Traveling Waves

PDEs · Axiom Academy

Understanding wave solutions to the wave equation and their fundamental properties 1. The Form of a Traveling Wave A traveling wave has the special form u(x,t) = f(x - ct) , where f is any differentiable function, c is the wave speed, and x - ct is called the phase of the wave. The key insight: the wave profile f depends only on the combination (x - ct) , not on x and t separately. This means the shape moves unchanged as time evolves. 2. Why This Satisfies the Wave Equation Let's verify that u(x,t) = f(x - ct) satisfies the wave equation u tt = c 2 u xx . Using the chain rule with s = x - ct : u tt = -c · f''(s) · (-c) = c 2 f''(s) Therefore: u tt = c 2 f''(s) = c 2 u xx The constant c determines both the speed and direction of wave propagation: u(x,t) = f(x - ct) travels to the right with speed c > 0 u(x,t) = g(x + ct) travels to the left with speed c > 0 To see why, track a point of constant phase: x - ct = x₀ means x = x₀ + ct , which moves right at speed c. A particularly important class of traveling waves are sinusoidal (harmonic) waves: A is the amplitude (maximum displacement) k is the wave number (spatial frequency, k = 2π/λ) ω is the angular frequency (temporal frequency, ω = 2π/T) λ is the wavelength, T is the period For a sinusoidal wave u = A sin(kx - ωt) to satisfy the wave equation, we need: Substituting into u tt = c²u xx : This gives us the dispersion relation: ω = ck 6. Superposition of Traveling Waves

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