16.4 Energy and power of a wave  (Page 4/6)

The equations for the energy of the wave and the time-averaged power were derived for a sinusoidal wave on a string. In general, the energy of a mechanical wave and the power are proportional to the amplitude squared and to the angular frequency squared (and therefore the frequency squared).

Another important characteristic of waves is the intensity of the waves. Waves can also be concentrated or spread out. Waves from an earthquake, for example, spread out over a larger area as they move away from a source, so they do less damage the farther they get from the source. Changing the area the waves cover has important effects. All these pertinent factors are included in the definition of intensity ( I )    as power per unit area:

where P is the power carried by the wave through area A . The definition of intensity is valid for any energy in transit, including that carried by waves. The SI unit for intensity is watts per square meter (W/m ² ). Many waves are spherical waves that move out from a source as a sphere. For example, a sound speaker mounted on a post above the ground may produce sound waves that move away from the source as a spherical wave. Sound waves are discussed in more detail in the next chapter, but in general, the farther you are from the speaker, the less intense the sound you hear. As a spherical wave moves out from a source, the surface area of the wave increases as the radius increases $\left(A=4\pi r^2\right).$ The intensity for a spherical wave is therefore

If there are no dissipative forces, the energy will remain constant as the spherical wave moves away from the source, but the intensity will decrease as the surface area increases.

In the case of the two-dimensional circular wave, the wave moves out, increasing the circumference of the wave as the radius of the circle increases. If you toss a pebble in a pond, the surface ripple moves out as a circular wave. As the ripple moves away from the source, the amplitude decreases. The energy of the wave spreads around a larger circumference and the amplitude decreases proportional to $\frac{1}{r},$ and not $\frac{1}{r^2},$ as in the case of a spherical wave.

Summary

• The energy and power of a wave are proportional to the square of the amplitude of the wave and the square of the angular frequency of the wave.
• The time-averaged power of a sinusoidal wave on a string is found by $P_{\text{ave}}=\frac{1}{2}\mu A^2{\omega }^{2}v,$ where $\mu$ is the linear mass density of the string, A is the amplitude of the wave, $\omega$ is the angular frequency of the wave, and v is the speed of the wave.
• Intensity is defined as the power divided by the area. In a spherical wave, the area is $A=4\pi r^2$ and the intensity is $I=\frac{P}{4\pi r^2}.$ As the wave moves out from a source, the energy is conserved, but the intensity decreases as the area increases.