The Physics Philes, lesson 104: Standing O for Standing Waves
Now that we have nodes, antinodes, and standing waves under our belt, it’s time to move on to something called normal modes.
But wait. Let’s step back a bit.
As we were learning about standing waves, we didn’t care so much about the length of the string or what was happening on the free end of the string. But now we need to think about a string of a finite length that is held fixed on both ends, like the strings of a violin or a piano. When one of those strings are plucked, the wave that is produced is reflected and re-reflected which makes a standing wave.
To understand the properties of the wave I just described, let’s first analyze what happens when we put a sinusoidal wave on that finite string. In that case, the resulting standing wave has to have nodes on both ends of the string. (They are both fixed, remember.) Adjacent nodes have to be half a wavelength apart (or λ/2), so the length of the string must be some integer multiple of that. If we translate that in math symbols, we get
where n = 1, 2, 3, etc. For a standing wave to exist on a string of length L that is fixed at both ends, it must satisfy this equation. If we solve that equation for lambda, we get
You notice that λ is labelled a bit differently. That little n just means that this represents all values of λ. And, as before, n is an integer starting from one.
Please don’t interpret this as meaning that waves can’t exist on the string if the wavelength does not satisfy this equation. They can, but the wave pattern cannot be steady with nodes and antinodes and it can’t be a standing wave.
Do you remember a while back we talked about the relationship of frequency to wavelength? Don’t worry if you don’t remember; it’s been a while. The frequency is equal to the wave velocity divided by the wavelength. That same relationship holds here. For a series of every possible standing-wave wavelengths is a series of possible standing-wave frequencies. The smallest frequency (the n = 1 situation), corresponds to the largest wavelength. That means that the wavelength will be equal to 2L by the previous equation. The resulting equation is
This is called the fundamental frequency. To get other standing-wave frequencies, you just multiply this by an integer. For example, if you want f2, the equation would be 2v/2L. If you want f3, the equation would be 3v/2L. If you want f50, the equation would be 50v/2L. (This is because, say for n = 2, the wavelength would be equal to 2L/2. When a fraction is in the denominator of another fraction, it’s just like multiplying by the inverse. So, for f2, our equation would be (v)(2/2L) = v2/2L.)
But I digress. The values for f2, f3, and so on are just integer multiple of f1, so we can represent all of the frequencies like this:
And, as ever, n = 1, 2, 3, etc. The frequencies represented here are called harmonics, and this equation is a series called a harmonic series. In music, f2, f3, and so on are called overtones. For example, f2 is the second harmonic and the first overtone. The first harmonic is the same as the fundamental frequency.
We learned last week about the wave function of a standing wave on a string that is fixed at the end x = 0. We can use that here, as well, were the string is fixed at x = 0 and x = L. The wave function of the nth standing wave is
Now, after all that, we can talk a bit about normal modes. A normal mode of an oscillating system is when all the particles move sinusoidally with the same frequency. For a system made up of a string with a finite length that is fixed at both ends, each wavelength is given by the equation for λ given above, and it corresponds to a possible normal-mode pattern and frequency. There are an infinite number of normal modes, each with a characteristic frequency and vibration pattern. This is different from a harmonic oscillator. Those have only one oscillating particle, only one normal mode, and only one characteristic frequency. A string fixed at both ends has an infinite number of normal modes because it’s made up of a very large number of particles.
So far we’ve only been talking about sinusoidal waves, which are relatively easy to understand. But what about something more complex? Well, you’ll just have to come back next week for that.
Featured image credit: Rocky Lubbers via Flickr