# Introduction to Mechanical Waves

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We shall begin our exploration of the phenomenon of waves. First, what is a wave? We know intuitively that when we drop say a pebble in a pond somewhere, it creates this ripple effect which propagates across the surface of the water: these are the waves in the water. When we wiggle a string up and down, this creates waves in the string. A sound wave is essentially a pressure wave that can propagate through air, water, or some other medium. If you lay a slinky down horizontally on a table and horizontally tug it back and forth at one end (doing so very rapidly), pressure waves will be sent down the length of the slinky. Or if you placed say a cylindrical tube filled with water (where the right end is closed, and the second end has a movable piston) horizontally down on a table and rapidly tugged and moved the piston back and forth, pressure waves would be sent down the length of the tube through the water.

More abstractly, there are such things as electromagnetic waves which appear in various forms: such as light, X-rays, microwaves, ultraviolet and inferred waves, gamma rays, and so on. These waves are a little more abstract than the previously mentioned waves because they do not require any medium to travel through; light for example can travel through the vacuum of empty space (see this video for additional information on the differences between mechanical and electromagnetic waves).

There are two main types of waves: there are mechanical waves which are essentially just waves that require a medium to be propagated through (for example, a wave which travels down the length of a string), and there are non-mechanical waves which are waves that do not require any medium at all to be propagated through (electromagnetic waves, for instance, can be propagated through the vacuum of empty space). In this chapter, we shall restrict our attention to just mechanical waves. There are essentially three different kinds of mechanical waves: there are transverse mechanical waves, there are longitudinal mechanical waves, and lastly there are some mechanical waves which are both transverse and longitudinal in nature. For this section, we will mainly focus on transverse mechanical waves and look at longitudinal waves only briefly; we will not go into much detail about what the physics of the third kind of mechanical wave but I will briefly comment on what it is.

Imagine that one took a pebble and dropped it upon the surface of still water in a pond. When this pebble collides with the water particles at the surface of the pond (here, we are modeling the water molecules as particles) and then passes through the surface coming into contact with the other water particles (below the surface), what is physically happening to these water particles? Well, what happens is that the pebble “hits” and “collides” with the water particles on the surface of the pond and below the surface of the pond; also, the pebble “pushes” these water particles out of its way. What this means is that the pebble is doing mechanical work $$W_{MA}$$ on these water particles which it “pushes”; the reason why it is doing $$W_{MA}$$ on the water particles is because when it “pushes” these mass elements (or water particles) off to the side, it is exerting a force on each of them over some distance. Because the $$W_{MA}$$ which system A (in this case, the pebble!) does on system B (in this case, the collection of water particles which the rock “pushes”) is defined as the energy transferred from system A to system B, this means that energy was transferred from the rock to the water particles which it “pushed against.” (As a reminder, this is because the value of $$W_{MA}$$ is positive; if this value were negative, energy would be transferred out of the water particles and into the rock.) Now, you could imagine that right when this pebble collided with and passed through the surface of the water, right at that instant, all of the energy transferred to the water particles was concentrated in that particular location. What happens next is very fascinating. There are restorative forces which make the water particles that were “pushed down” by the rock (below the rest of the surface of the still water in the pond) want to pop back up; the effect is that the water particles (the ones in the location where the pebble hit the water) accelerate upwards slightly above the surface of the water level of the pond and adjacent water particles, which are concentric around this region of collision, gets pulled downwards below the water level of the pond. This leads to the observed “ripple effect” on the surface of the still water. What is happening is this: all of the energy which was transferred to the water particles at the location where the pebble landed, all of that energy concentrated in that particular location get dispersed and “carried away” by the ripples traveling radially outwards along the surface of the water. This leads to a very precise definition of a mechanical wave: A mechanical wave is a disturbance (i.e. the ripples) in a medium (i.e. water) which transports energy from one location (i.e. where the pebble landed) to another location (i.e. a leaf floating on the surface of the water). What happens to the water particles when this ripple effect or this mechanical waves passes through them? In this particular example, when the disturbance passes by a water particle, the particle undergoes transverse, up-and-down displacements. By transverse, I mean that when the wave (or ripple) passes through the water particles, the water particles get displaced in an up-and-down direction perpendicular to the plane across which the wave travels—basically, the direction which the wave travels in is perpendicular to the direction in which the water particle is displaced. If the displacements of the particles were in the same direction as that which the wave is traveling, then we would call this a longitudinal wave; in the latter example, we are talking about a transverse wave.

