Preface

This section concerns about solutions to forced wave equation.

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Introduction to Linear Algebra with Mathematica

Glossary

When an elastic string is set into motion by an external force, such as fiddlestick or violin bow, we come to the nonhomogeneous wave equation.

Forced Wave Equation

Consider the initial value problem for the forced wave equation

\[ \square_c u \equiv \frac{\partial^2 u}{\partial t^2} - c^2 \,\frac{\partial^2 u}{\partial x^2} = F(x,t) , \qquad u(x,0) = \phi_0 (x) , \quad u_t (x,0) = \phi_1 (x) , \]
where F(x,t) is a prescribed function. Its solution is the sum
\[ u (x,t) = u_h (x,t) + u_p (x,t) , \]
where uh is a solution of the homogeneous equation (discussed in the previous section)
\[ \square_c u \equiv \frac{\partial^2 u}{\partial t^2} - c^2 \,\frac{\partial^2 u}{\partial x^2} = 0 , \qquad u(x,0) = \phi_0 (x) , \quad u_t (x,0) = \phi_1 (x) , \]
and up is the solution of the nonhomogeneous wave equation but is assumed to vanish on the initial timeslice:
\[ \square_c u \equiv \frac{\partial^2 u}{\partial t^2} - c^2 \,\frac{\partial^2 u}{\partial x^2} = F(x,t) , \qquad u(x,0) = 0 , \quad u_t (x,0) = 0 . \]
The differential operator \( \square_c \) is usually referred to as the d'Alambertian or the d'Alembert operator. We will usually drop index c (or consider the case when the wave speed is 1). Green’s method leads one to write
\[ u_p (x,t) = \iint G(x-y, t-\tau )\,F(y, \tau ) \, {\text d}y {\text d}\tau , \]
where the Green function is a solution of the equation
\[ \square_c G \equiv \frac{\partial^2 G(x,t)}{\partial t^2} - c^2 \,\frac{\partial^2 G(x,t)}{\partial x^2} = \delta (x)\,\delta (t) , \qquad G(x,0) = 0. \]
Such properties are possessed by (in particular) this close relative of the Heaviside function:
\[ G (x,t) = \begin{cases} \frac{1}{2} \left[ H(x+ct) - H(x-ct) \right] , & \ \mbox{for } \ t > 0 , \\ 0 , & \ \mbox{for } \ t < 0 . \end{cases} \]

 

 

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