Let's suppose you have equation of a dying oscillation such as:
a*e^(-b*t)*cos(d*x-f)
e = math constant for exponential functions
How do you determine how long will it take for the function to be always less than a particular value?
a*e^(-b*t)*cos(d*x-f)
e = math constant for exponential functions
How do you determine how long will it take for the function to be always less than a particular value?
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a*e^(-b*t)*cos(d*x -f) is an exponential dampened sinusoid
Let k be the particular value you want a*e^(-b*t)*cos(d*x-f) to be less than
ignore the cos (d*x-f) as this can never be greater than 1
k = a*e^(-b*t) then solve for t
ln(k/a ) = -b*t
say k is 0.5 a
ln (0.5) = -b*t
t = 0.693/b
This will work for any k as long as it is less than a
Let k be the particular value you want a*e^(-b*t)*cos(d*x-f) to be less than
ignore the cos (d*x-f) as this can never be greater than 1
k = a*e^(-b*t) then solve for t
ln(k/a ) = -b*t
say k is 0.5 a
ln (0.5) = -b*t
t = 0.693/b
This will work for any k as long as it is less than a