The Wave Equation

1The 1D Wave Equation and Its Physical Meaning

When you pluck a guitar string, you create a disturbance that travels along the string. The one-dimensional wave equation describes how this disturbance propagates:

Let's break down what each term means:

• y(x,t)The vertical displacement of the string at position x and time t. This is what we see when the string vibrates.
• ∂²y/∂t²The acceleration of the string at each point. This represents how quickly the velocity is changing. This is Newton's F = ma in disguise.
• ∂²y/∂x²The curvature of the string. A more curved section experiences a greater restoring force pulling it back toward equilibrium.
• cThe wave velocity, determined by the string's physical properties: c = √(T/μ), where T is tension and μ is linear mass density.

2Forming the General Solution

We solve the wave equation using separation of variables. We assume the solution can be written as a product of a function of space and a function of time:

Substituting this into the wave equation and dividing both sides by X·T:

Since the left side depends only on t and the right side only on x, both must equal the same constant (which we call -ω² for reasons that will become clear). This gives us two ordinary differential equations:

These are both simple harmonic oscillator equations with solutions:

• Spatial part: X(x) = A·sin(kx) + B·cos(kx)
• Temporal part: T(t) = C·sin(ωt) + D·cos(ωt)

3Boundary Conditions

A guitar string is fixed at both ends, at the nut (x = 0) and at the bridge (x = L). This gives us our boundary conditions:

Applying the first boundary condition y(0, t) = 0:

Applying the second boundary condition y(L, t) = 0:

We refer to k as the wave number, related to wavelength λ by k = 2π/λ. The boundary conditions mean that only certain wavelengths are allowed. The wave number must satisfy:

The longest allowed wavelength is λ₁ = 2L (the fundamental), then λ₂ = L, λ₃ = 2L/3, and so on. Each corresponds to a standing wave pattern with n antinodes.

Finally, looknig at the relationship k = ω/c, we see the allowed angular frequencies:

4Initial Conditions

To fully specify the string's motion after a pluck, we need two initial conditions, the initial displacement and initial velocity at t = 0:

For a plucked string:

• Initial shape f(x): Can be approximated as a triangle shape, zero at both ends, peaked where you pluck. If you pluck at position x₀ with height h, the shape is two line segments meeting at (x₀, h).
• Initial velocity g(x): Zero everywhere. When you first release the string, it's initially stationary before it starts moving.

5Normal Modes & Superposition

The solutions that satisfy the boundary conditions are the normal modes, the natural vibration patterns of the string:

The frequencies of these modes are:

Notice that the frequencies form a harmonic series: f₂ = 2f₁, f₃ = 3f₁, etc.

The general solution is a superposition of all modes:

The values of the specific coefficients A_n are determined by decomposing the specific initial shape f(x) using Fourier analysis:

And this is exactly what the simulation above does! It takes your pluck shape, computes the Fourier coefficients for each mode, and then sums up all the modes oscillating at their natural frequencies.