Transmission Line Odd Theory Concept

I need help with an odd concept in transmission line wave reflections. So what I'm reading is that when you have a transmission line, the voltage and current characteristics are repeated every λ/2, they change phase every λ/4, and the impedance is repeated every λ/2 as well. Apparently this idea means you can reflect specific impedances for matching, or that you can use the transmission line for tuning.
There are pictures that show the voltage and current 180° out of phase going through a transmission line, they peak every λ/4 and intersect every λ/4. I just don't understand how the impedance is repeated, or how you can use this idea for anything that was mentioned above. I'm pretty new to transmission line theory so maybe I'm missing something. Help is greatly appreciated

There are interesting things you can do with these properties. For example:

A quarter wavelength long transmission will look like a short if the far end is open circuited. This acts like a notch filter for frequencies at which the line is an odd-number of quarter wavelengths long.

If you short the far end of the line, the input will appear as an open circuit for odd-number f quarter wavelength frequencies.

The input impedance of a half wave long transmission line will equal the impedance at the far end, regardless of the characteristic impedance of the line.

One of the best books I've read on the topic is Reflections, by Walter Maxwell.

Remember, since these relationships are expressed as fractions of wavelength, they apply only to a sinusoidal signal at a particular frequency.
Once you apply a more complex signal, which must be a mixture of frequencies, then at any given point along the line it may be a multiple of half waves (say) at one of those frequencies. At other frequencies the phase may be at, say, 2/3 wave, 1/13 wave, 199/311 wave, or anything. So the impedance at these other frequencies may be dramatically transformed.

So there's nothing magic about transmission lines. These properties are simply a restatement of the physical fact that when a PURE SINUSOIDAL SIGNAL of CONSTANT AMPLITUDE (because ANY modulation adds new frequencies) travels down a transmission line at a finite speed, then there is cyclic variation in Voltage (and current) with respect to time at a fixed point and with respect to distance as you move along the line. In the latter case the Voltage and current must repeat exactly every wavelength and in a simple, predictable way at half and quarter wavelength steps, due to the symmetry of the signal.

As w2aew says, "A quarter wavelength long transmission will look like a short if the far end is open circuited", BUT you will not see a short with a multimeter for example, nor if you measure it at some low frequency like an audio frequency. For them, an open circuit line is still, more or less, an open circuit. Only when you get up to radio frequencies ( depending on the length of the line) will the open circuit line behave like a short circuit. So if the line were 1 metre long, it would be a short at about 75 MHz. To observe this at lower frequencies, the line has to be much longer because the wavelength is much longer: at an audio freq of 10kHz, wavelength is 300 km, so a quarter wave line is 75 km! Even if you played around with lines of this length, you probably wouldn't observe the predicted behaviour, because it would be very far from an ideal transmission line - the resistance of the conductors would not be negligible for a start.

I think the thing is to avoid thinking of impedance as if it were resistance. Perhaps think of a transmission line a bit like a tuned circuit, which can look like a short ( parallel LC) or open circuit (series LC) at DC and can look like something in between to AC signals, depending on their frequency.

Ooops! Forgot the velocity factor, so my rough numbers could be out a bit. Waves travel slower down transmission lines than in free space, so wavelengths are shorter by up to about 50%. So if we're talking common coax, my 75 MHz could be about 50 MHz and the 75 km about 50 km.
I think this does not affect the general point.