Power of a wave and Its intensity

The power or the rate at which energy is propagated along a string is P = ΔE /Δt.
Where ΔE is the total energy of an element of the string at the position x and at the time t; that passes entirely to its next neighberhood at the position x + Δx at the later time t + Δt.
We know that P can be expressed as follows:
P = ΔE/Δt = (ΔE /Δx) (Δx/Δt) = (ΔE /Δx) v . Where v is the speed of the wave.
Let's calculate ΔE for an element.
ΔE = ΔKE + ΔPE ( Kinetic energy + Potential energy)      (1)
ΔKE = (1/2)μ Δx v_e² = (1/2) μ Δx (∂y/∂t)²      (2)

From the figure 4, we can see that the element is streched from its initial position Δx to [(Δx)² + (Δy)²]^1/2
The related expansion is :
Δl = [(Δx)² + (Δy)²]^1/2 - Δx = ([1 + (Δy/Δx)²]^1/2 - 1) Δx
Assuming that the element is small (Δ → ∂) and using the binomial expansion ((1 + x )ⁿ = 1 + nx + ...) , we obtain:
Δl = ((1/2) (∂y/∂x)²) Δx = ((1/2) (∂y/∂x)²) Δx
The work done by the tension F to strech the element by Δl is the potential energy of the element at the position x and at time t.
Then:
ΔPE = F . Δl = F ((1/2) (∂y/∂x)²) Δx      (3)
P = ΔE /Δt = (ΔKE + ΔPE) /Δt = (1/2) μ (∂y/∂t)² (Δx/Δt) + ((1/2)F(∂y/∂x)²) (Δx/Δt)
Δx/Δt = v, that is the speed of the wave.
Then: P = (1/2) v μ (∂y/∂t)² + ((1/2) v F(∂y/∂x)²)
We know that :
∂y/∂t = - v(2π/λ) A cos[(2π/λ)(x - vt)]
∂y/∂x = A (2π/λ) cos[(2π/λ)(x - vt)]
Then: P = (v/2)[μ( v(2π/λ) A cos[(2π/λ)(x - vt)]) ² + F ((2π/λ) A cos[(2π/λ)(x - vt)])²] = (v/2)(μ v² + F)(2π/λ) A cos[(2π/λ)(x - vt)])²
We know from the equation (5.3) that : v = [F/μ]^1/2. Then:
P = F v (2π/λ) A cos[(2π/λ)(x - vt)])²      (4)

To calculate the related average power P; we use first the following relationship: cos²x = (1/2) + (1/2) cos2x
Then: cos²x = (1/T) ∫ dx cos²x (x: 0 → T) = (1/2) + 0 = 1/2. T is the period of the fonction cosx. It follows that:
P = (1/2) F v ((2π/λ)² A² = (1/2) F v (ω/v)² A² = (1/2) (F/v) ω² A² = (1/2) (μ v) ω² A²      (5)

P = (1/2) (μ v) ω² A²


We need to know how much energy, or power of a wave we have in a certain surface. The quantity that characterize the flow of energy is called intensity. The intensity I of a wave is the power of the wave propagated per unit area of a surface S that is normal to the propagation direction: I = P/ΔS (in watts per square meters: W/m²)
For a spherical wave that travels uniformly and that it is not attenuated; if P₀ is the power of the wave at the point source, the intensity of the wave at a distance r from the source is:
P = P₀/4πr²     (6)
4πr² is the spherical surface around the point source.