Standing waves — formation, nodes, antinodes: Foundations

1.6.18 · D1 · Physics › Oscillations & Waves › Standing waves — formation, nodes, antinodes

Yeh Standing waves — formation, nodes, antinodes ka ek "toolbox" page hai. Yahan hum standing wave derive nahi karenge (parent woh karta hai). Iske bajaye hum ensure karte hain ki jab tum $A$ , $x$ , $t$ , $k$ , $ω$ , $λ$ , $π$ , $sin$ , $cos$ , aur node, antinode, superpose, reflect ye shabd padho, inmen se koi bhi mystery na rahe.

0. "Rope pe wiggle" aakhir hoti kya hai

Ek lambi rope ko seedha leti hui picture karo. Use pluck karo, aur ek bump usmein se guzarta hai. Us bump ke baare mein numbers se baat karne ke liye humein do cheezein naam deni hogi:

kahan rope pe tum dekh rahe ho, aur
kab time mein tum dekh rahe ho.

Neeche jo bhi hai woh sab isi sawaal ka jawaab dene ke liye bana hai: "rope kitni oonchi hai, yahan, abhi?"

Figure — Standing waves — formation, nodes, antinodes

1. Amplitude $A$ , aur shabd node aur antinode

Kyunki node aur antinode shabd aage se use hone lagte hain, aaiye inhe abhi seedhe shabdon mein pin down karte hain (hum §12 mein formulas ke saath inse dobaara milenge, jab $sin$ aur $k$ exist karenge).

2. Number $π$ — kisi bhi circle ke around ek chakkar

Kisi bhi angle ya wave se pehle, humein ek special number chahiye jo bar-bar aata rehta hai.

3. Radians — angle measure karne ka honest tarika

Humein measure karna hai ki "circle ke around kitna dur" hum ghoom gaye hain. Hum ise directly $π$ se bane ek unit se karte hain.

4. Sine function $sin$ — har gentle wiggle ki shape

Ab jab hum ek angle naam de sakte hain (§3), hum wiggle shape define kar sakte hain. Ek point ko radius $1$ wale circle ke around ghoomao; centre line se uski oopar ki height $sin θ$ hai, jahan angle == $θ$ == (Greek "theta") hai woh kitna ghoom gaya hai, radians mein measure kiya gaya.

5. Cosine function $cos$ — sine ka quarter-turn partner

Standing-wave equation mein $cos (ω t)$ hai, toh cosine bhi humara hona chahiye. Yeh usi spinning spoke se aata hai — hum bas ek alag measurement padhte hain.

6. Wavelength $λ$ — space mein ek full repeat ki length

7. Wave number $k$ — distance ko angle mein convert karna

Sine sirf angles leta hai. Lekin rope par hum distance $x$ metres mein measure karte hain. Humein ek translator chahiye jo kahe "itni doori itne spoke-turn ke barabar hai." Woh translator $k$ hai.

8. Period $T$ , angular frequency $ω$ , aur time — wahi bridge, lekin clock ke liye

Pehle humein naam dena hoga ki ek point ka ek full up-and-down kitna time leta hai.

9. Bracket jodna: phase $k x - ω t$ — aur minus kyun

10. Superposition — waves bas add hoti hain

11. Reflection — doosri (left-moving) wave kahan se aati hai

Tum usually sirf ek wave rope down launch karte ho, phir bhi ek standing wave ko do chahiye, opposite ways mein chalti hui. Reflection missing partner supply karta hai.

12. Node aur antinode — ab formulas ke saath

Hum §1 mein inse plain words mein mile the. Ab jo $sin$ (§4) aur $k$ (§7) exist karte hain, hum inhe symbols se bol sakte hain pattern ki position-dependent amplitude $2 A sin (k x)$ use karke.

Yeh topic ko kaise feed karte hain

Equipment checklist

Recall Kya main ready hoon? (right side cover karo)

$y (x, t)$ plain words mein kya matlab rakhta hai? ::: Rope ki height position $x$ par time $t$ par — dono where aur when par depend karta hai. Amplitude $A$ kya hai aur uska sign kya hota hai? ::: Swing ki maximum height (metres mein ek distance); hamesha positive. Plain words mein, node kya hai? Antinode kya hai? ::: Node = ek spot jo kabhi nahi hilta (zero swing); antinode = sabse badi swing wala spot. Number $π$ kya hai? ::: Ek circle ki circumference aur uske diameter ka ratio ( $\approx 3.14159$ ); unit circle ki rim $2 π$ lambi hai. $sin θ$ geometrically kahan se aata hai? ::: Unit circle par angle $θ$ se ghume spoke ki tip ki vertical height. $cos θ$ kahan se aata hai, aur sine se kaise related hai? ::: Usi spoke ki tip ki horizontal reach; yeh sine ek quarter turn shifted hai, $cos θ = sin (θ + π /2)$ . Odd/even flip rules batao. ::: $sin (- θ) = - sin (θ)$ (odd); $cos (- θ) = cos (θ)$ (even). Kin angles par $sin θ = 0$ hota hai? ::: $π$ ke har integer multiple par: $0, π, 2 π, \dots$ (har full turn mein do baar). Kin angles par $∣ sin θ ∣ = 1$ hota hai? ::: Zeros ke beech mein: $\frac{π}{2}, \frac{3 π}{2}, \dots$ Radian kya hai? ::: Unit circle par arc length se measure kiya gaya angle; ek full turn $2 π$ hai. Wavelength $λ$ kya hai? ::: Rope ke saathe woh doori jisme shape ek baar repeat hoti hai. Wave number $k$ kya karta hai, aur uska formula kya hai? ::: Distance ko phase-angle mein convert karta hai; $k = 2 π / λ$ (radians per metre). Period $T$ kya hai? ::: Ek akele point ke ek complete cycle ke liye time (seconds mein). Angular frequency $ω$ kya karta hai, aur uska formula? ::: Time ko phase-angle mein convert karta hai; $ω = 2 π / T$ (radians per second). Phase $k x - ω t$ kya hai, aur minus kyun? ::: $sin$ ko diya gaya total angle; minus shape ko bade $x$ ki taraf slide karta hai jaise $t$ badhta hai, yaani right move karta hai. Superposition rule batao. ::: Jab waves overlap karti hain, heights add hoti hain: $y = y_{1} + y_{2}$ . Reflection kya hai, aur topic ko iska kyun zaroorat hai? ::: Ek boundary par wave ka wapas mudna; yeh opposite-direction doosri wave supply karta hai jo standing wave ko chahiye.

Connections

Standing waves — formation, nodes, antinodes — woh parent jiske liye yeh page tumhe taiyaar karta hai.
Travelling waves — §9 mein bana $A sin (k x - ω t)$ .
Superposition principle — §10 ka "waves add" rule.
Reflection of waves at boundaries — return wave banata hai (§11).
Wave number k and wavelength — §7 ka $k = 2 π / λ$ bridge.
Waves on a string / Sound in pipes — jahan yeh sab use hota hai.
Resonance and normal modes — foundations ke baad ka next step.