Ilka's lab notebook needs a drawing: the third harmonic of a string fixed at both ends. She has the two fixed ends and a faint rest line on the page, and nothing else yet.
Sketching any harmonic takes three decisions: where the nodes go, how many loops to draw, and how to space them. The harmonic's number settles all three.
Start at the ends. The string is held fixed at both ends, so the sketch shows a node at each end.
Loops meet at nodes. Three loops need two nodes between the ends: 3 - 1 = 2.
Space those two nodes evenly, so the three loops are equal.
Between the nodes, draw each loop as the mirrored pair of extreme curves, every loop the same width.
Check the sketch: a node at each end, two evenly spaced nodes between, three equal loops. That is the third harmonic.
Plan the sketch of the fifth harmonic of a string fixed at both ends: how many loops, and how many evenly spaced nodes between the ends?
The harmonic's number is the loop count: five loops.
5 - 1 = 4
Five equal loops, with four evenly spaced nodes between the fixed ends.
Joby checks a finished sketch: nodes at both ends, equal loops, and three evenly spaced nodes between the ends. Which harmonic is it?
Three nodes between the ends means four loops: 3 + 1 = 4.
The harmonic's number is the loop count.
The fourth harmonic.
Joby must sketch the ninth harmonic of a string fixed at both ends. How many evenly spaced nodes go between the fixed ends?
Kerstin sketches the second harmonic of a string fixed at both ends. Which sketch is correct?
Lazlo sketches the sixth harmonic of a string fixed at both ends. Counting both fixed ends, how many nodes does the whole sketch show?
Kerstin's correct sketch of a fixed-fixed string shows nine evenly spaced nodes between the fixed ends. Enter the number of the harmonic she sketched.
Lazlo's sketch of a harmonic has the right loop count, but the loops have different widths. Which one correction makes the sketch right?
A correct sketch of any harmonic on a fixed-fixed string shows a node at each fixed end. Why?