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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 .

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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

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.

pulse at the speed v. In this system, is no longer a function of time. As we move along with S’ we see a stationary profile with the same functional form as Eq. (2.2). Here the coordinate is x’, so that


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.