This question involves matching different wave-producing systems with the type of wave (longitudinal or transverse) and the wavelength of their fundamental frequency. Let's analyze each system step by step.

Key Concepts:

Standing waves are formed by the interference of two waves traveling in opposite directions. The fundamental frequency is the lowest natural frequency, and its wavelength (${\lambda }_f$) is related to the length of the system.

Analysis of Each System:

(A) Pipe closed at one end:

This system supports longitudinal waves (sound waves in air). For a pipe closed at one end, the fundamental mode has a node at the closed end and an antinode at the open end. The length L corresponds to one-quarter of the wavelength: $L=\frac{{\lambda }_f}{4}$, so ${\lambda }_f=4L$. Thus, it matches with (p) Longitudinal waves and (t) ${\lambda }_f=4L$.

(B) Pipe open at both ends:

This system also supports longitudinal waves. For an open-open pipe, the fundamental mode has antinodes at both ends. The length L corresponds to half the wavelength: $L=\frac{{\lambda }_f}{2}$, so ${\lambda }_f=2L$. Thus, it matches with (p) Longitudinal waves and (s) ${\lambda }_f=2L$.

(C) Stretched wire clamped at both ends:

This system supports transverse waves. For a string fixed at both ends, the fundamental mode has nodes at the ends and an antinode in the middle. The length L corresponds to half the wavelength: $L=\frac{{\lambda }_f}{2}$, so ${\lambda }_f=2L$. Thus, it matches with (q) Transverse waves and (s) ${\lambda }_f=2L$.

(D) Stretched wire clamped at both ends and at mid-point:

This system supports transverse waves. The additional clamp at the midpoint forces a node there. The fundamental mode will have the longest possible wavelength with nodes at both ends and the midpoint. This corresponds to the entire string being half of a full wavelength, meaning $L=\frac{{\lambda }_f}{2}$ or ${\lambda }_f=2L$. However, wait: with a fixed midpoint, the fundamental is actually the first mode where each half (length L/2) vibrates as a separate string fixed at both ends. For a string of length L/2, the fundamental wavelength is ${\lambda }_f=2\times (L/2)=L$. So the wavelength for the entire system is L. It matches with (q) Transverse waves and (r) ${\lambda }_f=L$.

Final Matching:

• (A) → (p), (t)
• (B) → (p), (s)
• (C) → (q), (s)
• (D) → (q), (r)

Related Topics and Formulae:

Standing Waves: Result from superposition of two identical waves traveling in opposite directions. Characterized by nodes (points of zero amplitude) and antinodes (points of maximum amplitude).

Fundamental Frequency: The lowest frequency at which a system resonates. For waves on strings or in pipes, it depends on boundary conditions (fixed/free ends).

Important Formulae:

• String fixed at both ends: ${\lambda }_n=\frac{2L}{n}$, n=1,2,3,...
• Pipe open at both ends: ${\lambda }_n=\frac{2L}{n}$, n=1,2,3,...
• Pipe closed at one end: ${\lambda }_n=\frac{4L}{n}$, n=1,3,5,... (odd harmonics only)