5-3 Vibrational Mode of a Stretched String

How are strings used in music to produce sound?

All string orchestra instruments (violin, viola, cello, and bass) are set up with a string placed over a bridge and end support. The distance between the endpoint and bridge is the string's length. Tension is placed on the string when using the finger to pluck a string or the bow to produce a sound.

What happens when a string is disturbed (bowed or plucked)?

1) The string is set into periodic motion.

2) The disturbance propagates along the string in one direction.

3) When the wave reaches the end of the string it reflects back along the string, producing a standing wave.

What affects the speed of the sound wave?

The speed of the sound wave can be found with the formula:

    velocity = frequency X wavelength

The speed of sound is also proportional to the amount of tension. (This is why a string instrument will sound louder when there is a quicker bow or more pressure on the bow.)

    velocity = the square root of (T/d)
    where T is the tension and d is the linear density

Experiment:: Try It Yourself"

" Slowly depress the C2 key on the piano and hold it down. In this way, the damper is lifted from the string, but the hammer does not strike it. Now strike C3 hard and staccato. After the initial sound has faded away, you will hear the C2 string vibrating clearly and beautifully in its second harmonic vibrational mode. Lift the C2 key and the damper will make contact with the string thereby terminating the sound. Repeat the above steps by striking G3, C4, E4, G4, and so on- all hard and staccato- and you will hear the C2 string vibrating in its third, fourth, fifth, and sixth harmonic modes, respectively. Now repeat the experiment, but this time strike D3 or F3. Nothing is heard on the C2 string because neither D3 nor F3 are harmonics of C2. This experiment is convincing evidence that a string can vibrate with many different frequencies provided that they are harmonically related."

Taken from: Physics and the Sound of Music, 2nd Edition, by John S. Rigden

How can you find the frequencies of a stretched string's harmonics?

The frequencies of a string's harmonics can be found with the formula:

    fn = (n(2L)) X (the square root of (T/d))
    where L is the length of the string, T is the tension, and d is the linear density

Wavelengths of the Natural Frequencies of a String in Terms of Its Length L

	Frequency	Wavelength	Wavelengths in Terms of L
Fundamental	f1	Wavelength 1	2L = (2L)/1
Second Harmonic	f2	Wavelength 2	L = (2L)/2
Third Harmonic	f3	Wavelength 3	(2/3)L = (2L)/3
...
...
...
nth harmonic	fn	Wavelength n	(2/n)L = (2L)/n

Practice Problems

1) A string with a linear density of 0.00082 kg/m is stretched so that the tension is 52 N. The string has a length of 0.7 m. What are the frequencies of the first three harmonics?

5-3: 1 SEE ANSWER

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