Consider the following wave equation,

$${{{\partial ^2}f(x,t)} \over {\partial {t^2}}} = 10000{{{\partial ^2}f(x,t)} \over {\partial {x^2}}}$$

Which of the given options is/are solution(s) to the given wave equation?

A waveguide consists of two infinite parallel plates (perfect conductors) at a separation of 10^$$-$$4 cm, with air as the dielectric. Assume the speed of light in air to be 3 $$\times$$ 10⁸ m/s. The frequency/frequencies of TM waves which can propagate in this waveguide is/are ___________.

In an electrostatic field, the electric displacement density vector, $$\overrightarrow D $$, is given by

$$\overrightarrow D (x,y,z) = ({x^3}\overrightarrow i + {y^3}\overrightarrow j + x{y^2}\overrightarrow k )$$ C/m²,

where, $$\overrightarrow i $$, $$\overrightarrow j $$, $$\overrightarrow k $$ are the unit vectors along x-axis, y-axis, and z-axis, respectively. Consider a cubical region R centered at the origin with each side of length 1 m, and vertices at ($$\pm$$ 0.5 m, $$\pm$$ 0.5 m, $$\pm$$ 0.5 m). The electric charge enclosed within R is __________ C (rounded off to two decimal places).

Consider a long rectangular bar of direct bandgap p-type semiconductor. The equilibrium hole density is 10¹⁷ cm^$$-$$3 and the intrinsic carrier concentration is 10¹⁰ cm^$$-$$3. Electron and hole diffusion lengthss are 2 $$\mu$$m and 1 $$\mu$$m, respectively. The left side of the bar (x = 0) is uniformly illuminated with a laser having photon energy greater than the bandgap of the semiconductor. Excess electron-hole pairs are generated ONLY at x = 0 because of the laser. The steady state electron density at x = 0 is 10¹⁴ cm^$$-$$3 due to laser illumination. Under these conditions and ignoring electric field, the closest approximation (among the given options) of the steady state electron density at x = 2 $$\mu$$m, is _____________.