# Work and mechanical energy relationship with wavelength

### Energy Transport and the Amplitude of a Wave

The relationship between photon energy and wavelength is . The additional potential energy possessed by the stone at B can be used for work or can be. We will find that some types of work leave the energy of a system constant, for example, whereas others . In equation form, the translational kinetic energy. Strictly speaking, that relationship only works for a particle in free space without any sources of potential energy. In this specific case, you can.

Large-amplitude earthquakes produce large ground displacements. Loud sounds have high-pressure amplitudes and come from larger-amplitude source vibrations than soft sounds. Large ocean breakers churn up the shore more than small ones. Consider the example of the seagull and the water wave earlier in the chapter Figure Work is done on the seagull by the wave as the seagull is moved up, changing its potential energy.

The larger the amplitude, the higher the seagull is lifted by the wave and the larger the change in potential energy. The energy of the wave depends on both the amplitude and the frequency. If the energy of each wavelength is considered to be a discrete packet of energy, a high-frequency wave will deliver more of these packets per unit time than a low-frequency wave. We will see that the average rate of energy transfer in mechanical waves is proportional to both the square of the amplitude and the square of the frequency.

## How does energy relate to wavelength and frequency?

If two mechanical waves have equal amplitudes, but one wave has a frequency equal to twice the frequency of the other, the higher-frequency wave will have a rate of energy transfer a factor of four times as great as the rate of energy transfer of the lower-frequency wave. It should be noted that although the rate of energy transport is proportional to both the square of the amplitude and square of the frequency in mechanical waves, the rate of energy transfer in electromagnetic waves is proportional to the square of the amplitude, but independent of the frequency.

Power in Waves Consider a sinusoidal wave on a string that is produced by a string vibrator, as shown in Figure The string vibrator is a device that vibrates a rod up and down. A string of uniform linear mass density is attached to the rod, and the rod oscillates the string, producing a sinusoidal wave. The rod does work on the string, producing energy that propagates along the string. As the energy propagates along the string, each mass element of the string is driven up and down at the same frequency as the wave.

Each mass element of the string can be modeled as a simple harmonic oscillator. A string vibrator is a device that vibrates a rod.

A string is attached to the rod, and the rod does work on the string, driving the string up and down.

**Simple Harmonic Motion, Mass Spring System - Amplitude, Frequency, Velocity - Physics Problems**

This produces a sinusoidal wave in the string, which moves with a wave velocity v. The wave speed depends on the tension in the string and the linear mass density of the string. The total mechanical energy of the wave is the sum of its kinetic energy and potential energy. To standardize the energy, consider the kinetic energy associated with a wavelength of the wave. This kinetic energy can be integrated over the wavelength to find the energy associated with each wavelength of the wave: Much like the mass oscillating on a spring, there is a conservative restoring force that, when the mass element is displaced from the equilibrium position, drives the mass element back to the equilibrium position.

This equation can be used to find the energy over a wavelength. The total energy associated with a wavelength is the sum of the potential energy and the kinetic energy: If the velocity of the sinusoidal wave is constant, the time for one wavelength to pass by a point is equal to the period of the wave, which is also constant.

For a sinusoidal mechanical wave, the time-averaged power is therefore the energy associated with a wavelength divided by the period of the wave. The tension in the string is When the string vibrator is turned on, it oscillates with a frequency of 60 Hz and produces a sinusoidal wave on the string with an amplitude of 4. What is the time-averaged power supplied to the wave by the string vibrator? Strategy The power supplied to the wave should equal the time-averaged power of the wave on the string.

If the same amount of energy is introduced into each slinky, then each pulse will have the same amplitude. But what if the slinkies are different? What if one is made of zinc and the other is made of copper? Will the amplitudes now be the same or different? If a pulse is introduced into two different slinkies by imparting the same amount of energy, then the amplitudes of the pulses will not necessarily be the same.

In a situation such as this, the actual amplitude assumed by the pulse is dependent upon two types of factors: Two different materials have different mass densities. The imparting of energy to the first coil of a slinky is done by the application of a force to this coil.

More massive slinkies have a greater inertia and thus tend to resist the force; this increased resistance by the greater mass tends to cause a reduction in the amplitude of the pulse. Different materials also have differing degrees of springiness or elasticity.

### Energy and Radiation

A more elastic medium will tend to offer less resistance to the force and allow a greater amplitude pulse to travel through it; being less rigid and therefore more elasticthe same force causes a greater amplitude. Energy-Amplitude Mathematical Relationship The energy transported by a wave is directly proportional to the square of the amplitude of the wave.

This energy-amplitude relationship is sometimes expressed in the following manner. This means that a doubling of the amplitude of a wave is indicative of a quadrupling of the energy transported by the wave. A tripling of the amplitude of a wave is indicative of a nine-fold increase in the amount of energy transported by the wave.

And a quadrupling of the amplitude of a wave is indicative of a fold increase in the amount of energy transported by the wave. The table at the right further expresses this energy-amplitude relationship. Observe that whenever the amplitude increased by a given factor, the energy value is increased by the same factor squared. For example, changing the amplitude from 1 unit to 2 units represents a 2-fold increase in the amplitude and is accompanied by a 4-fold 22 increase in the energy; thus 2 units of energy becomes 4 times bigger - 8 units.

As another example, changing the amplitude from 1 unit to 4 units represents a 4-fold increase in the amplitude and is accompanied by a fold 42 increase in the energy; thus 2 units of energy becomes 16 times bigger - 32 units.

Earthquakes and other geologic disturbances sometimes result in the formation of seismic waves. Seismic waves are waves of energy that are transported through the earth and over its surface. Earthquakes are given a Richter scale rating that indicates how intense the earthquake is. Use the Earthquake Energy widget below to explore the relationship between the Richter scale magnitude and the amount of energy transmitted by seismic waves.

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