Harmonics and overtones — on strings and in pipes: Question bank

1.6.19 · D5 · Physics › Oscillations & Waves › Harmonics and overtones — on strings and in pipes

Questions se pehle, teen anchors hain jinpar tu baar baar rely karega. Inhe ek baar dheeray se padh.

True or false — justify

Har answer mein kyun bataana zaroori hai, sirf "true/false" nahi.

T or F: Ek string jo dono ends par fixed hai, usse kisi bhi frequency par drive kiya ja sakta hai.

False. Sirf wahi frequencies survive karti hain jinki half-wavelength ek puri number of times fit ho; baaki frequencies apne khud ke reflection ke saath destructively interfere karke khatam ho jaati hain.

T or F: Ek pipe jo dono ends par open hai, usme har integer harmonic

n f_{1}

present hoti hai.

True. Open–open ka matlab hai dono ends par antinode, antinode-to-antinode spacing

λ /2

hoti hai, toh

L = nλ /2

saare integers

n

ke liye kaam karta hai — koi bhi inhe rokta nahi.

T or F: Length

L

ki ek closed pipe ka fundamental usi length

L

ki ek open pipe ke fundamental ke barabar hota hai.

False. Closed ko sirf quarter wave chahiye (

L = λ /4

), toh

λ = 4 L

hoti hai jabki open ke liye

λ = 2 L

; closed ki wavelength double ho jaati hai, isliye uska

f_{1} = v /4 L

open pipe ke

v /2 L

ka aadha hai.

T or F: "First overtone" aur "second harmonic" hamesha same frequency hote hain.

False. Yeh strings aur open pipes ke liye coincide karte hain, lekin closed pipe mein second harmonic exist hi nahi karta, toh first overtone third harmonic

3 f_{1}

hota hai.

T or F: Ek fixed–fixed string ki length double karne par uski har harmonic frequency aadhi ho jaati hai.

True.

f_{n} = n v /2 L

; agar

v

(tension aur density unchanged) fixed hai, toh

f_{n} \propto 1/ L

, toh

L

double karne par saari frequencies ek saath aadhi ho jaati hain.

T or F: Ek closed pipe ki harmonics

f_{1}

apart hoti hain, bilkul open pipe ki tarah.

False. Sirf odd multiples survive karte hain (

1, 3, 5, \dots

), toh lagaataar allowed frequencies

f_{1}

se nahi balki

2 f_{1}

se differ karti hain.

T or F: Clean formulas

f = v /2 L

aur

v /4 L

real pipes ke liye exactly sahi hain.

False. Antinode ek open end se thoda bahar baithta hai, toh effective length thodi lambi ho jaati hai (

L_{e f f} = L + 0.6 r

har open end ke liye), jo ideal se actual frequencies ko thoda neeche le aati hai.

T or F: Guitar string mein tension badhane se uski saari harmonics raise ho jaati hain.

True.

v = T / μ

tension ke saath badhta hai, aur

f_{n} = n v /2 L

v

ke saath scale karta hai, toh har harmonic ek hi factor se upar jaati hai.

T or F: "Node" aur "overtone" ek hi cheez describe karte hain.

False. Node standing wave par ek jagah hai jahan zero motion hota hai; overtone fundamental ke upar ek frequency hai. Ek spatial hai, doosra ek pitch hai.

Spot the error

Har line mein ek (flawed) reasoning di gayi hai; reveal mein galti ka naam aur uski repair hai.

"Ek closed pipe ek octave neeche sound karti hai, toh uski har harmonic matching open-pipe harmonic ki aadhi hoti hai."

Octave-lower fact sirf fundamental formula par apply hota hai (

v /4 L

v /2 L

). Lekin closed pipe sirf odd harmonics rakhti hai, toh uske higher modes open pipe ke saath half-for-half align hi nahi karte.

"Same length ki ek open pipe aur ek closed pipe apna third harmonic share karengi, kyunki dono

3 f_{1}

allow karti hain."

Dono apne alag-alag

f_{1}

3 f_{1}

allow karti hain. Open

f_{1} = v /2 L

3 v /2 L

milta hai; closed

f_{1} = v /4 L

3 v /4 L

milta hai — alag frequencies hain, toh yeh coincide nahi karte.

"Kisi bhi node aur kisi bhi antinode ke beech distance

λ /2

hoti hai."

Nahi — node se nearest antinode tak

λ /4

hoti hai. Node se node (ya antinode se antinode) ka span

λ /2

hota hai.

"Closed pipe ka 5th harmonic uska 5th allowed mode hai."

5th harmonic

5 f_{1}

hai, jo closed pipe ka sirf teesra allowed mode hai (allowed list:

1, 3, 5, \dots

). Harmonic number aur overtone/mode index yahan differ karte hain.

"Kyunki

v = f λ

hai, pipe ko lamba karne se wave tezi se travel karti hai."

v

medium se set hota hai (speed of sound), geometry se nahi. Lamba pipe allowed

λ

aur isliye

f

ko change karta hai, lekin

v

same rehta hai.

"String ka ek free end node hona chahiye kyunki end ek boundary hai."

