A standing wave is represented by $y = 2A\sin(kx)\cos(\omega t)$. The amplitude at a node is:

The distance between consecutive antinodes in a standing wave is:

A string fixed at both ends vibrates in 5 loops. The frequency is:

The first overtone of a closed pipe has the same frequency as the first overtone of an open pipe. The ratio of their lengths is:

An organ pipe closed at one end has length 25 cm. The first three resonance frequencies are ($v = 340$ m/s): <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 350 100" style="max-width:350px;margin:0.5em auto;display:block;"> <title>Closed pipe — node at closed end, antinode at open end, only odd harmonics</title> <rect width="350" height="100" fill="#faf9f5" rx="6"/> <rect x="40" y="20" width="4" height="50" fill="#14140f"/> <rect x="240" y="20" width="4" height="50" fill="#14140f"/> <line x1="36" y1="20" x2="44" y2="20" stroke="#14140f" stroke-width="2"/> <line x1="36" y1="20" x2="36" y2="70" stroke="#14140f" stroke-width="2"/> <line x1="36" y1="70" x2="44" y2="70" stroke="#14140f" stroke-width="2"/> <line x1="44" y1="45" x2="240" y2="45" stroke="#14140f" stroke-width="0.5" stroke-dasharray="3,3"/> <path d="M44,45 C80,18 140,18 244,45" fill="none" stroke="#c6613f" stroke-width="1.8"/> <circle cx="44" cy="45" r="3" fill="#14140f"/> <text x="48" y="38" font-size="7" fill="#14140f">N</text> <text x="244" y="20" font-size="7" fill="#c6613f">A</text> <text x="142" y="80" text-anchor="middle" font-size="8" fill="#14140f">L = λ/4 → f_{1} = v/(4L)</text> <text x="142" y="95" text-anchor="middle" font-size="8" fill="#788c5d">Odd harmonics: f_{1}, 3f_{1}, 5f_{1}...</text> </svg>