You should not doubt yourself . It took 2 hours pacing the floor but at the end your original assessment was correct , all waves are the same.
The infinitely long coax cable with regulated 10 volt power supply is the same as a infinitely long string for both longitudinal and transverse waves.
If I pull on a string with 10 ounces , 10 volts , my hand will move quickly. Quickly means high current smoking your nice 10 volt power supply. What went wrong. The natural series inductance of the coax cable went wrong. Must place lead marbles every inch on the string for the inductance of the cable. Now it works. When I pull on the string with constant 10 ounces , 10 volts , my hand does not move quickly as I now have to drag ever more and more lead balls as the string tightens up. This will lead to constant current , hand velocity , of 200 m amps for 10 volts at 50 ohms Z.
As to your transverse wave by pulling the string up with 10 ounces. I have a marked preference for side ways force to take gravity off the table. You can not just move the string sideways. You have to keep moving the string sideways forever with a 10 ounce force for a 10 volt power supply analogy of constant force. Done this way it ends up the same as the coax cable with constant velocity sideways at constant 10 ounce force as you start to accelerate more and more lead marbles on the string.
As for water we go from 1 dimension to 2 dimensions. The wave will attenuate faster. This complicates it. I will avoid this by asking another question. How can a wave in water with a parabolic reflector travel in a straight line in one dimension only. What keeps the sides of the wave crest from falling out sideways? And while I am at it why are the little wave crests on the side of a boat always the same angle regardless of the speed of the boat. I think it is 18 degrees but I could be wrong. If you go to a different plant with different gravity it is 12 degrees regardless of how fast their boats are going. In short the angle of the little water crests coming off a boat depends only on the amount of gravity. Why? No answers on my end. I wish there was an electrical analogy for this as I have a feeling something cool would come out of it.
All along I was considering
transverse waves/pulses on my stretched spring, so let’s assume transverse waves. I am happy to move the spring sideways, or do the experiment in deep space, so that gravity is not a player. As to the ‘lead marbles’ on the string, you don’t need to do that, because all springs or elastic rubber band already have a characteristic mass-per-length, so you don’t need to add additional masses.
What you say is exactly correct. I salute you, Sir.
I agree with you that the correct analogy is Force-Voltage and Velocity-Current.
Taking things even further, Mass-Inductance and Compliance-Capacitance.
A stretched spring is a distributed Mass-Spring system, so our analogy is that an electrical transmission line is a distributed LC system, which is correct.
Then it all works, just as you say, and I’ll agree that the math for one applies exactly to the math for then other, by swapping variables as described above.
That said, we are a long way from how waves/pulses on stretched springs are usually presented in textbooks. Generating waves/pulses on a slinky spring is a classic high-school experiment. Textbooks and videos love showing creation of a wave or pulse on a long spring. You move your hand back and forth with a displacement of say +- 200mm, and you get a wave with the same amplitude. The peak-peak amplitude of a wave on a long spring is taken as the transverse distance between peaks. I’m sure that’s what all the text books say. You just move the spring sideways to create a wave/pulse, and how far you move it sideways dictates the amplitude of the pulse. Can you show me a textbook or web tutorial that describes transverse waves on a spring differently to this? The tutorial video on this thread certainly shows waves in this way.
However, what we are saying is that if you want to treat a pulse propagating along a stretched spring as being analogous to a wave/pulse propagating along a transmission line, then the normal interpretation of the pulse on the spring is not really correct, but a simplistic ‘con job’ that appears OK at first sight, but ultimately fails to be analogous to wave propagation in a coaxial cable, for the reasons I gave in my previous posting.
Just as you say, the wave amplitude that is Volts on the electrical transmission line needs to be equated to Force on the stretched spring. This is curious but true. The observed shape of the stretched spring should not be regarded as the amplitude of the propagating wave at all, but actually represents the time integral of the current in the analogous model of a wave propagating along a coaxial cable. To put that another way, when you move your hand to produce a shape on the spring, it is the FORCE that your hand exerts that represents the
voltage amplitude of the equivalent coax wave, and the VELOCITY (dx/dt) of your hand represents the
current amplitude of the equivalent coax wave.
If you produce a steady sine-wave with the movement of your hand, then the velocity of your hand is a COS wave, which looks identical except for 90 degree phase shift. Thus, when all the textbooks/videos show a lovely sine wave on a stretched spring, then it’s OK to say that the visible sine wave on the spring is analogous to a sine wave propagating along a coax cable, because SIN and COS waves look the same.
However, if you want to produce a wave or pulse on your stretched spring that is equivalent to a
square wave in a coaxial cable then it’s a different story. To produce a wave on a stretched spring
that is equivalent to a square wave propagating along a coax cable, then your hand needs to produce a constant FORCE in one direction, followed by the same constant force in the other direction, and so on, with the result that your hand will move at constant speed in one direction, followed by constant speed in the other direction etc, so that the shape produced on the spring will actually look like a triangle wave.
Interesting.
I suppose it’s OK to still describe the moving shapes on a stretched spring as ‘propagating pulses/waves’, just so long as it is realized that such ‘waves’ are not directly analogous with waves/pulses propagating along an electrical transmission line.
As for water waves, I might talk about them later, because I'm thinking that the usual hand-waving that says they are 'just the same thing' isn't really right either.