Standing Waves and Resonance
المؤلف:
Professor John W. Norbury
المصدر:
ELEMENTARY MECHANICS & THERMODYNAMICS
الجزء والصفحة:
p 197
1-1-2017
2646
Standing Waves and Resonance
When waves travel down a string they can reflect back from the other end and interfere with the other waves. In this way standing waves of different wavelength can be produced. The wave of lowest frequency (longest wavelength) is called the fundamental harmonic. Higher frequencies are called higher harmonics. The relations between the wavelength ¸ and the length of the string L for the various harmonics are
etc. These can be written in general as
Now the wave speed is 
and writing 
gives 
or
This is an extremely important formula for the design of muscial instruments.
Example Middle C has a frequency of 262 Hz. What tension do we need to apply to a violin string to get this frequency for the fundamental harmonic? (Assume the string has a mass of about 10 gram and a length of 1/4 m.)
Solution The mass per unit length μ is
The frequency is given by 
. The fundamental harmonic corresponds to n = 1, giving
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