Calculating Standing Wave Diagrams and Wave Speeds for Closed Cylinders

[quaere verum]

Homework Statement

I was instructed to fill a graduated cylinder with approx. 90 mL of water and lower a marked tube into the water, taking note of the height of the air column above the water. Vibrating a 1024 Hz tuning fork above the opening of the column I found the shortest height of the air column where the sound reached maximum loudness, then the next shortest, then the next shortest. I found these to be 7.5 cm, 24.5 cm, and 41.5 cm - clearly, there is a difference of 17 cm between each height. Now I have been instructed to draw standing wave diagrams for each data point and calculate the wavelength corresponding wave speed measurement for each data point, averaging each of the wave speeds to approximate the speed of sound.

Homework Equations

v = λ * f

For a closed cylinder, the wavelength/length relationship is expressed by:

λ = (4/n) * L

where n represents the nth harmonic.

I also know that the fundamental frequency f1 = v/4L, but I don't know v...

The Attempt at a Solution

I am familiar with what the standing wave diagrams should look like in principle - 1 node and antinode in the 1st harmonic, 2 of each in the 3rd, etc.

I easily derived the equation v = (4/n) * L * f given the above equations, but I am confused as to where I use each measured quantity. I am assuming that I need to find the fundamental frequency of the wave using that of the tuning fork, and from there the frequency of the subsequent harmonics, but I am not sue how to do that - I am taking this course online and completing the assignment as make-up, so I feel a bit lost in this unit.

I apologize for the lack of my own calculations, but my problem is I have all of these images and formulas flying around in my head and I'm not sure which length and which frequency applies where - basically, I don't know where to start.

I feel as though I am missing something so if someone would be so kind as to point me in the right direction, I would greatly appreciate it. I desperately want to understand the relationships underlying the right method, not just get an answer, but I'm stumped.

Thank you so much in advance for the assistance.

 

[Ibix]

You've accurately described the standing wave diagrams, so you can draw them.

For each one, you can then write an expression for L in terms of $\lambda$. Plug in your numbers, and you get a set of values for $\lambda$.

What is the frequency in each case? Hint: it really is that obvious.

You appear to have all the formulae you need to finish off from here. Post your working if you get stuck.