Harmonics and overtones — on strings and in pipes

1.6.19 · Physics › Oscillations & Waves

Harmonics aur overtones KYA hote hain?

Crucial subtlety yeh hai: "harmonic" $f_{1}$ ke integer multiples count karta hai; "overtone" fundamental ke upar ke allowed modes count karta hai. Dono sirf tab same tarah line up karte hain jab har harmonic allowed ho (string, open pipe). Ek closed pipe mein, even harmonics forbidden hain, isliye numbering shift ho jaati hai.

Boundaries frequencies kaise choose karti hain (derivation from scratch)

Ek string par wave speed $v = T / μ$ par travel karti hai ( $T$ tension hai, $μ$ mass per length hai). Ek pipe mein, $v$ speed of sound hai. Kisi bhi medium ke liye, frequency, wavelength, speed yeh obey karte hain: $v = f λ \Rightarrow f = \frac{v}{λ} .$

Toh poora khel yeh hai: find karo ki boundary kaun sa $λ$ allow karti hai, phir $f$ mein convert karo.

Boundary ka rule

Ek fixed end (string clamp) ya closed pipe end ek node hona chahiye (wahan koi displacement nahi).
Ek free end ya open pipe end ek antinode hona chahiye (maximum displacement).

Ek node se agle node tak ki distance $λ /2$ hai; node se nearest antinode tak $λ /4$ hai.

Case 1 — String dono ends par fixed (node–node)

Hume dono ends par ek node chahiye. Sabse chhota fit ek loop hai: length $L = λ /2$ . Agla har baar ek full loop add karta hai, isliye: $L = n \frac{λ _{n}}{2}, n = 1, 2, 3, \dots \Rightarrow λ_{n} = \frac{2 L}{n} .$ $f = v / λ$ se convert karo: $f_{n} = \frac{n v}{2 L} = n f_{1}, f_{1} = \frac{v}{2 L}$ SAARE integer harmonics exist karte hain. Yahan overtone number = harmonic number $- 1$ .

Case 2 — Pipe dono ends par open (antinode–antinode)

String jaisi hi spacing! Antinode-to-antinode bhi $λ /2$ hai, isliye math identical hai: $λ_{n} = \frac{2 L}{n}, f_{n} = \frac{n v}{2 L} = n f_{1} .$ Saare harmonics present hain, bilkul string ki tarah.

Case 3 — Pipe ek end par closed (node–antinode)

Closed end = node, open end = antinode. Sabse chhota fit ek quarter wavelength hai: $L = λ /4$ . Har agla mode aadha wavelength add karta hai (node→antinode→node→antinode...), jisse odd quarters milte hain: $L = (2 n - 1) \frac{λ _{n}}{4}, n = 1, 2, 3, \dots \Rightarrow λ_{n} = \frac{4 L}{2 n - 1} .$ $f_{n} = \frac{( 2 n - 1 ) v}{4 L} = (2 n - 1) f_{1}, f_{1} = \frac{v}{4 L}$ Sirf odd harmonics ( $1, 3, 5, \dots$ ) exist karte hain. Fundamental same length ke open pipe se aadha hota hai — isliye ek stopped organ pipe ek octave neeche sunti hai.

Figure — Harmonics and overtones — on strings and in pipes

Worked examples

Apni galtiyon ko steel-man karo

Active recall

Ek fixed string end (ya closed pipe end) kaun si boundary condition impose karta hai?

Ek node (zero displacement).

Ek open pipe end (ya free end) kaun si boundary condition impose karta hai?

Ek antinode (maximum displacement).

Dono ends par fixed string ki fundamental frequency?

f_{1} = v /2 L

Dono ends par open pipe ki fundamental?

f_{1} = v /2 L

(string jaisi hi).

Ek end par closed pipe ki fundamental?

f_{1} = v /4 L

Ek closed (stopped) pipe mein kaun se harmonics exist karte hain?

Sirf odd wale:

f_{n} = (2 n - 1) f_{1}

Closed pipe mein 1st overtone kaun se harmonic ke barabar hai?

3rd harmonic,

3 f_{1}

Closed-pipe fundamental equal length ke open pipe se ek octave neeche kyun hai?

Closed ko sirf

λ /4

chahiye vs

λ /2

, isliye

λ

double ho jaata hai aur

f

aadha.

Ek standing wave mein adjacent nodes ke beech ki distance?

Aadha wavelength,

λ /2

String par wave speed tension aur linear density ke terms mein?

v = T / μ

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho ek rope ko wall se baandh ke skip kar rahe ho. Tum sirf neat saaf shapes mein wave bana sakte ho — ek bada hump, ya do humps, ya teen — kabhi ek-aur-aadha hump nahi, kyunki ends ko still rehna hai. Har neat shape apni khud ki pitch par hum karti hai. Ek pipe mein yahi trick hai lekin air ke saath: ek open end air ko freely swing karne deta hai, ek closed end use pinned rakhta hai. Ek end par closed pipe zyada fussy hai — woh har doosri shape skip karti hai — isliye yeh deeper sunti hai aur sirf "odd" pitches banati hai. Isliye ek flute aur ek closed organ pipe alag sunti hain.

Connections

Standing waves and superposition — woh mechanism jo nodes/antinodes create karta hai.
Reflection of waves at boundaries — kyun fixed/free ends phase invert ya preserve karte hain.
Speed of a wave on a string — $v = T / μ$ ka origin.
Speed of sound in gases — pipes ke liye $v$ set karta hai.
Beats and resonance — driven columns in frequencies par kaise lock karte hain.
Timbre and Fourier synthesis — kyun overtones ka mix ek instrument ki sound define karta hai.

System	$λ_{n}$	$f_{n}$	Allowed $n$
String / open pipe	$2 L / n$	$n v /2 L$	saare integers
Closed pipe	$4 L / (2 n - 1)$	$(2 n - 1) v /4 L$	sirf odd