How transmission line repeats after $\frac{\lambda}{2}$ distance?

Following two statements made me think that the transmission line repeates after $$\ \frac{ \lambda}{2} \$$ distance:

1. A complete revolution around the Smith chart corresponds to $$\ \frac{ \lambda}{2} \$$ distance on the line
2. Distance between two successive voltage Maxima is $$\ \frac{ \lambda}{2} \$$

In order to proof that the line repeates after $$\ \frac{ \lambda}{2} \$$ distance, my thoughts are as follows:
Consider a two conductor transmission line as shown below:

The instantaneous voltage expression at a distance $$\ z \$$ on the line is given as : $$v(z,t)= V_o^+ e^{(- \alpha z)}cos(wt - \beta z) + V_o^- e^{( \alpha z)}cos(wt + \beta z)$$ $$\implies V_s(z) = V_o^+ e^{(- \alpha z)}cos(-\beta z) + V_o^- e^{( \alpha z)}cos(\beta z) \quad \dots (1)$$ Now, $$\ cos(\beta z) \$$ repeats after a distance of $$\ \lambda \$$ , then $$\ V_s \$$ is also supposed to repeat after a distance of $$\ \lambda \$$;
but according to my conclusion (made in the first line) $$\ V_s \$$ should repeat after a distance of $$\ \frac{\lambda}{2} \$$ (as the line repeats after $$\ \frac{\lambda}{2} \$$ distance)
so, can anyone tell me how $$\ V_s \$$ repeats after a distance of $$\ \frac{\lambda}{2} \$$ ?
or any other approach which will prove that the line repeates after $$\ \frac{ \lambda}{2} \$$ distance

• The zero crossing wavelength.... Jan 24, 2021 at 15:38
• Who says the waveform repeats after a distance of $\frac{\lambda}{2}$? Jan 24, 2021 at 16:15
• You need to add where you're getting wrong information from. Perhaps there is some context there, (like |Vs| not Vs per Stefan's answer), or a clarification that you're measuring the sending voltage on a mis-terminated line (hence lambda/2 because you're seeing the out-and-back round trip) that might make this correct information. Jan 24, 2021 at 17:04
• @BrianDrummond A complete revolution around the Smith chart corresponds to $\frac{ \lambda}{2}$ ; also distance between two successive voltage Maxima is $\frac{ \lambda}{2}$ ; so can't we say the line repeats itself after $\frac{ \lambda}{2}$ distance? Jan 25, 2021 at 5:38
• It would be kind of the same as saying that a car wheel repeats itself when the car is moving. It is a repeating pattern, but not a repeating wheel. Jan 25, 2021 at 12:53

The absolute value of the voltage $$\V_s(z)\$$ repeats after $$\\lambda/2\$$, not $$\V_s(z)\$$.
• Could you please append to your answer that is it true to say that "the line repeats after $\frac{ \lambda}{2}$ distance " , if yes , then can you please show its derivation Jan 25, 2021 at 5:56