Changing medium - Standing Mechanical Waves

[IAmCurious]

I wanted to ask the following question: If we change the medium of the wave (lighter or heavier) , how it will affect the particles velocity of the standing mechanical wave ?

In my exam , there was a function's graph of one of the particles (of the standing wave) described with the coordinates y(displacement) and t(time). I claimed that the slope of the function's graph represents the velocity of the particles has to be changed because the mass of the medium changed.

My teacher claims that my answer is incorrect and the velocity of the particles won't be changed because of the fact that the question is about a specific kind of wave : a standing mechanical wave, so the particles velocity won't be affected by the change of the medium's mass.

So who's right and why?

 

[olivermsun]

If the graph of y(t) is unchanged then clearly y'(t) can't change either. So what do you mean by changing the medium in the problem? Are you replacing the string (or whatever) and then imposing the same initial condition of displacement? Do you have a standing wave which spans a string with two sections of different density?

 

[IAmCurious]

"Are you replacing the string (or whatever) and then imposing the same initial condition of displacement?"
Yes. I was asked if due to the change in the medium's mass (heavier) , does that it mean the graph supposed to be changed?

 

[olivermsun]

If the tension is kept the same and only the mass density per length changes, then the frequency of the standing wave should change (as evidenced by the behavior of real strings). Thus to "sweep" the same range of displacement per cycle, a heavier string should be moving more slowly and y(t) is "stretched" in time. So I believe your intuition is correct.

 

[IAmCurious]

Would you please give the equations you used to conclude this conclusion?

 

[AlephZero]

If you just consider the kinematics of the standing wave, your teacher is right.

But if you consider the physics of the system, you can't change only the mass and expect everything else to stay the same. The frequency and/or the amplitude of the wave would change, so in general you would be right.

But if the exam question didn't describe the complete system, you can't consider the physics without making some assumptions. Personally I think it's a poor exam question.

 

[BoilerUp85]

If the medium is a string, changing it likely changes the linear density to change, which would change the wave speed as the reciprocal of the square root of the factorial change in the linear density. I agree with AlephZero, it is a poorly worded question.

If it's not a string, it's then going to become a function of both elasticity and density.