-Resonance-

The Tacoma Narrows Bridge Disaster, 1940

Jimmy Hendrix, 1969

A Simple Physical System

-The Pendulum-

Kids on a Swing, Date: unknown

Pendulum

Questions:

What does the resonance frequency depend on?

What is the resonance frequency of a pendulum?

Is there a picture on the next overhead?

The resonance or natural frequency of a pendlum system is determined simply by the length of the string.

We can caltulate this natural frequency with the following equations:

For example, we will use a playground swingset as an example. The length of the chain is approx. L = 3.5 meters, and

g = 9.81 meters/second.

The period, T, is approx. 3.75 seconds. We get for our frequency:

Resonance: If a periodic force is applied to a system, the amplitude of the resulting motion is greater than normal when the frequency of the applied force is equal to or nearly equal to the natural frequency of the system.

What happens when you try drive a system at a faster speed than its natural frequency, like a swing set already in motion, like this man tried to do.

So how does resonance relate to sound, and the guitar that I have with me at the moment?

-Waves-

Because sound is propogated as waves, we need to firstly take a look at waves and their properties.

We know that sound waves have different frequencies, but how do we obtain these frequencies? How do they work?

We'll look a string of length L fixed at both ends (ie. Guitar strings).

Nodes, places on the wave with zero displacement, are present at each end of the string, where it is attatched. Strings have a number of natural patterns of oscillation, called Normal Modes. Each of these have their own natural frequency, depending on where the string is held down.

These normal modes of vibration form a series of 'harmonics'.

If we think of an open guitar string, being held down to the head and the fret board (at the node points), we look at the diagram of the 1st harmonic. Here, the length of the wave is twice the length of the string. This is known as the fundamental frequency. The anti node is right in the center of the string. The frequencies of these normal modes form a harmonic series. That means, when we distort a guitar string, (pluck, strum), the resulting vibration includes various harmonics, all at once. The normal modes are called harmonics. We can demonstrate a harmonic on the guitar.

-A Sample Problem-

Using these properties of harmonics and frequencies, there is a formula that is helpful when dealing with certain numerical problems. The formula is

where v is the speed of the wave on the string, L is the length of the string, n is a positive integer representing the nth normal mode of oscillation, and f is the fundamental frequency.

Problem: A guitar's high E string is 64 cm long, and has a fundamental frequency of 330 hz. If the first fret is pressed down, the string is now shorter, and the new frequency is now 350 hz, and it is now an F note. How far is the fret from the neck end of the string?

Using this useful formula, this is a simple problem to solve. We use n = 1, because it is the fundamental frequency initially, which gives us, with L = 0.64m, and f = 330 hz,

The speed of the wave on the string is 422 m/s. We do not adjust the tension in the string, so this wave speed remains constant. We again use the same equation, but fill in the values after the first fret is held down. And now we are solving for L. Turn the page for more excitement...

With our formula now shifted to solve for L, we get

We get 0.603 m. Finally, the distance between this length and our measured length of 64 cm is the distance from the fret to the neck side of the string. This calue is 3.7 cm. This is our answer.

We can now come back to our definition of what resonance is:

Resonance: If a periodic force is applied to a system, the amplitude of the resulting motion is greater than normal when the frequency of the applied force is equal to or nearly equal to the natural frequency of the system.

Our simple pendulum system only had one natural frequency, whereas now we have standing-wave systems, such as guitar strings, which can have a whole set of natural frequencies. We can drive these systems with a force equal or nearly equal to the natural frequencies to increase the amplitude of motion or sound volume.

We can demonstrate this with two tuning forks of equal frequency.

-Beats-

Beating is the periodic variation in intensity at a given point due to the superposition of two waves having slightly different frequencies.

The beat frequency, f is equal to

f = f – f

For example, if one tuning fork is vibrating at 438 Hz, and another is vibrating at 442 Hz, the sound wave, which is a combination of both, is 440 Hz (an A note), and the beat frequency is 4 Hz. This means that listening, you would hear that 440 Hz sound wave (an A note) go through an intensity maximum four times every second.

-Standing Waves in an Aluminum Rod-

If a rod is held in the center, and stroked on one side, there are oscillations of the particles in the rod that are longitudinal. When the vibrations reach one wavelength, the bar will resonate. Resonance amplifies the amplitude, the height, of the wave which is directly related to the volume we hear.

The center is our node point, and the ends are both anti-nodal points. This sound produced is the first normal mode. If the bar is held at L/4, it produces the second normal mode.

-The Ebow-

The Ebow is a device used for making unique guitar sounds. It is essentially a magnet that acts with the guitar's pick ups that causes the string to resonanate. It uses a focused feedback loop directed at a single string. The effect is like bowing a guitar string, hence the name, Ebow.

...and in theory...

Nameless lab assistant attempts to shatter glass using resonance. Feeble attempt is unsuccessful.

A wine Glass itself has a natural frequency, which can be heard by 'pinging' the glass. In theory, a loud pitch matching that of the resonance frequency pointed directly at the glass should be able to resonante the glass to the point where its structure breaks down, and it shatters.

(If anyone would like to attmept to scream at the glass, feel free.)

Graham Langridge

2005-03-01T18:19:10