... $P \equiv P_0=P_1$¹

The analogous result for waves on strings would be that the two applied tensions are in balance, $\tau_0=\tau_1$. For electromagnetic waves, there is no analogous balance: $\epsilon$ and $\mu$ may take any value on either side of the boundary.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

....²

Analogously, for strings we would have as the degrees of freedom $y_0(x\le 0,t)$, $y_1(x\ge 0,t)$ with the boundary condition $y_0(x=0,t)=y_1(x=0,t)$. For electromagnetic waves traveling along $x$, one finds as the degrees of freedom $\vec E_{0}(x\le 0,t)$, $\vec E_{1}(x\ge 0,t)$; $\vec B_0(x\le 0,t)$, $\vec B_1(x\ge 0,t)$, with constraint $\vec E_0(x=0,t)=\vec E_1(x= 0,t)$.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

....³

Analogously, for strings and electromagnetic waves, we find also our standard wave equations for all points $x \ne 0$.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... balance.⁴

For strings, the analogous quantity to pressure is the y-component of the tension $T_y$, and we find $\tau \partial y_0/\partial x = \tau \partial y_1/\partial x$ at $x=0$. In electromagnetic theory, we find that $(1/\mu_0) \vec B_0 = (1/\mu_1) \vec B_1$.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

....⁵

For sound we find the combination $Z \equiv \tau/c = (\mu c^2)/c = \mu c$ on each side of the equation. In electromagnetic theory, we find $Z \equiv (1/\mu)/c=(1/\mu)/\sqrt{1/(\epsilon \mu)} = \epsilon c$.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...eq:a1),⁶

Rather than writing the same two paragraphs twice, once for $a_0$ and once for $a_1$, we have written this paragraph just once for $a_0$, including everything that we would say differently for $a_1$ in parentheses.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.