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 .... $km$.

Diffraction of sound waves is more evident than light waves in daily life because:

Explain why (or how):
$(a)$ in a sound wave,a displacement node is a pressure antinode and vice versa,
$(b)$ bats can ascertain distances,directions,nature,and sizes of the obstacles without any eyes,
$(c)$ a violin note and sitar note may have the same frequency,yet we can distinguish between the two notes,
$(d)$ solids can support both longitudinal and transverse waves,but only longitudinal waves can propagate in gases,and
$(e)$ the shape of a pulse gets distorted during propagation in a dispersive medium.

The frequency of a stretched uniform wire under tension is in resonance with the fundamental frequency of a closed tube. If the tension in the wire is increased by $8 \ N$,it is in resonance with the first overtone of the closed tube. The initial tension in the wire is .... $N$.

An auditorium has a volume of $10^5 \ m^3$ and a surface area of absorption of $2 \times 10^4 \ m^2$. Its average absorption coefficient is $0.2$. The reverberation time of the auditorium in seconds is:

$A$ massless rod of length $L$ is suspended by two identical strings $AB$ and $CD$ of equal length. $A$ block of mass $m$ is suspended from point $O$ such that $BO$ is equal to $x$. Further,it is observed that the frequency of the $1^{st}$ harmonic in $AB$ is equal to the $2^{nd}$ harmonic frequency in $CD$. The value of $x$ is