Standing Wave (String)

A stretched string fixed at both ends can only vibrate in certain patterns, because each end must be a node where the displacement is always zero. Those allowed patterns fit an integer number of half-wavelengths into the string length, so the wavelength is \u03bb_n = 2L/n and the frequency becomes f_n = v/\u03bb_n = nv/(2L). Larger n means more half-waves on the string and a proportionally higher pitch.

Use this when analyzing harmonics of a string on instruments like guitars, violins, or a lab string under tension. It is also useful in physics problems that give the wave speed on a fixed string and ask for an overtone or harmonic frequency.

• Using the full wavelength instead of the half-wavelength condition for a string fixed at both ends
• Forgetting that n counts harmonics: n = 1 is the fundamental, n = 2 is the first overtone, and so on
• Mixing up string length L with wavelength \u03bb, or using the wrong wave speed units