A sigle wave
Superposition of waves
© The scientific sentence. 2010

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_{e}^{2} = (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 v^{2} + 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: cos^{2}x = (1/2) + (1/2) cos2x
Then: cos^{2}x = (1/T) ∫ dx cos^{2}x (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} A^{2} = (1/2) F v (ω/v)^{2} A^{2} =
(1/2) (F/v) ω^{2} A^{2} = (1/2) (μ v) ω^{2} A^{2} (5)
P = (1/2) (μ v) ω^{2} A^{2}
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/m^{2})
For a spherical wave that travels uniformly and that it is not attenuated; if
P_{0} is the power of the wave at the point source, the intensity of
the wave at a distance r from the source is:
P = P_{0}/4πr^{2} (6)
4πr^{2} is the spherical surface around the point source.
©: The scientificsentence.net. 2007.


