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[Note: The Java Applet for this animation can no longer be run in a web browser. Clicking this link, however, will download to your computer a Java Web Start file named PipesLaunch.jnlp. This file can launch the animation in a separate window on your computer. Downloading and executing the file may be blocked by local Java permissions, but persevere! You may finally have to launch the file by right-clicking (control-clicking) it and choosing Open, instead of simply double-clicking it.]
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This applet shows, in slow motion, air molecules in a cylindrical pipe of length L undergoing normal mode vibrations with selectable boundary conditions. In the diagram, the vertical green lines represent ideal planes of air molecules moving longitudinally in the pipe under the influence of a normal mode sound wave. Random thermal motion is ignored. In the diagram, ideal boundary conditions are used, with a pressure node exactly at the open ends of the pipe. That is, no end corrections have been applied.
A secondary but important purpose of this applet is to help relate the longitudinal motions of air molecules in standing waves to the transverse line graphs often drawn in textbooks to represent them.
Normal modes are vibrations in which all parts of the system execute simple harmonic motion at the same frequency, all parts passing through equilibrium at the same instant. Normal modes are also known as standing waves. Normal modes are not the actual vibrations of air in any real instrument. But they are fundamental because actual motions can be obtained by summing normal modes with appropriate relative amplitudes and phases.
For a given system, normal mode vibrations are possible only at specific frequencies. It happens that the frequencies of mode vibrations of air in a cylindrical pipe are integer multiples of the lowest mode frequency. This guarantees that arbitrary sums of mode vibrations will yield periodic disturbances and hence that cylindrical pipes will be integral components of musical instruments.
With this applet one can see why pressure antinodes occur at displacement nodes – air piles up or moves away from both sides of a displacement node. And one can also see why there are pressure nodes at displacement antinodes – at a displacement antinode the distances moved by all molecules in an extended region are about the same, hence there is no density or pressure change.
1. At the bottom of the window is a button labeled “Go/Stop.” Clicking the button at any time stops or restarts the motion.
2. At the top of the window is a menu bar that controls the display. Available menu choices and their effects are the following:
| Menu Selection | Action |
|---|---|
| Pipes > About Pipes | Reveals the name of the author and the year the applet was written. |
| Ends > Closed-Open | Selects the boundary condition for a pipe closed at the left end and open at the right end. This models a clarinet air column or an organ flue pipe closed at the top. |
| Ends > Open-Open | Selects the boundary condition for a pipe open at both ends. This models a flute air column or an organ flue pipe open at the top. |
| Ends > Closed-Closed | Selects the boundary condition for a pipe closed at both ends. Such an air column could only be excited by an internal energy source, and this example models air in one dimension of a closed room. |
| Mode > Mode 1 | The lowest frequency normal mode motion of air in the pipe. On the diagram the frequency of this mode is given in terms of v, the speed of sound in air, and L, the ideal length of the pipe. |
| Mode > Mode 2 | The second-lowest frequency normal mode motion of air in the pipe. The frequency of this mode, relative to that of the first mode, is shown on the diagram. |
| Mode > Mode 3 | The third-lowest frequency normal mode motion of air in the pipe. The frequency of this mode, relative to that of the first mode, is shown on the diagram. |
| Graph > Pressure | Selecting this turns on or off a dynamic graph of the deviation of the pressure from equilibrium at each point in the pipe at each instant. This and the displacement graph may be displayed simultaneously. |
| Graph > Displacement | Toggles a dynamic graph of the displacement of air molecules from their undisturbed positions in the pipe at each instant. This and the pressure graph may be displayed simultaneously. |