Standing Waves in Strings and Pipes

Vocabulary flashcards covering the harmonic modes, overtones, wavelengths, and frequencies for standing waves on strings fixed at both ends, open-open pipes, and closed-open pipes.

The 1st harmonic with no overtone, characterized by a length in wavelengths of $L = \frac{1}{2}\lambda$, a wavelength of $\lambda = 2L$, and a frequency of $f_1 = \frac{v}{2L}$.

Second harmonic (String fixed at both ends, or open-open pipe)

The 2nd harmonic and 1st overtone, characterized by a length in wavelengths of $L = \lambda$, a wavelength of $\lambda = L$, and a frequency of $2f_1 = \frac{v}{L}$.

Third harmonic (String fixed at both ends, or open-open pipe)

The 3rd harmonic and 2nd overtone, characterized by a length in wavelengths of $L = \frac{3}{2}\lambda$, a wavelength of $\lambda = \frac{2L}{3}$, and a frequency of $3f_1 = \frac{3v}{2L}$.

Fourth harmonic (String fixed at both ends, or open-open pipe)

The 4th harmonic and 3rd overtone, characterized by a length in wavelengths of $L = 2\lambda$, a wavelength of $\lambda = \frac{L}{2}$, and a frequency of $4f_1 = \frac{2v}{L}$.

General nth harmonic (String fixed at both ends, or open-open pipe)

The $n\text{th}$ harmonic and $(n-1)\text{th}$ overtone, characterized by $L = \frac{n\lambda}{2}$, $\lambda_n = \frac{2L}{n}$, and frequency $f_n = \frac{nv}{2L} = nf_1$.

Boundary conditions of a closed-open pipe

A resonant pipe configuration with a node at the closed end and an antinode at the open end.

Fundamental (Closed-open pipe)

The 1st harmonic with no overtone in a closed-open pipe, characterized by a length in wavelengths of $L = \frac{1}{4}\lambda$, a wavelength of $\lambda = 4L$, and a frequency of $f_1 = \frac{v}{4L}$.

Third harmonic (Closed-open pipe)

The 3rd harmonic and 1st overtone in a closed-open pipe, characterized by a length in wavelengths of $L = \frac{3}{4}\lambda$, a wavelength of $\lambda = \frac{4L}{3}$, and a frequency of $3f_1 = \frac{3v}{4L}$.

Fifth harmonic (Closed-open pipe)

The 5th harmonic and 2nd overtone in a closed-open pipe, characterized by a length in wavelengths of $L = \frac{5}{4}\lambda$, a wavelength of $\lambda = \frac{4L}{5}$, and a frequency of $5f_1 = \frac{5v}{4L}$.

Seventh harmonic (Closed-open pipe)

The 7th harmonic and 3rd overtone in a closed-open pipe, characterized by a length in wavelengths of $L = \frac{7}{4}\lambda$, a wavelength of $\lambda = \frac{4L}{7}$, and a frequency of $7f_1 = \frac{7v}{4L}$.

General harmonic (Closed-open pipe)

The $(2n - 1)\text{th}$ harmonic and $n\text{th}$ overtone in a closed-open pipe, characterized by $L = \frac{(2n - 1)\lambda}{4}$, $\lambda = \frac{4L}{2n - 1}$, and frequency $f = \frac{(2n - 1)v}{4L}$.