Vector Spaces in Physics 8/12/2010 8 - 1 Chapter 8. Standing Waves on a String The superposition principle for solutions of the wave equation guarantees that a sum of waves, each satisfying the wave equation, also represents a valid solution. In the next section we start with a superposition of waves going in both directions and adjust the superposition to satisfy certain requirements of the wave's surroundings. A. Boundary conditions and initial conditions. The wave equation results from requiring that a small segment of the string obey Newton's second law. This is not sufficient to completely specify the behavior of a given string. In general, we will find it necessary to specify initial conditions, given at a particular time, and boundary conditions, given at particular places on the string. These conditions determine which of the manifold of possible motions of the string actually takes place. The wave equation is a partial differential equation, and is second order in derivatives with respect to time, and second order in derivatives with respect to position. In general, a second-order differential equation requires two side conditions to completely determine the solution. For instance, the motion of a body moving in the vertical direction in the Earth's gravitational field is not determined until two conditions, such as the initial position and initial velocity, are specified. It is possible to vary the conditions - for instance, the velocity specified at two different times can replace position and velocity specified at t = 0. In the case of the wave equation, to determine the time dependence two conditions must be given, at a specified time and at all positions on the string. For instance, for a plucked guitar string, the initial conditions could be that initially the string has zero velocity at all points, and is displaced in a triangle waveform, with the maximum displacement at the point where it is plucked. Two additional conditions, the boundary conditions, are required to determine the spatial dependence of the solution. Each condition specifies something about the displacement of the string, at one particular point and for all time. For instance, for the guitar string, the displacement of the two endpoints of the string is required to be zero for all time. String fixed at a boundary. A very important type of boundary condition for waves on a string is imposed by fixing one point on the string. This is usually a point of support for the string, where the tension is applied. Imagine a traveling-wave pulse like that shown in figure 8-1, traveling from left to right and approaching a point of attachment of the string, where it cannot move up and down as the wave passes. The shape of this pulse obviously has to change as it passes over this fixed point,. But the wave equation says that this pulse will propagate forever, without changing its direction or shape. How can we get out of this impasse? The answer is shown in figure 8-1. Here the string occupies the x < 0 part of space, and is attached to a wall at x = 0, imposing the boundary condition ( 0, ) 0y x t= = (8.1)Vector Spaces in Physics 8/12/2010 8 - 2 The pulse traveling to the right can be represented by the function right( , ) ( )y x t f x vt= −, (8.2) where the functional form f(u) determines its shape, and the argument x-vt turns it into a wave traveling to the right. It is clear that for times when the pulse overlaps the fixed point of the string, yright alone is not the correct solution to the wave equation, since it does not vanish at the point where the string is attached. It is the superposition principle that saves us. The solution is illustrated graphically in figure 8-1, where a second pulse is shown, identical in shape to the first but (a) inverted, and (b) traveling in the opposite direction, from right to left. The figure shows the first pulse disappearing behind the wall, a region which we call "ghost space," and the second emerging from behind the wall, coming out of ghost space. The part of space with x > 0 is not really part of the problem - there is no string there. But for visualizing this problem it is helpful to imagine an invisible continuation of the string with positive x. We can then picture the erect and inverted pulses both propagating on an infinite string, of which we only see the part with x < 0. If the inverted pulse arrives at just the right time, its negative displacement cancels the positive displacement of the erect pulse, and the resulting zero displacement satisfies the boundary condition for the string. Figure 8-1. A wave pulse traveling from left to right has just started to impinge on a fixed point of the string. The condition that y = 0 at the fixed point is satisfied by the linear superposition of an inverted pulse traveling in the opposite direction.Vector Spaces in Physics 8/12/2010 8 - 3 How do we write this solution mathematically? If right( )y f x vt= − represents the shape f(u), right side up and traveling to the right, with velocity v, then left( )y f x vt= − − − represents the same shape f(u), inverted and traveling with velocity -v, to the left. So, a possible solution to the wave equation which satisfies the boundary condition at the fixed end is right left( , )( ) ( )y x t y yf x vt f x vt= += − − − −, (8.3) Here is how you convince yourself that this is the solution we want. Suppose that the pulse shape f(u) has its peak at u = 0, and vanishes except when u is fairly close to zero. Now consider the solution y(x,t) given above, for large negative times. Each term is zero except when its particular argument is near zero. So, the first pulse will be centered at a large negative x, in the "real world" part of the string, and the second pulse will be centered at large positive x, out of sight in the "ghost world." Thus, the initial conditions for this solution are that, at some large negative time, there is a pulse of shape f(u), with transverse velocity such that it travels to the right. Next, consider the solution for large positive times. Now yright peaks at positive x, out of sight in the "ghost world," and yleft peaks at negative x, where we can see it. Finally, check to see that the boundary condition is satisfied: Figure 8-2. A boundary between two parts of a string with different wave propagation velocities. Shown are a wave incident from the