APPENDIX

The solutions to the partial differential equations (7) and (8) are obtained using the method of characteristics. The characteristic curves are curves on which the partial differential equation in several independent parameters (in our case the parameters (x₀, t₀) ) can be transformed into a system of ordinary differential equations in a single parameter along the characteristic curve.

A partial differential equation of the form

$$u_x\:{a(x,t)} + u_t\: {b(x,t)}+u\: c(x,t)= d(x,t)$$

can be reduced to the form

$${\partial{u} \over \partial{\tau}} + u\:c(x,t)=d(x,t)$$

if we can find a parameter $\tau$ such that

$$\left \{ \begin{array} {l} \displaystyle{{\partial{x} \over ... ...partial{t} \over \partial {\tau}}}=b(x,t)\\ \end{array} \right.$$

so that we can express the variables (x, t) in the form

$$\left \{ \begin{array} {l} x=x(\tau)\\ t=t(\tau)\\ \end{array} \right. \; .$$

As a result we have

$${\partial{u(x,t)} \over \partial{\tau}}= {\partial{u[x(\tau)... ...{\partial{t} \over \partial{\tau}} ={u_x \:a(x,t)+u_t\: b(x,t)}$$

which we can introduce in the original equation.

For equation (7) the system of associated equations is  

$$\left \{ \begin{array} {l} \displaystyle{{\partial{x} \over ... ...al{u} \over \partial{\tau}} = f_1{(x,t)}}\\ \end{array} \right.$$ (9)

which after integration becomes  

$$\left \{ \begin{array} {l} \displaystyle{{{h^2-x^2} \over {x... ...\int{f_1[x(\tau),t(\tau)]\:d\tau} + c_3}}\\ \end{array} \right.$$ (10)

The constants c₁, c₂, c₃ are functions of the parameter given on the initial curve, and are determined by setting $\tau=0$in equations (10).

The initial condition is given along an ellipse, and has the form  

$$\left \{ \begin{array} {l} \displaystyle{ t={ {\tau_0^2} \ov... ...h^2p^2}}}}\\ \\ \displaystyle{ u=g_1(p)}\\ \end{array} \right.$$ (11)

As a result the solution for the function u is given in parametric form  

$$\left \{ \begin{array} {l} \displaystyle{t={ {\tau_0^2} \ove... ...^2)}} \over {2p\sqrt{\tau_0^2+h^2p^2}}}}\\ \end{array} \right.$$ (12)

For equation 8 the system of associated equations is  

$$\left \{ \begin{array} {l} \displaystyle{{\partial{x} \over ... ...al{u} \over \partial{\tau}} = f_2{(x,t)}}\\ \end{array} \right.$$ (13)

which after integration becomes  

$$\left \{ \begin{array} {l} \displaystyle{{\sqrt{h^2-x^2}} c_... ...\int{f_2[x(\tau),t(\tau)]\:d\tau} + c_3}}\\ \end{array} \right.$$ (14)

The constants c₁, c₂, c₃ are functions of the parameter given on the initial curve, and are determined by setting $\tau=0$in equations (14).

The initial condition is given along a curve of constant $\bf{p}$ parameter, and has the form  

$$\left \{ \begin{array} {l} \displaystyle{ t= { {p_0\sqrt{h^2... ...yle{ x= s} \\ \\ \displaystyle{ u=g_2(p)}\\ \end{array} \right.$$ (15)

As a result the solution for the function u is given in parametric form  

$$\left \{ \begin{array} {l} \displaystyle{t= { {p_0(h^2-s^2) ... ...ystyle{x={\sqrt{h^2-e^{2\tau}(h^2-s^2)}}}\\ \end{array} \right.$$ (16)

The amplitudes of the function u along the characteristic curves are found by calculating the integrals in equations (9) and (14).