Vector Spaces in Physics 8/12/2010 7 - 1 Chapter 7. The Wave Equation The vector spaces that we have described so far were finite dimensional. Describing position in the space we live in requires an infinite number of position vectors, but they can all be represented as linear combinations of three carefully chosen linearly independent position vectors. There are other analogous situations in which a complicated problem can be simplified by focusing attention on a set of basis vectors. In addition to providing mathematical convenience, these basis vectors often turn out to be interesting objects in their own right. The pressure field inside a closed cylindrical tube (an ''organ pipe'') is a function of time and three space coordinates which can vary in almost an arbitrary way. Yet there are certain simple patterns which form a basis for describing all the others, and which are recognizably important and interesting. In this case they are the fundamental resonance of the organ pipe and its overtones. They are the ''notes'' which we use to make music, identified by our ears as basis vectors of its response space. A similar situation occurs with the vibrating strings of guitars and violas (or with the wires strung between telephone poles), where arbitrary waves moving on the string can be represented as a superposition of functions corresponding again to musical notes. Another analogous situation occurs with quantum- mechanical electron waves near a positively charged nucleus. An arbitrarily complicated wave function can be described as a linear superposition of a series of ''states,'' each of which corresponds (in a certain sense) to one of the circular Bohr electron orbits. We will choose the stretched string to examine in mathematical detail. It is the easiest of these systems to describe, with a reasonable set of simplifying assumptions. The mathematical variety that it offers is a rich preview of mathematical physics in general. A. Qualitative properties of waves on a string Many interesting and intellectually useful experiments can be carried out by exciting waves on a stretched string, wire or n = 1 n = 2 Figure 7-2. Modes of resonance of a vibrating string. Figure 7-1. A pulse traveling down a telephone wire (solid curve), and a reflected pulse traveling back (dashed curve).Vector Spaces in Physics 8/12/2010 7 - 2 cord and observing their propagation. Snapping a clothesline or hitting a telephone line with a rock produces a pulse which travels along the line as shown in figure 7-1, keeping its shape. When it reaches the point where the line is attached, it reverses direction and heads back to the place where it started, upside down. The pulse maintains its shape as it travels, only changing its form when it reaches the end of the string. A string which has vibrated for a long time tends to "relax" into simpler motions which are adapted to the length of the string. Figure 7-2 shows two such motions. The upper motion has a sort of sinusoidal shape at any given moment, with a point which never moves (a node) at either end of the string. If the string is touched with a finger at the midpoint, it then vibrates as sketched in the lower diagram, with nodes at the ends and also at the touched point. It will be noticed that the musical note made by the string in the second mode of vibration is an octave higher than the note from the first mode. Another interesting motion can be observed by stretching the string into the shape of a pulse like that shown in figure 7-1, and then releasing it, so that the string is initially motionless. The pulse is observed to divide into two pulses, going in opposite directions, which run back and forth along the string. But after a time, the string "relaxes" into a motion like that in the upper part of figure 7-2. How can a pulse propagate without its shape changing? Why does it reverse the direction of its displacement after reflection? Why does the guitar string vibrate with a sinusoidal displacement? Why does the mode with an additional displacement node vibrate with twice the frequency? Why does a traveling pulse relax into a stationary resonant mode? We will try to build up a mathematical description of waves in a string which predicts all these properties. B. The wave equation. The propagation of a wave disturbance through a medium is described in terms of forces exerted on an element of mass of the medium by its surroundings, and the resulting acceleration of the mass element. In a string, any small length of the string experiences two forces, due to the rest of the string pulling it to the side, in the two different directions. Figure 7-3 shows an element of the string of length dx, at a point where the string is curving downwards. The forces pulling on each end of the string are also shown, and it is clear that there is a net downwards force, due to the string tension. We will write down Newton's law for this string element, and show that it leads to the Figure 7-3. Forces exerted on an element of string.Vector Spaces in Physics 8/12/2010 7 - 3 partial differential equation known as the wave equation. But first, we will discuss a set of approximations which makes this equation simple. First, we will make the small-angle approximation, assuming here that the angle which the string makes with the horizontal is small (1θ≪). In this approximation, cos 1θ≃ and sinθ θ≃. Secondly, we will assume that the tension T is constant throughout the string. Two of these assumptions (cos 1θ≃ and T constant) result in a net force of zero in the longitudinal (x) direction, so that the motion of the string is purely transverse. We will ignore a possible small longitudinal motion and assume that it would have only a small effect on the transverse motion which we are interested in. In the transverse (y) direction, the forces do not cancel. The transverse component of the force on the left-hand side of the segment of string has magnitude T sin θ. We will relate sin θ to the slope of the string, according to the relation from analytic geometry riseslope tan sinrundydxθ θ= = = ≃ . (7-1) The last, approximate, equality is due to the small-angle approximation. Thus the transverse force has magnitude approximately equal to dyTdx, and Newton's second law for the transverse motion gives ()( )22yx dx xF dm ay y yT T dxx x tµ+=∂ ∂ ∂− =∂ ∂ ∂