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In this Physics tutorial, you will learn:
|11.3||Energy and Power of Waves|
Do you know any example of waves carrying energy?
Have you ever heard of seismic waves? What do they cause?
Why it is not good to stay for long periods exposed to direct sunlight? Can you explain your opinion in terms of electromagnetic waves?
How do water waves chew up beaches?
How does loud sound damage the hearing?
All the above examples involve energy (and therefore power) of various kinds of waves which we will discuss in this tutorial. Also, the mathematical procedure to obtain the equations of waves energy and power will be explained.
As stated in the previous tutorials, 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. Look at the figure:
Energy of waves depends on two factors: amplitude and frequency. This is because when a particle of a wave as the one shown in the above figure 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).
For example, the strength of seismic waves, which cause up and down oscillations of the Earth surface during earthquakes, depends on the amplitude of oscillations. Stronger the earthquake, greater the amplitude of seismic waves. On the other hand, the energy of EM waves depends on their frequency. Thus, higher the frequency of EM waves, higher their penetrating ability and therefore, greater the energy such waves carry with them.
Now, let's find a formula for the energy and then for the power of waves based on the actual knowledge on waves and their properties.
Consider an oscillating spring of mass m as shown in the figure.
We chose such a spring as its motion is both a combination of transverse and longitudinal waves, so the study of energy will be more complete.
The string oscillated up and down, so its kinetic energy is
In the previous tutorial, we have shown that the oscillating speed vy of a wave is
Hence, we obtain for the kinetic energy of the oscillating spring:
As we know, 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:
In the tutorial "Simple Harmonic Motion", we have explained that the relationship between spring constant k, mass m and angular frequency ω is
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):
A 150 g rope shakes up and down as shown in the figure.
If the amplitude of wave caused by the shaking process is 20 cm and the up and down movement of the person who shakes the rope occurs every 2 seconds, what is the energy delivered by the rope's wave?
m = 150 g = 0.150 kg
A = 20 cm = 0.20 m
f = 1 oscillation / 2 s = 0.5 oscillation/ second = 0.5 Hz
E = ?
First, let's calculate the angular frequency ω. Thus, giving that
Now, let's apply the formula of wave's energy. Thus,
As expected, the energy delivered by such a wave is small. Otherwise, we would use shaking ropes as a source of energy.
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
Thus, we would simply divide this value by time t to obtain the formula of power, i.e.
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.
We define the linear mass density μ as the mass per unit length.
Given that the wave performs uniform motion, we have
If we consider the entire rope, we will obtain for the mass m
Substituting this equation in the equation we previously written for power of wave, we obtain
This expression is independent from the time t as this quantity does not appear in the formula of wave's power anymore.
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?
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,
Thus, the power of this wave is
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