Young's modulus can be thought of as the spring constant for solids. Consider an ideal rod (or bar) of length and cross-sectional area . Suppose we apply a force to the face of area , causing a displacement along the axis of the rod. Then Young's modulus is given by
For wood, Young's modulus is on the order of N/m. For aluminum, it is around (a bit higher than glass which is near ), and structural steel has .
Recall (§B.1.3) that Hooke's Law defines a spring constant as the applied force divided by the spring displacement , or . An elastic solid can be viewed as a bundle of ideal springs. Consider, for example, an ideal bar (a rectangular solid in which one dimension, usually its longest, is designated its length ), and consider compression by along the length dimension. The length of each spring in the bundle is the length of the bar, so that each spring constant must be inversely proportional to ; in particular, each doubling of length doubles the length of each ``spring'' in the bundle, and therefore halves its stiffness. As a result, it is useful to normalize displacement by length and use relative displacement . We need displacement per unit length because we have a constant spring compliance per unit length.
The number of springs in parallel is proportional to the cross-sectional area of the bar. Therefore, the force applied to each spring is proportional to the total applied force divided by the cross-sectional area . Thus, Hooke's law for each spring in the bundle can be written
We may say that Young's modulus is the Hooke's-law spring constant for the spring made from a specifically cut section of the solid material, cut to length 1 and cross-sectional area 1. The shape of the cross-sectional area does not matter since all displacement is assumed to be longitudinal in this model.
The tension of a vibrating string is the force used to stretch it. It is therefore directed along the axis of the string. A force must be applied at the endpoint on the right, and a force is applied at the endpoint on the left. Each point interior to the string is pulled equally to the left and right, i.e., the net force on an interior point is . (A nonzero force on a massless point would produce an infinite acceleration.)
If the cross-sectional area of the string is , then the tension is given by the stress on the string times .
Wave Equation for the Vibrating String