Fundamental frequency

Also called First harmonic

The fundamental frequency is the lowest frequency at which a standing wave can be set up in a given region. It corresponds to the longest wavelength the boundary conditions allow, and it is also called the first harmonic.

AP Physics 2 EK 14.6.B.2: a standing wave with the longest possible wavelength is called the fundamental or first harmonic. The second-longest wavelength is typically called the second harmonic, the third-longest the third harmonic, and so on.

Work it in two steps. Get $\lambda_1$ from the geometry, then convert with $\lambda = v/f$ from the AP Physics 2 sheet.

Region	$\lambda_1$	Harmonics present
String, both ends fixed	$2L$	every $n$
Pipe, both ends open	$2L$	every $n$
Pipe, one open one closed	$4L$	odd $n$ only

The odd-harmonic case is the exam point. EK 14.6.B.2 states it: for a standing wave with a node at one end and an antinode at the other end, only odd harmonics can be established. A pipe closed at one end has no second harmonic at all, and because its fundamental wavelength is $4L$ rather than $2L$ , its fundamental sits an octave below an open pipe of the same length.

No harmonic formula is printed. Neither the AP Physics 2 CED nor its equation sheet gives $f_n = nv/2L$ . You are meant to draw the pattern, read $\lambda_n$ off it and use $\lambda = v/f$ . EK 14.6.B.3 says as much: visual representations of standing waves are useful in determining the relationships between length of the region, wavelength, frequency, wave speed, and harmonic.

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