Читать книгу An Introduction to the Finite Element Method for Differential Equations - Mohammad Asadzadeh - Страница 29

The third equation in (1.5.2) is the wave equation: . Here, represents a wave traveling through an ‐dimensional medium; is the speed of propagation of the wave in the medium and is the amplitude of the wave at position and time . The wave equation provides a mathematical model for a number of problems involving different physical processes as, e.g. in the following examples (i)–(vi):

1 (i) Vibration of a stretched string, such as a violin string (one‐dimensional).

2 (ii)Vibration of a column of air, such as a clarinet (one‐dimensional).

3 (iii)Vibration of a stretched membrane, such as a drumhead (two‐dimensional).

4 (iv)Waves in an incompressible fluid, such as water (two‐dimensional).

5 (v)Sound waves in air or other elastic media (three‐dimensional).

6 (vi)Electromagnetic waves, such as light waves and radio waves (three‐dimensional).

Note that in (i), (iii) and (iv), represents the transverse displacement of the string, membrane, or fluid surface; in (ii) and (v), represents the longitudinal displacement of the air; and in (vi), is any of the components of the electromagnetic field. For detailed discussions and a derivation of the equations modeling (i)–(vi), see, e.g. Folland [62], Strauss [129], and Taylor [134]. We should point out, however, that in most cases, the derivation involves making some simplifying assumptions. Hence, the wave equation gives only an approximate description of the actual physical process, and the validity of the approximation will depend on whether certain physical conditions are satisfied. For instance, in example (i), the vibration should be small enough so that the string is not stretched beyond its limits of elasticity. In example (vi), it follows from Maxwell's equations, the fundamental equations of electromagnetism, that the wave equation is satisfied exactly in regions containing no electrical charges or current, which of course cannot be guaranteed under normal physical circumstances and can only be approximately justified in the real world. So an attempt to derive the wave equation corresponding to each of these examples from physical principles is beyond the scope of these notes. Nevertheless, to give an idea, below we shall derive the wave equation for a vibrating string.