Waves carry energy. This is manifested in the fact that laser waves can remove tumours, ultrasound waves can be used for medical treatment. Thus it is important to learn about the energy that a wave carries. Let us learn more about the wave energy along with the Energy of a Wave Formula. Let us begin!

## Energy

Energy is a quantity when given to a particular substance will heat it up or make it do work. The SI unit of energy is Joule. It gets its name from the British scientist James Prescott Joule.

Kinetic and potential are the two different kinds of energy. Water stored in the tank on the roof of a building will have potential energy. When the tap is released, the water which comes gushing out is due to the potential energy of the water being converted into kinetic energy.

## Mechanical Wave

An oscillating wave of matter transfers energy through a medium and hence is known as a mechanical wave. The waves can travel over long distances but the medium of transfer is stationary. This makes the oscillating material is also stationary. Energy is transported by the mechanical and electromagnetic waves. Energy and waves move in the same direction. Elasticity and inertia is required in a medium to produce mechanical waves. Mechanical waves and electromagnetic waves are very important concepts of modern physics. Several theories in Quantum electrodynamics and relativity are based on wave concepts.

## The Energy of a Wave Formula

Consider a sinusoidal wave as shown in the figure. The energy of a wave depends on the amplitude and the frequency of it. The components of the energy are Kinetic and Potential.

Consider a mass element of the string with a mass Δ m. Since the string has a constant linear density μ=ΔmΔx, each mass element of the string has the mass

Δ m = μΔx.

The total mechanical energy of the wave is the sum of its kinetic energy and potential energy. The kinetic energy comes out as,

K = 1/4(μA^{2}ω^{2}λ),

where A is the amplitude of the wave (in metres), ω is the angular frequency of the wave oscillator(in hertz), λ is the wavelength (in metres).

The potential energy also comes out as,

K = 1/4(μA^{2}ω^{2}λ),

where A is the amplitude of the wave (in metres), ω is the angular frequency of the wave oscillator(in hertz), λ is the wavelength (in metres).

Thus, the total energy,

U_{total} = U_{potential }+ U_{kinetic}

= 1/4(μA^{2}ω^{2}λ) + 1/4(μA^{2}ω^{2}λ)

= 1/2(μA^{2}ω^{2}λ)

where A is the amplitude of the wave (in metres), ω is the angular frequency of the wave oscillator(in hertz), λ is the wavelength (in metres).

## Solved Examples for Energy of a Wave Formula

1) Find the total energy of a wave with the values, A = 20 meters, ω = 40 Hz, λ = 50 meters and μ = 100 ? Use Energy of a Wave Formula.

Answer : U_{total }= 1/2(100 × 20 × 20 × 40 × 40 × 50)

= 1600000 Joules

= 1.6 MJ

2) Find the total energy of a wave with the values, A = 10 meters, ω = 1 Hz, λ = 1 meters and μ = 1 ?

Answer : U_{total }= 1/2(1 * 1 * 1 * 1 * 1 * 1)

= 0.5 Joules

3 ) A wave has the values , A = 100 meters, ω = 3 Hz, λ = 4 meters and μ = 7 , find its energy ?

Answer : U_{total }= 1/2(7* 4 * 100 * 100 * 3 * 3)

= 1260000 Joules

= 1260 Kilo Joules

= 1.26 mega Joules