A wave is represented by the equation

$\mathrm{y}=\mathrm{Asin}\left(10\mathrm{\pi x}+15\mathrm{\pi t}+\frac{\mathrm{\pi }}{6}\right)$

where x is in metres and t in seconds. The expression represents:

(1) a wave travelling in the negative x-direction with a velocity of 1.5 m/s.

(2) a wave travelling in the positive x-direction with a velocity of 1.5 m/s.

(3) a wave travelling in the positive x-direction with wavelength 0.2 m.

(4) a wave travelling in the negative x-direction with a velocity of 150 m/s.

A travelling wave in a string is represented by $y=3\sin \left(\frac{\pi }{2}t-\frac{\pi }{4}x\right)$. The phase difference between two particles separated by a distance of 4 cm is:

(Take x and y in cm and t in seconds)

(1) $\frac{\pi }{2}$ rad

(2) $\frac{\pi }{4}$ rad

(3) $\pi$ rad

(4) 0

A transverse wave is described by the equation $\mathrm{y}=\mathrm{Asin}2\mathrm{\pi }\left(\mathrm{nt}-\mathrm{x}/{\mathrm{\lambda }}_0\right)$. The maximum particle velocity is equal to 3 times the wave velocity if:

(1) ${\lambda }_0=\frac{\pi A}{3}$

(2) ${\lambda }_0=\frac{2\pi A}{3}$

(3) ${\lambda }_0=\pi A$

(4) ${\lambda }_0=3\pi A$

If $\stackrel{\rightharpoonup }{u}$ is the instantaneous velocity of the particle and $\overrightarrow{v}$ is the velocity of the wave, then:

(1) $\stackrel{\rightharpoonup }{u}$ is perpendicular to $\overrightarrow{v}$.

(2) $\stackrel{\rightharpoonup }{u}$ is parallel to $\overrightarrow{v}$.

(3) |$\stackrel{\rightharpoonup }{u}$| is equal to |$\overrightarrow{v}$|.

(4) |$\stackrel{\rightharpoonup }{u}$| = (slope of waveform)x|$\overrightarrow{v}$|.

In a simple harmonic wave, the minimum distance between the particles in the same phase always having the same velocity is:

(1) $\lambda /4$

(2) $\lambda /3$

(3) $\lambda /2$

(4) $\lambda$

The tension in a wire is decreased by 19%. The percentage decrease in frequency will be:

(1) 0.19%

(2) 10%

(3) 19%

(4) 0.9%

On the superposition of the two waves given as:

${\mathrm{y}}_1={\mathrm{A}}_0\sin (\mathrm{\omega t}-\mathrm{kx})$
and ${\mathrm{y}}_2={\mathrm{A}}_0 \cos \left(\mathrm{\omega t}-\mathrm{kx}+\frac{\mathrm{\pi }}{6}\right)$,

the resultant amplitude of oscillations will be:

(1) $\sqrt{3}{A}_0$

(2) $\frac{A_0}{2}$

(3) $A_0$

(4) $\frac{3}{2}{A}_0$

Two waves of amplitudes A₀ and xA₀ pass through a region. If x>1, the difference in the maximum and minimum resultant amplitude possible is:

(1) $(x+1)A_0$

(2) $(x-1)A_0$

(3) $2x{A}_0$

(4) $2A_0$

Which of the following represents a standing wave?

(1) $y=A \sin (\omega t-kx)$

(2) $y=A{e}^{-bx} \sin (\omega t-kx+\alpha )$

(3) $y=A \sin kx \sin (\omega t-\theta )$

(4) $y=(ax+b ) \sin (\omega t-kx)$

The equation of a standing wave in a stretched string is given by $\mathrm{y}=5 \sin \left(\frac{\mathrm{\pi x}}{3}\right) \cos \left(40\mathrm{\pi t}\right)$ where x and y are in cm and t is in seconds. The separation (in cm) between two consecutive nodes is:

(1) 1.5

(2) 3

(3) 6

(4) 4