11a: String Resonance

Recall that speed of a wave on a string is determined by the density and tension; v = (T/μ)1/2 where T is the tension in the string in Newtons and μ is the mass per length in kilograms per meter). We also know that once the speed is fixed, frequency (in Hertz) and wavelength (in meters) are inversely proportional; v = f λ. So the three parameters that determine the frequencies of a string are tension, density (mass per length) and length. The density of a string is determined by thickness and mass; a thick, heavy string is more dense so waves travel more slowly.

For standing waves on a string the ends are fixed and the string does not move. Places where the sting is not vibrating are called nodes. This limits the wavelengths that are possible which in turn determines the frequencies since the speed is fixed and v = f λ. The lowest frequency is called the fundamental or first harmonic. For a string, higher frequencies are all multiples of the fundamental and are called harmonics or partials. The more general term overtone is used to indicate frequencies greater than the fundamental which may or may not be harmonic. This can be a bit confusing because for strings there are only harmonics and the second harmonic is the first overtone, etc. The various harmonics (overtones) are also called the normal modes of vibration of the string.

What wavelengths will fit on a string of length L? There has to be a node on each end so it has to be the case that L = nλ/2 where n is a whole number. In other words, you can have half a wave on the string (n = 1), one wave (n = 2), one and a half waves (n = 3), etc. But you never have any other fraction of a wave because that would require not having a node at both ends.

harmonics

Notice that the higher harmonics have nodes in other locations besides just at the ends. In between two nodes is a region where the vibrations are a maximum. These are called anti-nodes. The fundamental has one anti-node the 2nd harmonic has two anti-nodes.

Once the string density and tension are chosen the speed is fixed and the frequencies will depend on the wavelength as shown in the following table. Notice that the harmonics are all multiples of the fundamental.

harmonicfreqs

In Chapter 4 we defined resonance to be the case when the amplitude of vibration got larger because the driving frequency matched the natural frequency. Here we see there are many natural frequencies. This means there are many resonance frequencies. For a string these are all harmonics (whole number multiples) of the fundamental. In a real string that is plucked or bowed, it is often the case that several of these resonance frequencies are present at the same time. This means that musical instruments usually produce not only a fundamental, but also harmonics that give the instrument its timbre.

Generally strings are either plucked or bowed. In both cases the string does not undergo the simple scenario of harmonics described above. Plucking a string at the center does not cause a nicely shaped sinusoidal wave; instead you start with a triangle shape on the string:

triangle2

But we know from Fourier’s work that any repeating shape can be formed from a series of sine waves. Plucking a string at the center emphasizes the fundamental but many other harmonics will be included. Plucking the string at a location one fourth of the way along the string makes the second harmonic a bit louder but other harmonics will still be present. The result of plucking (starting with a triangle shape) and plucking at different locations means the spectrum in not uniform; where you pluck the string determines which harmonics are emphasized.

In the plucked case the triangle shape immediately converts into a combination of sines and cosines, some of which die away quickly. If the string is bowed, however, the triangle wave is maintained since the bow continues to pull the string to one side at the point of contact. The triangle shaped wave travels to the bridge, reflects, and returns to the bow contact location. When the point of the triangle shape returns to the bow it causes the string to break loose from the bow. The wave continues and reflects off the fret end, returning to the bow again, now causing the string to stick to the bow. This slip-stick mechanism maintains a triangle shaped wave moving on the string, reflecting from each end. Once again, changing the location of the bow contact determines which harmonics are emphasized.

triangle3

Video of standing waves on a driven string.

Other resources:
  • Here is a simulation of a driven string. Try changing the driving frequency. What happens at 25 Hz? What happens at 50 Hz? What are the frequencies of other resonances?
  • If you want to see how the waves from a plucked string move along the string, here is a simulation of a plucked string from Wolfram (you may need to download their plug in to play with this demonstration).
  • Another Applet simulation from Paul Falstad of a plucked string that shows the Fourier components that make up the shape.
  • A simulation of a bowed string from Bruce Richards at Oberlin College: Vibrating Strings.
  • Wikipedia on string resonance (lists many types of stringed instruments) and stringed instruments.
  • A few stringed instruments, such as the Indian sitar have sympathetic strings. These are extra strings that are not normally played by the musician but vibrate due to resonance because they are tuned to the strings that are plucked or bowed.
Video/audio examples:

Slow motion of a bowed string.

Slow motion of a plucked string.

Simulation 11A (turn in answers on a separate sheet of paper): Driven String.
Simulation exercise 11B (turn in answers on a separate sheet of paper): Resonance on a String and in a Tube.

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