When a string is fixed at both ends, standing waves can be set up on the string by moving a portion of the string with simple harmonic motion of small amplitude. For certain frequencies of that motion, called resonance frequencies, characteristic vibration patterns called normal modes can be produced. Since the wave speed in the string, v, depends only on the tension in the string and the mass per unit length of the string, each resonant frequency, f, is related to the wavelength, lamda, of the stationary pattern set on the string by the relation f equals v divided by lamda. The lowest resonance of the string is called the fundamental mode or first harmonic. The fundamental mode has one antinode in the center of the length of string. This mode occurs when the wavelength lamda is equal to twice the length of the string: lamda one equals two L. The next lowest normal mode of the string is the second harmonic, which has two antinodes and one node in the middle of the string in addition to those at both ends of the strings. The wavelength associated with that case is lamda two equals L The third harmonic has three antinodes and two nodes. Accordingly, the wavelength is two-thirds of the length of the string: lamda three equals two thirds L. This pattern is repeated for higher harmonics. The general formula for normal modes of a string of length L is: lamda sub n equals two L divided by n. Where n equals a positive interger and refers to the nth mode of vibration. This result is known as the standing-wave condition. The frequency of the nth harmonic is given by f sub n is equal to v divided by lamda sub n, which is equal to v times n divided by two L, where the wave speed v is the same for all frequencies. Which of these Standing waves has a wavelength lambda equal to L, the length of the string? ... The middle wave has two half wavelengths and thus one full wavelength on the string. Standing waves can also be set up on a string that is fixed only at one end. In that case, the free end is an antinode and the standing-wave condition must be modified. The fundamental mode or first harmonic of a string fixed at one end and free at the other is produced when the wavelength lambda is equal to four times the length of the string, written lambda one equals 4 L The next harmonic corresponds to four-thirds of the length of the string, which is written lambda three equals the product of 4 thirds and L ... The standing-wave condition for a string fixed at one end only can be written: lambda sub n equals 4L divided by n, where n is an odd positive integer, 1, 3, 5, and so on. Note that the even harmonics are missing because the free end is an antinode. Consequently, the resonant frequencies are given by: f sub n is equal to v divided by lambda sub n which is equal to the product of v and n divided by 4 L, where again n is an odd positive integer, 1, 3, 5, and so on.

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