spend some time on this
    topic. You can't use the refractive index for optical frequencies for
    the radio frequency range.

    Is your challenge related to pure water, or for example seawater?

    Best regards,
  12. Guest

    On Thu, 04 Apr 2013 13:35:54 +1100, wrote:

    Can someone please explain the difference in methodolgies suggested by
    Martin Brown and Wimpie?

    The result for the same frequency of 900MHz in water appears to differ
    substantially, eg. 38mm and 120mm.

    Which one applies to my OP?

    Martin Crawford
  13. Wimpie

    Wimpie Guest

    El 04-04-13 11:25, escribió:
    Hello Martin,

    If you are familiar with complex calculus it is not that difficult to
    calculate the wavelength and attenuation. However you need to know
    the complex permittivity of the material at the operating frequency.
    When using magnetic material, you also need the complex permeability.
    You generally can't use tables that are valid for optical frequencies.

    Complex permittivity is mostly specified as relative epsilon in the
    form of eps.r' and eps.r''

    Complex permittivity = eps = 8.854e-12(eps.r' - j*eps.r'')

    Eps.r''/eps.r' = tan(delta).

    The complex material properties fit into the formula for the complex
    propagation constant (gamma).

    Gamma = Alpha + j*Beta = j*omega*sqrt(u*eps)

    For non magnetic materials u = 4*pi*e-7 H/m,
    eps = complex permittivity.

    You may know that: Beta = 2*pi/lambda and

    36% Penetration depth (skin depth) = 1/alpha [m]

    After some calculus, you will arrive at:

    Lambda = 1/( f*Re{sqrt(u*eps)} )

    Only when the material is low loss (so Eps.r''/eps.r' <<1), the
    formula will convert to:

    Lambda = (lambda.freespace)/sqrt(eps.r), where eps.r is a just a real
    number because we assumed "low loss" material.

    If you do some google search you may find graphs that show wavelength
    versus frequency for water, but knowing the formulas may give you a
    means to check third party data.

    Best regards,
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