2D String Physics: Wave Equation & Real FFT Harmonics

What is the Wave Equation?

The wave equation describes how waves propagate in space and time. For a vibrating string, it models the displacement of each point along the string as a function of position (x) and time (t). The general form for a 1D string is given by ∂²y/∂t² = v²∂²y/∂x², where y(x,t) represents the vertical displacement at any point x on the string at time t, and v is the wave velocity. This equation captures how disturbances travel along the string, with solutions representing various modes of vibration.

The wave equation can be solved numerically using finite-difference methods, as demonstrated in this simulation, where each point on the string updates based on its neighbors' displacements over time.

How Does Real FFT Work?

Real Fast Fourier Transform (FFT) is a computational algorithm that efficiently computes the frequency components of a signal. In the context of this simulation, it analyzes the sampled motion of the string to identify which harmonics are present in the wave pattern. By transforming the time-domain data into the frequency domain, FFT reveals the specific frequencies contributing to the overall waveform.

The real FFT is particularly useful here because the input signal (string displacement over time) is real-valued, and its output provides a spectrum of harmonic amplitudes that correspond to different modes of vibration.

Why Does Plucking Suppress Specific Overtones?

When you pluck a string at a specific point, it creates a disturbance that excites certain harmonics while suppressing others. This is due to the boundary conditions imposed by the fixed ends and the location of the pluck. For example, if you pluck exactly in the middle of the string, you create a node (zero displacement) at that point, which means no vibration can occur there. Consequently, all odd harmonics are suppressed because they would require nodes at integer multiples of half the length of the string.

This phenomenon is analogous to standing waves on a guitar or piano string, where certain frequencies resonate while others do not.

How Does Tension and Linear Density Affect Wave Speed?

The wave speed (v) in a string depends on the tension (T) and linear density (μ) of the string. The relationship is given by v = √(T/μ). This means that increasing the tension or decreasing the linear density will increase the wave speed, causing higher frequency harmonics to become more prominent.

For instance, tightening a guitar string increases its tension, thereby raising its fundamental frequency and all subsequent harmonics, making it sound higher-pitched.

Frequently asked questions

Try it live

Everything above runs in your browser — open 2D String Physics — Wave Equation & Real FFT Harmonics and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.