All waves carry energy. The energy of some waves, such as the energy of earthquakes, can be directly observed. Similarly, the churning of the shore by the water waves and the pulverisation of nerve cells due to loud sounds are manifestations of wave energy. In this article, we shall be examining the quantitative expression of energy in waves, i.e, energy carried by a wave. Let us also understand what gives waves their energy and how amplitude and frequency relate to wave energy.
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How is the Energy of a Wave Related to Amplitude?
The amount of energy carried by a wave directly depends on its amplitude. A high amplitude defines a high energy wave. Likewise, a low energy wave is characterized by a low amplitude. To better understand the energy-amplitude relationship, let us consider a slinky. A given slinky is stretched in a horizontal direction, and a transverse pulse is introduced into it by providing the first coil of the slinky an initial amount of displacement. The given displacement is due to the force applied by the person to displace it from rest. The more force imparted to the coil, the more displacement it will exhibit; as a result, the more amplitude it will have. Concluding, we can say that the amplitude of a transverse pulse is directly related to the energy with which the pulse transports through the medium. The energy of the transverse pulse will not change the wavelength, the frequency or the speed of the pulse; it will only affect the amplitude of the pulse.
Wave Energy-Amplitude Relationship
The energy transported by a wave is directly proportional to the square of the amplitude of the wave. This energy-amplitude relationship is mathematically expressed as follows:
E ∝ A2
Energy of a Wave Formula
To obtain the mathematical expression for the energy of a wave, consider a sinusoidal wave on a string that is produced by a string vibrator, as shown in the figure below:
The string vibrator is a device that vibrates a rod up and down. A string of uniform linear mass density is fastened to the rod. The rod oscillates the string, creating a sinusoidal wave. The rod does a certain amount of work on the string, generating energy that travels along the string. Consider a mass element of the string with a mass Δm, as shown in the figure. As the energy travels along the string, each mass element of the string is driven up and down at the same frequency as the wave. The mass element of the string has the mass
The total mechanical energy of the wave is the sum of its kinetic energy and potential energy.
The kinetic energy associated with the wave can be represented as:
A is the wave amplitude, ω is the angular frequency of the wave oscillator, λ is the wavelength, and µ is the constant linear density of the string.
The following equation represents the potential energy of the string:
A is the wave amplitude, ω is the angular frequency of the wave oscillator, λ is the wavelength, and µ is the constant linear density of the string.
Thus, the total energy of the wave is given as:
UTotal = UPotential + UKinetic
In the video, you will learn to derive the expression for the power transmitted by a wave.
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Frequently Asked Questions – FAQs
Do all waves include energy?
Yes, energy is carried by all waves.
How much overall energy does a wave have?
Total energy associated with a wavelength is the sum of potential and kinetic energy.
What is the relationship between wave energy and amplitude?
The quantity of energy carried by a wave is proportional to its amplitude. As a result, a high amplitude indicates a high energy wave. A low energy wave, similarly, has a small amplitude.
What happens to the wave’s energy when the frequency increases?
The greater the frequency, the more energy the wave carries.
What is the Wave Energy-Amplitude Relationship?
The frequency of a sound wave is defined as the number of back-and-forth vibrations of particles per unit of time, as well as the number of compressions or rarefactions that pass through a specific point per unit of time.
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