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| Wave equations for a lossless electronic signal cable: $ ~ ~ ~ L ~ { \Large { { \partial^2 I } \over { \partial t^2 } } } ~ = ~ { \Large { 1 \over C } ~ { { \partial^2 I } \over { \partial x^2 } } } ~ ~ ~ ~ ~ { \Large { 1 \over C } ~ { { \partial^2 V } \over { \partial t^2 } } } ~ = ~ { L ~ \Large { { \partial^2 V } \over { \partial x^2 } } }$ | '''Wave equations for a lossless electronic signal cable:''' $~~~ L C ~ { \Large {{ \partial^2 I } \over { \partial t^2 }} } ~=~ { \Large {{ \partial^2 I } \over { \partial x^2 }} } ~~~~~ L C ~ { \Large {{ \partial^2 V } \over { \partial t^2 }} } ~=~ { \Large {{ \partial^2 V } \over { \partial x^2 }}} $ |
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| . wavenumber $ ~ k ~ = ~ \pm \sqrt{ L C } \omega ~ ~ $ radians per meter . speed $ ~ v ~ = ~ \omega / k ~ = ~ \pm 1 / \sqrt{ L C } ~ ~ $ a large fraction of the speed of light, a function of materials. |
. wavenumber $ ~ k ~ = ~ \pm \sqrt{ L C } ~ \omega ~ ~ $ radians per meter . speed $ ~ v ~ = ~ \omega / k ~ = ~ \pm 1 / \sqrt{ L C } ~ ~ $ in the +x or -x direction, a large fraction of the speed of light . a function of the materials used, typically around 0.5'''c''' or 150 million meters per second |
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| . a function of the materials used and cross section, typically around 50 ohms but can be higher than 100 ohms and lower than 10 ohms | |
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| . MoreLater |
Acoustic Climber for Space Elevator
THe current reference space elevator design assumes solar-powered climbers. This assumes that vast areas of solar panels can cantilever from the sides of a climber - in gravity - and provide megawatts of climb power, while being lightweight and affordable. The example pictured is a DLR solar sail, intended for microgravity, NOT an array of solar cells. Solar sails are ultrathin plastic films covered with just enough shiny metal to reflect light. Aluminum conductivity is 2.8e-8 ohm/meter; a film with 10 ohm per square resistivity (95% reflective) is 2.8e-9 meters thick - a few atomic layers. The density is 2700 kg per cubic meter, works out to 8 kilograms per square kilometer on top of the plastic. This is far less than actual satellite solar panels (300 W/m2, 1 kg/m2), which are designed for microgravity, not to deploy in a gravity field. These arrays are stabilized by gravitational gradients in orbit, and they are test-deployed on the ground hanging sideways from heavy structure.
Instead, a superstrong tether can carry megawatts of subkilohertz acoustic power, which can be impedance-matched and mechanically rectified (mad handwaving here) to produce climber thrust. The acoustic transmitters on the ground and at GEO node can provide 2 MW and 10 MW of climb power respectively, more by trading off climber mass, gravitational weight, and climber speed. Climbers will have a mechanical receiver that transforms vibration to rotary wheel motion.
That's the old stuff. This is morphing into a pair of low inertia electrical motors separated by a quarter wave, with electrical power conversion at each motor, with power cable and stiff tether between.
Analogy Between an Electronic Signal Cable and a Stiff Tether
All units MKS: meters, kilograms, seconds, volts, amperes (amps), radians
( FYI: if you don't think radians are a unit, you are turned around, and can't distinguish energy from torque. 
)
Light speed and sound speed
Wave equations for a lossless electronic signal cable: ~~~ L C ~ { \Large {{ \partial^2 I } \over { \partial t^2 }} } ~=~ { \Large {{ \partial^2 I } \over { \partial x^2 }} } ~~~~~ L C ~ { \Large {{ \partial^2 V } \over { \partial t^2 }} } ~=~ { \Large {{ \partial^2 V } \over { \partial x^2 }}}
C = capacitance per unit length, farads per meter
L = inductance per unit length, henrys per meter
Sinusoidal solutions (many others are possible): ~~~I = I_0 \sin( \omega t + k x ) amps ~~~~~ V = V_0 \sin( \omega t + k x ) volts
frequency ~ \omega ~ ~ radians per second
wavenumber ~ k ~ = ~ \pm \sqrt{ L C } ~ \omega ~ ~ radians per meter
speed ~ v ~ = ~ \omega / k ~ = ~ \pm 1 / \sqrt{ L C } ~ ~ in the +x or -x direction, a large fraction of the speed of light
a function of the materials used, typically around 0.5c or 150 million meters per second
impedance ~ Z ~ = ~ \sqrt{ L / C } ~ = ~ V_0 / I_0 ~ ~ ohms
- a function of the materials used and cross section, typically around 50 ohms but can be higher than 100 ohms and lower than 10 ohms