Phase DifferenceDate: 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/ |
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