Adding or removing energy from a guitar string.

[Spinnor]

Say we have a guitar with electrical pickups near each end of one string. Say this guitar string is ideal in that it looses no energy to any form of "friction". Say the electrical pickups can act to both remove and add energy to the guitar string in the two dimensions perpendicular to the string. Say we can add energy to produce both the first and second excited states of the string. Consider vibrations where the string rotates about the axis of rest. Let us add energy to the guitar string at rest and produce the first rotating harmonic. As the pickups can act in reverse and remove energy let us do this and bring the string to rest. For fun calculate both the forcing and "anti-forcing" function that will work.

With the above setup can I now do the following. Excite the first harmonic and latter both remove energy from first harmonic and add energy to the second harmonic? Could my idealized pickups do "two things as once", destroy one state of the string and create another state at the same time?

Would the following function approximate what could go on,

ψ =
exp(-δt)sin(x)exp(+or-iωt) + [1 - exp(-δt)]sin(2x)exp(+or-i[2ωt+α])

What does the second rotating harmonic look like from the point of view of someone rotating "with" the first rotating harmonic and visa-verse (edit, when equal parts of both waves)?

Thanks for any help!

 

[Spinnor]

...

ψ =
exp(-δt)sin(x)exp(+or-iωt) + [1 - exp(-δt)]sin(2x)exp(+or-i[2ωt+α])

What does the second rotating harmonic look like from the point of view of someone rotating "with" the first rotating harmonic and visa-verse (edit, when equal parts of both waves)?

Thanks for any help!

Just multiply ψ by the right factor exp(+or-iωt) or exp(+or-i2ωt)?

Say ψ = .5sin(x)exp(iωt) + .5sin(2x)exp(i[2ωt+α]) -->

ψ = .5sin(x) + .5sin(2x)exp(i[ωt+α])

We would "see" .5sin(2x)exp(i[ωt+α]) rotating about .5sin(x)?

 

[Rap]

With the above setup can I now do the following. Excite the first harmonic and latter both remove energy from first harmonic and add energy to the second harmonic? Could my idealized pickups do "two things as once", destroy one state of the string and create another state at the same time?

If the ideal setup is linear, then yes, you can. Linear means that if $\psi_1$ and $\psi_2$ are possible vibrations of the string then so is $C_1\psi_1+C_2\psi_2$ where $C_1$ and $C_2$ are constants. If $V_1$ is the voltage to the pickups that you use to destroy the second wave, and $V_2$ is the voltage you use to create the second, then $V_1+V_2$ will both destroy the first and create the second.