The Wave Equation

CHAPTER

3

3.1 The Vibrating String

A simple and historically important example of a problem that includes the
wave equation is provided by the study of the vibration of a string, like a violin
or guitar string. We set up a coordinate system as shown in Fig. 1. The un-
known is the transverse displacement,u(x,t),measuredupfromthex-axis.
The situation is similar to that of the hanging cable discussed in Chapter 0,
but here the string is taut, and of course motion is allowed. In order to find the
equation of motion of the string, we consider a short piece whose ends are at
xandx+ xand apply Newton’s second law of motion to it.
First, we must analyze the nature of the forces on the string. We assume
that the only external force is the attraction of gravity, acting perpendicular
to thex-direction. Internal forces are exerted on the segment by the rest of the
string. We willassume that the string is perfectly flexibleand offers no resistance
to bending. Then the only force that can be transmitted by the string is a pull
or tension, which acts in a direction tangential to the centerline of the string.
Its magnitude we denote byT(x,t).
The forces on the small segment of string are shown in Fig. 2. We shall fur-
therassume that each point on the string moves only in the vertical direction.
Thus, the horizontal component of acceleration is zero. Application of New-
ton’s second law for the horizontal direction to the segment leads to the equa-
tion

−T(x,t)cos

(

φ(x,t)

)

+T(x+ x,t)cos

(

φ(x+ x,t)

)

= 0 ,

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