16.8:
Kinetic and Potential Energy of a Wave
The energy carried by a wave has two components, namely kinetic energy and potential energy.
In a sinusoidal wave formed in a string, consider an element with mass Δm. If the string has a constant linear density, the mass of each element equals density times the string’s length.
Using the linear mass density relation, the kinetic energy of each mass element can be calculated.
Since the wave is sinusoidal, the position of each mass element can be described by the wave function.
The wave function velocity is then substituted in the kinetic energy expression. Integrating this equation over the wavelength gives the kinetic energy of the wave.
The potential energy of each mass element oscillating in a simple harmonic motion can be calculated by considering the string's restoring force.
Replacing the spring constant by the expression for angular frequency and integrating over the wavelength gives the wave's potential energy.
16.8:
Kinetic and Potential Energy of a Wave
All forms of waves carry energy; this is directly visualized in nature. For instance, the waves of earthquakes are so intense that they can shake huge concrete buildings, causing them to fall. Loud sounds can damage nerve cells in the inner ear, causing permanent hearing loss. The waves of the oceans can erode beaches.
In mechanical waves, the amount of energy is related to their amplitude and frequency. In the context of the above examples, large-amplitude earthquakes produce large ground displacements. Loud sounds have high-pressure amplitudes and come from larger amplitude source vibrations than quiet sounds. Large ocean waves can agitate the shore more than small ones.
The energy carried by a wave has two major components: kinetic energy and potential energy. Consider a seagull floating on the waves of the sea. The waves do the work of moving the seagull up and down. The larger the wave's amplitude, the higher the seagull is lifted, therefore the more significant the change in potential energy. Remember that the energy of the wave depends on frequency too; if the energy of each wavelength is assumed to be a discrete packet of energy, a high-frequency wave will deliver more of these packets per unit of time than a low-frequency wave.
This text is adapted from Openstax, University Physics Volume 1, Section 16.4: Energy and Power of a Wave.