Waves and harmonic frequncy

Firstly, let's assume that these are the first two modes of vibration.

For a tube like this (one closed and one open end) that means they look like the first two patterns on this diagram


Now we need to use the relationship that connects Lamda to L, the length of the tube, for these two modes.

These are, for the first one Lamda = 4L and for the second, Lamda = 4L/3 (if you can't remember where that comes from, let me know)

So, we can find the wavelength if we know the length of the tube and the end correction* so for now let's say the end correction is x.

*In practice the antinode of the wave is NOT exactly at the end of the tube, it's usually a couple of cm outside the tube - this is what's mean by end correction, it's the distance of the antinode from the end of the tube

That means that for the first pattern, the length we need to use is not 0.176 but (0.176 + x) and so wavelengh is then 4(0.176 + x)

Now, do the same for the 2nd mode and remember that these two wavelengths must be the same. So, now you have two expressions for wavelength, set them equal to each other, rearrange and solve for x.

Once you've got x, you can subs it back in to find the wavelength, and once you've got that, you can use it with the frequency to find the wavespeed.

If you need more step by step help, let me know.

FWIW I got the end correction to be 0.0275 which gave me a wavelength of 0.814 and a speed of sound of 342 to 3sf.

I think the 2L/n thing you are thinking of is for a wave with either two fixed/closed ends or two open ends.
The patterns are a bit messier when there's one fixed end and one open.

The 420Hz you only need once you get the wavelenth, and that's to work out the speed using the wave equation.

This isn't the neatest diagram but is meant to help you see how the lamda and L are related

That's a distance, so it's in metres, (like the 0.176 and 0.583 you used to calculate it).

but is the value right? does this mean that the antinode is just before the tube ends rather than adfter?

Yes, it's right. I kept it as 0.0275 (I got 2x = 0.055)

The antinode is just outside the tube, not inside it. I have difficulty seeing why this happens but know it's true!

Very many questions on this ignore the end effects I think.

But my value is a negative, does it mattwer?

No, it doesn't (it would, but it shouldn't be negative here and the antinode is always outside, not inside). I don't think mine was negative - it might be just a slip when rearranging the equations?

4(0.176 + x) = ((4(0.583+x))/3) - this is your first line
Here I cancelled the 4s and multiplied by 3 to get rid of the fraction bit

So 3 ( 0.176 + x ) = 0.583 + x

0.528 + 3x = 0.583 + x
3x - x = 0.583 - 0.528
2x = 0.055