1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Wave decay rate

  1. Sep 25, 2005 #1
    hi all.
    Some clarification on this would be helpful to get me going in the correct direction.
    For a specified system, I'm trying to prove that the time it takes for the system to decay to 1/e of its original value (which works out to ~36.8%), takes a certain amount of time. The actual values are unimportant but the process is.
    I have gone through my classical mech book-- 4th ed of Marion Thornton, as well as my diff/eq book-- 5th ed of Nagle, Saff, Snider, and of course my waves and oscillation text-- A.P.French, and cannot decipher what seems-- or I thought would be-- a fairly straightforward problem.
    I'm not schooled/skilled in latex, so please bear with my "hand version."
    I've taken the time derivative of the energy, and get a m/s^3 function for my acceleration value. With the values for b, k, and m, I do not get the time I'm looking to prove.
    m*x_dbldot + b*x_dot + k*x = 0
    Where x(t) = (A*exp(omega*t) +B*exp(-omega*t)
    The rate given for decay to 1/e is:
    Where delta_E is given by -b*E/(m*nu)
    where nu is given by omega_o/2pi.
    I've also tried the quality value Q for this. I know I'm missing something, but can't quite identify it.
    A detailed explanation of this would be deeply appreciated.
    Best regards,
    Thank you.
    Last edited: Sep 25, 2005
  2. jcsd
  3. Sep 25, 2005 #2


    User Avatar
    Homework Helper

    I'm not clear what you want ...

    You know that for this viscous-damped oscillator
    the envelope function is exponential, and you
    even have the right exponent!

    You already have t in the envelope function.

    OK, did you forget that the E envelope
    and the x-envelope are related by E = ½k x^2 ?

    for reasonably small values of "b", the solution is
    x approx. A exp(-omega*t)*sin(w_o*t) ,
    because the natural frequency isn't changed much.

    You don't want exponential growth curve, do you?
    (I mean, set your A=0 and rename B=Amplitude)
  4. Sep 25, 2005 #3
    that's part of what I meant when I said that I'd taken the time derivative of the energy equation.
    E= m/2 (x_dot)^2 + k/2 *x^2
    Based on my energy of the system, I need to then proof that the time only takes a certain amount of time to decay to 1/e.
    E(t)= E(0)/e
    Sounds like I'm not the only one that's struggling with the decay function.
    I don't know how to explain it any better. that's part of what's confusing me, and why I posted.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook