Click here to go to the applet.
This java applet is a simulation that demonstrates wave motion
in a perfectly elastic rectangular
membrane (like a drum head, except rectangular instead of round),
showing its various vibrational modes.
An ideal continuous membrane has an infinite number of modes of oscillation,
each with its own frequency. The lowest-frequency mode is called the
fundamental, and involves the entire membrane vibrating up and down
at a frequency determined by the membrane's size, tension, and mass.
The other modes involve parts of the
membrane (called nodal lines) standing still while the rest of the membrane
vibrates. When a membrane is vibrating, more than one mode is typically
present at once.
At the top of the applet on the left you will see the membrane.
To set it in motion, click Fundamental.
If you click Clear, it will be at rest again.
Below the membrane you will see a graph showing each
normal mode's contribution to the membrane's vibration.
The modes are laid out in the following order:
| 1,1 |
2,1 |
3,1 |
... |
| 1,2 |
2,2 |
3,2 |
... |
| 1,3 |
2,3 |
3,3 |
... |
| ... |
... |
... |
... |
where m,n refers to the mode having a shape proportional to
sin(mx)sin(ny), where x ranges from 0 to pi. So, mode 1,1 has one "bump" and does not have
any nodal lines. Mode 3,2 has six bumps (3 in the x direction and
2 in the y direction), so it has two nodal lines crossing the x axis
and one nodal line crossing the y axis. In general, mode m,n
has m*n bumps, m-1 nodal lines crossing the x axis,
and n-1 nodal lines crossing the y axis.
Normally the edges of the membrane are fixed.
If the Fixed Edges checkbox is turned off, then the edges are
free to vibrate, and this causes a new set of modes to be available:
|
1,0 |
2,0 |
... |
| 0,1 |
1,1 |
2,1 |
... |
| 0,2 |
1,2 |
2,2 |
... |
| ... |
... |
... |
... |
where m,n refers to the mode having a shape proportional to
cos(mx)cos(ny). Modes 0,n are constant in the x direction and
modes m,0 are constant in the y direction. The slot for 0,0 is
left empty because this is simply a constant factor added to the height
of the membrane. This constant is always 0 in this applet.
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 1,2 has a positive coefficient, mode 3,1 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 1,3 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 2,2 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
up and down. 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.
You can modify the membrane in one of two ways. You can
modify the normal modes. Or you can
click on it directly and drag the mouse up or down;
it will be "poked" at that point.
The Mouse popup controls what happens when you click on the membrane.
The default setting is Poke membrane, which causes the membrane to be
deformed at the point where you click.
If you set the popup to Shape membrane, you can edit
the displacement of each individual point.
If you set it to Adjust view angle or Adjust view zoom, you can
adjust the orientation or size of the 3-d view.
And if you set it to Show mode, and then click on one of the modes
in the frequency grid, it will show the shape of that mode.
The Display popup can be used to control whether the 3-d or 2-d
view is shown. By default it is set to Display 3-d Only. To show a
2-d representation, you can select Display 2d Only or Display 3d+2d.
The 2-d representation often gives you a clearer picture of what is
going on. Also the Mouse = Shape membrane popup menu item can be
used to alter multiple points on the membrane at once by drawing
on the 2-d representation.
The 3d View popup can be used to control the look of the 3-d view.
By default it is set to Solid, but you can also see a wireframe
view by selecting one of the other options.
The Stopped checkbox can be used to stop the motion of the membrane
for a moment.
The Show Frequencies checkbox can be used to show or hide
the frequency grid. By default, it is shown.
The Color checkbox can be used to turn off color. This may
look better and/or it may make the applet run faster.
The Sound checkbox allows you to hear the sound the membrane
would make.
The Simulation Speed slider controls how fast the simulation will
proceed.
The Damping slider controls how much damping there is. Damping is
a force that slows the membrane down. High harmonics are damped more than lower ones.
The default
setting is set to zero, so you may want to set this higher in order to
get more realistic behavior.
The Brightness slider controls the brightness, just like on a
TV set. This can be used to view faint waves more easily.
The Resolution slider will adjust the fineness of the
rendering. This can be set lower to speed up the applet.
The Aspect Ratio slider will adjust the width of the membrane.
The Base Frequency slider will adjust the fundamental frequency of the
membrane, when the Sound checkbox is on. Normally this would be
determined by the tension, size, and mass of the membrane.
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 aspect ratio of the membrane, which is its width relative
to its length. This is controlled by the Aspect Ratio slider.
By default it is equal to one, which means the membrane is square.
In this case, all the m,n modes are degenerate, because
they have the same frequency
as the n,m modes.
If the aspect ratio is not equal to one,
there usually is no degeneracy. There may be some if the aspect ratio
is equal to an integer or a simple fraction. The aspect ratio slider
ranges from 1/2 to 3, so in either of those cases there will be some
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 membrane will change shape
at a rate equal to the frequency of the modes.
Click here to go 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. Fuscus sees his object, and being mischievously determined to defeat it, answers: ‘Yes, I remember perfectly, but we must wait for some better opportunity; this is the thirtieth Sabbath, do you wish to insult the circumcised Jews?’ ‘I have no scruples on that point,‘ replies the impatient poet. ‘But I have,’ rejoins Fuscus,—‘a little weak-minded, one of the many, you know—excuse me, another time.‘331 Nor were the Jews content with the countenance thus freely accorded them. 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. In presenting the money bills in the month of May, Speaker Onslow could not avoid remarking that there were two circumstances which tended to create alarm—foreign subsidies and foreign troops introduced, and nothing but their confidence in his Majesty could allay their fears, or give them confidence that their burdens would be soon reduced. There was, in fact, no chance for any such reduction, for wars, troubles, and disgraces were gathering around from various quarters. The first reverse came from the Mediterranean. MUCH to their amazement, the boys waked up the next morning in Nashville, and found that they had passed through the "dark and bloody ground" of Kentucky absolutely without adventure. After drawing and dividing the rations and cartridges. 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. He walked straight, and his roll was natural to him, while though he was exceedingly cheerful, and often burst into song, his words were not jumbled, and he generally seemed to have a fair idea of what he was saying. "But I heard what the doctor said to you." "A purty accident—wud them stacks no more dry than a ditch. 'Twas a clear case of 'bustion—fireman said so to me; as wicked and tedious a bit o' wark as ever I met in my life." Calverley stept from the shadow of the cliff, and beheld a meteor in the sky, brightening and expanding, as the clouds opened, until it assumed the appearance of a brilliant star, of astonishing magnitude, encircled by dazzling rays, which, in a singular manner, were all inclined in one direction, and pointing to that part of the horizon where lay the rival of England—France. The foreman's face assumed a deeper hue than usual: he looked fiercely at the galleyman, but there was a determination in the weather-beaten face that made him pause ere he spoke. "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.