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In addition to the revision notes for Energy and Power of Waves on this page, you can also access the following Waves learning resources for Energy and Power of Waves
| Tutorial ID | Title | Tutorial | Video Tutorial | Revision Notes | Revision Questions | |
|---|---|---|---|---|---|---|
| 11.2 | Energy and Power of Waves |
In these revision notes for Energy and Power of Waves, we cover the following key points:
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 × v2 / 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
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:
Also, we know that the potential energy PE of an oscillating spring is calculated by the equation
and giving that
we obtain for the potential energy of spring:
Given that
Thus,
Substituting this value of k in the equation of potential energy, we obtain
Like in the cosine function, the square of sine function is equal to 1/2 as well. Remember the fundamental equation of trigonometry
This means each of terms is equal to 1/2 because 1/2 + 1/2 = 1. Therefore, we have
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):
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:
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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