Wave Equation

The Equation

$\frac{\partial^2 y}{\partial t^2} = c_0^2 \frac{\partial^2 y}{\partial x^2}$

Or: $y_{tt} = c_0^2 y_{xx}$

$c_0$ is the wave speed.

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Physical Derivation

Consider a stretched string with:

• Tension $T$ (constant for small displacements)
• Mass per unit length $\mu$

For small $y$ and small slope:

Newton’s second law on element:

$\mu\Delta x \frac{\partial^2 y}{\partial t^2} \approx T\left[\left(\frac{\partial y}{\partial x}\right)_{x+\Delta x} - \left(\frac{\partial y}{\partial x}\right)_x\right]$

Dividing by $\Delta x$ and taking limit:

$\mu\frac{\partial^2 y}{\partial t^2} = T\frac{\partial^2 y}{\partial x^2}$

Wave speed: $c_0^2 = T/\mu$

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D’Alembert’s Solution

The general solution is:

$y(x, t) = f(x - c_0 t) + g(x + c_0 t)$

where $f$ and $g$ are arbitrary functions.

Interpretation:

• $f(x - c_0 t)$: wave travelling right at speed $c_0$
• $g(x + c_0 t)$: wave travelling left at speed $c_0$

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Standing Waves on a Finite String

Consider string of length $L$ with fixed ends:

Boundary conditions:

• $y(0, t) = 0$
• $y(L, t) = 0$

Separation of variables: Try $y(x, t) = X(x)T(t)$

Substituting: $\frac{T''}{c_0^2 T} = \frac{X''}{X} = -\omega^2$

Spatial part: $X(x) = A\sin(\omega x/c_0) + B\cos(\omega x/c_0)$

$X(0) = 0 \Rightarrow B = 0$

$X(L) = 0 \Rightarrow \sin(\omega L/c_0) = 0$, so $\omega L/c_0 = n\pi$

Normal modes:

$y_n(x, t) = \sin\left(\frac{n\pi x}{L}\right)\left[C_n\cos\left(\frac{n\pi c_0 t}{L}\right) + D_n\sin\left(\frac{n\pi c_0 t}{L}\right)\right]$

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Initial Conditions

Given $y(x, 0) = Y(x)$ and $\dot{y}(x, 0) = V(x)$:

Expand in Fourier sine series:

$Y(x) = \sum_{n=1}^{\infty} A_n \sin\left(\frac{n\pi x}{L}\right)$

$V(x) = \sum_{n=1}^{\infty} B_n \left(\frac{n\pi c_0}{L}\right) \sin\left(\frac{n\pi x}{L}\right)$

The solution combines position and velocity terms.

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Standing Waves

The normal modes are standing waves with:

• Wavelength: $\lambda_n = 2L/n$
• Frequency: $f_n = \frac{nc_0}{2L}$

$n = 1$: fundamental mode
$n = 2$: first harmonic
$n = 3$: second harmonic

The general solution is a superposition of all modes.

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Energy Conservation

Total energy of the string:

$E = \frac{1}{2}\int_0^L \left[\mu\left(\frac{\partial y}{\partial t}\right)^2 + T\left(\frac{\partial y}{\partial x}\right)^2\right] dx$

For free vibrations, energy is conserved.

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Why the Wave Equation Matters

• Describes vibrations (strings, membranes, acoustics)
• Electromagnetic waves (light, radio)
• Seismic waves (earthquakes)
• One of the fundamental equations of physics

The separation of variables method generalises to many other PDEs.