Click here to go to the applet.
This java applet is a simulation that demonstrates waves on a vibrating string.
A loaded string is a very light (massless) string with a number of small
masses attached to it at equal intervals along the string. As the number of
masses increases, the behavior of the loaded string more closely approximates
the behavior of a real string where the mass is distributed continuously
along its length.
A loaded string with n loads has n normal modes of oscillation.
The first mode
is called the fundamental, and involves the entire string vibrating up
and down at a frequency determined by the string's length, tension, and
mass. The other modes are called harmonics, and involve parts of the
string (called nodes) standing still while the rest of the string vibrates.
The frequency of the harmonics is some multiple of the fundamental
frequency. With a real (continuous) string, the frequency of the
second harmonic is twice the frequency of the fundamental, and the
third harmonic is three times the fundamental, and so on. With a
loaded string with a finite number of masses, that's not quite correct,
but it's reasonably close, at least for the first few harmonics.
When a string is vibrating, more than one mode is typically present at once,
but the fundamental is typically the loudest.
At the top of the applet on the left you will see the string.
By default, the number of loads is set to 60.
You won't see any loads on the string,
because if the number of loads is more than 39, the applet does not show
them, to avoid clutter.
To set the string in motion, click "Center Pluck" or
"Fundamental".
If you click "Fundamental" then the string will vibrate at the
fundamental frequency.
If you click "Center Pluck", the string will be plucked in the center;
this will cause some of the harmonics to be excited as well, although
the fundamental will still dominate.
If you click "Clear", the string will be at rest again.
Below the string you will see a graph showing each
normal mode's contribution to the string's vibration.
There are two sets of terms; on top are the magnitude terms, which
shows the amplitude of each normal mode, and on
the bottom are the phase terms. The fundamental is on the left and
higher harmonics are on the right. Since the higher harmonics oscillate
at a higher frequency than the fundamental, the phase terms will move
faster on the right.
If you move the mouse over one of the harmonics, it will turn
yellow, and the corresponding harmonic will be
drawn on top of the string in yellow (unless it's too small to see). So if you move the mouse
over all the harmonics, you can see each of the terms individually.
You can modify the string in one of two ways. You can
click on it directly; in this case,
it will be plucked at that point. Or, you can modify the normal modes.
The "Mouse" popup controls what happens when you click on the string.
The default setting is "Pluck string", which causes the string to be plucked
where you click. If you set the popup to "Shape string", you can edit
the shape of the string directly.
The "Display" popup can be used to display some additional information.
By default it is set to "Display Phases", which shows the magnitudes
and phases of the harmonics.
If you set it to "Display Phase Cosines", you will see the cosines of
the phases rather than the phases themselves. If you set it to
"Display Phasors", it will show phasors for the first three modes.
If you set it to "Display Modes", it will display graphs of the
first twelve active modes.
If you set it to "Display Left+Right",
it will decompose the wave on the string into two components, one
travelling
left and one travelling right. The wave on the string is called a standing
wave because it is not moving in any direction. But if
you add up the displacements of the travelling waves at each load,
you get the standing wave. Notice that the travelling waves always
have equal and opposite displacements at the edges because they have to
add up to zero there.
The travelling waves will often have a fixed shape, but in some cases
they will change shape slightly as they move. Part of this is damping,
but even without damping there may be a shape change. The shape change is
caused by the fact that not all the harmonics move at the same speed
along the string.
The lower modes move at about the same speed but the higher modes
move slower.
This is called dispersion. If the string were continuous, with the
mass distributed evenly along its entire length, there would be no
dispersion. But the mass is localized at a fixed number of points
(at the loads), so the string is dispersive. To see an example of this,
turn off damping, turn on "Display Left+Right", and turn up one of
the middle harmonics and one of the highest harmonics (by "turn up"
I mean set the magnitude of that harmonic to the maximum). Notice that
the travelling waves are changing shape as they move. If you increase
the number of loads without changing anything else, this effect will
be reduced. (Dispersion will still be present, but you will need to
select higher modes to see it.)
The "Stopped" checkbox allows you to stop or start the simulation.
The "Driving Force" checkbox allows you to push the string with a
periodic driving force that acts at the center of the string.
The magnitude of the force is shown with an arrow that oscillates
back and forth. The frequency of the driving force's oscillation can
be controlled with the "Force Frequency" slider. The driving force
won't accomplish very much unless the force frequency is close to the string's
resonance frequency (the fundamental). To set it there, click the
"Resonance Frequency" button.
The "Sound" checkbox allows you to hear the sound the string would
make. To make this more realistic, move the damping slider to the middle.
The "Log View" checkbox uses a logarithmic scale to show the
magnitudes of each harmonic.
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 string down. The default
setting is fairly low so you may want to set this higher in order to
get more realistic behavior.
The "Number of Loads" slider will adjust the number of loads on the
string. This can be set as low as one. If you change the number of
loads then you also change the number of normal modes.
The "Tension" slider will adjust the tension. Lowering the tension
will reduce the frequency of the string's vibration but will increase
the amplitude.
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.