Everything You Need To Know About Wave Power

Introduction to Wave Power

All Waves carry Power. Wave Power is defined as the amount of energy that is obtained from wind waves. This energy carried by waves can be utilized to serve many purposes such as laser waves are used to remove tumours in the body, ultrasound waves, and sound waves are used for everyday verbal communication.

In this article, we will discuss what is wave power, how energy is transferred through waves, and what is the relation between wave energy and amplitude and frequency.

Factors that Impact Wave Power

The wave power factors are the amplitude and frequency of the wave. Large amplitude results in large displacements such as large displacements in earthquakes, loud sounds that have large amplitudes, and so on.

Let’s see it this way, Energy is stored in waves in form of discrete packets of energy. High-frequency waves will transfer more of these packets whereas low-frequency waves will transfer less of these energy packets, and thus will carry less energy.

However, In electromagnetic waves, the rate of energy transfer through waves is directly proportional to the square of amplitude but independent of frequency.

Derivation to find Energy associated Wave or Wave Power:

Consider the sinusoidal wave produced by a string vibrator which puts the string attached to it in a wave motion. A string vibrator has a rod that moves the rod up and down, which in turn oscillates the string attached to it, producing a sinusoidal wave.  It is assumed that the string attached to the string vibrator has uniform linear mass density. Transfer of energy occurs from the string vibrator to the string and along the string, as each mass element (Δm) moves up and down depending on the frequency and amplitude of the wave.

As string has uniform linear mass density along its length, μ = Δm/ Δx

Each mass element of string has mass, Δm = μ.Δx

To find total mechanical energy, we have to find kinetic energy and potential energy as,

Total Mechanical energy = Kinetic Energy + Potential Energy

Kinetic Energy (K.E) = ½ mv²

Kinetic Energy for mass element Δm: ΔK=1/ 2 (μΔx)v².

As Δx approaches zero, we get a differential equation,

dK = lim_Δx→0 ½ (μΔx)v² = ½ (μdx)v²

As it is a travelling/ sinusoidal wave with an angular frequency of ω,

We can model each mass element in terms of displacement using the following equation,
y = A. sin(kx−ωt)

And therefore, the velocity with which the string oscillates is given by
v = ∂y/ ∂t = −A ω cos(kx−ωt).

Putting the values of v and Δm, in the equation we get,

dK = (μ dx)(−A ω cos(kx−ωt))²

= ½ (μ dx) A² ω² cos² (kx−ωt)

We considered that kinetic energy is associated with the wavelength of the wave. As a wave consists of many wavelengths, we integrate kinetic energy over wavelength to find kinetic energy associated with each wavelength of the wave.

Deriving potential energy of the waveform

Similarly, we can find the potential energy of the wave power. As the mass element oscillates on the spring, a conservative restoring force will act to bring the mass element to the equilibrium position.

Potential Energy stored in spring is written as, U = ½ kx²

The Mass Element of the spring oscillates in SHM (Simple harmonic motion), and then we can write angular frequency as

ω = √k/m, where k = spring constant

The potential energy of a mass element is given by,

ΔU = ½ k x² = ½ Δm ω² x²

To find the energy over wavelength, we integrate it over wavelength,

dU = ½ k x² = ½ μ ω² x² dx,

U = ½ μ ω² A² ∫_0^λ sin^2 kx dx

     = ¼ μ ω² A²

Then, Total Energy = Potential Energy + Kinetic Energy

E _λ = K _λ + U = ¼ μ ω² A² + ¼ μ ω² A²

= ½ μ ω² A²

Formula to find the Wave Power

Wave Power = ½ μ ω² A²

To find the average rate at which wave power is transferred, we take the total energy of the wave and divide it by the time it takes to transfer the energy. If velocity is constant, then the time it takes one wavelength to pass also becomes constant.

P _avg = ½ μ ω² A² / T = ½ μ ω² A² v

From the above equation, we can see that Power is directly proportional to the square of amplitude (A) of the wave.

Power is also directly proportional to the square of angular frequency (ω) of the wave.

Relation between Wave Power or Energy and Amplitude

If Amplitude is more then displacement will be more.

P _avg = ½ μ ω² A² / T = ½ μ ω² A² v

From the above equation, we can see that Power is directly proportional to the square of amplitude (A) of the wave.

Relation between Wave Power or Energy and Amplitude

From the above equation, we can also conclude that Power is also directly proportional to the square of angular frequency (ω) of the wave.

The significance of both relations lies in the fact that if we double the angular frequency and amplitude of the wave, the Power associated with the wave will also increase by 4 times.

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