San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

The d'Alembert Solution to the Wave
Equation with Initial Conditions

Consider the wave equation

(∂²y/∂t²) − C²(∂²y/∂x²) = 0

Let f(x) equal y(x, 0) and g(x) equal ((∂y/∂t)t=0. Then the d'Alembert solution to the wave equation with the given initial conditions (at t=0) is

y(x, t) = ½[f(x+Ct) + f(x−Ct)]
+ (1/2C)∫x−Ctx+Ctg(z)dz

Demonstration that this d'Alembert Solution
Does Indeed Satisfy the Wave Equation and Initial Conditions

The partial derivatives with respect to t are

(∂y/∂t) = ½[f'(x+Ct)C − f'(x−Ct)C]
+ (1/2C)[g(x+Ct)C + g(x−Ct)C]
(∂²y/∂t²) = ½[f"(x+Ct)C² + f"(x−Ct)C²]
+ (1/2C)[g'(x+Ct)C² + g'(x−Ct)C²]
which reduces to
(∂²y/∂t²) = C²{½[f"(x+Ct) + f"(x−Ct)C]
+ (1/2C)[g'(x+Ct) + g'(x−Ct)]}

On the other hand

(∂y/∂x) = ½[f'(x+Ct) + f'(x−Ct)]
+ (1/2C)[g(x+Ct) + g(x−Ct)]
(∂²y/∂x²) = ½[f"(x+Ct) + f"(x−Ct)]
+ (1/2C)[g'(x+Ct) + g'(x−Ct)]

Thus the wave equation is satisfied for all x and t; i.e.,

(∂²y/∂t²) −C²(∂²y/∂x²) = 0

For t=0

y(x, 0) = ½[f(x) + f(x)] + (1/2C)∫xxg(z)dz
which reduces to
y(x,0) = f(x)

Also for t=0

∂y/∂t)t=0 = ½[f'(x)C − f'(x)C]
+ (1/2C)[g(x)C + g(x)C]
which reduces to
∂y/∂t)t=0 = g(x)

Thus the initial conditions are satisfied.

The Wave Equation and Sinusoidal Solutions

Wave equations are thought to involve sinusoidal solutions but the wave equation is just part of an initial values problem. However the d'Alembert solution shows that a sinusoidal profile will be propagated, as would any initial profile, but a sinusoidal solution will not be created when it does not exist under the inital conditions.

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