Broad/Descriptive Questions: 1. Derive the equation of a progressive wave or prove that y(x,t)=Acos(kx-omega t) 2. Explain Standing wave Find the position of Node and antinode in a standing eave. 3. Find the fundamental frequency of a standing wave or prove that f_(n)=(n)/(2L)sqrt ((r)/(mu ))

1. The equation of a progressive wave can be derived from the wave equation, which states that the rate of change of displacement with respect to time is equal to the wave speed times the Laplacian of displacement. By assuming a sinusoidal form for the displacement, we can simplify the wave equation to obtain the equation of a progressive wave: $y(x,t)=Acos(kx-\omega t)$, where $A$ is the amplitude, $k$ is the wave number, $\omega$ is the angular frequency, and $x$ and $t$ are the position and time coordinates, respectively.<br /><br />2. A standing wave is formed when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other. The resulting wave pattern has points of maximum and minimum displacement, called nodes and antinodes, respectively. Nodes occur at points where the waves cancel each other out, while antinodes occur at points where the waves reinforce each other. The position of nodes and antinodes can be determined by solving the wave equation for the given boundary conditions.<br /><br />3. The fundamental frequency of a standing wave is the lowest frequency at which the wave can vibrate. It can be found by setting the wave equation equal to zero and solving for the frequency. The general formula for the fundamental frequency of a standing wave is given by $f_{n}=\frac {n}{2L}\sqrt {\frac {r}{\mu }}$, where $f_{n}$ is the fundamental frequency, $n$ is the mode number, $L$ is the length of the waveguide, $r$ is the radius of the waveguide, and $\mu$ is the permeability of the medium.