4.7.10 · D5 · HinglishPartial Differential Equations
Question bank — Wave equation (hyperbolic) 1D — derivation
4.7.10 · D5· Maths › Partial Differential Equations › Wave equation (hyperbolic) 1D — derivation
Shuru karne se pehle, ek line ka reminder taaki koi bhi symbol anjaana na lage:
- = position par, time pe string ka sideways displacement.
- = slope (string kitni tilted hai); = curvature (slope kitna bend karta hai).
- = string ke ek point ka vertical acceleration.
- = wave speed, tension aur linear density se bana.
- = ek point par string ka horizontal se bana angle; iska slope hai.
- aur = do initial conditions: hai starting shape, aur hai har point ki starting velocity.
True or false — justify karo
Wave ki speed badh jaati hai agar string zyada zor se pluck karo.
False. Linear model mein sirf medium par depend karta hai (tension aur density); zyada zor se pluck karne se amplitude badhti hai, speed nahi.
Wave equation energy conservation se derive hoti hai.
False. Yeh seedha Newton's second law se aati hai jo ek tiny element par apply hota hai — tension se vertical force mass times vertical acceleration ke barabar hota hai. Dekho Newton's second law.
Kisi point par ka matlab hai string wahan steep hai.
False. curvature hai, steepness nahi. ka matlab hai concave-up (ek dip), aur equation ke hisaab se woh point upar ki taraf accelerate karta hai.
Agar string ke ek point ka slope zero ho () toh uska acceleration bhi zero hona chahiye.
False. Acceleration se set hota hai, se nahi. Ek hump ki crest par hota hai lekin bahut bada negative hota hai, isliye woh strongly downward accelerate karta hai.
form ka har function wave equation solve karta hai.
True. Operator factor hota hai ke roop mein, aur right factor se annihilate ho jaata hai, isliye yeh automatically ek rightward-travelling solution hai.
Wave equation heat equation jaisi hi type ki PDE hai.
False. ke saath wave equation hyperbolic hai (); heat equation mein koi nahi, isliye aur yeh parabolic hai (). Alag type, alag behaviour — dekho Classification of second-order PDEs.
Tension double karne se wave speed double ho jaati hai.
False. , isliye double karne se , se multiply hoti hai, se nahi.
Ek rightward pulse apni exact shape hamesha ke liye maintain karta hai.
True (is ideal model mein). Argument sirf shift karta hai; pure wave equation mein koi dispersion ya damping nahi hone se, profile rigidly transport hoti hai.
Tension ka horizontal component string ko upar-neeche move karne mein kaam karta hai.
False. Sirf vertical component (jahan horizontal se string ka angle hai) transverse motion drive karta hai; horizontal components cancel ho jaate hain (isliye ke saath constant rehta hai).
Factorisation sirf ek formal trick hai jiska koi justification nahi.
False. Product ko term by term expand karne par, ; do mixed terms cancel ho jaate hain (partial derivatives commute karte hain, ), aur exactly bachta hai.
Error dhundo
"Newton's law deta hai ."
Net force vertical components ka difference hai, sum nahi: . Dono tensions string ke saath opposite senses mein khichti hain. (Yahan horizontal se string ka angle hai.)
"Small angles ke liye , aur slope ke barabar hai."
string ka horizontal se angle hone par, slope hai, khud nahi. Small angles ke liye sab ek saath coincide karte hain, lekin aapko par pahunchna zaroori hai taaki ban sake.
" ko se divide karke aur lene par milta hai."
ki limit second derivative hai, nahi. Sahi result hai .
"Kyunki ki units hain, dimensionless hai."
N kg·m/s expand karne par m/s milta hai. Yeh ek genuine speed hai.
"D'Alembert's formula mein, ka integral zero ho jaata hai jab bhi string rest se shuru hoti hai, isliye hum ise hamesha drop kar dete hain."
Integral term initial velocity carry karta hai. Yeh sirf tab zero hota hai jab ; ek struck (plucked nahi) string mein hoti hai aur ise zaroori hai. Dekho d'Alembert solution.
" mein change karne par PDE ban jaata hai."
