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Tutorial ID | Title | Tutorial | Video Tutorial | Revision Notes | Revision Questions | |
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11.2 | Energy and Power of Waves |

In these revision notes for Energy and Power of Waves, we cover the following key points:

- What are the factors affecting the energy of waves?
- What is the equation of kinetic energy of waves?
- The same for the potential energy of waves.
- How do these energies compare?
- How can we calculate the total energy of waves?
- What is power of a wave?
- How can we calculate the power of a mechanical wave?

Waves do not carry matter but only energy. This is because in all waves (although this is more visible in transverse waves) a particle oscillates around an equilibrium position giving a zero resultant displacement, while the wave spreads in a certain direction.

Energy of waves depends on two factors: **amplitude** and **frequency**. This is because when a particle of a wave oscillates up and down, its gravitational potential energy depends on the amplitude, i.e. how far it displaces from the equilibrium position. Therefore, a greater amplitude means a greater gravitational potential energy for this particle when it reaches the maximum position.

On the other hand, it is a known fact that kinetic energy depends on the moving speed (KE = m × v^{2} / 2) and the latter depends on the wave frequency if wavelength is taken as constant (v = λ × f=>v ~ f for constant λ).

The kinetic energy of the oscillating spring is

KE = *m × [-A × ω × cos(k × x - ω × t) ]*^{2}*/**2*

=*m × A*^{2} × ω^{2} × cos^{2} (k × x - ω × t)*/**2*

=

Cosine values vary from -1 to + 1 but when they are raised in power two, they become always positive. Hence, they vary from 0 to 1. This means the average value of all cosines at power two is 1/2. Therefore, we obtain for the kinetic energy of spring:

KE = *1**/**2* × *m × A*^{2} × ω^{2}*/**2*

=*m × A*^{2} × ω^{2}*/**4*

=

Also, we know that the potential energy PE of an oscillating spring is calculated by the equation

PE = *k × y*^{2}*/**2*

and giving that

y(x,t) = A × sin(k × x - ω × t)

we obtain for the potential energy of spring:

PE = *k × [A × sin(k × x - ω × t) ]*^{2}*/**2*

=*k × A*^{2} × sin^{2} (k × x - ω × t)*/**2*

=

Given that

ω^{2} = *k**/**m*

Thus,

k = ω^{2} × m

Substituting this value of k in the equation of potential energy, we obtain

PE = *ω*^{2} × m × A^{2} × sin^{2} (k × x - ω × t)*/**2*

Like in the cosine function, the square of sine function is equal to ** 1/2** as well. Remember the fundamental equation of trigonometry

cos^{2} x + sin^{2} x = 1

This means each of terms is equal to ** 1/2** because

PE = *1**/**2* × *ω*^{2} × m × A^{2}*/**2*

=*m × A*^{2} × ω^{2}*/**4*

=

This is the same result as the result obtained for kinetic energy. Hence, we can write for the total (mechanical) energy of the oscillating spring (here we have a spring but this approach can be applied in all situations involving waves):

Total Energy = ME

= KE + PE

=*m × A*^{2} × ω^{2}*/**4* + *m × A*^{2} × ω^{2}*/**4*

=*m × A*^{2} × ω^{2}*/**2*

= KE + PE

=

=

Given that power is the work done (or the energy delivered) by a system in the unit of time, we obtain for power of waves:

P = *E**/**t*

=*m × A*^{2} × ω^{2}*/**2t*

=*μ × v × t × A*^{2} × ω^{2}*/**2t*

=*μ × v × A*^{2} × ω^{2}*/**2*

=

=

=

where μ is the linear mass density in kg/m.

This expression is independent from the time t as this quantity does not appear in the formula of wave's power anymore.

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