Speed of a wave on a string w.r.t. which reference frame

[Pushoam]

Speed of a wave in a string is given by √(τ/μ) .
But this speed is with respect to which reference frame?
Since, the speed depends on τ and μ( which are independent of reference frame ), I can consider speed of wave independent of reference frame.
But this is not so. From experiment, we know that if the Speed of a wave in a string is v = √(τ/μ) with respect to string ,then with respect to a reference frame moving with speed v with respect to string along the direction of wave propagation, the wave's speed will be 0 m/s.
So, experimentally , I can't consider speed of wave independent of reference frame.
Can anyone please resolve this contradiction?

 

[pixel]

It's the speed with respect to the frame of reference in which the string is at rest.

 

[Pushoam]

It's the speed with respect to the frame of reference in which the string is at rest.

This is what I said here

From experiment, we know that if the Speed of a wave in a string is v = √(τ/μ) with respect to string

I want to know what is wrong with the following argument?

Since, the speed depends on τ and μ( which are independent of reference frame ), I can consider speed of wave independent of reference frame.

 

[Dale]

I want to know what is wrong with the following argument?

How was the equation derived? What assumptions were made in the derivation of the equation?

 

[Pushoam]

Speed of light in a medium depends on the electric and magnetic properties of the medium i.e. mu and epsilon and since these properties do not depend upon the frame , speed of light in medium should be independent of frames.
Is this so?
If speed of light depends on the frames ,then does the wave equation imply it ?

 

[PeterDonis]

Is this so?

Yes, but not for the reason you give. Spacetime is not a "medium", and the "electric and magnetic properties of the medium" you refer to are artifacts of a particular choice of units, not physical properties of anything. The speed of light is frame independent because, like $\mu_0$ and $\epsilon_0$, it is really just an artifact of a particular choice of units; it is a conversion factor between our choice of time units and our choice of distance units, because we did not, historically, use the same units for these things. In "natural" units, the speed of light is $1$.

 

[Nugatory]

speed of light in medium should be independent of frames.

As PeterDonis says, spacetime is not a medium. But with that said... Yes, the speed of light in a medium relative to the medium is indeed frame-independent, and this does follow from the frame-independent values of the permittivity and permeability of the medium.

However, relativistic velocity addition still applies. If the speed of light relative to the medium is $u$ and the medium is moving at speed $v$ in some frame, then in that frame the light will be moving at speed $(u+v)/(1+uv)$ according to the relativistic velocity addition rule. This was first observed by Fizeau in 1851 when he was measuring the speed of light in moving water; his result was an underappreciated mystery until Einstein discovered relativity more than half-century later.

 

[Pushoam]

As PeterDonis says, spacetime is not a medium.

I don't understand "spacetime" and why are we using it here?

Yes, the speed of light in a medium relative to the medium is indeed frame-independent,

To illustrate the above quote using an example:
Let's say that I have two transparent boxes A and B filled with water .
B moves with speed v w.r.t. A along +ve x-axis.
light wave moves in both boxes along +ve x-axis.
Now , speed of light wave (which passes through A) w.r.t. A = speed of light wave (which passes through B) w.r.t. B
You say that this is true because of

the frame-independent values of the permittivity and permeability of the medium.

speed of light wave (which passes through B) w.r.t. A ≠ speed of light wave (which passes through B) w.r.t. B
Here,relativistic velocity addition applies.
This I don't understand.
According to me,
even if a person sitting on A wants to calculate speed of light passing through B, he will write the wave equation and he will get v = 1/(sqrt με) = speed of light wave (which passes through B) w.r.t. B
Why should relativistic addition come here when speed of light depends only on ε and μ and both of these are independent of frame?

 

[Nugatory]

even if a person sitting on A wants to calculate speed of light passing through B, he will write the wave equation and he will get v = 1/(sqrt με) = speed of light wave (which passes through B) w.r.t. B

That gives the speed of the light flash with respect to B. If B is moving with respect to A, then the speed of the light flash relative to A is given by the relativistic velocity addition formula.

 

[PeterDonis]

I don't understand "spacetime"

I apparently misunderstood your OP. "Spacetime" is a basic concept in relativity, and my post was talking about light propagation through vacuum, with no material medium present; under those conditions, the properties of spacetime are the only thing governing light propagation. However, you appear to be asking about the case where there is a material medium (like air or water) present; Nugatory's posts address that case.

 

[Pushoam]

That's gives the speed of the light flash with respect to B. If B is moving with respect to A, then the speed of light relative to A

O.K.
So,if the person sitting on A wants to calculate speed of light passing through B,
1) just by using wave equation, what modification will he have to do in his wave equation? Is this a right approach?
I mean can he get (u+v)/(1+uv) as the speed of light from the wave equation itself?
2)or
The wave equation always gives speed of light only w.r.t. medium and so he should calculate speed of light w.r.t. medium i.e.B using wave equation and then apply relativistic velocity addition to get the speed of light w.r.t. his frame i.e. A.

 

[Pushoam]

I apparently misunderstood your OP.

Will you please tell me what is meant by OP?

 

[PeterDonis]

Will you please tell me what is meant by OP?

The Original Post in the thread.

 

[Dale]

Moderators note: I merged your two threads about the invariance of the wave equation since they are fairly similar topics.

The recommendation that I gave you before still applies. What are the assumptions, and do those assumptions still hold?

Also, for EM you need to make sure that you understand how EM quantities transform between reference frames.

 

[Pushoam]

The recommendation that I gave you before still applies. What are the assumptions, and do those assumptions still hold?

I am attaching here derivation of speed of wave in a string.
This is from University Physics , 12 th edition, Sears and Zemansky.
I don't see here any mention of reference frame or any assumption which will lead me to the answer of my question( post 1).

 

[Nugatory]

I am attaching here derivation of speed of wave in a string.
This is from University Physics , 12 th edition, Sears and Zemansky.
I don't see here any mention of reference frame or any assumption which will lead me to the answer of my question( post 1).

People often do not mention the reference frame in their derivations when it is obvious from the context which one they're using; I could pick up any first-year textbook and find hundreds of examples and exercises that don't mention a reference frame for every one that does. Here the context makes it clear that they're working in a frame in which the string is at rest.

So the answer to the question in your original post

Speed of a wave in a string is given by √(τ/μ) .
But this speed is with respect to which reference frame?

is the one that we've already given above: That will be the speed using a frame in which the string is at rest. The speed of the wave relative to an observer at rest in some other frame is calculated using the appropriate velocity addition rule.

 

[Pushoam]

o.k.
Thank you.

 

[Dale]

I am attaching here derivation of speed of wave in a string.
This is from University Physics , 12 th edition, Sears and Zemansky.
I don't see here any mention of reference frame or any assumption which will lead me to the answer of my question( post 1).

The file wouldn't open, so I cannot check this specific derivation. Most derivations explicitly assume that the material points on the string move only in the transverse direction. This implies that the velocity in the longitudinal direction is always 0. This condition is not met for a moving string.

 

[Pushoam]

Most derivations explicitly assume that the material points on the string move only in the transverse direction. This implies that the velocity in the longitudinal direction is always 0. This condition is not met for a moving string.

O.K. Now I understood that while deriving the wave equation , it is assumed that the we are working in the reference frame in which the medium is at rest and so, the wave equation gives speed of the wave w.r.t. this reference frame.
Thank you.
Thank you, all for helping me to come to this conclusion.

 

[Pushoam]

The file wouldn't open

In my laptop, it is opened.
By the way, leave it. Now it doesn't matter.