Unit 04 - Sound

Vibrating Strings  Each string on a guitar or violin has a distinct frequency when set in motion.  The frequency or pitch of a string is determined by four factors: 1) Length 2) Diameter 3) Tension 4) Density of the material ○ (nylon or steel strings).

1) Length  1. Frequency increases as the string length decreases, so frequency varies inversely with the length. f = frequency L = length of string  Example: halving the length produces double the frequency.

2) Diameter  2. Frequency increases as the string diameter decreases, so frequency varies inversely with the diameter. f = frequency d = diameter  Example: halving the diameter of the string produces double the frequency.

3) Tension  3. Frequency increases as the tension (tightness of string) increases. Frequency varies directly with the square root of the tension. f = frequency (in Hz) T = tension (in Newtons, “N”)  Example: increasing the tension 4 times produces double the frequency.

4) Density of Material  4. Frequency increases as the density of the string decreases. Frequency varies inversely with the square root of the density. f = frequency D = density  Example: decreasing the density to 1/4 of its value produces double the frequency.

Using the String Relationships  Usually two lengths (diameters, tensions, densities) are compared to determine an unknown frequency.  We should be able to derive the following expressions from the above information.  Length:  Diameter:  Tension:  Density:

Examples  1. A violin string 30.0 cm long produces a frequency of 288 Hz. How long does the string need to be to produce a frequency of 384 Hz?  Given: L 1 = 30.0 cm f 1 = 288 Hz f 2 = 384 Hz L 2 = ?

Example  2. A piano string with a frequency of 440. Hz is under a tension of 140. newtons. What tension is needed if the frequency is 523 Hz? (A newton is a unit of force usually written as just N.)  Given: f 1 = 440.Hz T 1 = 140.N f 2 = 523Hz T 2 = ?

Harmonics and Overtones  Recall our study of standing waves with nodes and loops.  Sound also behaves in standing waves.  When a string vibrates between two fixed points (like a guitar string) nodes are present at both ends.  Depending on the number of nodes and loops, different frequencies may form.

Harmonics and Overtones  When a string vibrates in its simplest form, it vibrates in one segment producing its lowest pitch or frequency. This is called the fundamental frequency (f) or 1st harmonic. fundamental (f) - eg. 220 Hz Nodes at each end and a loop in the middle. Nodes are areas of no movement - in this case, no sound.

Harmonics and Overtones  Here is the same string vibrating in two segments. 2nd harmonic (2f) - eg. 440 Hz  This is called the first overtone or 2nd harmonic. Since the string always has nodes at each end, the frequency of the overtones is a multiple of the fundamental (f).  The above frequency is 2f (440 Hz) since it vibrates in 2 segments. Notice that 440 Hz is one octave above the fundamental.

Harmonics and Overtones  Here is the same string vibrating in three segments.  3rd harmonic (3f) - eg. 660 Hz  This is the second overtone or 3rd harmonic. This frequency is 3f (660 Hz) since the string vibrates in 3 segments.

Example  What would be the frequency of the 4th harmonic? f = 220 Hz 4 th harmonic = 4 f 4 (220Hz) 880 Hz  String instruments vibrate with a complex mixture of overtones superimposed or stacked over the fundamental.  That's why a guitar sounds different than a piano when they play exactly the same frequency.