1. Natural Frequency & Resonance
AP Physics 1

2. Natural Frequency
Nearly all objects when hit or struck or plucked or strummed or somehow disturbed, will vibrate.
if you drop a pencil or a meter stick on the floor is will vibrate.
If you pluck a guitar string, it will begin to vibrate.
If you blow over the top of a soda bottle, the air inside will vibrate.
When each of these objects vibrate they tend to vibrate at a particular frequency or a set of frequencies known as the natural frequency.
All objects have a natural frequency or set of frequencies at which they vibrate.

3. Timbre
The quality or timbre of a sound produced by a vibrating object is dependent upon the natural frequencies of the sound waves produced by the objects.

4. Pure Tone
Some objects tend to vibrate a single frequency are are said to produce a pure tone.
Example: Flute
200 Hz

5. Rich Tone
Other objects vibrate and produce more complex waves with a set of frequencies that have a whole number mathematical relationship between them and are said to produce a rich sound.
A tuba tends to vibrate at a set of frequencies that are mathematically related by whole number ratios; it produces a rich tone.

6. Noise
Still other objects will vibrate at a set of multiple frequencies that have no simple mathematical relationship between them. These objects are not musical at all and the sounds that they create could be described as noise.
When a meter stick or pencil is dropped on the floor, it vibrates with a number of frequencies, producing a complex sound wave that is clanky and noisy.

7. Natural Frequencies Produced

8. Factors Affecting Natural Frequency
The actual frequency at which an object will vibrate at is determined by a variety of factors. Each of these factors will either affect the wavelength or the speed of the object. Since
frequency = speed/wavelength
Consider a guitar as an example. There are six strings ,each having a different linear density (the wider strings are more dense on a per meter basis), a different tension (which is controllable by the guitarist), and a different length (also controllable by the guitarist).

9. Guitar example
The speed at which waves move through the strings is dependent upon the properties of the medium - in this case the tightness (tension) of the string and the linear density of the strings. Changes in these properties would affect the natural frequency of the particular string.
The vibrating portion of a particular string can be shortened by pressing the string against one of the frets on the neck of the guitar. This modification in the length of the string would affect the wavelength of the wave and in turn the natural frequency at which a particular string vibrates at. Controlling the speed and the wavelength in this manner allows a guitarist to control the natural frequencies of the vibrating object (a string) and thus produce the intended musical sounds. The same principles can be applied to any string instrument - whether it is the harp, harpsichord, violin or guitar.

10. Wind Instrument Example
Consider the trombone with its long cylindrical tube that is bent upon itself twice and ends in a flared end. The trombone is an example of a wind instrument. The tube of any wind instrument acts as a container for a vibrating air column. The air inside the tube will be set into vibration by a vibrating reed or the vibrations of a musician's lips against a mouthpiece. While the speed of sound waves within the air column is not alterable by the musician (they can only be altered by changes in room temperature), the length of the air column is.
For a trombone, the length is altered by pushing the tube outward away from the mouthpiece to lengthen it or pulling it in to shorten it. This causes the length of the air column to be changed, and subsequently changes the wavelength of the waves it produces. And of course, a change in wavelength will result in a change in the frequency. So the natural frequency of a wind instrument such as the trombone is dependent upon the length of the air column of the instrument. The same principles can be applied to any similar instrument (tuba, flute, wind chime, organ pipe, clarinet, or soda bottle) whose sound is produced by vibrations of air within a tube.

11. Forced Vibration
If you were to take a guitar string and stretch it to a given length and a given tightness and have a friend pluck it, you would hear a noise; but the noise would not even be close in comparison to the loudness produced by an acoustic guitar.
On the other hand, if the string is attached to the sound box of the guitar, the vibrating string is capable of forcing the sound box into vibrating at that same natural frequency. The sound box in turn forces air particles inside the box into vibrational motion at the same natural frequency as the string. The entire system (string, guitar, and enclosed air) begins vibrating and forces surrounding air particles into vibrational motion.
The tendency of one object to force another adjoining or interconnected object into vibrational motion is referred to as a forced vibration. In the case of the guitar string mounted to the sound box, the fact that the surface area of the sound box is greater than the surface area of the string means that more surrounding air particles will be forced into vibration. This causes an increase in the amplitude and thus loudness of the sound.

12. Resonance with Tuning Forks
Video: Resonance with tuning forks
In this demonstration, one tuning fork forces another tuning fork into vibrational motion at the same natural frequency. The two forks are connected by the surrounding air particles. As the air particles surrounding the first fork (and its connected sound box) begin vibrating, the pressure waves that it creates begin to impinge at a periodic and regular rate of 256 Hz upon the second tuning fork (and its connected sound box). The energy carried by this sound wave through the air is tuned to the frequency of the second tuning fork. Since the incoming sound waves share the same natural frequency as the second tuning fork, the tuning fork easily begins vibrating at its natural frequency.

13. Resonance
resonance - when one object vibrating at the same natural frequency of a second object forces that second object into vibrational motion.

14. Resonance Videos
Water Glass Resonance Mechanical Universe: Resonance