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Understanding the wave equation is crucial for comprehending how sound waves, such as the one presented in the exercise, propagate in space. In essence, the wave equation is a mathematical model that describes the behavior of waves; it is a second-order partial differential equation that relates the wave's spatial structure to its temporal evolution.

In the context of sound waves, the wave equation can be expressed as \[\begin{equation}(abla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}) p(\mathbf{r}, t) = 0\text{,}\end{equation}\] where \(p(\mathbf{r}, t)\) represents the pressure variation due to the wave at a point \(\mathbf{r}\) in space and time \(t\), \(abla^2\) is the Laplacian operator describing the spatial variation, \(c\) is the speed of sound, and \(\frac{\partial^2}{\partial t^2}\) is the second derivative with respect to time, denoting temporal changes.

A plane wave is a particular solution to the wave equation where the wave fronts are planes of constant phase and travel in a uniform direction. These waves are described by an equation showing how the wave phase depends on position and time. Moreover, they are fundamental to understanding many physical phenomena, including sound.

The wave equation solved in the step-by-step solution \[\begin{equation}p(\mathbf{r}, t)=p_{m} \exp i(\mathbf{x} \cdot \mathbf{r}-\omega t)\text{,}\text{,}\text{,}\text{,}\end{equation}\] represents a plane sound wave. The phase \(\phi = \mathbf{k} \cdot \mathbf{r} - \omega t\), depends on the dot product of the wave vector \( \mathbf{k} \) and the position vector \( \mathbf{r} \), as well as the angular frequency \( \omega \) and time \( t \), suggesting a uniform wavefront structure.

Understanding Particle Displacement

Particle displacement in waves refers to the deviation of particles from their equilibrium position as a wave passes through the medium. Displacement is directly related to the physical properties of both the particle and the wave, such as the pressure variation in sound waves.

The formula derived from the exercise \[\begin{equation}\boldsymbol{\rho} = -\frac{1}{\omega^2 \rho_0} abla p\text{,}\text{,}\end{equation}\] indicates how the pressure gradient can influence particle displacement. Here, \( \omega \) is the wave's angular frequency, \( \rho_0 \) is the medium's density, and \( abla p \) is the pressure gradient.

Irrotationality is a concept in fluid dynamics and wave theory, referring to a flow or wave field where rotation at any point within the field is zero. For a wave to be irrotational, its velocity field must have no vorticity, which means the curl of the velocity field is zero.

In the exercise, after computing the curl of the velocity, we found that \[\begin{equation}abla \times \dot{e} = 0\text{,}\text{,}\end{equation}\] confirming the irrotational nature of the sound wave. This property is essential in various applications, including acoustics and aerodynamics, as it simplifies the analysis of wave motion.

Longitudinal waves, such as sound waves, have particle motion parallel to the direction of wave propagation. These waves transfer energy through the medium by compressing and rarefying it, as opposed to transverse waves, where particle motion is perpendicular to the direction of wave propagation.

The justification for referring to the plane sound wave in the exercise as 'longitudinal' stems from the velocity field's parallel alignment to the wave vector \(\kappa\). This orientation means that the displacements and velocities of the medium's particles are along the direction of wave travel, characteristic of longitudinal waves.