Chapter 2 The Mathematics of Wave motion

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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,
(2-1)
(2-2)
Represents the shape or profile of the wave at
t0. 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.
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• 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
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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
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If f(x-vt) and g(xvt) 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.
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Fig 2.4 A harmonic wave
Fig 2.3 A progressive wave at three different
time
From the definition of we have

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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.
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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 of polar 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
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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
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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,
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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.