pension333: Why are harps harp-shaped? | Jon Butterworth

The physics behind it is part of the beauty, of the shape and of the sound

On Tuesday at a concert I found my mind drifting to physics. No criticism implied of the music, which was stunning.

In fact, when stunned, my mind often wanders off. It considers the fact that the LSO logo does in fact look like a conductor with a baton. It admires my wife's cheekbones. It wonders what Szymanowski did with the rest of his vowels, and what it felt like to be Bartok, creating beauty while Europe headed towards horror.

And, very often, it wonders about physics.

Waves are possibly the best bit of physics. They pop up everywhere. Music is of course a particularly patterned arrangement of pressure waves in the air.

The orchestra had two harps. Beautiful sweeping things, nearly triangular but with a distinctive curve in the top side. Why is that curve like that, I wondered? Maybe because it looks nice, but since the curve affects the length of the strings, and the length of the strings affects the note played, probably not just ornamental.

Four quantities characterise a wave. Speed, frequency, wavelength and amplitude. You could add a fifth maybe, the shape of the wave, which is where all the tone and other subtleties come in. That's where Stradivarius made his cash. But being a physicist I want to keep things simple. Think spherical cows.

Amplitude is a bit boring. It's the height of the peaks of the wave, or the depths of the troughs. So for sound it is basically the pressure difference between the highest pressure bit of a wave and the lowest pressure bit. It's the volume, in the turn-it-up-to-11-Spinal-Tap sense. I'm not sure why we use the word volume for loudness really, but at least amplifiers do increase amplitude, so that's nice.

Speed, frequency and wavelength are related; speed is equal to the wavelength multiplied by the frequency. And in fact the speed is a property of the medium through which the wave travels. The speed of sound in air is a fixed 1,236 km per hour. So for a sound wave, once you have fixed the frequency, you have also fixed the wavelength.

The frequency is what we hear as the pitch of the note. When a harp string is plucked, it vibrates with a certain frequency, compressing and decompressing nearby air* and making sound waves of the same frequency.

The frequency of the vibration in the string is set by the length of the string, the tension in the string, and the material it is made of. If you want to have all your strings made of the same stuff (so they have similar tone) and at the same tension (so they take the same effort to pluck), you have to increase the length to get deeper notes. Unfortunately, you have to double the length every octave. This means an exponential growth in length, and would lead to an unfeasibly big harp.

A triangular-shaped harp, without the curve in the upper frame, gives only a linear growth in string length, not an exponential one. The curve in the top of the frame looks to me like it is there to allow the lengths of the strings to grow exponentially as far as this is feasible. So for the shortest strings, they can have the same tension, be made of the same stuff, and be about equally spaced along the frame. But if you extrapolate that curving shape as you go away from the harpist, you can see the harp would get too big before you got to the next octave. So the curvature in the frame changes, and the type of string has to change too, to keep the frequency dropping exponentially and so allow more octaves.

I said waves are everywhere in physics. Another place they show up is in quantum mechanics. The energy, wavelength and frequency of quanta are all related, and amongst other things this is why we learn about small things like quarks from big things like the Large Hadron Collider. I often use double bass strings as an analogy to try and explain this in talks on the LHC, for example at the Royal Institution (about 22:20). I might write that up a some point too but this is enough for a Saturday.

Anyway: the graceful curves are functional as well as ornamental.

I'm no expert on harps, but I found this very useful. I loved the music in that concert. And I recommend listening to the harpist in Sunday Driver.

* Of course it does this mostly via the soundbox which is on the bottom of the harp - thanks to the friend who pointed this out. Worth clarifying.

Source: http://www.guardian.co.uk/science/life-and-physics/2012/may/12/harp-physics

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