What does an SWR of 2:1 actually cost you? Slide the SWR and power to see how much comes back, how much reaches the antenna, and why a long run of lossy coax can make a poor antenna look good.
Against a 50 Ω line. Use a minus sign for capacitive X.
Rough guide at 2 m for 30 m (100 ft): RG-58 about 6 dB, RG-8X about 4.5 dB, LMR-400 about 1.5 dB.
Under 1.5:1: don't chase it. Less than 4% comes back, and you'd never notice it on the air.
1.5 to 2:1: fine for most radios. You lose about half a decibel at most, less than a tenth of an S-unit.
2 to 3:1: many modern radios start cutting back power to protect themselves. A tuner keeps the radio happy, but it doesn't fix the antenna.
Over 3:1: something is off: wrong length, a bad connector, water in the coax, or the wrong band. Find it before you run high power.
The coax trap: lossy coax eats the reflected power twice, once going up and once coming back. So the meter at the radio reads a nicer SWR than the antenna really has. A "perfect" SWR on a long run of thin coax can mean the coax is just hot. Tom Buchanan VE6ARG's articles Antenna Tuner and A Transmatch Tuner in the VE6ARG Library dig into this.
Maths: Γ = (SWR − 1) / (SWR + 1); reflected = Γ²; return loss = −20 log Γ; mismatch loss = −10 log(1 − Γ²). Line loss with mismatch uses the standard formula with a = 10^(loss/10). "Delivered" assumes nothing re-matches the reflected power; with a tuner, most of it is sent back up the line.