Explain the construction of a spherical wavefront by using Huygens' principle.
Huygens' principle states that every point on a wavefront acts as a source of secondary spherical wavelets. As these wavelets spread outward, they form a new wavefront. For a spherical wavefront, the wavelets originate from a point source, and as they travel, they create spherical surfaces around the source. The radius of each spherical wavefront expands over time, and the new wavefront is tangent to the wavelets.
Derive an expression for the equation of stationary wave on a stretched string. Show that the distance between two successive nodes or antinodes is \( \frac{\lambda}{2} \).
Two tuning forks having frequencies 320 Hz and 340 Hz are sounded together to produce sound waves. The velocity of sound in air is 340 m/s. Find the difference in wavelength of these waves.
The slope of the tangent to the curve \( x = \sin\theta \) and \( y = \cos 2\theta \) at \( \theta = \frac{\pi}{6} \) is ___________.
Solve the following L.P.P. by graphical method:
Maximize:
\[ z = 10x + 25y. \] Subject to: \[ 0 \leq x \leq 3, \quad 0 \leq y \leq 3, \quad x + y \leq 5. \]