Stationary waves and its graphical representation - class-XII

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 $y=0$

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.

Let two waves be $y _1=A \sin\ (wt-kx)$
$y _2=A \sin\ (wt+kx)$
$y=y _1+y _2$
$=(2A \cos\ kx)\sin\ wt.$ 
For all point in one loop i.e as $x$ varies in $2A \cos k x$, the phase is same. The phase changes only after crossing a node.

Standing wave produces when two waves of identical frequency interfere with one another while travelling in opposite directions and this coincidence directions and this coincidence is not possible in fluids or gases.

In a string which is connected at both ends (similarity to sine wave), anti-nodes appear at odd multiples of $\dfrac{\lambda}{4}$.

Equ of standing wave is $y=2A \sin wt \cos Kx$. The quantity $\dfrac { W }{ K } $ always represent the speed of the wave.

Stationary wave is also known as standing wave. It remains in a constant position. Two opposing waves combine to form a standing wave. Hence energy is not propagated in stationary wave.

The main difference between stationary and progressive waves is: Progressive waves transfer energy from one place to another, without transferring matter and Stationary waves do not transfer energy from one place to another. Clearly only one option has stationary waves.

Stationary waves do not carry energy with it as it is stationary or does not change position.

a ) Two particles in alternate loops refers to particles having same phase and direction
$\therefore $ phase difference is $2\pi $
b ) Two particles in successive loops differs in phase by $\pi $ 
c ) Two particles in same loop are always in phase $\Rightarrow $ phase difference is $0$.
d ) $y _1= a  \sin  (\omega t-kx)$
$y _2= a  \cos  (\omega t-kx)$
$= a  \sin (\omega t-kx +\dfrac{\pi}{2})$
$\Rightarrow $ phase difference is $ \dfrac{\pi }{2}$.

(A) Magnitude of strain is maximum at antinode because medium particles at antinodes have minimum possible velocity.
(B) Nodes and antinodes form in case of travelling waves also.
(C) In case of stationary waves maximum pressure change occurs at node.
(D) Due to propagation of longitudinal wave in air maximum pressure change is equal to $2\pi f s _{0}\rho V$.

In stationary wave, total energy associated with it is twice the energy of each of incident and reflected wave. large amount of energy are stored equally in standing wave and became trapped with wave. Hence there is no transmission of energy through the waves.

Standing waves are obtained when two waves with same angular frequencies and velocity are superimposed, (They are however moving in the opposite directions).
$x(t) = A\sin(\omega t - kx) + A\sin(\omega t + kx + \delta)$
$x(t) = 2A\cos(kx)\sin(\omega t +\dfrac{\delta}{2})$
For all particles to be simultaneously at rest, the value of the sine function must be equal to zero.
i.e. $\omega t + \dfrac{\delta}{2} = n\pi$
$\Rightarrow$ $t = \dfrac{1}{\omega}(n\pi - \dfrac{\delta}{2})$
$\omega = \dfrac{2\pi}{T}$
$\Rightarrow$ $t = \dfrac{T}{2\pi}(n\pi - \dfrac{\delta}{2})$
$t _{1} =  \dfrac{T}{2\pi}(n\pi - \dfrac{\delta}{2})$
$t _{2} =  \dfrac{T}{2\pi}((n+1)\pi - \dfrac{\delta}{2})$
$t _{2} - t _{1} = \dfrac{T}{2}$

$y=1A\sin\omega t\cos kx\k=\cfrac{2\pi}{\lambda}\ \Rightarrow \cfrac{\omega}{K}=\cfrac{\omega\times \lambda}{2\pi}=\cfrac{\lambda}{2\pi/\omega}=\cfrac{\lambda}{T}=\lambda f=V$