Past exam of the mathematical methods II course of the University of Oxford / 2013 / 11

Let $\mathbf{r}(s)$ be any smooth curve in the level surface $\varphi =c$. Differentiating the constant value along it givesThe curve tangent can be any tangent vector to a regular level set, soThe nonzero-gradient condition matters for determining a normal direction. For example, $\varphi =z^3$ has the plane $z=0$ as a level set, but its gradient vanishes there and supplies no direction, even though the geometric plane has a well-defined normal.

Use the ambient position coordinates $\mathbf{r}$ and a fixed point $\mathbf{a}$. The distance is $R=|\mathbf{r}-\mathbf{a}|$, so $R^2=(x-a_x{)}^2+(y-a_y{)}^2+(z-a_z{)}^2$. Taking its gradient gives $2R\nabla R=2(\mathbf{r}-\mathbf{a})$. Therefore, away from $P=A$,It points from $A$ to $P$. The gradient is taken in the surrounding space and then evaluated on the surface; the intrinsic surface gradient of restricted distance would instead be its tangential projection and need not have unit magnitude.

For reflection between the foci of an ellipse, assume $s>|A-B|$ and define $\Phi (P)=R_A(P)+R_B(P)$. From part (b),On the ellipse this sum is normal to the tangent. Thus a unit tangent $\mathbf{\tau }$ obeys $\mathbf{\tau }\cdot {\mathbf{u}}_A=-\mathbf{\tau }\cdot {\mathbf{u}}_B$. The magnitudes of the projections of the two unit focal directions on the tangent are equal, provingfor the corresponding acute, unoriented line angles.

More explicitly, let $\mathbf{n}=({\mathbf{u}}_A+{\mathbf{u}}_B)/|{\mathbf{u}}_A+{\mathbf{u}}_B|$. The law of reflection maps an incoming unit ray ${\mathbf{u}}_A$ towhich points from $P$ toward $B$. Thus the equal-angle property has the correct incoming/outgoing ray orientation, rather than merely an algebraic bisector assertion.

In a homogeneous medium and the short-wavelength ray approximation, a smooth reflecting elliptical boundary sends a ray from focus $A$ to focus $B$ after one specular reflection, by the equal-angle property just proved. Moreover, every such route has the same path length $AP+PB=s$, hence the same propagation time $s/c_s$ for constant sound speed $c_s$.

An ideal two-dimensional elliptical reflector, or the corresponding ellipsoidal geometry, therefore concentrates sound near the other focus; equal travel times also support coherent arrival for a monochromatic source with a consistent reflection phase. Finite wavelength, imperfect surfaces, absorption and three-dimensional room geometry limit the focusing. The result predicts a finite physical focal region, not an infinite intensity at a mathematical point.