This is an English translation of the Basque original for a course on mechanics, oscillations and waves.

It requires Java 1.5 or newer and was created by Juan M. Aguirregabiria with Easy Java Simulations (Ejs) by Francisco Esquembre.

I, lookang also thank Fu-Kwun Hwang, Wolfgang Christian and Francisco Esquembre for their invaluable help for which i have learnt so much from their work.!

closed-closed pipe system in 5th harmonic |

http://weelookang.blogspot.sg/2012/01/ejs-open-standing-wave-in-pipes-model.html https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejs_pipewee01.jar author: Juan Aguirregabiria and lookang (this remix version) |

changes made:

- 9 jan 2012 Design layout to usual bottom
- A = 0.49 instead of 0.4 previously to make same as lecture notes
- add Nodes texts and extra
- add Antinodes text and extra
- add v = f*lambda assuming speed of sound is 330 m/s
- 10 Jan 2012 add modeling component through drop-down menu and input field for learners to key in equations to understand the standing waves formed
- readjusted position of pipes and everything to start at y = 0 instead of the previous y =0.5 for ease of modeling
- 3 October 2013 reduce the number of air molecules representation (RED) to draw to 19 and thicken the lines
- added text into actual frames "u(t,x) displacement" and "p(t,x) pressure"
- add text pipe side view
- add text "pressure variation BLACK=-1, BRIGHT=+1 "
- add dt= slider to allow slowing slow of the simulation
- added boundary or envelope of the amplitude in dark-grey
- made u(x,t) and p(x,t) appears as the check-box is selected

closed-closed pipe system in 1st harmonic |

closed-closed pipe system in 2nd harmoni |

closed-closed pipe system in 3rd harmonic |

closed-closed pipe system in 4th harmonic |

closed-closed pipe system in 5th harmonic |

closed-closed pipe system in 1st harmonic random molecules representation |

closed-closed pipe system in 2nd harmonic random molecules representation |

closed-closed pipe system in 3rd harmonic random molecules representation |

closed-closed pipe system in 4th harmonic random molecules representation |

closed-closed pipe system in 5th harmonic random molecules representation |

closed-closed pipe system in 1st harmonic |

closed-closed pipe system in 2nd harmonic |

closed-closed pipe system in 3rd harmonic |

closed-closed pipe system in 4th harmonic |

closed-closed pipe system in 5th harmonic |

closed-closed pipe system in 1st harmonic |

closed-closed pipe system in 2nd harmonic |

closed-closed pipe system in3rd harmonic |

closed-closed pipe system in 4th harmonic |

closed-closed pipe system in 5th harmonic |

Standing waves in a pipe by Juan Aguirregabiria

Let us consider a narrow pipe along the OX axis. Each end may be open or closed. The simulation will display the first 5 normal modes, which are

u(t,x) = A sin(n π x) cos(ω t + δ) when both ends are closed.

u(t,x) = A sin((n-1/2) π x) cos(ω t + δ) when the left end is closed and the right end open.

u(t,x) = A cos((n-1/2) π x) cos(ω t + δ) when the left end is open and the right end closed.

u(t,x) = A cos(n π x) cos(ω t + δ) when both ends are open.

Units are arbitrary.

Below you may choose the mode n = 1, ...,5, as well as the animation step Δt.

The upper animation shows the displacement field u(t,x) and the pressure p(t,x) as functions of x at each time t.

In the lower animation you may see the evolution of the position x + u(t,x) of several points and a contour plot of p(t,x) (lighter/darker blue means higher/lower pressure).

Optionally one can see the nodes where the displacement wave vanishes at all times.

Scale has been arbitrarily enhanced to make things visible; but keep in mind that we are considering very small displacements and pressure changes in a narrow pipe.

Put the mouse point over an element to get the corresponding tooltip.

Activities by Juan Aguirregabiria

Compute the position of the nodes for mode number n in the four considered cases.

Use the simulation to check your calculation.

Where are the pressure nodes in the different cases?

Which is the relationship between the displacement and pressure waves? How does it appears in the animation?

This is an English translation of the Basque original for a course on mechanics, oscillations and waves. It requires Java 1.5 or newer and was created by Juan M. Aguirregabiria with Easy Java Simulations (Ejs) by Francisco Esquembre. I thank Wolfgang Christian and Francisco Esquembre for their help.

lookang also thank Juan M. Aguirregabiria for sharing such a useful computer model!

customization is below.

1. Musical instruments make use of stationary waves to create sound.

2. All strings (or pipes) have a natural frequency also known as the resonant frequency, which is related to the length of the string (or pipe).

3. The resonant frequencies can be determined using the following rules:

a. The two ends of a guitar string do not move and hence they must always be nodes.

b. The air molecule at any closed end of the pipe does not move and hence it must always be a node.

c. However, if the end of the pipe is open, the air molecule has the room to vibrate about the equilibrium position at maximum amplitude. This location is an antinode.

contribution to wikipedia:

## Harmonics and overtones

The tight relation between overtones and harmonics in music often leads to their being used synonymously in a strictly musical context, but they are counted differently leading to some possible confusion. This chart demonstrates how they are counted:

Harmonics are not overtones, when it comes to counting. Even numbered harmonics are odd numbered overtones and vice versa

Frequency | Order | Name 1 | Name 2 | Wave Representation | Molecules Representation |
---|---|---|---|---|---|

1 · f = 440 Hz | n = 1 | fundamental tone | 1st harmonic | ||

2 · f = 880 Hz | n = 2 | 1st overtone | 2nd harmonic | ||

3 · f = 1320 Hz | n = 3 | 2nd overtone | 3rd harmonic | ||

4 · f = 1760 Hz | n = 4 | 3rd overtone | 4th harmonic |