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"Standing wave" describes one of its properties, but I was seeking a name for the curve.

Soon after being plucked or struck, a string vibrating at its fundamental frequency adopts a characteristic shape. Does this shape have a name? Is it a vesica Pisces, some sort of catenary, a tropskein or something else?

so if you pluck the string in the middle second fourth and other even harmonics are suppressed and first third fifth and odd harmonics are strongest.

And it's not quite correct to say it's A sine wave for the fundamental. It's actually two added together:sin(x-wt) + sin(x+wt)

Quote from: wolfekeeper on 20/08/2011 15:48:38And it's not quite correct to say it's A sine wave for the fundamental. It's actually two added together:sin(x-wt) + sin(x+wt)I'm probably thick, but I don't see why it necessarily follows that the shape of the string is sinusoidal at the fundamental (although I suspect it is).The thing that is sinusoidal is the orthogonal displacement of any element of the string (assuming, of course, that the oscillation of the string never decays) and it is that displacement, or really those displacements, that produce the note.

Geezer, it's a sinusoid because it's defined that way. If you solve for the physics of a string, the displacement of the string is a wave equation. If you solve the wave equation subject to the conditions that the ends of the string are fixed, you can of course get a bunch of funky shapes that aren't sinusoids. But these shapes aren't modes, either. The modes are defined to be sinusoids, and the fundamental mode is the lowest frequency sinusoid allowed by physics (aside from an unmoving string, which is technically a sinusoid of infinitely long wavelength :p ).The reason why we bother defining modes at all has to do with the way the physics can be described mathematically. No matter what shape the vibrating string takes, it should be the sum of a bunch of modes. To specify the more complex shape, you only need to specify the coefficients of the modes, which is a set of numbers. It's often easier to deal with a set of numbers than to try to find an equation for an arbitrary wave. This is especially true if you're representing waves in a computer, where you can figure out how many modes actually are important, and truncate your set of numbers there, making a very efficient way of storing that wave's complex shape.

Geezer, it's a sinusoid because it's defined that way.

Quote from: JP on 22/08/2011 03:24:53Geezer, it's a sinusoid because it's defined that way.No, that's not quite right I think. Sinusoids aren't defined in terms of this, nor vice versa.It's pretty much an accident of physics.I suppose it's because there's two modes of energy storage here: potential energy (energy stored in the tension) and kinetic energy (energy of motion), and the energy oscillates between the two in a similar way that distances change along the x and y axes as you spin around a circle.

No. Nature doesn't use maths. Humans use maths to predict what nature does. Maths is just a collection of patterns that humans have noticed about the world and about patterns in patterns. And we give these patterns symbols and we write them down, and we call that maths.But nothing, ever, in nature is due to our human definitions.

The fact that the fundamental must be a sine wave is because of the relationship between the energy modes and the x/y dimensions of something spinning around a circle. It's the same pattern. That's why it's sinusoidal. There's probably other ways to see it as well.

Your argument that you just have to solve the wave equation doesn't really solve anything, since ANY waveform can travel obeying the wave equation. Square, triangular, anything. You need to show why sinusoids are so common.

No, sorry. Point of fact you can analyse vibrations using many different bases, including various orthogonal square waves. You're essentially implicitly using fourier analysis, but you absolutely don't have to.

The answer, I think, is because the tension in the string varies sinusoidally due to the sinusoidal orthogonal displacement.

If you pluck it as a triangle shape you don't get a nicely behaved sine wave. I believe that if you let the vibration go on for a while, the higher frequency harmonics that make up that triangle will be damped, so you will end up with something that looks like the sine wave, since that's the most resistant to air resistance.