Energy of a Wave Formula

Work and energy are two different units that are closely associated with one another. In Physics, for two objects, the work done is achieved because of the transfer of energy from the first acting object to the second object. Physics defines the capability of doing work as a concept called energy. Energy also follows the law of conservation that is “energy can not be created and can not be destroyed”. Heat, light, and sound all are forms of energy. Energy also follows some rule to move from one body to another. 

Whenever energy has been transferred, it is always designated according to its nature. This states that energy always changes its form if required like electric energy got converted into light when a bulb is lightened, similarly wind energy can be converted to mechanical and then electrical in windmills. 

What is a Wave?

A wave is a disturbance/ movement of particles in a medium that transports energy without causing net particle movement. Elastic deformation, pressure variations, electric or magnetic field, electronic potential, or temperature variations are all examples.

Formula for the Energy of a Wave 

Whenever a matter oscillates it transfers energy through a medium and such waves are known as a mechanical waves. Also, waves can travel over long distances keeping the medium stationery. Although the wave travels from one point to another it keeps the oscillating material stationary. Energy is transported by mechanical and electromagnetic waves. Energy and waves both have the same direction.

The amount of energy during a wave is said to be its amplitude and its frequency. Large-amplitude earthquakes produce large ground displacements. Loud sounds have high-pressure amplitudes and are available from larger-amplitude source vibrations than soft sounds. If the energy of any wavelength is taken into account to be a discrete packet of energy, a high-frequency wave will deliver more of those packets per unit of time than a low-frequency wave. If two mechanical waves have equal amplitudes, but one wave features a frequency adequate to twice the frequency of the opposite, the higher-frequency wave will have a rate of energy transfer an element of four times as great because the rate of energy transfer of the lower-frequency wave.

The main components of wave energy are Kinetic energy and Potential energy.

 A string is attached to the rod of the string vibrator, which produces a sinusoidal wave in the string, with a wave velocity v. A section of the string with mass Δm oscillates at the same frequency as the wave.

• Kinetic Energy Component

The Formula of Kinetic energy is,

U_kinetic = mv² / 2 

Let v be the velocity of the wave.

Since, velocity has two component v_x (horizontal component in direction of motion of wave) and v_y (perpendicular component perpendicular to motion of wave).

So, the kinetic energy of each mass element of the string is, 

ΔU_kinetic = 1/2 (Δm) v_y²

as the mass element oscillates perpendicular to the direction of the motion of the wave. 

If the density of string is μ, then the mass of element (Δx) of string, 

Δm = μΔx

Hence, Kinetic energy is:

ΔU_kinetic = 1/2 (μΔx)v_y²

For total kinetic energy of wave we have,

U_kinetic = 1/4(μA²ω²λ)

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).

• Potential Energy Component

In Oscillations, the potential energy stored in a spring with a linear restoring force is,

U = 1/2k_sx²

where the equilibrium position is defined at x = 0 m.

The potential energy of the mass element is,

U = 1/2k_sx² 

= 1/2 Δmw²x²

= 1/4 (μA²ω²λ)

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).

• Hence, the Total Wave Energy

U_total = U_potential + U_kinetic

=  1/4(μA²ω²λ) + 1/4(μA²ω²λ)

U_total  =  1/2(μA²ω²λ)

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

Sample Problems

Problem 1: For a wave with given values, amplitude A = 10 m, angular frequency, ω = 50 Hz, wavelength λ = 10 m, and string density μ = 200. Find the wave energy by using Wave Energy Formula.

Solution:

U_total = 1/2 (200 × 10 × 10 × 50 × 50 × 10)

= 2500000 J

= 2.5 MJ

Problem 2: Describe the components of wave energy.

Solution: 

Wave energy has two components kinetic energy of wave particles and potential energy. 

Wave energy, U = U_potential + U_kinetic = 1/4(μA²ω²λ) + 1/4(μA²ω²λ) =  1/2(μA²ω²λ)

where A is the amplitude of the wave (in meters), ω the angular frequency of the wave oscillator (in Hertz), and λ the wavelength (in meters).

Problem 3: With what factor should the amplitude be increased, increase the intensity of a wave by a factor of 64?

Solution: 

Since, according to the Wave energy formula, the intensity of a wave is directly proportional to the square of the amplitude of the wave.

So, the relation can be written as:   

I ∝ X²

Here I is the intensity and X is the amplitude.

To increase the intensity by a factor of 64 we need to increase the amplitude by a factor of (64)^1/2  which is equal to 8.

Problem 4: With what factor should the intensity of a wave be increased, increase amplitude by a factor 5?

Solution: 

Since, according to the Wave energy formula, the intensity of a wave is directly proportional to the square of the amplitude of the wave.

X²  ∝ I 

Here I is the intensity and X is the amplitude.

To increase amplitude by a factor of 5 we need to increase the intensity by a factor of (5)² which is equal to 25.

Problem 5: Find the Amplitude of a wave of 0.5 J of energy with, ω = 1 Hz, λ = 1 m, and μ = 1.

Solution: 

Using wave energy formula :

U_total = 1/2(μA²ω²λ))

0.5 J = 1/2 (1 × A² × 1² × 1) J

A^{2 =} 1 m²

or

A = 1 m

Problem 6: Find the wavelength of a wave of 16 J of energy with, ω = 1 Hz, A (amplitude) = 1 m, and μ = 2.

Solution: 

Using wave energy formula :

U_total = 1/2(μA²ω²λ))

16 J = 1/2 (2 × 1² × 1² × λ) J

λ = 16 m