Harmonics and Telegrapher's Equation

[harshey]

Homework Statement

Consider a small (sub-threshold) signal on a nerve fiber or a telegraph cable. The signal can be written as a sum of "harmonics"

V(x,t) = $$\sum(V_{n}cos(k_{n}x)f_{n}(t)$$

where n is some index identifying the terms, the V$$_{n}$$ are constants, the k$$_{n}$$ are wave numbers for the harmonics, and the f$$_{n}$$ are time-dependence functions to be determined [with f$$_{n}$$ = 1 for all n]. We have seen that a uniform signal (k$$_{n}$$ = 0) decays exponentially in time. For the above more general signal, if the harmonics all decay at the same rate, the signal preserves its shape as time passes. If harmonics with larger k$$_{n}$$ values decay faster, the signal smears out with time; if those with smaller k$$_{n}$$ values decay faster, the shape of the signal sharpens with time. Which happens?

Sorry, I accidentally put all the subscripts as superscripts in the paragraph above, I'm not sure how to change it so I'm sorry. So please assume all the superscripts in the above paragraph only are subscripts. Thanks.


Relevent equations

I think the telegrapher's equation is relevent, but I'm not sure how to manipulate it.


Attempt at a solution

I have tried solving this problem but the only way I have been able to so far is to read it over and over and try and understand everything that is asked. I'm sorry I haven't come up with a definite track to try and solve the problem because I honestly haven't been able to do it.

 

[Jhero]

I think the first step would be to rewrite the equation in terms of the telegrapher's equation, which is typically used to describe the propagation of signals on a transmission line or nerve fiber. This equation is given as:

\frac{\partial^{2}V}{\partial x^{2}} = LC\frac{\partial^{2}V}{\partial t^{2}} - RC\frac{\partial V}{\partial t}

where V is the voltage, L is the inductance per unit length, C is the capacitance per unit length, and R is the resistance per unit length.

From the given equation, we can see that each term is a harmonic with its own wave number and time-dependence function. This means that each term can be written in the form of a traveling wave, with a certain amplitude and phase. The telegrapher's equation deals with the propagation of these waves, so we can use it to analyze the behavior of the signal as time passes.

To determine whether the signal sharpens or smears out with time, we need to look at the behavior of the different harmonics. If the harmonics with larger wave numbers (k_n) decay faster, this means that their amplitudes decrease more quickly over time. This would result in a decrease in the overall amplitude of the signal, causing it to smear out or become less defined with time. On the other hand, if the harmonics with smaller wave numbers decay faster, this means that their amplitudes decrease more slowly over time. This would result in an increase in the overall amplitude of the signal, causing it to sharpen or become more defined with time.

In terms of the telegrapher's equation, this can be seen by looking at the RC term. The resistance (R) is related to the decay rate of the signal, with a higher resistance leading to a faster decay. So if the harmonics with larger wave numbers have a higher resistance, they will decay faster and the signal will smear out. If the harmonics with smaller wave numbers have a higher resistance, they will decay slower and the signal will sharpen.

Overall, the behavior of the signal depends on the specific values of the wave numbers (k_n) and the resistance (R) for each harmonic. It is possible for both scenarios to occur, depending on the specific values chosen. So to determine exactly what happens, we would need to plug in values for these variables and solve the telegrapher