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