Mechanical waves

The work of a curious fellow

Where up and down or back and forth do not add up to zero...

To this point in this course we have discussed the motion of material objects, particles or collections of particles moving through otherwise empty space so that they are in different places at different times. Now we want to study a related but different sort of motion. Imagine a long string stretched between a fixed point and a point that can be moved up and down. If we pulse the movable point quickly up and down once, then hold it in place, we will see something move down the length of the string. Run the Stretched String Pulse display to see this. In this display we only look at part of the string. It is attached way off the screen to the right. The Action button produces the initial impulse.

devilish hard What this display illustrates is that although the particles that make up the string only move in the y dimension, the disturbance moves along the string in the x dimension. This idea of a "disturbance", as opposed to a particle or other object, moving through space is something new to us. What is actually being transported down the string is energy, in the form of kinetic energy of the string's mass and potential energy due to displacement of the string. Energy transmitted in this way is said to be carried by a traveling mechanical wave.

In order for a mechanical wave to exist there must be some source of disturbance, a medium that can be disturbed, and some physical connection through which adjacent particles in the medium can influence one another. We distinguish here between mechanical waves where there is a tangible medium in which the wave travels, and other types of waves where the medium is not so evident. Sound waves, water waves and waves in solid objects are all mechanical waves. For the rest of this section I will use the term waves to mean mechanical waves.

stretched string pulse
pulse superposition The kind of traveling wave we show above is called a pulse. It is a disturbance that does not repeat itself. This particular pulse is very well behaved in that it does not change its shape as it travels along the string. Real pulses on real strings tend to spread and flatten out over time. Another characteristic of this pulse is that it may be combined with other similar pulses by simply adding the amplitudes at any instant in time. The amplitude is the height of the pulse. This sort of combination of waves is called superposition. The Pulse Superposition display shows two pulses, one starting at the left and one starting at the right, traveling in opposite directions. Waves that add in this fashion are called linear waves. Linear meaning in this case that the waves follow a simple superposition rule, not that the waves themselves involve straight lines.

Waves may be transverse, as the pulsed string example was, or longitudinal, as is the case of a sound wave where the disturbance is in the form of changing density of the medium. An example of a longitudinal wave is the disturbance we saw traveling the length of the little nine atom string in the Atomic Solid display when Action was first clicked. Look at that once again if you please.

Notice that when you halt the display with the Cut button you may find some of the atoms bunched together and others spread apart. Since density is the number of atoms per unit length in this system, what you are seeing is variations in density as the energy of the initial poke travels down the string of atoms and bounces back and forth between the ends. In longitudinal waves, the regions of high density are called condensations, those of low density called rarefactions. In our transverse wave example we avoided the complication of the energy being reflected at the ends of the medium by dealing with a very long string and only looking at the left 20 meters of it. In this case our medium is only nine atoms long so the energy is trapped in that short length and must be reflected at the ends, greatly complicating the motion of the atoms.

atomic solid
inverted reflected pulse If we use a string that is attached to a rigid support at the right end of the screen we will see this reflection phenomena take place on the string as well. Look at the Inverted Reflected Pulse display to see this effect. Notice that the reflected wave has a negative amplitude whereas the original wave had a positive amplitude.

If the right end of the string was attached to a frictionless massless ring on a vertical post so that it was free to move up and down, then the reflection would be without the phase inversion. The reflected wave would come back with the amplitude of the same sign as the initial pulse. The Uninverted Reflected Pulse display illustrates this situation.

uninverted reflected pulse
pulse time slice

Both the examples we have looked at so far were of one dimensional waves. That means that the energy moved in only one dimension, even though in the case of the transverse wave the string motion actually involved a second dimension. Let's go back to the pulsed string example and look at it in a slightly different way. At any instant in time, we have y depending on x, y=f(x). Run the Stretched String Pulse display and halt the run with the Cut button when the pulse is near mid screen. Then you may use the cursor to show the value of y for any x at the instant you stopped time.

In the Pulse Sliced Along Time display what you see is a whole series of y vs. x pulses at different values of time, as though 20 snapshots were taken and laid out along the time axis in accordance with the time they were taken. Click on the Action button to see the plot develop. You can change the viewing angles to better visualize the (x,t) surface.

This display is sort of a "Where is the pulse?" view. It shows for the selected values of t, how far along the x-axis the pulse has moved. Notice that the displacement of the pulse between any pair of time values is the velocity of the pulse times the difference in time. By replacing x in f(x) with x-v*t like this, y=f(x-v*t), we can include the time dependence in our pulse function. In fact any function, f(x-v*t), represents a traveling wave. In the case of a single pulse the function must be one that yields a single peak. The actual function used here is y=3/((x-v*t)2+1). Whenever (x-v*t) is zero, the function has its maximum value of 3.0. When x is much less than or greater than v*t, the denominator becomes large and the function approaches zero.

Now if instead of freezing time at several points and examining y=f(x) over the range of x values, we will fix several values of x and look at y as a function of time at that x position. At any position on the x-axis we have y depending on time, y=f(t). A series of plots of y vs. t pulses corresponding to different values of x is shown in the Pulse Sliced Along x display. This is sort of a "When is the pulse?" view. If I take a fixed position on the x axis, when does the pulse come by?

pulse x slice
pulse space-time surface

All together then y is a function of both position, x, and time, t, y=f(x,t). As we have already seen, the particular function of x and t in this case is y=3/((x-v*t)2+1).

If you imagine an (x,t) plane, then for every point in that space-time plane there is a unique y so that the values of y form a surface over the plane. The Pulse Space-Time Surface display illustrates that idea.

