Every Starting Shape Is a Sum of the Allowed Ones
Last timeThe Shapes a String Is Allowed to Hold
A plucked string starts with a corner in it, which is not an allowed shape. It is a sum of allowed shapes, and that single fact turns the list of shapes into a complete method.
The previous lesson left a problem. A real string gets plucked, and a pluck leaves a
corner in it where the finger was. No allowed shape has a corner anywhere: they are
all smooth wiggles. So the string has been put into a shape it is not allowed to
hold, and the list of shapes appears to have nothing to say about the one thing
strings actually do.
The way out is the most useful idea in the course, and it is not a special fact
about strings.
The corner is a sum
Add together allowed shapes. The slowest one alone is a single smooth hump, which is
a poor corner. Add a small amount of the shape with three half waves and the hump
develops a point. Add a smaller amount of the one with five and the point sharpens
further.
Notice what is absent from the list of ingredients. Only the odd-numbered shapes
appear. A string plucked at the exact middle contains none of the shape with two
half waves, none of the one with four, and none of any shape that has a still point
where the finger was. This is the first hint that the mixture is a property of the
pluck rather than of the string.
The lesson stops here
6 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