Phase Difference

Date: 05/30/2003 at 01:13:38
From: Mark Meyer
Subject: Phase Difference

Does the sine wave lag or lead the cosine wave by pi/2?

They are obviously 90 degrees out of phase, but half of the resources 
I have seen say the sine leads and the others say the sine lags the 
cosine wave.

Date: 05/30/2003 at 13:20:06
From: Doctor Douglas
Subject: Re: Phase Difference

Hi Mark,

Thanks for writing to the Math Forum.

There are some common conventions here that can cause confusion.
Because "lead" and "lag" have to do with one thing happening before
or after another, I will use t as the independent variable, so that
it reminds us that everything happens as a function of time.
 
First, a crude drawing of sin(t) and cos(t) near the origin:

       y
       |                          S:  y = sin(t) goes thru the
  1 -- C    S             C           origin at O
    C  | SC    S        C 
       |S  C    S      C          C:  y = cos(t) goes thru the
   ----O----C----S----C-----t         point (0,1).
      S|     C    S  C  
     S |      C    SC    S
 -1 -- +         C    S

If we let time progress, we move to the right on this graph. So we see 
that C hits its maximum 1/4 cycle before S hits its maximum (e.g. C 
has a maximum at t=0 and S has a maximum at t=pi/2).  Thus, we say 
that C leads S. I think that most people follow this convention.

Now, suppose you have a sine "wave" in space, and you let it propagate 
to the right with speed v. The equation for this curve, as a function 
of both position x and time t is

   y1 = sin(x - vt).

You can see that as time t increases, x must also increase (by an
amount vt), so that the graph of the curve slides to the right as t 
increases. If we had a similar cosine wave

  y2 = cos(x - vt),

which of y1 and y2 are the leader and lagger now? If you consider both 
functions y1 and y2 at a fixed point x (say x=0), you can convince 
yourself that it is now the sine wave (y1) that leads the cosine (y2):  
at t=0, y1 is crossing (downward) through zero, and y2 is at its 
maximum. Therefore, the cosine will, in a quarter-period, also have to 
cross downward through zero. This is because the variable t enters the 
argument of the trig functions y1 and y2 with a negative coefficient.

So you see it matters what you consider to be the "wave" - is it 
simply the value of y(t) as t increases? or is it the graph of 
y=f(x,t) as t increases?

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/