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The wave equation

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The wave equation
20.4 THE WAVE EQUATION
20.4 The wave equation
We have already found that the general solution of the one-dimensional wave
equation is
u(x, t) = f(x − ct) + g(x + ct),
(20.26)
where f and g are arbitrary functions. However, the equation is of such general
importance that further discussion will not be out of place.
Let us imagine that u(x, t) = f(x − ct) represents the displacement of a string at
time t and position x. It is clear that all positions x and times t for which x − ct =
constant will have the same instantaneous displacement. But x − ct = constant
is exactly the relation between the time and position of an observer travelling
with speed c along the positive x-direction. Consequently this moving observer
sees a constant displacement of the string, whereas to a stationary observer, the
initial profile u(x, 0) moves with speed c along the x-axis as if it were a rigid
system. Thus f(x − ct) represents a wave form of constant shape travelling along
the positive x-axis with speed c, the actual form of the wave depending upon
the function f. Similarly, the term g(x + ct) is a constant wave form travelling
with speed c in the negative x-direction. The general solution (20.23) represents
a superposition of these.
If the functions f and g are the same then the complete solution (20.23)
represents identical progressive waves going in opposite directions. This may
result in a wave pattern whose profile does not progress, described as a standing
wave. As a simple example, suppose both f(p) and g(p) have the form§
f(p) = g(p) = A cos(kp + ).
Then (20.23) can be written as
u(x, t) = A[cos(kx − kct + ) + cos(kx + kct + )]
= 2A cos(kct) cos(kx + ).
The important thing to notice is that the shape of the wave pattern, given by the
factor in x, is the same at all times but that its amplitude 2A cos(kct) depends
upon time. At some points x that satisfy
cos(kx + ) = 0
there is no displacement at any time; such points are called nodes.
So far we have not imposed any boundary conditions on the solution (20.26).
The problem of finding a solution to the wave equation that satisfies given boundary conditions is normally treated using the method of separation of variables
§
In the usual notation, k is the wave number (= 2π/wavelength) and kc = ω, the angular frequency
of the wave.
693
PDES: GENERAL AND PARTICULAR SOLUTIONS
discussed in the next chapter. Nevertheless, we now consider D’Alembert’s solution
u(x, t) of the wave equation subject to initial conditions (boundary conditions) in
the following general form:
initial displacement, u(x, 0) = φ(x);
initial velocity,
∂u(x, 0)
= ψ(x).
∂t
The functions φ(x) and ψ(x) are given and describe the displacement and velocity
of each part of the string at the (arbitrary) time t = 0.
It is clear that what we need are the particular forms of the functions f and g
in (20.26) that lead to the required values at t = 0. This means that
φ(x) = u(x, 0) = f(x − 0) + g(x + 0),
∂u(x, 0)
= −cf (x − 0) + cg (x + 0),
ψ(x) =
∂t
(20.27)
(20.28)
where it should be noted that f (x − 0) stands for df(p)/dp evaluated, after the
differentiation, at p = x − c × 0; likewise for g (x + 0).
Looking on the above two left-hand sides as functions of p = x ± ct, but
everywhere evaluated at t = 0, we may integrate (20.28) between an arbitrary
(and irrelevant) lower limit p0 and an indefinite upper limit p to obtain
1 p
ψ(q) dq + K = −f(p) + g(p),
c p0
the constant of integration K depending on p0 . Comparing this equation with
(20.27), with x replaced by p, we can establish the forms of the functions f and
g as
p
1
K
φ(p)
−
(20.29)
ψ(q) dq − ,
f(p) =
2
2c p0
2
p
φ(p)
K
1
g(p) =
ψ(q) dq + .
+
(20.30)
2
2c p0
2
Adding (20.29) with p = x − ct to (20.30) with p = x + ct gives as the solution to
the original problem
x+ct
1
1
u(x, t) = [φ(x − ct) + φ(x + ct)] +
ψ(q) dq,
(20.31)
2
2c x−ct
in which we notice that all dependence on p0 has disappeared.
Each of the terms in (20.31) has a fairly straightforward physical interpretation.
In each case the factor 1/2 represents the fact that only half a displacement profile
that starts at any particular point on the string travels towards any other position
x, the other half travelling away from it. The first term 12 φ(x − ct) arises from
the initial displacement at a distance ct to the left of x; this travels forward
arriving at x at time t. Similarly, the second contribution is due to the initial
displacement at a distance ct to the right of x. The interpretation of the final
694
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