This is problem Problem 47 from Chapter 16 from the Fundamentals of Physics 9th

Question

This is problem Problem 47 from Chapter 16 from the Fundamentals of Physics 9th Edition by Halliday, Resnick, Walker. I don't understand why the velocities of each harmonic are the same, or constant. Please explain.

Here's the question: "One of the harmonic frequencies for a particular string under tension is 325 Hz. The next higher harmonic frequency is 390 Hz. What harmonic frequency is next higher after the harmonic frequency 195 Hz?"

I understand how to do the problem if the velocities for each harmonic are equal. I just want to know why they're equal, and I want to see the math to prove it.

Explanation / Answer

Because of the formula that v = f, we can see that frequency and wavelength are inversely proportional.

When you have harmonics, the wavelength will decrease by a whole number for each higher order harmonic. When that happens the frequency increases by that exact same number.

If the wavelength goes down by a factor of two, the frequency increases by that same factor. Plugging that into the formula shows that the amount of increase of frequency cancels out the amount of wavelength decrease and thus v = f does not change overall.

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