Video: Harmonics of Vibrating Strings

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Harmonics of Vibrating Strings

 
Every string has a set of frequencies at which it naturally vibrates. Each frequency is associated with a standing wave pattern. Standing wave patterns, like this one, have nodes and antinodes. A node is a location along the string that appears to be standing still – a point of "no des"-placement. Get it? An antinode is the opposite – a point that is vibrating wildly from a maximum positive displacement to a maximum negative displacement.
 
Each frequency at which a string naturally vibrates is known as a harmonic frequency. The lowest-frequency harmonic is the first harmonic or fundamental frequency. Other frequency values are whole number multiples of the fundamental frequency and are referred to as the second harmonic, third harmonic, fourth harmonic, etc.
 
This is the standing wave pattern for the first harmonic. It has one anti-node and one-half a wavelength in the string. The wavelength of the first harmonic is ALWAYS twice the string length. If the string is 60 cm in length, then the wavelength would be 120 cm. Other harmonics have wavelengths that are fractions of this wavelength - like one-half, one-third, one-fourth the wavelength of the first harmonic.
 
Compared to the 1^st harmonic, the 2^nd harmonic has twice the frequency, one-half the wavelength, and two anti-nodes. It looks like this. The 3^rd harmonic has three times the frequency and one-third the wavelength of the first harmonic. You can tell the pattern for the 3^rd harmonic because there are three anti-nodes. The 4^th harmonic has four times the frequency and one-fourth the wavelength of the 1^st harmonic. And you know it’s the 4^th by its four antinodes.
 
You likely noticed a pattern in all these patterns. This table summarizes the relationships associated with the pattern, the frequency (f) and wavelength (l) values, and the number of antinodes for the first several harmonics. In general, if you know the frequency (f) of any harmonic, you can determine the frequency of all the harmonics. And if you know the string length (L), you can determine the wavelength of the first harmonic (l₁) and all other harmonics.
 
Here's the first of two examples: A vibrating string has a first harmonic of 100 Hz. What would the standing wave pattern look like for the same string when vibrating at 500 Hz?
The first harmonic is 100 Hz. Other frequencies are a whole-number multiples of 100 Hz. The 500 Hz harmonic has five times the frequency so it is the 5^th harmonic. The pattern would have five antinodes … like this one...
 
Here’s our second example: A string is 75 cm long. What would the standing wave pattern look like for a standing wave with a wavelength of 50 cm?
The first harmonic of such a string would have a wavelength of 150 cm – always twice the string length. Other harmonics would be a fraction of 150 cm, like 1/2, 1/3, 1/4, 1/5, etc.  The 50-cm wavelength is one-third of 150 cm; so it represents the third harmonic. Its standing wave pattern has three antinodes … like this one ...
 
I'm Mr. H, letting you know … You got this!



 
 

 

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