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.

(N/A) displacement node is a point where the amplitude of vibration is zero,which corresponds to the point of maximum pressure variation (pressure antinode). Conversely,a displacement antinode is a point of maximum amplitude,which corresponds to the point of minimum pressure variation (pressure node).
$(b)$ Bats emit high-frequency ultrasonic waves. These waves reflect off obstacles. By analyzing the time delay,intensity,and frequency shift of the reflected waves,bats can determine the distance,direction,nature,and size of obstacles.
$(c)$ Although the fundamental frequencies may be the same,the quality or timbre of the sound differs because the number and relative intensities of the overtones (harmonics) produced by a violin and a sitar are different.
$(d)$ Solids possess shear modulus,allowing them to support shearing stress,which is required for transverse waves. Gases lack shear modulus and cannot support shearing stress,thus they cannot support transverse waves. Both solids and gases possess bulk modulus,allowing them to support longitudinal waves.
$(e)$ $A$ pulse is a superposition of waves with different wavelengths. In a dispersive medium,the velocity of a wave depends on its wavelength. Since different components of the pulse travel at different speeds,the pulse shape becomes distorted over time.