$A$ stretched uniform wire of length $L$ under tension $T$ is vibrating with fundamental frequency $n$. $A$ closed pipe of the same length $L$ is also vibrating with the same fundamental frequency $n$. If the tension $T$ is increased by $16 \,N$, the wire resonates with the $2^{\text{nd}}$ harmonic of the same closed pipe. The initial tension in the wire is: (in $\,N$)

$A$ rod of length $L$ and negligible mass is suspended by two identical strings $AB$ and $CD$ as shown in the figure. $A$ mass $M$ is suspended from point $O$ which is at a distance $x$ from $B$. If the frequency of the first harmonic of $AB$ is equal to the frequency of the second harmonic of $CD$, then the value of $x$ is

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.

An air column in a tube $32 \,cm$ long, closed at one end, is in resonance with a tuning fork. The air column in another tube, open at both ends, of length $66 \,cm$ is in resonance with another tuning fork. When these two tuning forks are sounded together, they produce $8$ beats per second. Then the frequencies of the two tuning forks are (Consider fundamental frequencies only):

$A$ massless rod 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 (fundamental frequency) in $AB$ is equal to the $2^{nd}$ harmonic frequency in $CD$. Then,the length of $BO$ is:

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 .... $N$.