18.2.3.1 Solution stanexw-a

Question:

Derive the wave equation for small transverse vibrations of a string by considering a little string segment of length ${\Delta}x$.

Answer:

A sketch of the string segment is shown below.

The vertical position of the string segment is $u$ and the horizontal coordinate is $x$. Call the mass of the string per unit length $\rho$. Call the tension force in the string $T$. The magnitude of this tension force can be assumed to be constant because there is no significant longitudinal motion. However, the direction of the force varies and that cannot be ignored. The varying direction produces net vertical forces. While small, they are big enough to produce the small vertical vibrations of the string.

Now find the net vertical force on the segment by taking vertical components of the tension forces on the two end points of the segment. Then apply Newton’s second law. Note that the tension force must be in the direction of the string at each point. Otherwise you get into trouble with the momentum equation for infinitesimal segments; an infinitely thin string has no bending stiffness nor a moment of inertia proportional to the length of string.

Note also that for small angles $\theta$,

$$\sin(\theta) \approx\theta\qquad\tan(\theta) \approx\theta$$

Also remember from calculus that

$$\frac{\partial u}{\partial x} = \tan(\theta)$$

where $\theta$ is the angle between the tangent of the curve $u$ versus $x$ and the $x$-axis for a given time.

You should be able to find arguments like the above in many books on engineering mathematics.