1.6.13 · D1Mechanical waves — transverse and longitudinal
Intuition Is poore topic ke peeche ek hi idea haiEk wave ek aisi hilaan hai jo travel karti hai jabki hilane wali cheez wahin rehti hai . Baaki sab — sine, speed, wavelength — yahi track karta hai: "abhi har particle kya kar raha hai, aur apne neighbour se kitna peeche hai?"
Is page par koi assumption nahi hai . Parent note mein jo bhi letter aur symbol aata hai, woh sab yahan ek-ek karke, ek aisi tartib mein banaya gaya hai jahan har piece pichle par tikta hai.
Kisi bhi formula se pehle, humein ek particle ke apni rest jagah se jaane aur wapas aane ka idea chahiye.
Ek akele dot ki picture socho jo ek hi height par baithna pasand karta hai — yahi uska equilibrium (rest) position hai. Use thoda sa dhakka do, aur woh dur chala jaata hai, phir ek restoring force use wapas kheenchti hai, woh overshoot karta hai, phir wapas aata hai... woh oscillate karta hai. Yahi bilkul Simple Harmonic Motion hai — yahi engine har wave ke andar chupta rehta hai.
Definition Displacement — symboly Plain words: ek particle abhi apni rest position se kitni door hai, abhi is waqt , aur kis direction mein (upar = positive, neeche = negative).
Picture: figure mein upar wala lal vertical arrow, dashed rest line se dot tak.
Topic ko iski zaroorat kyun hai: ek wave inhi displacements se bani hai. Agar tum har waqt har particle ka y
Displacement akele kaafi nahi. Humein batana hoga kaun sa particle aur kab .
x Plain words: ek particle string (ya hawa ki line) mein kitni door baitha hai, ek chosen zero se naapke.
Picture: neeche wale figure mein horizontal ruler — x
t Plain words: clock ki reading. Wahi particle, alag t Picture: ek stopwatch jo dots ke hilne ke dauran chalta rehta hai.
Kyunki y dono par depend karta hai — kahan dekh rahe ho aur kab dekh rahe ho — hum ise y ( x , t ) "y as a function of x and t" . Brackets ka bas matlab hai "y in chezon par depend karta hai."
Intuition Ek hi wave ko slice karne ke do tarike
Time freeze karo (ek photograph): y x shape dikhti hai.Position freeze karo (ek dot dekho): y t ek hi wave hain, do tarike se slice ki gayi. Yeh picture yaad rakho — yehi wajah hai ki humein do variables chahiye.
A Plain words: ek particle jo sabse bada displacement kabhi paata hai (hamesha positive).
Picture: figure s01 mein, rest line se particle ke swing ke bilkul top tak ki door.
Kyun: yeh batata hai ki wave kitni "oonchi" hai, aur (baad mein) kitni energy carry karti hai.
Parent note achanak sin
Intuition Sine kyun, koi aur wiggly curve kyun nahi?SHM mein ek particle ko ek aisi force wapas kheenchti hai jo uske hatne ki door se proportional hai. Woh mathematical curve jo "jitna dur jaao, utni tezi se wapas aata hai" aur hamesha repeat karta rehta hai woh sine hai (aur uska twin, cosine). Toh SHM hi sine motion hai — yeh coincidence nahi hai, yeh physics ki majboori hai.
sin ( θ ) Plain words: ek point ko radius 1 wale circle par ghoomte hue socho; sin height hai.
Picture: neeche wale figure mein lal height marker jab point ghoomta jaata hai.
Topic ko iski zaroorat kyun hai: ek-ghoomte-hue-point-ki-height hi ek oscillating particle ka upar-neeche hai. Circular motion ek line par project karo = SHM.
Woh angle θ sin phase kehte hain — yeh batata hai circle mein kitni door (dance mein kitna andar) particle abhi hai.
Sine repeat karta hai. Humein words chahiye ki yeh kitni tezi se repeat karta hai. Teen symbols, teeno ek hi baat teen alag tarike se keh rahe hain.
T Plain words: ek complete wiggle ke liye lagney wala time — upar, neeche, aur wapas start par.
Picture: s03 mein spinning point ka ek poora chakkar.
f Plain words: ek second mein kitne complete wiggles hote hain.
