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Z
x 1
Tu x
x 1
= ρu tt dx transverse.
p
1 + u 2 x 0
x x 0
The right sides are the components of the mass times the acceleration integrated over
the piece ofstring. Since we have assumed that the motion is purely transverse, there is
no longitudinal motion.
Now we also assume that the motion is small — more specifically, that |u x l is quite
p
2
small. Then 1 + u may be replaced by 1. This is justified by the Taylor expansion,
x
actually the binomial expansion,
p 1
2
2
1 + u = 1 + u + . . .
x x
2
where the dots represent higher powers of u x . If u x is small, it makes sense to drop the
2
even smaller quantity u and its higher powers. With the square roots replaced by 1, the
x
first equation then says that T is constant along the string. Let us assume that T is
independent of t as well as x. The second equation, differentiated, says that
(Tu x ) x = ρu tt .
That is,
s
T
2
u tt = c u xx where c = . (4.2)
ρ
This is the wave equation. At this point it is not clear why c is defined in this manner,
but shortly we’ll see that c is the wave speed.
There are many variations of this equation:
1) If significant air resistance r is present, we have an extra term proportional to the
speed u t , thus:
2
u tt − c u xx + ru t = 0 where r > 0.
2) If there is a transverse elastic force, we have an extra term proportional to the
displacement u, as in a coiled spring, thus:
2
u tt − c u xx + ku = 0 where k > 0.
3) If there is an externally applied force, it appears t/s an extra term, thus:
2
u tt − c u xx = f(x, t),
which makes the equation inhomogeneous.
Our derivation of the wave equation has been quick but not too precise. A much
more careful derivation can be made, which makes precise the physical and mathematical
assumptions.
The same wave equation or a variation of it describes many other wavelike phenomena,
such as the vibrations of an elastic bar, the sound waves in a pipe, and the long water
waves in a straight canal. Another example is the equation for the electrical current in a
transmission line,
u xx = cLu tt − (CR + GL)u t + GRu,
where C is the capacitance per unit length, G the leakage resistance per unit length, R
the resistance per unit length, and L the self-inductance per unit length.

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