Standing Waves on a StringJEE Main

Concept

A wave reflecting between two fixed ends interferes with itself. Only wavelengths that fit whole half-wavelengths into the length survive: $L = n\lambda/2$. The result is a standing wave — points of permanent stillness (nodes) and maximal swing (antinodes), with every point oscillating in step but with position-dependent amplitude.

Key formula

Derivation

Superpose incident and reflected waves: $A\sin(kx-\omega t) + A\sin(kx+\omega t) = 2A\sin(kx)\cos(\omega t)$.

Fixed ends force $\sin(kL) = 0$, i.e. $kL = n\pi$ → the harmonic ladder $f_n = nf_1$. All harmonics of a string are present — that's why plucked strings sound rich.

Scenarios to explore

• Standing Waves — Harmonics on a string — nodes, antinodes, fₙ = n·f₁.

Real-world applications

• Every string instrument: pitch = f₁, timbre = harmonic mix.
• Melde's experiment measures μ or T from the loop count.
• Microwave-oven hot spots are antinodes of a standing EM wave.

JEE exam tips

• Adjacent node–antinode distance = λ/4; node–node = λ/2.
• String fixed at both ends → ALL harmonics; compare organ pipes (closed pipe: odd only).
• f₁ = (1/2L)√(T/μ): tension ↑ 4× doubles the pitch.

Common mistakes

• λ = L/n instead of 2L/n.
• Counting loops as nodes — n loops means n antinodes and n+1 nodes.
• Thinking standing waves transport energy — they don't (no net flux).

Exam traps to avoid

• Points between two adjacent nodes move IN PHASE; across a node the phase flips by π.
• A finger lightly at the midpoint kills odd harmonics — the string jumps to n = 2 (octave).