What a Clamped String Is Allowed to Do
Last timeTwo Pendulums and a Weak Spring
A string is many small pieces each pulling on its neighbours, so it has many of the combinations from the last lesson. Clamping both ends decides exactly which shapes are allowed.
A string is not a new kind of object. It is the previous lesson with more pieces:
each little stretch of it pulls on the stretches either side, exactly as the spring
pulled on the two pendulums.
So the same question applies. Are there combinations that move at a single rate and
ignore each other? There are, and for a string they have a name outside physics as
well as in it, because a musician can hear them.
The clamps do the choosing
Both ends of the string are held still. That single fact is what turns a continuous
range of possible shapes into a list.
A smooth wiggle that leaves zero at the left end comes back to zero at regular
intervals along its length. If the right-hand clamp happens to fall at one of those
places, the shape is possible. If it falls anywhere else, the shape would require the
clamped end to move, and the clamp does not move.
- the distance between the clamps, which is the only length in the problem
- the wavelength of the shape, meaning the distance over which the wiggle completes a full up-and-down
- which shape it is, counted by how many half waves fit. This is forced to be a whole number by the clamps and it is the only thing distinguishing one allowed shape from another
An allowed shape keeps its shape
The claim is stronger than it sounds. In one of these shapes, every point of the
string oscillates at the same rate and they all cross the straight line at the same
instant. The string swells to one side, flattens completely, and swells to the
other, without the shape ever changing or anything moving along it.
Rather than assert that, here is a chain of three beads on springs between two
walls, which is a string with the detail turned down far enough to watch. The beads
have been placed in the slowest allowed shape, with the middle one further out than
the other two by a factor of the square root of two, and released from rest.
The lesson stops here
3 more paragraphs to go
You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.
The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.
See the planThe contentsThis is the reading half
Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.
The contents