Yeh tak reduce hota hai (ek mixed second derivative). Use do baar integrate karne par milta hai, do travelling waves.
Why questions
Derivation kaam karne ke liye hum small slopes () kyun assume karte hain?
Small slopes se hum aur (jahan horizontal se angle hai) dono ko same se replace kar sakte hain aur treat kar sakte hain, geometry linearise ho jaati hai isliye tension constant rehti hai aur PDE linear rehti hai.
Equation ek travelling shape kyun predict karti hai, static kyun nahi?
Curvature har point ko straightness ki taraf drive karta hai (); jaise point overshoot karta hai, uska neighbour ab sabse zyada curved hota hai, isliye "bulge of curvature" — aur hence shape — aage move karta hai.
PDE classify karne ke liye discriminant test kyun use hota hai?
ko ke coefficients maanke, yeh conic-section test ko mirror karta hai: sign decide karta hai ki characteristic directions real aur distinct hain ya nahi. Yahan do real characteristic families deta hai, wave propagation ki pehchaan — dekho Method of characteristics.
Hume do initial conditions ( aur par) kyun chahiye lekin sirf time ke liye?
Equation mein second order hai (ek ), isliye ki tarah ise motion fix karne ke liye ek initial position aur initial velocity dono chahiye; boundary conditions direction ko alag handle karte hain.
Zyada bhaari string (bada ) waves ko slow kyun carry karti hai?
Length per zyada mass ka matlab hai ki same tension force kam acceleration produce karta hai (), isliye disturbances slower propagate karti hain — exactly yehi kehta hai.
Separation of variables aur d'Alembert dono same equation kyun solve kar sakte hain?
Ye dono same solution space ki do alag representations hain — d'Alembert travelling waves deta hai, separation standing-wave sums deta hai; ek standing wave sirf do opposite travelling waves ka superposition hai.
Edge cases
Agar tension ho (ek slack string) toh ka kya hoga?
: koi restoring tension nahi hone se koi bhi element ko wapas nahi kheenchta, isliye koi wave travel nahi kar sakti — model degenerate ho jaata hai.
Agar initial shape flat line ho (, jahan ) aur zero velocity (, jahan ), toh string kya karti hai?
Yeh hamesha ke liye flat rehti hai. D'Alembert deta hai; koi curvature nahi hone se koi acceleration nahi, isliye kuch bhi kabhi shuru nahi hota.
Initial shape mein ek sharp corner (kink) par, undefined hai — kya wave equation phir bhi meaningful hai?
Strong (pointwise) form kink par break kar jaata hai, lekin d'Alembert ka corner ko weak/travelling solution ke roop mein transport karta rehta hai; corner sirf speed se move karta hai.
Exactly us instant kaisa dikhta hai jab ek rightward aur leftward pulse poori tarah overlap karte hain?
Ye linearly superpose hote hain (add hote hain), momentarily cancel ya double ho sakte hain, phir ek dusre se ungchanged guzar jaate hain — yeh directly equation ke linear hone ka consequence hai.
(infinitely stiff/light string) ki limit mein, information kaise travel karti hai?
Characteristics instantaneous horizontal lines mein flatten ho jaati hain, isliye ek disturbance ek saath saare points tak pahunch jaati — ek physically forbidden idealisation jo dikhata hai ki finite kyun matter karta hai.
Agar aap initial velocity is tarah set karo ki poore space par ek nonzero constant tak integrate kare, kya yeh problem hai?
Local wave equation ke liye nahi — d'Alembert sirf finite interval par integrate karta hai, isliye kisi bhi finite ke liye value finite rehti hai.
Semi-infinite string ke liye jo par fixed hai, kya ek pulse wall tak pahunchne par simply vanish ho jaata hai?
Nahi — yeh inverted reflect hota hai. Fixed end force karta hai, jo (ek odd image extension ke through) ek flipped pulse wapas bhejta hai; energy conserve hoti hai, lost nahi.
Recall Ek-line self-test
Agar koi kahe "curvature = steepness," toh tumhara instant correction kya hai? ::: Steepness hai (slope); curvature hai (slope kitna change hota hai) — aur acceleration drive karta hai curvature, steepness nahi.