Two waves represented by $y_1 = a \sin \frac{2\pi}{\lambda} (vt - x)$ and $y_2 = a \cos \frac{2\pi}{\lambda} (vt - x)$ are superposed. The resultant wave has an amplitude equal to

The amplitude and phase of a wave that is formed by the superposition of two harmonic travelling waves,$y_1(x, t) = 4 \sin(kx - \omega t)$ and $y_2(x, t) = 2 \sin(kx - \omega t + \frac{2\pi}{3})$,are (Take the angular frequency of initial waves same as $\omega$):

If the phase difference between two waves is $2\pi$ during superposition,then the resultant amplitude is

Sound waves are passing through two routes—one in a straight path and the other along a semicircular path of radius $r$—and are again combined into one pipe and superposed as shown in the figure. If the velocity of sound waves in the pipe is $v$,then frequencies of resultant waves of maximum amplitude will be integral multiples of

Two loudspeakers ($L_1$ and $L_2$) are placed with a separation of $10 \ m$,as shown in the figure. Both speakers are fed with an audio input signal of the same frequency with constant volume. $A$ voice recorder,initially at point $A$,at equidistance to both loudspeakers,is moved by $25 \ m$ along the line $AB$ while monitoring the audio signal. The measured signal was found to undergo $10$ cycles of minima and maxima during the movement. The frequency of the input signal is . . . . . . $Hz$. (Speed of sound in air is $324 \ m/s$ and $\sqrt{5} = 2.23$)

Three waves of equal frequency having amplitudes $10 \,\mu m, 4 \,\mu m$ and $7 \,\mu m$ arrive at a given point with successive phase difference of $\frac{\pi}{2}$. The amplitude of the resulting wave in $\mu m$ is given by