Chapter 2 The Mathematics of Wave motion. 2.1 One-dimensional waves. The essential aspect of a propagating wave is that it is a self-sustaining disturbance of the medium through which it travels. For example, a pulse travels along a stretched string, as shown in Fig. 2.1 .
2.1 One-dimensional waves
The essential aspect of a propagating wave is that it is a self-sustaining disturbance of the medium through which it travels. For example, a pulse travels along a stretched string, as shown in Fig. 2.1
Wave is a function of both position and time, and thus it can be written,
Represents the shape or profile of the wave at t=0. Figure 2.2 is a “double exposure” of a disturbance taken at the beginning and at time t. Introduce a coordinate system S’, which travels along with the
Fig. 2.1 A wave on a string.
it represents the most general form of the one-dimensional wave function, traveling in the positive x direction
Similarly, if the wave were traveling in the negative x-direction, Eq. (2.3) would become
Fig. 2.2 Moving of wave.
We may conclude therefore that, regardless of the shape of the disturbance, the variables x and t must appear in the function as a unit, i.e., as a single variable in the form
Using the information above, we can derive the general one-dimensional deferential wave equation. Take the partial derivative of with respect to x. Using
The partial derivative with respect to time is (note: )
Since two constants are needed to specify a waveform, we can anticipate a second-order wave equation. The second partial derivatives of Eqs. (2..7) and (2.8) yield
Combining these equations, we obtain the one-dimensional differential wave equation
If f(x-vt) and g(x+vt) are separate solutions to Eq. (2.9), the general solution of Eq. (2.9) is
Where C1 and C2 are constants determined by initial conditions of the wave.
2.2 harmonic Waves
The simplest wave form is a sine or cosine wave, that is called harmonic wave. The profile is
Where k is a positive constant known as the propagation number. A is known as the amplitude of the wave Replace x with x-vt, we have a progressive wave traveling to the positive x-direction with a speed of v
Holding either x or t fixed results in a sinusoidal disturbance, so the wave is periodic in both space and time. The spatial period is the wave length and is denoted by as in fig 2.3 The temporal period is the amount of time it takes for one complete wave to pass a stationary observer and is denoted as , as in Fig 2.4.
Fig 2.4 A harmonic wave
Fig 2.3 A progressive wave at three different time
From the definition of we have
The inverse of the period is called the frequency f, which is the number of cycles per unit of time. Thus
Angular frequency, , is defined as the number of radians per unit of time. thus
The harmonic wave Eq.2.12 can also be written as
The most general harmonic wave equation is
The whole quantity in the bracket is called the phase, , with the initial phase.
The wavelength, period describe aspects of the repetitive nature of a wave in space and time. These concepts are equally well applied to waves that are not harmonic, as long as each wave profile is made up of a regularly repeating pattern (Fig 2.5)
Fig. 2-5 Some anharmonic waves.
2.3 Complex representation
The complex number has the form,
where Both and are themselves real numbers, and they are the real and imaginary parts of , respectively.
In terms ofpolar coordinate , we have
The Euler formula, , allows us to write
where is the magnitude of , and is the phase angle of .
It is clear that either part of could be chosen to describe a harmonic wave. In general, we shall write the wave function as
2.4 Plane waves
The plane wave is perhaps the simplest example of a three-dimensional wave. It exists at a given time, when all the surfaces upon which a disturbance has a constant phase form a set of planes, each generally perpendicular to the propagation direction.
The following equation
defines a plane harmonic wave.
In Eq. 2.23, the vector , whose magnitude is the propagation number , is called the propagation vector.
The plane harmonic wave is often written in Cartesian coordinates as
Fig. 2-6 Wavefronts for a harmonic plane wave.
At given time, i.e., , the shape of the plane harmonic wave is
It is clear that is constant when . The surfaces joining all points of equal phase are known as wavefronts or wave surfaces.
2.5 Three-dimensional differential wave equation
The differential equation
is so-called three-dimensional differential wave equation.
In Eq. 2.26, is the Laplacian operator,
2.6 Spherical waves
Consider now an idealized point light source. The radiation emanating from it streams out radially, uniformly in all direction. The resulting wavefronts are concentric spheres that increase in diameter as they expand out into the surrounding space. The obvious symmetry of the wavefronts suggests that it might be more convenient to describe them mathematically in terms of spherical polar coordinates. In doing so, the differential wave equation 2.26 can be written as
A special solution of the Eq. 2.28 is the harmonic spherical wave
Wherein the constant A is called the source strength. Each wavefront is given by
Fig. 2.7 Spherical wavefronts.