This java applet
demonstrates standing sound waves
in a 3-d rectangular box.
At the top of the applet on the left you will see the box,
oscillating in its fundamental mode.
Below the box you will see a series of grids showing each
normal mode's contribution to the vibration.
In each grid, the modes are laid out in the following order:
| 0,0,p |
1,0,p |
2,0,p |
... |
| 0,1,p |
1,1,p |
2,1,p |
... |
| 0,2,p |
1,2,p |
2,2,p |
... |
| ... |
... |
... |
... |
and each grid has a different value of p, as shown below:
| p=0 |
p=1 |
p=2 |
p=3 |
| p=4 |
p=5 |
p=6 |
p=7 |
So for example, the top left corner in the top left grid is the
0,0,0 mode, and the top left corner in the bottom right grid is the
0,0,7 mode.
The notation m,n,p refers to the mode having an acoustic pressure
proportional to
cos(mx)cos(ny)cos(pz), where x ranges from 0 to pi.
Each element of the grid has a color which indicates the presence or
absence of the mode it represents. Black means the mode is not present;
green means the mode is present with a positive coefficient, and red means
it is present with a negative coefficient. So in this example,
mode 0,1,0 has a positive coefficient, mode 2,0,0 has a negative one,
and the rest of the modes are absent:
In addition, each mode may have a phase shift, which indicates that its
oscillatory cycle leads or lags the others. This is indicated by
a blue line. In this example, mode 0,2 has a blue line about 1/3 of
the way across the width of the box, which means a phase shift of
about 1/3 the maximum possible. The maximum is pi*2, so this shift is
pi*2/3. Mode 1,1 is also present but does not have a phase shift.
A phase shift is normally hard to detect. If the applet is stopped,
then by altering the phase of a mode you can see the mode oscillate.
If the applet is running it is much harder to tell,
except in a case which will be explained later.
You can add or remove a mode by clicking on it. If you click and drag
up and down you can fine-tune the magnitude of the mode. If you drag
left and right you can alter the phase shift.
For performance reasons, only 10 modes can be active at a time.
The Mouse popup controls what happens when you click on the box.
If you set it to Adjust view angle or Adjust view zoom, you can
adjust the orientation or size of the 3-d view by clicking and
dragging on the box.
The Stopped checkbox can be used to stop the oscillation
for a moment.
The Show Frequencies checkbox can be used to show or hide
the spectrum of mode frequencies. Each mode is represented on
the spectrum by a vertical line. Higher frequencies are on the right.
In some cases, multiple modes will have the same frequency. This is
called degeneracy. When that happens, the line for that frequency
will be taller than normal. The more modes have the same frequency,
the taller the line.
The Alternate Rendering checkbox is used to speed up
rendering, but it actually slows things down on some machines.
Also for large windows and low resolutions it is usually slower.
(Internally, it uses the MemoryImageSource class instead of drawing
a bunch of rectangles.)
The Simulation Speed slider controls how fast the simulation runs.
The Brightness slider can be used just like the brightness on
a TV, to brighten dim areas of the box.
The Image Resolution slider can be used to generate a higher
quality image. If you move it to the right, it will increase the
resolution of the image, but it will also slow things down,
so be careful.
The Width and Height sliders will adjust the width or height
of the box. This changes the frequency spectrum. The depth can't be
changed.
Normally each mode has a different frequency. However in some cases,
several modes can have the same frequency. This is called
degeneracy. The presence or absence of degeneracy
depends on the dimensions of the box.
By default the width, height, and depth are equal, which means it is
a cube.
In this case, all the m,n,p modes are degenerate, because
they have the same frequency
as the n,m,p modes, and the m,p,n
modes, and the p,n,m modes, and so on.
If the width, height, and depth are not all equal,
there usually is no degeneracy. There may be some if two or more of
the dimensions
have a ratio equal to an integer or a simple fraction.
The width and height slider
ranges from 1/2 to 3, so if either slider is set to one of those extremes,
there will be degeneracy.
When two or more modes have the same frequency, each one is shown in
yellow, like so:
The interesting thing about degenerate modes is that if you combine
them, you get a new mode that oscillates with simple harmonic motion,
just like any other mode. Since both modes are oscillating at the
same rate, you can't tell them apart. If one of the modes has a phase
shift relative to the other one, then the wave will change shape
at a rate equal to the frequency of the modes.
The frequencies are proportional to:
where Lx is the width of the box,
Ly is the height, and
Ly is the depth.
