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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).
- 16.11: Energy in Waves- Intensity
Calculate the intensity and the power of rays and waves. All...
- 16.11: Energy in Waves- Intensity
Learning Objectives. By the end of this section, you will be able to: Explain how energy travels with a pulse or wave. Describe, using a mathematical expression, how the energy in a wave depends on the amplitude of the wave. All waves carry energy, and sometimes this can be directly observed.
The conservation of energy provides a straightforward way of showing that the solution to an IVP associated with the linear equation is unique. We demonstrate this for the wave equation next, while a similar procedure will be applied to establish uniqueness of solutions for the heat IVP in the next section. Example 7.1.
There are many solutions to this 3D wave equation. Important solutions are plane waves A(x,y,z,t)=A0cos ~k·~x −ωt+φ (34) for some amplitude A0, frequency ω and fixed vector ~k called the wavevector. For a plane wave to satisfy the wave equation, its frequency and wavevector must be related by ω=v ~k (35)
The point where » = +A is typically called the \crest" of the wave and the point where » = ¡A is called the \trough" of the wave.2 The distance from crest to crest (or trough to trough) is called the \wavelength," the distance between points on the wave which have the same phase at the same instant of time.
Calculate the intensity and the power of rays and waves. All waves carry energy. The energy of some waves can be directly observed. Earthquakes can shake whole cities to the ground, performing the work of thousands of wrecking balls. Loud sounds pulverize nerve cells in the inner ear, causing permanent hearing loss.
We have remarked that any plane-wave disturbance which moves with a constant velocity $v$ has the form $f(x - vt)$. Now we have to see whether $\chi(x,t) = f(x - vt)$ is a solution of the wave equation. When we calculate $\ddpl{\chi}{x}$, we get the derivative of the function, $\ddpl{\chi}{x} = f'(x - vt)$.
