Energy of a Wave Formula

Wave energy refers to the energy carried by waves, including both the kinetic energy of their motion and the potential energy stored in their amplitude or frequency. This energy plays a vital role in natural phenomena such as ocean currents and seismic activity, and it also holds great potential as a source of renewable energy.

Wave

A wave is a disturbance or motion of particles in a medium that transports energy from one point to another without causing any net movement of the particles themselves. Waves can arise from various sources, such as elastic deformations, pressure variations, electric or magnetic fields, electronic potentials, or temperature differences.

Formula for the Energy of a Wave 

When matter oscillates, it transfers energy through a medium as mechanical waves, which can travel long distances without moving the medium. Both mechanical and electromagnetic waves carry energy, which depends on the amplitude and frequency. Large amplitude waves like strong earthquakes or loud sounds carry more energy. Higher-frequency waves deliver more energy per unit time. Example: a wave with twice the frequency of another (same amplitude) transfers energy four times faster.

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

• Kinetic Energy Component

The Formula of Kinetic energy is,

U_{\text{kinetic}} = \frac{1}{2} m v^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, 

\Delta U_{\text{kinetic}} = \frac{1}{2} (\Delta m) v_y^2

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:

\Delta U_{\text{kinetic}} = \frac{1}{2} (\mu \Delta x) v_y^2

For total kinetic energy of wave we have,

U_{\text{kinetic}} = \frac{1}{4} \mu A^2 \omega^2 \lambda

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 = \frac{1}{2} k_s x^2

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

The potential energy of the mass element is,

U = \frac{1}{2} k_s x^2

U = \frac{1}{2} \Delta m \, w^2 x^2

U = \frac{1}{4} \mu A^2 \omega^2 \lambda

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

=  U_{\text{total}} = \frac{1}{4} \mu A^2 \omega^2 \lambda + \frac{1}{4} \mu A^2 \omega^2 \lambda

U_{\text{total}} = \frac{1}{2} \mu A^2 \omega^2 \lambda

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

Question 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

Question 2: 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

Question 3: 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

Question 4: A wave travelling on a string has amplitude A=2 m angular frequency ω = 10 Hz, wavelength λ=5 m and linear density μ = 4 kg/m. Find the total energy of the wave.

Solution:

U_{\text{total}} = \frac{1}{2} \mu A^2 \omega^2 \lambda

Substituting the given values

U_{\text{total}} = \frac{1}{2} (4 \times 2^2 \times 10^2 \times 5)

= \frac{1}{2} (4 \times 4 \times 100 \times 5)

= \frac{1}{2} (4 \times 4 \times 100 \times 5)

= 4000 \, J

\boxed{U_{\text{total}} = 4000 \, J}

Unsolved Problems

Question 1: A wave travelling on a string has amplitude A = 2 m, angular frequency ω = 4 rad/s wavelength λ = 3m, and linear density μ=5 kg/m. Find the total energy of the wave.

Question 2: A wave carries total energy 64J on a string having linear density μ=4 kg/m, angular frequency ω=4 rad/s and wavelength λ=2 m. Find the amplitude of the wave.

Question 3: A wave travelling on a string of linear density μ=2 kg/m has amplitude 3 m and wavelength 4 m. If the total energy of the wave is 144 J, find the angular frequency ω.

Question 4: Two waves travel on different strings. Wave 1 has amplitude 2 m, angular frequency 5 rad/s, wavelength 4 m, and linear density 3 kg/m. Wave 2 has amplitude 3 m angular frequency 4 rad/s, wavelength 5 m and linear density 2 kg/m. Determine which wave carries more energy.

Question 5: A wave travelling on a string of linear density 3 kg/ and wavelength 6 m initially has total energy 50 J. If its amplitude is doubled and angular frequency is tripled while other quantities remain constant, find the new total energy of the wave.

Related Article

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