FIGURE 5-3: Examples of standing waves, demonstrating the first three normal modes of a string fixed at both ends. The top wave represents the fundamental frequency.

      If the children get ambitious, however, and begin putting more energy into the wave motion of their jump rope, a curious thing happens. Eventually, the children will pump enough energy into the rope that instead of one large antinode, two smaller antinodes are created, and the center of the rope seems to be at rest, as shown in Figure 5-3b. It’s almost as if someone grabbed the middle of the rope and is gingerly, but firmly, holding it in place! (If you are a musician, you may recognize this as the second harmonic or first overtone of the rope.)

Another type of standing wave can be considered if instead of a child holding each end of the rope, one end is mounted on a ring around a pole. The child holding one end begins the wave motion, but the end on the pole is now unconstrained and moves up and down. Instead of having a node on each end, one end is a node (held by the child), and the other is an antinode (moving up and down on the pole).

      A similar situation happens when a musician uses a pipe that’s closed at one end and open at the other, like pipes in an organ. A node forms at the closed end of the pipe, but the open end of the pipe is always an antinode.

      A third type of standing wave has an antinode at each end. This would be represented by either a pipe that’s open on both ends or a rope that’s free to move on both ends.

      The more energy that’s pumped into the standing wave, the more nodes form (see Figure 5-3c). The series of frequencies that cause new nodes to form are called harmonics or, in music, overtones. The waves that correspond to harmonics are called normal modes or vibrational modes.

      Music works because of the manipulation and superposition of harmonic overtones created by these normal modes of vibration. The first three normal modes are shown in Figure 5-3, where a string is fixed on both ends.

Newton’s Revolution: How Physics Was Born

      Many see Sir Isaac Newton’s discoveries as the start of modern physics (along with a bit of help from his predecessor Galileo Galilei). Newton’s discoveries dominated two centuries of physics, until Albert Einstein took his place at the apex of scientific greatness.

Newton’s accomplishments are diverse, but he’s known largely for four crucial discoveries that define the realm of physics even today:

      Each of these discoveries has elements that will prove important as you attempt to understand the later discoveries of string theory.

      Force, mass, and acceleration: Putting objects into motion

      Newton formulated three laws of motion, which showed his understanding of the real meaning of motion and how it relates to force. Under his laws of motion, a force creates a proportional acceleration on an object.

      This understanding was a necessary foundation upon which his law of gravity was built (see the next section). In fact, both were introduced in his 1686 book, Philosophiae Naturalis Principia Mathematica, a title that translates to Mathematical Principles of Natural Philosophy. This book has become known in physics circles by the shorter title Principia.

      The second law of motion says that the force required to accelerate an object is the product of the mass and acceleration, expressed by the equation F = ma, where F is the total force, m is the object’s mass, and a is the acceleration. To figure out the total acceleration on an object, you figure out the total forces acting on it and then divide by the mass.

NEWTON MAKES SOME LAWS ABOUT MOTION

      Newton’s second law of motion, and the way it relates force, acceleration, and mass, is the only law of motion relevant to a string theory discussion. However, for true Newton-o-philes, here are the other two laws of motion, paraphrased for ease of understanding: