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.

I have also posted this in the advanced physics section because to me it seems like it's advanced physics, but maybe to others it may be introductory physics, so i posted it here as well.

Thanks!

 

[anathema]

Thank you for your interesting question. As you correctly pointed out, the telegrapher's equation is relevant to this problem. This equation describes the propagation of signals along a transmission line, such as a nerve fiber or telegraph cable.

In this case, the signal can be written as a sum of harmonics, each with their own wave number and time dependence function. The key to understanding how the signal will behave with time is to look at how these harmonics decay.

If all the harmonics decay at the same rate, then the signal will preserve its shape as time passes. This is because each harmonic will decay at the same rate, meaning that the amplitudes of all the harmonics will decrease proportionally and the overall shape of the signal will remain the same.

On the other hand, if the harmonics with larger wave numbers decay faster, then the signal will smear out with time. This is because the higher frequency components of the signal will decay faster, leading to a decrease in the overall sharpness of the signal.

Similarly, if the harmonics with smaller wave numbers decay faster, the signal will sharpen with time. This is because the lower frequency components of the signal will decay faster, resulting in a higher proportion of higher frequency components and a sharper overall signal.

To determine which of these scenarios will occur, we can look at the telegrapher's equation, which relates the wave number and time dependence of a signal to its decay rate. By manipulating this equation, we can determine the relationship between the wave number and decay rate of each harmonic, and thus predict how the signal will behave with time.

I hope this helps to answer your question. If you have any further questions or need clarification, please don't hesitate to ask.

 

[ogb p]

I can provide some insight into this topic. Harmonics and the Telegrapher's Equation are both important concepts in the study of wave propagation, and they have applications in various fields such as telecommunications, neuroscience, and acoustics.

Firstly, let's define some terms. Harmonics refer to the different components or frequencies that make up a complex wave. In the context of the given problem, the harmonics are represented by the terms V_{n}cos(k_{n}x)f_{n}(t), where n is the index for each harmonic. These harmonics can have different wave numbers (k_{n}) and time-dependence functions (f_{n}). The Telegrapher's Equation, on the other hand, is a partial differential equation that describes the propagation of electrical signals along a transmission line or nerve fiber.

Now, let's consider the given scenario of a small sub-threshold signal on a nerve fiber or telegraph cable. The signal is represented by the sum of harmonics, and we are interested in how the shape of the signal changes with time. The key point to note here is that the behavior of the signal is dependent on the decay rate of the harmonics. If all the harmonics decay at the same rate, the signal will preserve its shape as time passes. This means that the amplitude of each harmonic decreases at the same rate, and the overall shape of the signal remains unchanged.

If the harmonics with larger wave numbers (k_{n}) decay faster, the signal will smear out with time. This is because the higher frequency components of the signal will decay faster than the lower frequency components, causing the overall shape of the signal to change. On the other hand, if the harmonics with smaller wave numbers decay faster, the signal will sharpen with time. This is because the lower frequency components will decay faster, causing the signal to become more dominated by the higher frequency components.

In terms of the Telegrapher's Equation, the decay rate of the harmonics is related to the attenuation constant, which determines how quickly the signal loses its amplitude as it travels along the transmission line or nerve fiber. Therefore, the behavior of the signal can be predicted by solving the Telegrapher's Equation and determining the attenuation constant.

In conclusion, the behavior of a small sub-threshold signal on a nerve fiber or telegraph cable is dependent on the decay rate of its harmonics. If all the harmonics decay at the same rate, the signal