Picture: ek second mein spinning point ke chakkaron ko gino.
Link: zyada wiggles per second = har wiggle mein kam time, toh
f = T 1 .
ω Plain words: spinning point kitni tezi se angle sweep karta hai, radians mein naapa gaya.
Picture: s03 mein arm ki speed, "circle-fractions per second" mein.
Teesra symbol kyun chahiye: kyunki sin angle chahiye, time nahi. Ek full turn 2 π T ω = T 2 π = 2 π f .
Radian bas yeh hai ki "angle kitna bada hai, radius ko rim par wrap karke naapa gaya." Ek poora circle = 2 π
Toh source particle ki motion, y ( 0 , t ) = A sin ( ω t ) "height = amplitude baar sine of (time t
Time freeze karo (photograph slice). Shape space mein bhi repeat karta hai.
λ Plain words: shape par do matching points ke beech ki door — crest se agley crest tak.
Picture: neeche wale figure mein lal span.
k Plain words: wave ke saath ek metre chaloge toh kitna phase (angle) milta hai.
Picture: ek wavelength λ 2 π k = λ 2 π . Iski zaroorat kyun hai: jaise ω time ko angle mein badalta hai sin k distance ko angle mein badalta hai. Isi liye travelling wave sin ( ω t − k x )
Intuition Time aur space ke beech perfect mirror
Time slice
Space slice
period T
wavelength λ
ω = 2 π / T k = 2 π / λ
Ek hi idea, do axes. Jab bhi ω t k x
v Plain words: shape kitni tezi se aage slide karta hai — koi bhi particle kitni tezi se move karta hai woh nahi .
Picture: travelling-wave picture mein ek akele crest ko daayein travel karte dekho.
Kyun: ek period T λ v = T λ = f λ = k ω .
Parent T μ ρ Y B tug-of-war ka aadha hissa hai: kuch jo particle ko wapas kheenchta hai (elasticity) versus kuch jo motion badlne se rokta hai (inertia).
Definition Restoring vs inertial properties
T tension (newtons): string ke saath woh khichav jo ek displaced bit ko wapas snap karta hai. (Restoring — dhyan do: period ke saath same letter! Context decide karta hai.) μ mass per unit length (kg/m). (Inertia.) ρ density (kg/m³), mass per unit volume. (Inertia.) Y Young's modulus , ek solid kitni sakhti se stretch hone ka resist karta hai. (Restoring.) B bulk modulus , ek material kitni sakhti se squeeze hone ka resist karta hai. (Restoring.)
Picture / pattern: har wave speed hai v = heaviness (inertia) stiffness (restoring)
Parent v p = ∂ y / ∂ t Steepness ke do ideas.
∂ t ∂ y Plain words: displacement y jab time guzarta hai , position fixed rakhke.
Picture: ek dot ki upar-neeche speed — middle cross karte waqt sabse tezi, crest par momentarily still.
Curly ∂ d "y t
∂ x ∂ y Plain words: ek point par frozen wave-shape kitna steep hai.
Picture: photograph slice mein curve ka tilt.
Topic ko dono kyun chahiye: yeh v p = − v ( slope ) sideways move karta hua shape kaisa upar-neeche move karte hue particle mein badalta hai.
Simple harmonic motion = sine dance
Displacement y of x and t
Spinning point on a circle
Time slice gives T f omega
Space slice gives lambda k
Wave speed v equals f lambda
Particle velocity from d y d t
Daayein side cover karo. Agar tum bina dekhey har ek bata sako, toh parent note ke liye ready ho.
Symbol y ( x , t ) Position x t
Ek wave sine use kyun karta hai, koi aur curve kyun nahi? Kyunki har particle SHM karta hai, aur SHM exactly ek circle par ghoomte hue point ki height hai — ek sine.
sin ( θ ) Unit circle par ghoomte hue point ki centre ke upar height.
T f ω f = 1/ T ω = 2 π f = 2 π / T
λ k λ k = 2 π / λ
ω t k x Dono angles hain — ω k
Har wave-speed formula ke peeche ka pattern batao. v = restoring stiffness / inertia .
∂ y / ∂ t v ∂ y / ∂ t v shape ki constant speed hai.
Slope aur particle velocity ko kya connect karta hai? v p = − v ( ∂ y / ∂ x )
