Change the wavelength in a driven rope?

[Edel Crine]

Homework Statement

You and a friend each have one rope. You tie the two ropes together
and stand as far apart as possible, each holding one end
of the new longer rope and pulling to put it under tension. You
then begin moving your arm in such a way as to produce a harmonic
wave with a wavelength of 1.0 m. Your friend looks at
the waves as they reach her arm. Is it possible that she measures
a wavelength of (a) 0.8 m, (b) 1.0 m, or (c) 1.2 m?

Relevant Equations

λ=cT

I think I know what this problem means, but not sure how can I do quantitative work to solve it.
I think only a would work because of friction of string, the reflected waves from the junction between two strings like that.
Although, I am not sure how can I build the process, but this was my guess...

 

[BvU]

because of friction of string

You may assume "no friction" (nothing was said about that).

the reflected waves from the junction between two strings like that

You may assume the knot is behaving as an infinitely short piece of rope

how can I do quantitative work

There is only one quantitative piece of information given !

However, nothing is said in the sense of "both ropes are identical in mass per length..."

 

[Steve4Physics]

Hi. Don't worry about reflection - the problem is about possible change of wavelength as a wave moves along joined strings. There are no calculations needed - you are just asked if each of the 3 answers is possible. You need to know how different factors (e.g. tension) affect wave speed.

Call the ropes A and B.

Q1. Can the frequency (or period if you prefer) of waves on A be different to the frequency (or period) of waves on B? (Hint: what happens at the junction?)
Q2. Can the tensions of A and B be different? (Hint: what happens at the junction?)
Q3. Can the wave speeds on A and B be different (Hint: what factors affect wave speed on a string?)

Using λ = c/f (= cT), and using your answers to Q1-3, can you answer the question?

 

[Edel Crine]

You may assume "no friction" (nothing was said about that).

You may assume the knot is behaving as an infinitely short piece of rope

There is only one quantitative piece of information given !

However, nothing is said in the sense of "both ropes are identical in mass per length..."

Then think all of them could be an answer...??

 

[Edel Crine]

Hi. Don't worry about reflection - the problem is about possible change of wavelength as a wave moves along joined strings. There are no calculations needed - you are just asked if each of the 3 answers is possible. You need to know how different factors (e.g. tension) affect wave speed.

Call the ropes A and B.

Q1. Can the frequency (or period if you prefer) of waves on A be different to the frequency (or period) of waves on B? (Hint: what happens at the junction?)
Q2. Can the tensions of A and B be different? (Hint: what happens at the junction?)
Q3. Can the wave speeds on A and B be different (Hint: what factors affect wave speed on a string?)

Using λ = c/f (= cT), and using your answers to Q1-3, can you answer the question?

Q1. Since it does not tell that those ropes are identical, the frequency could be different if the ropes are different (different medium?)
Q2. If those ropes are different, yes; no if they are same..?
Q3. The wave speed is changed by tension and mass per unit length, so if two ropes are different, yes; no when they are same...?

Or probably it could be different because of junction...?

 

[etotheipi]

The frequency is unchanged when a wave passes from one medium to the other. The speed of the waves in each section is $\sqrt{T/\mu}$, and the wavelength in the medium thus $\lambda = \frac{1}{f}\sqrt{\frac{T}{\mu}}$. The tensions in both sections are the same, since you require equilibrium in the horizontal direction at the knot. So the functional dependence on wavelength is just on the linear density, $\lambda = \lambda(\mu)$.

 

[Edel Crine]

I really appreciate every one of you who helped me to grasp the concept... Soooo thank you...! It's sad that I can't give like multiple times...

 

[Steve4Physics]

Hi. I'd just like to add:

The frequencies of the 2 strings *MUST* be the same or the junction would be torn apart. Having the same frequencies comes from the requirement for continuity (no breaks in the strings).

The tensions of the 2 strings *MUST* be equal. Or the junction would have a (horizontal) resultant force and accelerate horizontally - which it clearly can't do.

The ropes may or may not be identical. If identical, the wave speeds are the same and wavelength is unchanged. If one rope has a higher linear density, the wave speed on it is slower, so its the wavelength is shorter (since λ=v/f and f is unchanged).