Let us further investigate d’Alembert’s formula. Suppose that a disturbance u travels in the positive x-direction with a constant positive velocity v. The specific nature of the disturbance is unimportant at the moment. Because the disturbance, or pulse, is moving, it must be a function of both position and time and therefore can be written as . The shape of the disturbance at any instant, say, , can be found by holding time constant at that value. In this case,
represents the shape or profile of the wave at time . The process is analogous to taking a photograph of the pulse as it travels by. For the moment, we will limit ourselves to a wave that does not change its shape as it progresses through space. Figure 3 is a double exposure of such a disturbance, taken at the beginning and end of a time interval t. In a fixed frame of reference, the pulse has moved along the x-axis a distance vt, but in all other respects, it has remained unaltered.
We now introduce a coordinate frame that travels along with the pulse at speed v. In this moving frame, u is no longer a function of time but instead is a stationary constant profile with the same functional form as the waveform at time 0. In other words, can be written as
This equation represents a general form of the 1D wave function. To be more specific, we have only to choose a shape and then substitute for x in . The resulting expression describes a wave traveling in the positive x-direction and having the desired profile. We can check the form of equation 5 by examining it after an increase in time of and a corresponding increase of in x. We find
which shows that the profile is unaltered.
Similarly, if the wave is traveling in the negative x-direction - that is, to the left - the equation becomes
where, as before, we assume that the quantity v is a positive number. We can conclude, therefore, that regardless of the shape of the wave, the variables x and t must appear in the function as a unit, i.e., as a single variable in the form . If we require that v be a positive quantity, then both of the above equations for u can be combined into the single equation
with the negative sign indicating propagation in the positive x-direction and the positive sign indicating propagation in the negative x-direction.
Because the wave equation is a linear partial differential equation, it follows that if two wave functions and are each separate solutions, then is also a solution. Accordingly, the wave equation is satisfied by a wave function that has the form
where and are constants. This is clearly a sum of two waves traveling in opposite directions along the x-axis with the same speed but not necessarily the same profile. The superposition principle is inherent in this equation.
Many physical systems involve simultaneous propagation of two or more waveforms. Examples are especially common in acoustics. The sounds we hear represent a complicated combination of various wave motions that result in some overall pattern. Within normal limits, the following basic assumption holds: The resultant of two or more individual waves is simply the sum of the individual waves.
The behavior of a single pulse traveling in one direction is easily visualized. But what happens when one pulse moves from right to left at the same time that another pulse moves from left to right? As we know, the two pulses superimpose when they meet. The best way to understand this phenomenon is by means of a motion picture. Suppose two pulses start at opposite ends of the same string at the same time and head toward each other. The pulses approach each other as if each had the string to itself. As they cross each other, the two pulses additively combine to form a composite shape. But after they cross, they again assume their original shapes and travel along the string as if nothing had happened. If we perform this experiment over and over with different pulses, we always get the same result.
The fact that two pulses pass through each other without either pulse being altered is a fundamental property of linear waves. However, when two tennis balls are thrown at each other in opposite directions, their motion is changed violently if they hit. Thus, the crossing of waves is a very different process from the crossing of streams of balls made from solid matter.
Let us now take a closer look at the superposition that occurs when two pulses cross each other. Often, the shape of the combined pulse does not resemble the shape of either of the two original pulses. We can visualize each of the original pulses at the position it would occupy if it were alone, and then we add the displacements of both original pulses to get the resultant pulse. In other words, the principle of superposition says that the resultant displacement of any point on the string at any instant is equal to the sum of the displacements that would have been produced by the two pulses independently. As a matter of fact, the principle works for more than two pulses; the resulting displacement for any number of pulses is the sum of the displacements for the individual pulses.
The principle of superposition can be summarized as follows. To find the form of the total wave displacement at any time, add at each point the displacements corresponding to each pulse that is passing through the medium. This simple addition gives the actual displacement in the medium.
Let us now apply the superposition principle to the case of two equal symmetrical pulses that have opposite polarity. The two pulses are assumed to have exactly the same shape and size, and each is symmetrical. Suppose that the one that displaces the string upward is the one that travels to the right. The pulse that displaces the string downward travels to the left. There is some moment in their crossing when the addition of equal displacements upward (plus) and downward (minus) leaves us with a net displacement of zero. Thus, at the moment when the pulses pass each other, the whole string appears to be undisplaced. In other words, there is complete cancellation at that particular moment. How is this situation different from the case of a string at rest? In such a case - that is, when the string carries no wave motion - all the various pieces of the string stand still at all times. On the other hand, when two symmetrical equal and opposite waves travel along the string, the string passes through its rest position for only a single instant, yet at that instant, the string is still moving. At that particular instant, all of the moving string’s energy exists purely as kinetic energy.
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Also in this chapter
- Wavefronts and raypaths
- d’Alembert’s solution
- Sinusoidal waves
- Phase velocity
- Wave pulses
- Geometric seismology
- The speed of light
- Huygens’ principle
- Reflection and refraction
- Ray theory
- Fermat’s principle
- Fermat’s principle and reflection and refraction
- Appendix A: Exercise