Source: https://commons.wikimedia.org/wiki/File:2006-01-14_Surface_waves.jpg. Author: https://commons.wikimedia.org/wiki/User:Roger_McLassus_1951
Mechanical waves are a fundamental physical aspect of the movement of physical objects, whether these objects are solids, liquids, or gases. Every time an object moves under the influence of a force, waves are involved. In some cases the waves are more obvious, such as sound waves traveling through the air (which we can hear). But other times the presence of waves is not as obvious, such as in solid objects moving under the influence of a force.
A mechanical wave is really nothing more than a disturbance traveling through a medium (such as a solid, liquid, or gas). More precisely, a mechanical wave is the transmission of motion through matter, like a ripple moving across a lake. The speed, size, and shape of the wave depends on the mechanical properties of the medium, and on the force that caused the wave to occur in the first place.
A mechanical wave is caused by a force, and the movement of this wave is directly related to Newton's second law equation (F = ma). We are mostly familiar with this equation in the context of moving objects that we can see, such as a ball being acted on by a force and then moving as a result. But it also applies to matter in general.
For example, when a solid object is acted upon by a force, a wave emanates out from the point(s) of contact, and this wave travels through the body of the object. When it reaches the boundary of the object it reflects off of it and travels in the reverse direction. Different waves moving through the body may then combine with other waves moving through the body, and interfere with each other, either constructively or destructively. This behavior is fully predicted by applying Newton's second law equation, and combining it with equations that describe the mechanical properties of the solid object (these are called constitutive equations).
As the waves move through the solid object the object itself may begin to move as a whole. This will happen if the object is free to move and not constrained in some way. So for example, imagine you are picking up a ball, and for visualization purposes imagine this is occurring in extreme slow motion. At the instant that you touch it you cause waves to form inside the ball, which emanate out from the points of contact, and then these waves travel through the ball in the manner previously described. At the same time the ball begins to move and eventually lifts off the floor.
Wave speed through solids is very fast, much faster than wave speed through liquids and gases. For example, wave speed through air (the speed of sound) is about 340 m/s, wave speed through water is about 1500 m/s, and wave speed through a solid, such as iron, is 5100 m/s. Note that the 1500 m/s speed of waves traveling through water refers to the speed at which disturbances travel through the body of the water. The waves that we typically see on the surface of water do not travel at nearly this speed (far from it). These are called surface waves and are a different type of wave phenomenon. The motion of these waves occurs much more slowly than body wave motion. This is somewhat similar to how the lifting up of the ball, described in the previous example, occurs much more slowly than the speed of waves traveling through the inside of the ball. The ball motion occurs on a "bulk" (or macroscopic) level and as a result is subject to the entirety of whatever forces act on it, which will in general mean that it will move much, much slower than mechanical waves propagating through the ball interior.
It is interesting to note that the dynamics equations for rigid body motion are just a simplified mathematical representation for wave motion through bodies that are very stiff. If we were to solve Newton's second law equation (F = ma) along with the constitutive equations for a solid body that is very stiff, the result would be almost identical to the result we would get if we were to solve rigid body dynamics equations. The reason we apply rigid body dynamics equations to solid bodies instead of wave equations is because they are accurate enough and are much easier to solve.
When a solid body oscillates on its own, at its natural frequency, such as a vibrating ruler, it gives you a clue on how fast waves travel through it. The time it takes for a moving point on the ruler to undergo a complete oscillation cycle is the time that it takes for a mechanical wave to travel through the body of the ruler for one complete cycle. However, the ruler will not oscillate back and forth at the same linear speed as the wave speed. It will oscillate at a much slower speed than that.
The oscillation period of the ruler will be equal to the time it takes for a wave to travel through the body of the ruler for one complete cycle. Since the ruler oscillates due to the effect of mechanical waves moving through its body, one complete cycle of ruler oscillation must correspond to one complete travel cycle of the wave(s) moving through the body of the ruler.
Lastly, it is interesting to mention that no prior knowledge of mechanical waves is required to solve a problem involving mechanical waves, provided that the basic fundamental equations (Newton's second law and constitutive equations) are applied to the problem. Knowledge of wave behaviour for solving wave problems is only required when we don't start with the fundamental equations, in which case we have to apply known wave concepts to solve the problems. Sometimes this is best (and easiest) since the fundamental equations are quite complicated, and would be too time consuming to solve from scratch every time you want to solve a wave problem.
If you want to see more wave related material, including the mathematics of it, look up the theory behind the following well-known topics: Vibrations of a Circular Membrane, and Vibrating Strings.
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