x = 0 +-

𝑣𝑝=𝑑𝑥𝑑𝑡

>0 𝑇=𝜆𝑣𝑝

=1𝑓=

2𝜋𝜔

𝜔=2𝜋 𝑓 =2𝜋𝑣𝑝

𝜆=𝛽𝑣𝑝

𝑦=cos𝜃𝜃 (𝑥 )=2𝜋𝑥𝜆

=β 𝑥

Think of a train carrying sinusoids. Each flatcar carries one sinusoid having length l.

If the train is not moving, the phase at any point x is:

If the train is moving:

𝛽=2𝜋𝜆

x = 0 +-

𝑦=cos𝜃𝜃 (𝑥 )|𝑡=0=𝛽 𝑥

Consider the position x = 0.

At time , the point on the train passing x = 0 will be the point on the train which was at when t = 0.

The phase associated with that point is :𝜃 (0 , �̂� )=𝜃 ( �̂� , 0 )=2𝜋 (−𝑣𝑝 �̂�𝜆 )=− 𝛽𝑣𝑝𝑡=−𝜔𝑡

For 𝑥 ≠ 0 ,𝜃 (𝑥 , 𝑡 )=𝛽 𝑥−𝜔𝑡

𝛽=2𝜋𝜆

𝜔=𝛽𝑣𝑝

�̂�=−𝑣𝑝 �̂�

𝑦=cos (𝛽 𝑥−𝜔𝑡 )

𝑦=cos (−𝜔�̂� )

x = 0 +-

𝜃 (𝑥 )|𝑡=0=𝛽 𝑥𝛽=2𝜋𝜆 𝜔=𝛽𝑣𝑝 𝑦=cos (𝛽 𝑥−𝜔𝑡 )

Our choice for the position of the origin, x = 0, was totally arbitrary!!

𝑦=sin ( 𝛽 𝑥−𝜔𝑡 )=−sin (𝜔𝑡−𝛽 𝑥 )

𝑦=− cos (𝛽 𝑥−𝜔𝑡 )=− cos (𝜔𝑡−𝛽 𝑥 )

𝑦=− sin (𝛽 𝑥−𝜔𝑡 )=sin (𝜔𝑡−𝛽 𝑥 )

¿cos (𝜔𝑡− 𝛽 𝑥 )

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:

𝑦=sin ( 𝛽 𝑥+𝜔𝑡 )=−sin (−𝜔𝑡− 𝛽 𝑥 )

𝑦=− cos (𝛽 𝑥+𝜔𝑡 )=−cos (−𝜔𝑡− 𝛽 𝑥 )

𝑦=− sin (𝛽 𝑥+𝜔𝑡 )=sin (−𝜔𝑡− 𝛽 𝑥 )

𝑦=cos (𝛽 𝑥+𝜔𝑡 )¿cos (−𝜔 𝑡−𝛽 𝑥 )

The Cowboy WayA real cowboy uses complex exponentials. The preferred form for voltage waveforms is:

~𝑉 𝐹=𝑽+¿𝑒 𝑗(𝜔 𝑡− 𝛽𝑥 )¿

~𝑉 𝑅=𝑽−𝑒 𝑗 (𝜔 𝑡+𝛽 𝑥 )

… 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 = xs

Dxs DS = -2Dxs

You

How many cars are in the station at any time?

𝑁 𝑆=𝑆𝜆

What do you see, standing at the station entrance?

You see each car coming out exactly nS (the fractional part of NS) 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 nS .

𝜙  𝑆=2𝜋𝑁 𝑆≡ 2𝜋 𝑛𝑆

What has changed?

Only the observer’s position!

Each car coming out is exactly NS cars ahead of each car going in.

Δ𝜙𝑆=− 2 𝛽 Δ𝑥𝑆

Voltage MaximaVoltage Minima

𝜆2

𝜆2𝜆4