# Why am I getting an error in my calculations for End Correction in an Open-Open Pipe?

I've been spending a rather large amount of time looking into the relationship between a pipe's diameter and its fundamental frequency. As a result, I discovered the phenomenon of End Correction, and discovered that f = v / 2(L + 0.6d) in an open-open pipe.

I designed an experiment where I could find the fundamental frequencies of four different pipes with different diameters but equal lengths, and conducted said experiment. My experiment consisted of placing a phone speaker at one end of a pipe and a microphone at the other end of the pipe. I would play different frequencies from the speaker and measure the amplitude of the sound wave with the microphone. Using this information, I would then determine the fundamental frequency of the pipes. At the end of the experiment, my data showed that f actually equals v / 2(L + 1.2d).

At first, I assumed that error from uncertainty was at play here. However, after spending a couple hours calculating my uncertainties and their effect on this equation, I've concluded that there's not enough uncertainty to explain such a large difference.

In addition, my data is consistently showing 1.2d, which suggests that my methodology is correct.

My question is, what could possibly be causing this error? I've determined it's not pipe length, pipe diameter, speed of sound, the frequency emitted by the speaker, and the sound wave recorded by the microphone.

• An obvious guess (and just a guess) is that your formula is for a pipe open at one end. For musical instruments, this would be usual. Your pipe might be behaving as two ends open. Commented Dec 23, 2020 at 6:50
• your two answers are off by a factor of two. have you confused diameter with radius somewhere in your analysis? Commented Dec 23, 2020 at 8:02
• Are you able to publish your data? Commented Dec 23, 2020 at 8:46
• I've looked at all my data and I'm sure I haven't confused radius and diameter. As for the pipes I'm using, they are open at both ends, and from the research I've gathered it should be 0.6d and not 1.2d. My data is, in this format: Pipe Diameter (cm)--Frequency (Hz): 1.45--324.349; 2.00--322.365; 2.57--319.750; 3.38--312.762. The length of the pipe is 0.508m. Commented Dec 23, 2020 at 23:03

First, I thought I would define all of the quantities in the original post for clarity. The length of the tube is $$L$$, the radius of the (presumably circular) tube is $$d$$, and the wave speed in the tube is $$v$$. Ignoring end corrections, the fundamental frequency would be given by $$f_0=v/2L$$.
The end correction is $$\alpha d$$ per end, where $$\alpha\approx0.6$$ for an unflanged pipe (I assume this is your case). In the equation for the fundamental frequency you gave above, you have only included one end. Since you have an open-open pipe, your fundamental frequency should be given by $$f_0=\frac{v}{2(L+2\alpha d)}=\frac{v}{2(L+1.2d)}.$$