Arrrgh!I know this seems like a lot of bother, studying a one-dimensional wave in three dimensions... and not even proper dimensions at that, one of them being measured in seconds rather than meters. This idea of a function of more than one variable is a logical extension of the function discussion we had early in the course. It will be convenient for us to be able to visualize a wave as a function of position and time when it comes to understanding some of the underlying principles that make waves work.

Let's look at another example of our wave on a string, one perhaps more in line with our intuition about waves. The pulsed string served its purpose in making clear the traveling aspect of the wave but we are accustomed to waves which are periodic, which actually look wavelike. Instead of giving the string one shake and holding it still, now we will move the end continually up and down in simple harmonic motion. Run the String Wave display to see this. This time the Action button turns on an oscillator to which the string is attached. You will see that our string wave travels down the string just as the single pulse did, with a certain velocity. The difference here is that there is a whole train of pulses alternating between negative and positive y values.

Again by replacing x in y=A*sin(x), by (x-v*t) we get the function y=A*sin(x-v*t), which describes the traveling sine wave.
string wave
continuous wave space-time surface

The Continuous Wave Space-Time Surface display illustrates the continuous wave started at time zero. Notice that part of the space-time plane is not covered by the disturbance since distant x at early time was not accessible to the wave.

Clearly continuous waves of the sort illustrated here have a definite frequency , f, established by the driving mechanism. One cycle of the driver produces one cycle of the wave. The velocity, v, of the wave is determined by the nature of the medium. We will work on that relationship some more later. The period, period , T, of the wave is obviously 1/f based on the definition of f. The wavelength , l , of the wave is determined by the period and the velocity. The wavelength must be the distanced traveled by the wave in one period. In the form of equations we have T = 1/f and l = T * v.

Now if we look at a wave on the space-time surface which has a higher velocity we would expect to see more of the plane covered because the wave could reach distant x earlier. Also we should see that the number of fluxuations seen at a fixed x are more per unit time since the wave peaks and valleys are passing that spot faster. Look at the Fast Wave (x,t) Surface display.

Now, getting back to the velocity of the wave, Newton's second law tells us that acceleration is proportional to force. If we think of a wave as a disturbance in some medium, like a string, then the more force a certain displacement produces in the medium, the more quickly things happen. In other words, the undisturbed portion of the string will get the news about the disturbance in less time if the string is stretched more tightly. If a second string which is more massive than the first had the same tension on it, the wave would travel more slowly down that string. For waves in strings then the velocity depends inversely on the mass per unit length (m) and directly on the tension (F) in the string. So a candidate expression for the velocity would be v = F / m. But look at the units of F / m. They are mass times length over time squared divided by mass over length, or M*L/T2*L/M, or L2/T2. In order to get units of L/T or velocity, it is necessary to take the square root of our candidate expression. If F and m are the only quantities involved, as we suspect then v = (F / m)0.5.

devilish hard You should be noticing a pattern developing here. Dynamical system properties that are inversely dependent on time, like velocity, frequency and angular frequency always seem to be proportional to the square root of springiness over inertia. For the pendulum w = (l/m)0.5. For the simple harmonic oscillator w = (k/m)0.5. For a wave traveling down a string v = (F / m)0.5. I used the term "springiness" here because it captures the idea of stiffness or resistance to displacement without being so specific that it applies in only one situation. Likewise the term "inertia" in general represents the persistence of motion. These two concepts cut across the boundary between mechanics and electrodynamics so that the ideas we are developing here can be applied later.

fast wave
water waves

Now let's go a bit farther with the mathematics of traveling waves on strings. We have talked about the first and second derivative of functions of one variable. How would those concepts apply to functions of two variables such as the functions representing traveling waves? It turns out that we can apply the derivative concept one variable at a time. We can hold time constant and look at the rate change of y with respect to x, and the rate of change of that rate of change with respect to x. Likewise we can work with the first and second derivative of y with respect to t, holding x constant. Derivatives like these are called partial derivatives.

Go back to the Fast Wave (x,t) Surface display and follow one of the red lines, imagining its slope and rate of change of slope. That would be the first and second partial derivative of y with respect to x. Then follow one of the green lines looking at the slope and rate of change of slope. That would be the first and second partial derivative of y with respect to t. The symbol for the first partial derivative of y with respect to t is partial derivativey/partial derivativet, similar to the ordinary derivative being dy/dt. For the second partial derivative you see partial derivative 2y/partial derivativet2.

In the case of our string, the second partial derivative of y with respect to t is the acceleration in the y direction of the bit of string located at a particular point on the x-axis. We know from Newton's second law that this acceleration is proportional to the tension in the string that we called F. The second partial derivative of y with respect to x is a measure of the sharpness of the peak of y=f(x) at a particular instant in time. If the mass per unit length of the string were very small, the shape of the y=f(x) curve would be rather broad and flat since distant parts of the string would be responsive to small forces. With a very massive string, the peak would be rather narrow and sharp since distant parts of the string would be unresponsive to small forces.

By sort of a hand waving argument we have arrived at a point where it at least seems plausible that the velocity of a wave on a string could be expresses in terms of partial derivative2y/ partial derivative t2 and partial derivative2y/ partial derivativex2, since partial derivative 2y/ partial derivativet2 sort of plays the role of springiness and partial derivative2y/ partial derivativex2 sort of plays the role of inertia. In fact it is true that for any linear wave, the wave function will be a solution to the partial differential equation v2 = F / m = partial derivative2y/partial derivativet2 / partial derivative 2y/partial derivativex2. If you feel up to it you can verify that the two wave functions we have been using, y=3/((x-v*t)2+1) and y=A*sin(x-v*t), both satisfy this equation.

We will save more detailed discussion of traveling waves for another course. This introduction to the topic will serve as an anchor point, tying those later courses back to mechanics.

For the last time, are there any questions?

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