Standing Waves in Tubes and Strings

Vocabulary flashcards covering boundary conditions, harmonic formulas, overtones, and physical properties of standing waves in tubes and strings.

Equation and Unit for Frequency in terms of Velocity ($v$) and Wavelength ($\lambda$)

$f = v/\lambda$ in units of Hz or 1/s

Equation for the speed of sound in air (in ºC) at a given temperature $T$

$$v = 331 + 0.6T$$

Node (N)

A point of no motion in a standing wave displacement envelope; always located at a closed end.

Antinode (A)

A point of maximum motion in a standing wave displacement envelope; located at an open end.

Speed of sound in air (at $20\,^\circ\text{C}$)

Approximately $343\,\text{m/s}$.

Wavelength Equation for Tube Open at Both Ends

$\lambda_{n}=2L/n$ where $L$ is the length of the tube and $n$ is the fundamental ($n = 1$) or harmonic number

Frequency Equation for Tube Open at Both Ends

$f_n = nv/2L$ where $L$ is the length of the tube, $n$ is the fundamental ($n = 1$) or harmonic number, and $v$ is the speed of the wave

Wavelength Equation for Tube Closed at Both Ends or a String connected to a Surface at Both Ends

$\lambda_n = 2L/n$ where $L$ is the length of the tube and $n$ is the fundamental ($n = 1$) or harmonic number

Frequency Equation for Tube Closed at Both Ends or a String connected to a Surface at Both Ends

$f_n = nv/2L$ where $L$ is the length of the tube, $n$ is the fundamental ($n = 1$) or harmonic number, and $v$ is the speed of the wave

Allowable values for $n$ in an open or completely closed tube

$$n = 1, 2, 3, 4, \text{etc}$$

Wavelength Equation for Tube Closed at One End

$\lambda_{n}=4L/n$ where $L$ is the length of the tube and $n$ is the fundamental ($n = 1$) or harmonic number

Frequency Equation for Tube Closed at One End

$f_n = nv/4L$ where $L$ is the length of the tube, $n$ is the fundamental ($n = 1$) or harmonic number, and $v$ is the speed of the wave

Allowable values for $n$ in a tube closed at one end

$n = 1, 3, 5, 7, \text{etc}$ (i.e. $n$ can only be an odd number)

Definition of Fundamental Frequency

The lowest frequency of a periodic waveform

Definition of Natural Frequency

The rate at which an oscillation system or wave tends to oscillate without any external disturbance

Definition of Resonant Frequency

A frequency that generates a maximum amplitude response in the system

Definition of Resonance

A phenomenon that occurs when an object or system is subjected to an external force or vibration whose frequency matches a resonant frequency; also occurs when a forced vibration matches an objects natural frequency

Definition of an Overtone

Any resonant frequency above the fundamental frequency (hit the speaker icon to hear an example of a fundamental pitch and its overtones)

Doppler Effect

The change in the frequency or, equivalently, the period of a wave in relation to an observer who is moving relative to the source of the wave (hit the speaker icon to hear an example)

What does the fundamental ($n = 1$) waveform look like for an open, closed, and closed at one end system?

What does the 2nd harmonic/1st overtone ($n = 2$) waveform look like for an open, closed, and closed at one end system?

Doesn’t exist for the closed at one end system

What does the 3rd harmonic/2nd overtone ($n = 3$) waveform look like for an open, closed, and closed at one end system?

What does the 4th harmonic/3rd overtone ($n = 4$) waveform look like for an open, closed, and closed at one end system?

Doesn’t exist for the closed at one end system (needs an odd $n$)

What does the 5th harmonic/4th overtone ($n = 5$) waveform look like for an open, closed, and closed at one end system?