Physics Lesson 11.3.2 - Power of Waves

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Welcome to our Physics lesson on Power of Waves, this is the second lesson of our suite of physics lessons covering the topic of Energy and Power of Waves, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.

Power of Waves

As we know, power is the work done (or the energy delivered) by a system in the unit of time. In this regard, it would be very easy to find a formula for power of a wave giving that

E = m × A² × ω²/2

Thus, we would simply divide this value by time t to obtain the formula of power, i.e.

P = m × A² × ω²/2t

However, this is not always possible as in most cases we don't have any info about the time, or worse, the time is infinity as the wave is standing. Thus, we must find another formula for power of waves that is independent from the time.

Let's consider again the rope of the previous exercise. We take a small piece of it, with mass Δm and length Δx ash shown in the figure.

Physics Tutorials: This image provides visual information for the physics tutorial Energy and Power of Waves

We define the linear mass density μ as the mass per unit length.

μ = ∆m/∆x

Thus,

∆m = μ × ∆x

Given that the wave performs uniform motion, we have

∆x = v × ∆t

Hence,

∆m = μ × v × ∆t

If we consider the entire rope, we will obtain for the mass m

m = μ × v × t

Substituting this equation in the equation we previously written for power of wave, we obtain

P = m × A² × ω²/2t
= μ × v × t × A² × ω²/2t
= μ × v × A² × ω²/2

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

Example 2

A rope of mass density 0.05 kg/m shakes at 30 cm amplitude. The wave produced, propagates at 2 m/s along the rope. The wave is produced by shaking the rope at 1.2 cycles per second. What is the power delivered by the wave?

Solution 2

Clues:

μ = 0.05 kg/m

A = 30 cm = 0.30 m

v = 2 m/s

f = 1.2 cycles/s = 1.2 Hz

P = ?

Let's calculate the angular frequency ω first. Thus,

ω = 2 × π × f
= 2 × 3.14 × 1.2
= 7.536 rad/s

Thus, the power of this wave is

P = μ × v × A² × ω²/2
= 0.05 × 2 × 0.30² × 7.536²/2
= 0.256 W

You have reached the end of Physics lesson 11.3.2 Power of Waves. There are 2 lessons in this physics tutorial covering Energy and Power of Waves, you can access all the lessons from this tutorial below.

More Energy and Power of Waves Lessons and Learning Resources

Waves Learning Material
Tutorial ID	Physics Tutorial Title	Revision Notes	Revision Questions
11.3	Energy and Power of Waves
Lesson ID	Physics Lesson Title	Lesson	Video Lesson
11.3.1	Energy in Waves
11.3.2	Power of Waves

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Continuing learning waves - read our next physics tutorial: Interference of Waves

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