When an ultrasonic wave travels through a solid medium, such as a horn, the relationship between the area of the medium and the wave amplitude can be explained using the principle of conservation of energy.
When the cross-sectional area of the medium decreases along the propagation path of the wave, the same amount of energy carried by the wave is now concentrated in a smaller area. As a result, the energy density (energy per unit area) of the wave increases.
According to the principle of conservation of energy, the total energy of the wave must remain constant along its propagation path, assuming no energy losses. Therefore, as the cross-sectional area decreases, the energy density increases to compensate for the reduction in area.
The amplitude of the wave is directly related to the energy density. Higher energy density corresponds to a higher amplitude, while lower energy density corresponds to a lower amplitude.
To illustrate this, consider a scenario where an ultrasonic wave is traveling through a horn with a gradually decreasing cross-sectional area. As the wave enters a narrower section of the horn, the same amount of energy that was spread out over a larger area is now confined to a smaller area. This concentration of energy increases the energy density and, consequently, the amplitude of the wave.
It's important to note that the conservation of energy principle assumes ideal conditions without any energy losses due to factors like absorption, scattering, or reflection. In real-world situations, these factors can influence the behavior of the ultrasonic wave, and the relationship between the area and amplitude may be more complex.