Resonance

max for wave from both side at same point so A is max...

in other condition maxima is not at same point...

but how to prove mathematically....not graphically

its just that at resonanse waves interfere constructively at all places

@celestine..... The thing u r saying is true for transverse waves.
Prove this for longitudinal waves just like the oscillations of a mass connected to a spring..

@ sunil. Please post the solution.....

I'll post later on. Let others try too.....

Guys here is the way I solved it.

consider the oscillation of a spring having mass m.
let a force F = F₀cosωt is applied to the system.

therfore:-

ma = -kx + F₀cosωt

x^{" "} + (k/m)x = F₀cosωt / m ......... (1) ( x^{" "} denotes double differentiation of x)

At the steady state where the system begins to oscillate with the frequency of the force other than its natural frequency.

x = Acosωt where a = amplitude
→ x^{" "} = -Aω²cosωt

Putting this in (1), we've
-Aω²cosωt + (k/m)Acosωt = F₀cosωt / m
→ A (k/m - ω²) = F₀ / m .............(2)

Now the natural frequency of the system is given by
ω_o = √ (k/m)

putting in (2) :-

A = (F₀ / m) / ( ω_o - ω²) .......... (3)

From equation 3:-
If ω = ω_o
denominator becomes 0 and A → ∞.
This is resonance.