An earthquake generates both transverse $(S)$ and longitudinal $(P)$ waves in the Earth. The speed of $S$ waves is about $4.5 \, km/s$ and that of $P$ waves is about $8.0 \, km/s$. $A$ seismograph records $P$ and $S$ waves from an earthquake. The first $P$ wave arrives $4.0 \, min$ before the first $S$ wave. The epicenter of the earthquake is located at a distance of about .... $km$.

Four different independent waves are represented by $(i)$ $y_1 = a_1 \sin \omega t$,$(ii)$ $y_2 = a_2 \sin 2\omega t$,$(iii)$ $y_3 = a_3 \cos \omega t$,and $(iv)$ $y_4 = a_4 \sin (\omega t + \frac{\pi}{3})$. With which two waves is interference possible?

$A$ vibrating string of certain length $l$ under a tension $T$ resonates with a mode corresponding to the first overtone (third harmonic) of an air column of length $75 \ cm$ inside a tube closed at one end. The string also generates $4$ beats per second when excited along with a tuning fork of frequency $n$. Now,when the tension of the string is slightly increased,the number of beats reduces to $2$ per second. Assuming the velocity of sound in air to be $340 \ m/s$,the frequency $n$ of the tuning fork in $Hz$ is:

Two identical flutes produce fundamental notes of frequency $300 \ Hz$ at $27 \ ^oC$. If the temperature of air in one flute is increased to $31 \ ^oC$,the number of beats heard per second will be

Two uniform strings of mass per unit length $\mu$ and $4 \mu$,and length $L$ and $2 L$,respectively,are joined at point $O$,and tied at two fixed ends $P$ and $Q$,as shown in the figure. The strings are under a uniform tension $T$. If we define the frequency $v_0=\frac{1}{2 L} \sqrt{\frac{T}{\mu}}$,which of the following statement$(s)$ is(are) correct?
$(A)$ With a node at $O$,the minimum frequency of vibration of the composite string is $v_0$
$(B)$ With an antinode at $O$,the minimum frequency of vibration of the composite string is $2 v_0$
$(C)$ When the composite string vibrates at the minimum frequency with a node at $O$,it has $6$ nodes,including the end nodes
$(D)$ No vibrational mode with an antinode at $O$ is possible for the composite string

$A$ wave equation is given as $y = \cos(500t - 70x)$,where $y$ is in $mm$,$x$ is in $m$,and $t$ is in $s$. Which of the following statements is correct?