Wave equation and progressive waves

The equation of a plane progressive wave is given by $y=5\cos\pi\left(200t-\dfrac x{150}\right)$ where $x$ and $y$ are in cm and $t$ is in second. The velocity of the wave is ________ m/s.

A transverse wave on a string is described by $y=3\sin(36t+0.018x+\pi/4)$, where $x$, $y$ are in cm and $t$ in seconds. The least distance between the two successive crests in the wave is ________ cm. (Nearest integer)
($\pi=3.14$)

Consider a system of three connected strings, $S_1$, $S_2$ and $S_3$ with uniform linear mass densities $\mu$ kg/m, $4\mu$ kg/m and $16\mu$ kg/m, respectively, as shown in the figure. $S_1$ and $S_2$ are connected at the point $P$, whereas $S_2$ and $S_3$ are connected at the point $Q$, and the other end of $S_3$ is connected to a wall. A wave generator O is connected to the free end of $S_1$. The wave from the generator is represented by $y = y_0\cos(\omega t - kx)$ cm, where $y_0$, $\omega$ and $k$ are constants of appropriate dimensions. Which of the following statements is/are correct:

A longitudinal wave is represented by $x=10\sin2\pi\left(nt-\dfrac x\lambda\right)$ cm. The maximum particle velocity will be four times the wave velocity if the determined value of wavelength is equal to :

Which of the following equations correctly represents a travelling wave having wavelength $\lambda=4.0$ cm, frequency $\nu=100$ Hz and travelling in positive $x$-axis direction ?

A transverse wave is represented by $y=2\sin(\omega t-kx)$ cm. The value of wavelength (in cm) for which the wave velocity becomes equal to the maximum particle velocity, will be :

In the wave equation

$$y=0.5\sin\frac{2\pi}\lambda(400t-x)\text{ m}$$

the velocity of the wave will be :

A standing wave is formed by the superposition of two waves travelling in opposite directions. The transverse displacement is given by
$$y(x,t)=0.5\sin\left(\frac{5\pi}4x\right)\cos(200\pi t).$$
What is the speed of the travelling wave moving in the positive $x$ direction ?

($x$ and t are in meter and second, respectively.)

A transverse wave is represented by :
$$y=\frac{10}{\pi}\sin\left(\frac{2\pi}{T}t-\frac{2\pi}{\lambda}x\right)$$
For what value of the wavelength the wave velocity is twice the maximum particle velocity ?

A person blows into open-end of a long pipe. As a result, a high-pressure pulse of air travels down the pipe. When this pulse reaches the other end of the pipe,

A transverse sinusoidal wave moves along a string in the positive $x$-direction at a speed of 10 cm/s. The wavelength of the wave is 0.5 m and its amplitude is 10 cm. At a particular time $t$, the snap-shot of the wave is shown in figure. The velocity of point P when its displacement is 5 cm is