Question about Waves on a String

[Abdullah Almosalami]

Summary:: Could you send a second wave pulse down a string that would overtake an earlier wave pulse?

Got this question in my physics textbook. Ignoring reflection (i.e., you had a very long string), say you send a transverse wave pulse down a string fixed on its other end to a wall. Could you somehow send another wave pulse that overtakes this initial pulse?

My Initial Answer
My thinking is the second wave pulse obviously must move faster through the string than the first pulse, and at least as far as the book has derived, the speed of a transverse wave on a string is given by $v = \sqrt \frac{F_T}{\mu}$, where $F_T$ is the tension in the string and $\mu$ is the string's mass per unit length. The only way then that the speed of the second string could be higher would be to increase the tension in the string. So I'm imagining ok, you send the first pulse then you pull the string tighter. I think longitudinal waves travel faster than transverse waves, and pulling the string tighter I think amounts to doing that (sending a longitudinal wave pulse). If the longitudinal pulse is much faster, then you would be able to increase the tension quickly and then send another wave pulse down that now moved at a higher speed.

Oopsies
I'm stupid. I forgot that both waves are moving through the same medium, so their speed is set. I guess it isn't possible in this situation. Anybody else?

 

[berkeman]

Very interesting question. It sounds like the question is asking if there is any way that you could make one transverse wave on the string completely catch up to the lead one, right? I can see some variations in linear mass density that would help the trailing tranverse wave to catch up and get close, but that would not make it catch the lead pulse.

But you mention an interesting thing about longitudinal waves, since that could alter the tension in the string locally -- do you have an equation for the propagation speed of longitudinal waves, or is it the same as for transverse waves?

 

[LCSphysicist]

I think is interesting to think about if the string is a dispersive medium. Or if it is with uniform density.
If it is dispersive, we can find different velocities.
If it is non uniform too.

λ = 2L/n
f = 2*fo*sin(n*π/(2(N+1))

Pure sinusoidal waves on the string with high frequencies, and longer wavelength travels faster than short wavelengths.