Sound and Music - Mission SM5 Detailed Help

In a rare moment of artistic brilliance, a physics teacher clamps a metal plate to a table, shakes a load of salt onto it, pulls out a violin string and draws it across the plate. The plate vibrates in a pure tone at 600 Hz. The salt moves around and finally settles when the plate is done vibrating. Quite miraculously (perhaps), the salt forms the pattern shown in Diagram A. Pushing his luck, the teacher tries again, producing the even more spectacular pattern shown in diagram B. What frequency did the plate vibrate at to produce the pattern shown in Diagram B?


 
The metal plate vibrates at one of its natural frequencies. The frequency is associated with a standing wave pattern characterized by nodes and antinodes. As the plate vibrates, the salt vibrates off the antinodal positions and settles onto the nodal positions. The diagram shows how the salt has settled on the metal plate. The locations of the salt are locations of the nodal positions.


 
Natural Frequencies and Standing Wave Patterns:
All objects have a natural frequency or a set of natural frequencies at which they vibrate at. Each frequency in the set is referred to as a harmonic frequency and is associated with a unique standing wave pattern. The standing wave pattern is characterized by the presence of nodes and antinodes that are always present at the same position along the medium. The fundamental frequency or first harmonic has the smallest possible number of nodes and antinodes. The standing wave patterns for the other harmonics - second, third, fourth, etc. - have an increasing number of nodes and antinodes.


 
The vibration of a two-dimensional plane (like the metal plate in this question) is significantly more complicated than the vibration of a one-dimensional string or wire. The analysis of the pattern is best done if an effort is made to visualize looking along the plane of the metal plate and viewing the vibrations of the plate's edge. Diagram A shows the fundamental frequency of the plate - it is the lowest natural frequency at which the plate can vibrate. Looking along the plane of the plate, there would be nodes on the two ends of one edge and an antinode in the middle. Diagram B shows a higher harmonic frequency. Looking along the plane of the plate, there would be four nodes (two on the ends of the edge) and three antinodes. There are three times the number of antinodes in diagram B compared to diagram A; thus, the frequency associated with diagram B would be three times that associated with diagram A.