Preface

This section concerns about two dimensional wave equation.

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Introduction to Linear Algebra with Mathematica

Glossary

2D Wave Equations

The wave equation for a function u(x1, …... , xn, t) = u(x, t) of n space variables x1, ... , xn and the time t is given by
\[ \square u = \square_c u \equiv u_{tt} - c^2 \nabla^2 u = 0 , \qquad \nabla^2 = \Delta = \frac{\partial^2}{\partial x_1^2} + \cdots + \frac{\partial^2}{\partial x_n^2} , \]
with a positive constant c (having dimensions of speed). The operator □ defined above is known as the d'Alembertian or the d'Alembert operator. The wave equation subject to the initial condisions is known as the initial value problem:
\[ \square u = 0, \qquad u({\bf x}, 0) = f_0 ({\bf x}), \quad u_t ({\bf x}, 0) = f_1 ({\bf x}) , \]
where f0(x) and f1(x) are given (smooth) functions in n-dimensional space ℝn. For n = 2, the solution of the initial value problem for wave equation is
\[ u(x_1 , x_2 , t) = \frac{1}{2\pi c} \iint_{r < ct} \frac{f_1 (y_1 , y_2 )}{\sqrt{c^2 t^2 - r^2}}\,{\text d} y_1 {\text d} y_2 + \frac{\partial}{\partial t} \, \frac{1}{2\pi c} \iint_{r < ct} \frac{f_0 (y_1 , y_2 )}{\sqrt{c^2 t^2 - r^2}}\,{\text d} y_1 {\text d} y_2 , \]
where
\[ r = \sqrt{\left( x_1 - y_1 \right)^2 + \left( x_2 - y_2 \right)^2} . \]
We observe that the domain of dependence of the point (x1, x2, t) on the initial data consists of the solid disk r ≤ ct in the (y1y2)-plane. So disturbances will continue indefinitely, as exhibited by water waves.

Example: We consider vibrations of an elliptical drumhead with vertical displacement \( u = u(x, y,t) \) governed by the wave equation

\[ \frac{\partial^ u}{\partial t^2} = c^2 \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right) , \]
where the velocity squared \( c^2 = T/\rho \) with tension T and mass density ρ is a constant. We first separate the harmonic time dependence, writing
\[ u(x,y,t) = v(x,y)\,w(t) , \]
where \( w(t) = \cos \left( \omega t + \delta \right) , \) with ω the frequency and δ the constant phase. Substituting this function into the wave equation, we get
\[ \frac{1}{v} \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} \right) = \frac{1}{c^2 w^2} \frac{\partial^2 w}{\partial t^2} = - \frac{\omega^2}{c^2} = -k^2 , \qquad \mbox{a constant}. \]
It is the two-dimensional Helmholtz equation for the displacement v We now use this equation to convert the Laplacian \( \nabla^2 \) to the elliptical coordinates, where we drop the u-coordinate. This gives
\[ \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + k^2 v = \frac{1}{h_\xi^2} \left( \frac{\partial^2 v}{\partial \xi^2} + \frac{\partial^2 v}{\partial \eta^2} \right) + k^2 v =0 , \]
that is, the Helmholtz equation in elliptical ξ,η coordinates
\[ \frac{\partial^2 v}{\partial \xi^2} + \frac{\partial^2 v}{\partial \eta^2} + c^2 k^2 \left( \cosh^2 \xi - \cos^2 \eta \right) v =0 . \]
Lastly, we separate ξ and η, writing \( v(\xi , \eta ) = \Xi (\xi)\,\Phi (\eta ) , \) , which yields
\[ \frac{1}{\Xi} \, \frac{{\text d}^2 \Xi}{{\text d} \xi^2} + c^2 k^2 \cosh^2 \xi = c^2 k^2 \cos^2 \eta - \frac{1}{\Phi}\,\frac{{\text d}^2 \Phi}{{\text d} \eta^2} = \lambda + \frac{1}{2} \, c^2 k^2 , \]
where \( \lambda + c^2 k^2 /2 \) is the separation constant. Writing \( \cosh 2\xi , \ \cos 2\eta \) instead of \( \cosh^2 \xi , \ \cos^2 \eta \) (which motivates the special form of the separation constant in the differential equation), we find the linear, second-order ODE
\[ \frac{{\text d}^2 \Xi}{{\text d} \xi^2} - \left( \lambda - 2q\,\cosh 2\xi \right) \Xi =0, \qquad q= \frac{1}{4}\, c^2 k^2 . \]
which is also called the radial Mathieu equation, and
\[ \frac{{\text d}^2 \Phi}{{\text d} \eta^2} + \left( \lambda - 2q\,\cos 2\eta \right) \Phi =0, \]
the angular, or modified, Mathieu equation. Note that the eigenvalue λ(q) is a function of the continuous parameter q in the Mathieu ODEs. It is this parameter dependence that complicates the analysis of Mathieu functions and makes them among the most difficult special functions used in physics.    ■

 

 

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