Energy of a Wave Formula

Wave energy, often referred to as the energy carried by waves, encompasses both the kinetic energy of their motion and the potential energy stored within their amplitude or frequency. This energy is not only essential for natural processes like ocean currents and seismic waves but also holds significant promise for renewable energy generation.

Energy

Physics defines the capability of doing work as a concept called energy. Energy also follows the law of conservation that is “energy cannot 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. 

Also Check: Energy

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.

Check: Introduction to Waves

The Relationship Between Wave Energy and Amplitude

• The relationship between wave energy and amplitude is crucial for understanding wave behavior.
• Amplitude refers to the maximum displacement or intensity of a wave.
• It represents the height of a wave from its crest to its trough.
• Wave energy encompasses both kinetic and potential energy carried by a wave.
• Greater amplitudes result in waves carrying more energy.
• Understanding this relationship aids in predicting and manipulating wave behavior for various applications.
• Applications include earthquake monitoring, ocean wave energy conversion, and telecommunications.
• Studying changes in amplitude provides insights into optimizing technologies and harnessing wave power effectively.

Formula for the Energy of a Wave 

• When matter oscillates, it transfers energy through a medium, forming mechanical waves.
• These waves can travel long distances while keeping the medium stationary.
• Energy is carried by both mechanical and electromagnetic waves, flowing in the same direction as the waves.
• The energy of a wave is determined by its amplitude and frequency.
• Large-amplitude earthquakes result in significant ground displacements.
• Loud sounds have high-pressure amplitudes, originating from larger-amplitude source vibrations than softer sounds.
• High-frequency waves deliver more energy packets per unit of time compared to low-frequency waves.
• If two mechanical waves have equal amplitudes but one has twice the frequency of the other, the higher-frequency wave transfers energy at a rate four times greater.

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).

Related Article

• What is a Wave?
• Light Energy
• Wave Motion
• Properties of Waves
• Types of Waves

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

Energy of a Wave-FAQs

What is wave energy?

Wave energy refers to the energy carried by waves, encompassing both kinetic and potential energy associated with their motion and amplitude.

How do waves transfer energy?

Waves transfer energy through a medium by oscillating matter. In mechanical waves, particles of the medium oscillate as the wave passes through.

What are mechanical waves?

Mechanical waves are waves that require a medium to propagate. Examples include water waves, sound waves, and seismic waves.

Can waves travel long distances without moving the medium?

Yes, waves can travel over long distances while keeping the medium stationary. This is observed in phenomena such as light waves traveling through space.

How is wave energy related to amplitude and frequency?

The energy of a wave is determined by its amplitude (related to the wave’s intensity) and its frequency (related to the number of wave cycles per unit of time).

What are some real-world examples of wave energy?

Earthquakes with large amplitudes result in significant ground displacements, demonstrating the energy carried by seismic waves. Similarly, loud sounds have high-pressure amplitudes, originating from larger-amplitude source vibrations.

How does the frequency of a wave affect its energy transfer?

High-frequency waves deliver more energy packets per unit of time compared to low-frequency waves. This means that waves with higher frequencies can transfer more energy over a given period.

What happens when two waves with different frequencies but equal amplitudes interact?

If two mechanical waves have equal amplitudes but one has a frequency twice that of the other, the higher-frequency wave will transfer energy at a rate four times greater than the lower-frequency wave.