Back to the applet.
java@falstad.com
A far higher place must be assigned to Judaism among the competitors for the allegiance of Europe. The cosmopolitan importance at one time assumed by this religion has been considerably obscured, owing to the subsequent devolution of its part to Christianity. It is, however, by no means impossible that, but for the diversion created by the Gospel, and the disastrous consequences of their revolt against Rome, the Jews might have won the world to a purified form of their own monotheism. A few significant circumstances are recorded showing how much influence they had acquired, even in Rome, before the first preaching of Christianity. The first of these is to be found in Cicero’s defence of Flaccus. The latter was accused of appropriating part of the annual contributions sent to the temple at Jerusalem; and, in dealing with this charge, Cicero speaks of the Jews, who were naturally prejudiced against his client, as a powerful faction the hostility of which he is anxious not to provoke.330 Some twenty years later, a great advance has been made. Not only must the material interests of the Jews be respected, but a certain conformity to their religious prescriptions is considered a mark of good breeding, In one of his most amusing satires, Horace tells us how, being anxious to shake off a bore, he appeals for help to his friend Aristius Fuscus, and reminds him of217 some private business which they had to discuss together. 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The same poet elsewhere intimates that whenever they found themselves in a majority, they took advantage of their superior strength to make proselytes by force.’332 And they pursued the good work to such purpose that a couple of generations later we find Seneca bitterly complaining that the vanquished had given laws to the victors, and that the customs of this abominable race were established over the whole earth.333 Evidence to the same effect is given by Philo Judaeus and Josephus, who inform us that the Jewish laws and customs were admired, imitated, and obeyed over the whole earth.334 Such assertions might be suspected of exaggeration, were they not, to a certain extent, confirmed by the references already quoted, to which others of the same kind may be added from later writers showing that it was a common practice among the Romans to abstain from work on the Sabbath, and even to celebrate it by praying, fasting, and lighting lamps, to visit the synagogues, to study the law of Moses, and to pay the yearly contribution of two drachmas to the temple at Jerusalem.335 Jeff’s hand was quietly coming down. “What happened to you?” begged Sandy. “Something new has come up, sir. I was waiting there by my ship a good while back, and I heard another one cruising and spiraling, shooting the field, I guess, because he came in and set down. My crate, just the way you ordered, was down by the grove, not in plain sight in the middle of the course. But Jeff set his ship down, left the engine running, and went off. I stayed hid to see what would happen, but when he didn’t come back, I thought I’d better go and find you—and see if it meant anything to you.” She stood alone, with the sticky, wet knife in her hand, catching her breath, coming out of the madness. Then she stooped, and pushing the branches aside felt about for her pistol. It lay at the root of a tree, and[Pg 80] when she had picked it up and put it back in the holster, there occurred to her for the first time the thought that the shot in the dead stillness must have roused the camp. And now she was sincerely frightened. If she were found here, it would be more than disagreeable for Landor. They must not find her. She started at a swift, long-limbed run, making a wide detour, to avoid the sentries, bending low, and flying silently among the bushes and across the shadowy sands. The year 1756 opened with menaces to England of the most serious nature. The imbecility of the Ministry was beginning to tell in the neglect of its colonies and its defences. France threatened to invade us, and a navy of fifty thousand men was suddenly voted, and an army of thirty-four thousand two hundred and sixty-three of native troops; but as these were not ready, it was agreed to bring over eight thousand Hessians and Hanoverians. To pay for all this it was necessary to grant excessive supplies, and lay on new duties and taxes. 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Si gave the boys the necessary instruction about having their things ready so that they could get them in the dark the next morning, and ordered them to disregard the bonfires and mirth-making, and lie down to get all the sleep they could, in preparation for the hard work of the next day. Then, like the rest of the experienced men, who saw that the campaign was at length really on, and this would be the last opportunity for an indefinite while to write, he sat down to write short letters to his mother and to Annabel. "Bully for the Wild Wanderers of the Wabash," Shorty joined in. "They're the boss regiment in the army o' the Cumberland, and the Army o' the Cumberland's the boss army on earth. Hooray for US Co. Le's have a speech. Where's Monty Scruggs?" "Bring a light, do—I can't abide this dark." Albert suddenly began to look uneasy. After all he was not really drunk, only a little fuddled. 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"Galleyman," he at length said, "you knew the business before you came: if you be so fond of saving old witches' lives, why didn't you say so, that I might not now be in this dilemma?" "No, no, not the boy," replied Merritt, rather impatiently.