Wave Equation via Fourier Transform

PDEs · Axiom Academy

LESSON Wave Equation via Fourier Transform Solving the wave equation on the infinite line using Fourier transforms and recovering d'Alembert's solution 1. The Wave Equation on the Infinite Line We begin with the one-dimensional wave equation on the infinite spatial domain: where u(x,t) represents displacement at position x and time t , and c is the wave speed. We require initial conditions: 2. Taking the Fourier Transform We apply the Fourier transform in the spatial variable x . Define: The key property: differentiation in x becomes multiplication by ik in frequency space: Applying the Fourier transform to both sides of u tt = c²u xx : This is a simple harmonic oscillator ODE for each frequency mode k ! 4. General Solution in Frequency Space The ODE û tt + c²k²û = 0 has the general solution: where A(k) and B(k) are determined by initial conditions. Equivalently, we can write: This form shows that each frequency mode k oscillates at angular frequency ω = |ck|. 5. Determining Coefficients from Initial Conditions Apply initial conditions to find A(k) and B(k) : 6. Inverse Transform and d'Alembert's Formula Taking the inverse Fourier transform of our solution: Using the property that multiplication by e ±ikct corresponds to spatial shift by ±ct: This is precisely d'Alembert's solution , showing that the solution consists of two waves traveling in opposite directions at speed c .

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