Free end ek antinode hota hai — yeh maximally move karne ke liye free hai. Sirf ek fixed clamped end node hone ke liye force hota hai.

"String mein mass per length

μ

badhane se uski pitch raise hoti hai."

Iska ulta hota hai:

v = T / μ

μ

badhne par girta hai, toh

f_{n} = n v /2 L

drop karta hai. Isliye moti bass strings neeche sound karti hain.

Why questions

String ka fixed end node kyun hona chahiye?

Clamp physically move nahi kar sakta, toh wahan displacement har waqt zero hone ke liye forced hai — yahi ek node ki exact definition hai.

Fixed–fixed string par sirf puri numbers of half-wavelengths hi kyun fit hoti hain?

Har end par ek node hona chahiye, aur nodes

λ /2

apart hoti hain; beech mein wave ke saath dono ends par nodes fit karne ke liye inhi

λ /2

segments ki ek integer count chahiye.

Closed pipe even harmonics kyun skip karti hai?

Isse closed end par ek node aur open end par ek antinode chahiye. Yeh node–antinode geometry sirf tab fit hoti hai jab

L

quarter-wavelengths ki odd number ho, jo frequencies ko

f_{1}

ke odd multiples hone par force karta hai.

Ek string aur ek open pipe exactly same frequency formula kyun share karte hain?

Dono ke dono ends par matching boundary conditions hain (string ke liye node–node, pipe ke liye antinode–antinode), aur dono boundary types har

λ /2

par repeat karte hain, toh fitting condition

L = nλ /2

identical hai.

Lambi wavelength ka matlab lower pitch kyun hota hai?

Kyunki

f = v / λ

mein

v

medium se fixed hai: denominator mein bada

λ

per second kam oscillations deta hai, yaani lower frequency.

End correction observed frequency ko kyun neeche laati hai?

Yeh effective length ko lamba karti hai, jo fitting wavelength ko lamba karta hai, aur lamba

λ

se chota

f = v / λ

milta hai.

"Harmonic" systems compare karte waqt "overtone" se zyada safe word kyun hai?

n

-th harmonic" ka matlab hamesha fixed number

n f_{1}

hota hai, jabki "

n

-th overtone" ka matlab sirf

f_{1}

ke upar

n

-th allowed mode hai, aur kaunse modes allowed hain yeh open aur closed pipes mein badalta rehta hai.

Edge cases

Agar string ko exactly do harmonics ke beech ki frequency par drive kiya jaaye toh kya hoga?

Koi stable standing wave nahi banega — reflected wave incoming wave ko reinforce nahi karta, toh pattern khud se interfere karta hai aur die away ho jaata hai; sirf true harmonics resonate karte hain.

Kya koi "zeroth harmonic" (

n = 0

) hota hai?

Nahi.

n = 0

f = 0

milta hai, matlab koi vibration nahi (ek infinite wavelength), toh yeh ek real oscillation nahi hai — ladder

n = 1

se shuru hoti hai.

Length

L

ki ek fixed–fixed string par jo sabse badi wavelength fit ho sakti hai, woh kya hai?

λ = 2 L

, fundamental

L = λ /2

se. Koi bhi lambi wavelength dono ends par nodes nahi rakh sakti, toh

2 L

ceiling hai.

Closed pipe ke liye sabse lamba allowed wavelength kya hai?

λ = 4 L

L = λ /4

se — yeh fundamental hai aur pipe ki sabse gehri note deta hai.

Agar closed pipe ki length double kar do, toh uske fundamental ka kya hoga?

Yeh aadha ho jaata hai, kyunki

f_{1} = v /4 L \propto 1/ L

; pipe ek octave neeche sound karti hai.

Khud ek node par standing wave ki frequency kya hoti hai?

Node par har waqt zero displacement hota hai, lekin woh phir bhi usi oscillation ka hissa hai — mode ki frequency poore pattern ki property hai, kisi ek point ki nahi; node sirf kabhi move nahi karta.

String par harmonic number

n

bahut bada ho jaane par nodes ke beech spacing ka kya hoga?

Wavelength

λ_{n} = 2 L / n

shrink hoti hai, toh nodes ek doosre ke paas aa jaate hain (

λ /2

apart) — bahut high harmonics mein kaafi closely packed nodes hote hain.

Recall Ek-line survival kit

Koi bhi overtone name karne se pehle puchho kaun se modes allowed hain; $λ /2$ (node–node) aur $λ /4$ (node–antinode) apne fingertips par rakho; yaad rakho $v$ medium ka hai, geometry sirf $λ$ pick karti hai.

Connections

Standing waves and superposition — kyun unfit frequencies khud ko destroy kar leti hain.
Reflection of waves at boundaries — kyun fixed aur free ends ek doosre se opposite behave karte hain.
Speed of a wave on a string — tension traps mein use hone wale $v = T / μ$ ki origin.
Speed of sound in gases — har pipe question mein $v$ kya set karta hai.
Beats and resonance — kaise ek driven column exactly inhi frequencies par lock karta hai.
Timbre and Fourier synthesis — kyun allowed overtones ka set ek instrument ki sound shape karta hai.