What do I mean though when I say that this energy gets “carried away” or “transported”? What I am implicitly saying here is that the energy will not “stay” at the location where the pebble landed. The question is: Why and how does the energy transport from where the pebble landed, to somewhere else? To think about this, imagine that I take a string (which is tied to a stationary wall) and hold onto it with a constant tension force parallel to the ground beneath me. Now imagine that we ran a little experiment. If I jerked my hand upwards (and then rapidly came to a stop) and then I jerked my hand back downwards (suddenly coming to a stop again at that initial position), we would observe a pulse being sent down the length of the string like the one shown in figure 2.2. In this experiment, if one looked closely, one would observe that this pulse travels down the length of the string at a constant velocity; one could perform some type of empirical measurement to show that the pulse indeed does travel down the length of the string at a constant velocity. For this example, only think about the experimental aspect of things, not the theoretical aspect—for now at least. For now, we accept the fact that the final effect of the jerking of the hand up and down is to create a pulse which is sent down the string at a constant velocity—we, for now, do not concentrate on answering why or how this is. The reason why this happens will be answered shortly. Now, if you keep jerking your hand up and down, the final effect is to create a sinusoidal wave that travels across the string at a constant speed. We, for now, accept this on the basis of observation and experiment; we will concern ourselves with the theoretical aspect of things in order to answer the why and how, later on. Just like the ripple effect in the pond, this mechanical wave is a transverse wave: meaning that as the wave passes through the string to the right or left, the mass elements (or particles) making up the string are displaced up-and-down perpendicular to the direction that the wave travels in. I will now answer the initial question we started with: How is the energy transported from one location to another? We will think about this by considering the string example. While holding onto the end of the string, when you jerk the string upwards (and then suddenly stop) you are “tugging” the string upwards through some vertical displacement (in the same direction as the tug or applied force); then you “tug” the string back downwards through a vertical displacement (in the same direction as the exerted force). This means that you did $$W_{MA}$$ on the mass elements (or particles) in the string; this means that energy was transferred from your body and hand and into the

mass elements making up the string. Now, right after you “tug” your hand back downwards coming to an abrupt stop, you will observe the mass elements on the end of the left side of the string accelerate upwards forming a little “bump” or a pulse. What can we deduce from this observation? Well, we can deduce that the energy transferred to the mass elements on the end of the left side of the string went into changing their $$KE_y$$ and $$PE_g$$. (As a reminder, because this is a transverse wave, the particles are displaced in a direction perpendicular to the direction that the wave or pulse travels in; this means that the particles do not have any left-and-right motion; all the motion caused by the disturbance (the pulse) is up-and-down; so this means that $$KE_x$$.) One will also observe that what happens next is that this pulse will travel down the length of the string at a constant speed. Similar to the pebble and the pond example, right after I tug my hand back down, right at that moment when the “bump” or pulse in the string is formed, all of the transferred energy is concentrated right around that particular location in the string (namely, at the end of the left side of the string); when this pulse moves to the right down the length of the string, the energy contained in it (which looks to be mostly $$PE_g$$) moves the right down the string. This means that when the pulse moves from the left end of the string to say the middle of the string, there is no longer any energy contained in the mass elements on the left end of the string; all that energy was transported somewhere else—namely, at the middle of the string. The same exact thing is happening with the pebble and the pond example; initially, all of the energy in the water particles is concentrated where the pebble landed; afterwards through that energy gets “transported” or “carried away” by the water ripples.