In many cases, waves are caused by physical vibrations. With string instruments and percussions, you can actually see the parts of instruments such as wires, skins, wood, or metal trembling. When the vibration traverses through the air, it becomes a sound wave.


Spring equation


Let’s take a look at a spring as a simple model of vibration. The equation of a spring (Hooke's law) is $F=-kx$, which means that when it’s extended or compressed, a force of the object to return to its original shape is proportional to the amount of deformation. If we ignore other factors, stretching and releasing a spring will create a perpetual oscillation. And if we plot this against time, we get a sine curve. This is called simple harmonic motion.

振動の簡単なモデルとしてバネをみてみましょう。バネの方程式(フックの法則)$F=-kx$ は、バネを伸ばしたり縮めたりしたときに物体が元の形に戻ろうとする力が変形された距離に比例していること示します。他の要因を無視すると、バネを伸ばして手を離せば永遠に続く振動が生まれることになります。この振動を時間軸にプロットするとサインカーブになります。これは単振動と呼ばれています。

Although the actual musical instruments are not as simple as this, spring is a good entry point for understanding the physics behind vibrations, such as the force of a string being pulled and returning with tension.


Wave equation


Next, let’s simulate the propagation of the vibration using the wave equation. We could use harmonic motion, but let’s use a noise function to make it slightly