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Irregular Sinusoidal Curves

Date: 08/24/2004 at 23:39:08
From: David
Subject: Irregular sinusoidal curves

Is there a function to graph a sinusoidal curve where the periodicity
remains constant but the distance between the maximum and minimum
values is non-equidistant?  For example, is there a function to plot a
sinusoidal curve with a period of 24 hours but where the time span
from the maximum value to the minimum value is 16 hours and the span
from minimum value to maximum value is 8 hours?

Date: 08/26/2004 at 16:01:54
From: Doctor Douglas
Subject: Re: Irregular sinusoidal curves

Hi, David.  Thanks for writing to the Math Forum.

Yes, you can construct a function that does this by varying parameters 
in the basic sinusoidal function to achieve what you want.  There are 
many ways to do this, and I’ll demonstrate one method below.  Strictly 
speaking, the resulting function is not sinusoidal, but it does have a 
fundamental period T. 

We start from a simple sinusoidal function of the form below, and let 
the phase phi be time-dependent so as to make the maximum and minimum 
occur when we want:

  y = A*sin[2*pi*t/T + phi(t)],      period  T = 24 hr.

Clearly, if phi is a constant, all we do is shift the graph of y 
horizontally—-it is still a sine wave with a half-wavelength between
maximum and minimum and between minimum and maximum.

We know that the sine function takes on its (first) maximum when the
argument of the sine is equal to pi/2 [using radians as the angular
unit of measure].  Let’s force this to occur at t = tmax = +8 hr 
(i.e. t/T = 1/3) by demanding that the argument [**] of the sine above
reach pi/2 when t = tmax:

  pi/2 = 2*pi*(1/3) + phi(T/3).
We also enforce a similar condition on the argument at t = tmin = -8
hr, where we want the sine function to go thru its minimum:

  -pi/2 = 2*pi*(-1/3) + phi(-T/3).

These two equations specify the values of phi at the maximum and 

  phi(T/3) = -pi/6
  phi(-T/3) = +pi/6

and while so far we have not required anything about the behavior of 
phi at other values of t, we might as well choose a simple function 
phi that passes through these two points, and is T-periodic:

         = +pi*(t/T + 1/2)           -T/2 <= t < -T/3
  phi(t) = -pi*t/(2*T)               -T/3 <= t <= T/3
         = -(pi/6) + pi*(t/T - 1/3)   T/3 <  t <= T/2

By using this expression for phi(t) in the equation for y, you will
force the maximum to occur at t = tmax = 8 hr, and the minimum to 
occur at t = tmin = -8 hr, so that the durations between min-max and
max-min are indeed imbalanced, as required.  To reverse the intervals
so that the max-min event is the longer of the two, we can simply
invert the function by multiplying it by –1:

  y = -sin[2*pi*t/T + phi(t)].

I think that this function will serve your needs.  Note that there are 
all sorts of functions phi(t) that will work, as long as they pass 
through the given points at t = T/3 and t = -T/3 and are T-periodic.

The method above is an example of “phase modulation” (PM), and is a 
technique that is exploited in communication systems.  You can also
vary other parameters in the sinusoidal wave

  y = A(t) sin[2*pi*f(t)*t + phi(t)].

If you let A vary with t, it is called “amplitude modulation”.  If you
let f vary with t, it is called “frequency modulation”.  You may have
heard of these terms in the context of AM and FM radio.

- Doctor Douglas, The Math Forum 
Associated Topics:
High School Functions
High School Trigonometry

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