Wave Propagation
Speed at which the phase of a monochromatic signal travels through a medium characterizes the elastic properties of the material. The phase velocity in paperboard testing determines the stiffness and density of the sheet by measuring how quickly ultrasonic waves move between sensors. It is a fundamental parameter for non-destructive testing and quality control on the paper machine.
This metric varies depending on the direction of the fibres and the moisture content.
Dispersion Relation
Ultrasonic waves in thin sheets exhibit dispersion, meaning that different frequencies travel at different speeds. The phase velocity is distinct from the group velocity, which represents the speed of the overall wave packet. In a paper mill, sensors measure the time of flight for a specific frequency to calculate the internal bond strength.
If the thickness of the board changes, the relationship between frequency and speed also shifts. This behavior is captured in a dispersion curve where the speed is plotted against the product of frequency and thickness. Analysis uses these curves to identify the specific modes of vibration, such as the fundamental symmetric or antisymmetric lamb waves.
These modes interact with the internal structure of the paper differently, revealing defects or variations in the fibre orientation.
Elastic Constant
Calculations of the Young modulus in the machine direction and cross direction rely on the measured speed of the sound. The phase velocity provides a direct link to the mechanical performance of the board under stress. It allows operators to adjust the refining process.
High speeds indicate a more tightly bonded fibre network.
