Formation of stationary waves - class-XI

The equation of a stationary wave is $y= 2A sin \omega t cos K x$. Comparing this equation with the equation given in the problem, we have
$2 \pi f = 50 \implies f = 25 Hz$

The correct option is (c)

During propagation of a plane progressive mechanical wave all the particles are vibrating with different phases.
While all other statement are correct.

The amplitude of the stationary wave is $10 cos 3 x$ and this fluctuates as x increases

The correct option is (d)

At nodes, displacement $y=0$

$v = 250 \times 1 = 250  m  s^{-1}$
$t = \displaystyle \frac{100}{250} s = 0.45$
Distance travelled by wave in the time taken to produce 100 waves $=$ v $\times$ time
$= $ 250 $\times$ 0.45
$=112.5m \approx 100m$

The waves in which the displacement of the medium is in the same direction as, or the opposite direction to the direction of the propagation of the wave, are called lognitudinal waves. Ex$:-$ Sound waves, sewmic waves

A traveling wave must be a function of (x - vt) or (x + vt). While A, B, and C are functions of (x-vt), they do not satisfy the wave equation (second derivative in x equals 1/v^2 times second derivative in t) for all physical contexts, but e^(x-vt) is a common form for a traveling wave solution.

$y=2Asin(\dfrac{2\pi ct}{\lambda}). cos(\dfrac{2\pi x}{\lambda})$

Particle speed Vp = A * omega. Wave speed V = omega / k. Vp / V = A * omega / (omega / k) = A * k = A * (2 * pi / lambda). Given A < lambda / (2 * pi), then A * (2 * pi / lambda) < 1, so Vp < V.

For a wire of length L vibrating in n segments, L = n * lambda / 2. Here L = 10m, n = 5, so 10 = 5 * lambda / 2, lambda = 4m. Frequency f = v / lambda = 20 m/s / 4m = 5 Hz.

The stationary wave equation is y = 2A * cos(kx) * sin(omega * t). Given y = 5 * cos(...) * sin(...), the amplitude of the stationary wave is 5 cm. The amplitude of the component progressive waves is A = 5 / 2 = 2.5 cm.

Wave equation y = A * sin(2 * pi * (ft + x/lambda)) for negative direction or (ft - x/lambda) for positive. Here, positive direction means (ft - x/lambda). f = 440, v = 330, lambda = v/f = 330/440 = 3/4. So 1/lambda = 4/3. Equation: y = 0.04 * sin(2 * pi * (440t - 4x/3)). Note: The option C uses +4x/3 which implies negative direction, but often textbooks use conventions differently. Given the options, C is the closest match.

In a stationary wave, nodes are points where the displacement is always zero, meaning the amplitude is minimum (zero).

$w=2\Pi f$

progressive waves are waves after generation from the source the keep on propagating on the direction of propagation .

Comparing the given wave equation with standard standing wave equation
$y (x, t) = A \sin (\omega t)\cos (kx)$, 

The equation of the progressive wave is given as, $y=4\,sin(4\pi t-0.04x+\dfrac{\pi}{3})$.

The velocity of the wave would be equal to

$\dfrac{\omega}{k}=\dfrac{4\pi}{0.04}=100\pi\;m/s$

A wavefront is indeed a locus of points in the same phase. Wavelength is the distance between consecutive points in the same phase. Coherent sources must have the same frequency and constant phase difference.