2D case

If we use the same type of considerations as in the 3D case, we receive

$$U^{(-)}({\bf r},t)={1 \over \sqrt{\pi d^{(-)}}}{wA_0H[\tau({\bf r})-t] \over \sqrt{\tau^{(-)}({\bf r})-t}}+\psi({\bf r})$$

where $\psi ({\bf r})$ is a smooth function,

$$d^{(-)}={\partial^2 T({\bf r},x) \over \partial x}-{\partial^2 \tau_0(x) \over \partial x}.$$

Let us expand the class of discontinuities

$$R^{(+)}_q(t) = {t^q_+ \over \Gamma(q+1)}{\:}\ \ \ \ q \neq -1$$

with arbitrary (noninteger) q, and

R^(-)_q(t) = R⁽⁺⁾_q(-t).

We must also expand the notion of q-equivalence (which was introduced in the Chapter 1) to a noninteger q: $f_{1}(t) \stackrel{q}{\sim } f_{2}(t)$ if ${\bf D}^{q}(f_{1}(t)-f_{2}(t))\in {\rm C}$.The operator of noninteger differentiation ${\bf D}^q$ was considered above for q=1/2 (see Chapter 8). For arbitrary q it can be defined by spectra response $(i\omega)^q=\vert\omega \vert^q \exp(-{i\pi \over 2} q \hbox{sgn}(\omega))$.

It can be shown that

$$R^{(-)}_q(t) \stackrel{q}{\sim} -R^{(+)}_{q,-2q} \ \ ;\ \ R^{(+)}_q(t) \stackrel{q}{\sim} -R^{(-)}_{q,2q}(t)$$

$${\bf D}^p_{\pm t}R^{\pm}_q(t) \stackrel{q-p}{\sim} R^{(\pm)}_{q-p}(t)$$

It is easy to see that

$$U^{(-)}({\bf r},t) \sim {wA_0 \over \sqrt{d^{(-)}}} R^{(-)}_{-{1 \over2}} [t-\tau^{(-)}({\bf r})]$$

and

$$u({\bf r},t) \sim {wA_0 \over \sqrt{d^{(-)}}} R^{(-)}_{q+{1 \over2}} [t-\tau^{(-)}({\bf r})]$$

(it is proposed that d^(-)>0).

If d^(-)<0, then

$$U^{(-)} \sim {wA_0 \over \sqrt{\vert d^{(-)}\vert}} R^{(+)}_{-{1 \over2}} [t-\tau^{(-)}({\bf r})]$$

and general formula

$$U^{(-)} \sim \kappa {wA_0 \over \sqrt{\vert d^{(-)}\vert}} R^{(-)}_{-{1 \over2},{1- \kappa \over 2}} [t-\tau^{(-)}]$$

where $\kappa = \mbox{sgn}(d^{(-)})$.

Let d^(-)=0. We introduce the order of touching of curves $T({\bf r},x)$and $\tau_0(x)$:the order = p if

$${\partial^kT \over \partial x^k} - {\partial^k \tau_0 \over \partial x^k} = 0, \ \ \ \ k=0,1,\ldots,p$$

and

$$d^{(-)}_p = {\partial^{p+1} T \over \partial x^{p+1}}- {\partial^{p+1}\tau_0 \over \partial x^{p+1}} \neq 0.$$

If the order p is even, then

$$u^{(\pm)} \sim A_p[R^{(+)}_{q+{1 \over p+1}}(t-\tau^{(-)})+R^{(-)}_{q+{1 \over p+1}} (t-\tau^{(-)})]$$

$$A_p \sim {1 \over \vert d_p\vert^{1 \over p+1}}$$

If the order is uneven

$$u^{(\pm)} \sim A_p R^{(-)}_{q+{1 \over p}}(t-\tau_{(-)}).$$

The point ${\bf r}$ is a special one if, for given ${\bf r}$ and ${\bf r}^\ast_0 =\omega({\bf r})$,$\ \ d^{(-)} = 0$.Each special point of the order p=2 is the point on caustics.