For sound waves, if the number of nodes for the 5^th harmonic of an open-ended pipe is n and that for the 9^th harmonic of the same pipe with one of its ends closed is m, the ratio (n)/(m) is:

Solution & Explanation

Principle: count the pressure NODES (displacement nodes) of the standing wave in each pipe. Open pipe (both ends open): • Both ends are displacement antinodes; nodes sit in between. • The p-th harmonic has p loops, and the number of displacement nodes equals the harmonic number. • So the 5th harmonic has n = 5 nodes. Closed pipe (one end closed): • Only odd harmonics exist; the closed end is a node, the open end an antinode. • Counting nodes: 1st harmonic → 1 node, 3rd → 2, 5th → 3, 7th → 4, 9th → 5 nodes. • So the 9th harmonic has m = 5 nodes. Ratio: n/m = 5/5 = 1, which matches option C. Trap: don't divide the harmonic numbers (5/9). For node counting, the open pipe's node count equals its harmonic number, while the closed pipe's node count for the (2k-1)th harmonic is k — the 9th gives 5, exactly matching the open pipe's 5.