# A longitudinal wave as a particular example of mechanical waves

The concept of ‘wave’ appertains to those undoubtedly particularized conceptions that remain largely incomprehensible to a lion’s share of students, even despite the fact that they receive the knowledge about the main characteristics of waves during first school classes in physics. In modern physics, a wave is determined as an oscillation, which occurs with a transfer of energy through a medium. In fact, a wave process may have very different physical characteristics; therefore, it is possible to divide all kinds of waves in several groups: mechanical, chemical, electromagnetic, gravity waves, spin waves, etc. According to the more specific characteristic of waves, they are divided into several separated groups. Here is a concise list of these classes in accordance with standard classification, which is generally accepted in modern physics:

• According to the type of distribution in space, waves are divided into standing and running.
• According to the nature of waves: oscillatory and secluded (solitons).
• According to the type of waves: transverse, longitudinal and mixed (water waves, Rayleigh surface waves, etc.).
• According to the mathematical laws that describe a wave process: linear and nonlinear.
• According to the properties of a substance: waves in discrete structures and waves in continuous substances.
• From a point of view of geometry: spherical (spatial), one-dimensional (flat), and spiral waves.

Doubtlessly, a variety of diverse wave processes leads to the fact that there exist no such thing as general properties that are common for all types of waves. In truth, the idea of ‘wave’ per se cannot be realized by students eschewing examination of the elementary examples of waves, which surround them in their mundane life. Thereby, in order to ensure a gradual increase in the complexity of the educational material it is recommended to start by studying the main principles and characteristics of mechanical waves, and, more particularly, of a longitudinal wave.

A wave process is accompanied by a transfer of energy without a mass transfer. A characteristic feature of all types of mechanical waves is that they propagate only in a material medium (solid, liquid or gaseous). However, a few types of waves can propagate in a vacuum (for example, light waves). In fact, mechanical waves can occur only in a medium, which can reserve kinetic and potential energy. Thus, the medium in which mechanical waves can be traced must have elastic and inert properties. Using different mathematical models, we can create various types of these mediums; however, in real environments, these properties are distributed throughout all the volume of an object. For example, any element of a solid object has a mass and an elasticity. The simplest one-dimensional model of a solid object can be represented as a set of balls and springs. In this model, elastic and inert properties of the medium are separated: the balls have a mass (M), whereas the springs – stiffness (K). Using this elementary model, we can describe the propagation of longitudinal and transverse waves in solids. During a longitudinal wave, which moves through the solid object, balls move along the chain, whereas the springs are stretched or compressed. Such a deformation is called the stretching deformation (compressive deformation). In liquids and gases, this deformation is accompanied by sealing and rarefaction.

Therefore, a longitudinal wave can be defined as a wave motion in which the particles of the medium oscillate about their mean positions in the parallel direction with respect to the initial directions of the wave’s propagation. In other words, a longitudinal wave is a process of a wave motion in which a particle displacement is parallel to the direction of wave propagation. In addition, to this portion of materials, it is also wise to understand the concept of a transverse wave with an eye to achieving a systematic and complete understanding of the most significant peculiarities of mechanical waves. A transverse wave is a wave motion in which the particles of a medium move in a perpendicular direction with respect to the direction in which the wave itself moves. Thereby, it is obvious that these two types of waves differ in a direction in which the oscillations are with respect to a direction a wave itself: a perpendicular direction in case of a transverse wave and a parallel direction in case of a longitudinal wave. However, despite this difference, it would be a gross mistake to consider that there exist no processes, which can be characterized as a combination of these types of wave propagation processes. For example, waves that occur in a liquid medium with sufficient density, such as water, supply us with a classic example of a wave process that combines two different types of wave motion. In fact, one can easily verify this statement just by performing a thought experiment with water waves. Indeed, as a wave travels through the waver, the particles move in clockwise circles. Thereby, we can postulate that water waves involve two completely different types of wave processes that are characteristic for both transverse and longitudinal waves.

Another bright example of waves that are characterized by both longitudinal and transverse motion are Rayleigh surface waves. These waves occur in solids in contradistinction to wave processes that occur in liquids. Thereby, they differ from water waves in one significant detail. As it was shown in the previous examples, in a water wave process particles move in clockwise circles. However, the particles in a solid that are involved in a Rayleigh surface wave move in elliptical paths and the major axis of the ellipse is always perpendicular to the surface of the solid. Therefore, whereas all particles in a water wave move clockwise, the particles at the surface of a solid trace out a counter-clockwise ellipse and the particles at a depth of more than 20% of a wavelength trace out clockwise ellipses. In other words, particles of a solid in which a Rayleigh surface wave occurs are involved in both up-down and side-to-side motions. Additionally, it has to be emphasized that in both a transverse wave and a longitudinal wave a mass transfer in the direction of wave propagation does not occur. In a process of wave propagation, particles only oscillate about their equilibrium positions. However, waves carry energy fluctuations from one point of a medium to another.

## Sound as a longitudinal wave: a brief theoretical explanation

Obviously, one of the most demonstrative and, simultaneously, ubiquitous wave processes, which may be used as obvious examples of a longitudinal wave is a sound wave. In fact, from the point of view of physics, a sound wave can be regarded as a specific example of mechanical waves, which is created by a specifically vibrating object. The oscillations of an object set particles in a surrounding medium in vibrational motion. Therefore, a sound wave, as well as all other types of waves maintain a transport of energy through a medium. A sound wave can be described as a particular example of a longitudinal wave. Let us prove this statement by studying the main characteristics of a sound wave. Sound wave in air (and any other fluid medium) is a longitudinal wave because particles of a fluid medium through which a sound wave moves oscillate parallel according to its direction. For example, a vibrating string creates longitudinal waves. Let us present the string and these waves in two-dimensional form in order to simplify considerably the description of the mechanism of their movement. As the vibrating string moves in the forward direction, it pushes upon surrounding molecules of a fluid medium, moving them to the right towards their nearest neighbors (from the point of view of an experimenter). As a result, the neighboring molecules of a medium that are located to the right of the vibrating string become compressed into a small region of space. According to this statement, we can predict that as the vibrating string moves in the reverse direction (leftward), it causes a decrease in the pressure of the fluid medium to its right. As a result, the molecules of the medium move back leftward. Thus, the molecules, which are located to the right of the vibrating object, move to the right of the string, occupying the vacant place. In fact, the molecules adopt the motions of the string, moving rightward as the string moves rightward and then leftward as the string moves leftward. Due to the particle-to-particle interaction, these back and forth oscillations are imparted to adjacent neighbors. As a result, more and more particles that surround those molecules that were involved in the initial oscillations become involved in this process, transporting a sound wave to the right. Therefore, we can summarize that since the particles of the medium are moving in the direction that is parallel to the direction in which the sound wave moves, the sound wave must be referred to as a longitudinal wave. The result of such longitudinal vibrations is the creation of diverse rarefactions and compressions within a fluid medium.

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