x = Phase velocity: the velocity of a point of constant phase on the traveling waveform. Think of a train carrying sinusoids. Each flatcar carries one sinusoid having length. If the train is not moving, the phase at any point x is: If the train is moving:

x = 0 + -

+ - Our choice for the position of the origin, x = 0, was totally arbitrary!! Any of these forms are valid for expressing a traveling wave moving in the positive x direction!

For traveling waves moving in the negative x direction, the sign on one of the terms of the phase expression must be reversed:

The Cowboy Way A real cowboy uses complex exponentials. The preferred form for voltage waveforms is: … for traveling waves moving in the positive x direction. … for traveling waves moving in the negative x direction. Complex constants representing the magnitudes and reference phases of the traveling waves.

S Train Station x = x s xsxs  S = -2  x s You How many cars are in the station at any time? What do you see, standing at the station entrance? You see each car coming out exactly n S (the fractional part of N S ) cars ahead of each car going in. The phase lead of the sinusoid coming out with respect to the phase of the sinusoid going in is equal to two pi times n S. What has changed? Only the observer’s position! Each car coming out is exactly N S cars ahead of each car going in.

Voltage Maxima Voltage Minima