How to prove that velocity of wave= sqrt(tension/denisty)?

[MARK STRETERS]

Homework Statement

Here is the lab overview: In this experiment standing waves are produced in a string using a vibrator of variable frequency. The vibrator is placed under one end of the string which is tied to a rod and the other end of the string passes over a pulley and holds a suspended mass of 550g. The tension in the string can be found from multiplying the mass of the weight (.55kg) by the force of gravity(9.81) and the density of the string is 7.95 x 10^-4 kg/m. The velocity of the wave can be calculated using the equation v= √ Tension/density. So I recorded the frequencies needed for the string to reach its fundamental, 1st, 2nd and 3rd harmonics. The goal of this lab is to prove that velocity=√ Tension/ Density.
Here is the data I recorded

Fundamental= frequency of 18.76
1st Harmonic= 37.34
2nd Harmonic= 55.38
3rd harmonic= 69.81

Homework Equations

velocity=√ Tension/ Density

The Attempt at a Solution

So the problem with this lab is that I'm really not sure how to go about starting the process to prove that velocity equation so if anyone could provide any insight it would be much appreciated. What I'm confused about is that it seems like the velocity of the waves would be changing with increasing frequency however I don't see how velocity could be anything but a constant using that function seeing that density will be constant. The only thing that I can think of is that tension somehow changes with increasing frequency but I'm really not sure because it seems like Tension would be a constant given that the mass of the object at the end of the string isn't changing and the force of gravity on the mass certainly isn't changing. Maybe I'm overlooking some simple mistake so any insight would be much appreciated. Thanks!

 

[MARK STRETERS]

Update: I now realize that the greater the frequency the lower the velocity. Could the fact that at lower frequencies the string has a larger amplitude be a factor? As in at larger amplitudes the tension created by the mass at the end of the string will have to be greater in order to counteract the greater displacement in the string which would then contribute to a larger velocity using the equation v= √ Tension/density? Just theorizing here I still don't think that I have a solid argument and any help would be much appreciated.

 

[berkeman]

Homework Statement

Here is the lab overview: In this experiment standing waves are produced in a string using a vibrator of variable frequency. The vibrator is placed under one end of the string which is tied to a rod and the other end of the string passes over a pulley and holds a suspended mass of 550g. The tension in the string can be found from multiplying the mass of the weight (.55kg) by the force of gravity(9.81) and the density of the string is 7.95 x 10^-4 kg/m. The velocity of the wave can be calculated using the equation v= √ Tension/density. So I recorded the frequencies needed for the string to reach its fundamental, 1st, 2nd and 3rd harmonics. The goal of this lab is to prove that velocity=√ Tension/ Density.
Here is the data I recorded

Fundamental= frequency of 18.76
1st Harmonic= 37.34
2nd Harmonic= 55.38
3rd harmonic= 69.81

Homework Equations

velocity=√ Tension/ Density

The Attempt at a Solution

So the problem with this lab is that I'm really not sure how to go about starting the process to prove that velocity equation so if anyone could provide any insight it would be much appreciated. What I'm confused about is that it seems like the velocity of the waves would be changing with increasing frequency however I don't see how velocity could be anything but a constant using that function seeing that density will be constant. The only thing that I can think of is that tension somehow changes with increasing frequency but I'm really not sure because it seems like Tension would be a constant given that the mass of the object at the end of the string isn't changing and the force of gravity on the mass certainly isn't changing. Maybe I'm overlooking some simple mistake so any insight would be much appreciated. Thanks!

Welcome to the PF.

How did you measure the velocity of the fundamental and harmonics? What were the results?

And the causality of the relationship is that tension and string linear density determine the velocity of the waves. The frequency of the waves does not affect their velocity (at least not in this setup).

 

[MARK STRETERS]

Welcome to the PF.

How did you measure the velocity of the fundamental and harmonics? What were the results?

And the causality of the relationship is that tension and string linear density determine the velocity of the waves. The frequency of the waves does not affect their velocity (at least not in this setup).

Well we never actually measured the velocity we were just asked to prove the equation however the velocity of the fundamental= 37.52 1st= 37.34 2nd= 39.92 3rd= 34.905
and to find those figures we used the equation velocity= wavelength x frequency

 

[berkeman]

Well we never actually measured the velocity we were just asked to prove the equation however the velocity of the fundamental= 37.52 1st= 37.34 2nd= 39.92 3rd= 34.905
and to find those figures we used the equation velocity= wavelength x frequency

Have a look at the wikipedia page on the Wave Equation. There is a section there that should help you understand the derivation...

 

[MARK STRETERS]

Have a look at the wikipedia page on the Wave Equation. There is a section there that should help you understand the derivation...

Ok thank you. I don't know if you saw my earlier comment about the amplitudes of waves but could you check that out and see if there is any sense in it?

 

[berkeman]

Update: I now realize that the greater the frequency the lower the velocity. Could the fact that at lower frequencies the string has a larger amplitude be a factor? As in at larger amplitudes the tension created by the mass at the end of the string will have to be greater in order to counteract the greater displacement in the string which would then contribute to a larger velocity using the equation v= √ Tension/density? Just theorizing here I still don't think that I have a solid argument and any help would be much appreciated.

I don't know if you saw my earlier comment about the amplitudes of waves but could you check that out and see if there is any sense in it?

Oops, nope, I missed it. I did a Google search on non-linear wave velocity in a string, and got lots of good hits. As the amplitude grows, it becomes more likely that you will get some non-linear effects that can influence the propagation velocity...

 

[MARK STRETERS]

Oops, nope, I missed it. I did a Google search on non-linear wave velocity in a string, and got lots of good hits. As the amplitude grows, it becomes more likely that you will get some non-linear effects that can influence the propagation velocity...

I really couldn't find anything too specific that related amplitude to anything in this scenario. Do you think you could give me a specific website?

 

[berkeman]

I really couldn't find anything too specific that related amplitude to anything in this scenario. Do you think you could give me a specific website?

I found some things in the hit list that would be of help to you. Do you have a textbook for this course? Look for "dispersion" in it, or add it as a keyword to your Google search.