Look up or derive the frequencies of the standing waves of a length of wire, L, clamped at the ends and under tension.
What are the similarities and differences between a vibrating string and a resonating pipe?
Does the diameter of the wire matter?
Does the diameter of the pipe matter?
I don't understand any of these questions and how to answer them.
Wavelength at resonance is dependent on the length of the wire, or pipe, and to a minor extent, the diameter. Your text has formulas for this.
To understand and answer these questions, we need to explore the concepts of standing waves, resonating pipes, and vibrating strings.
1. Frequencies of Standing Waves in a Wire:
When a wire is clamped at both ends and under tension, it can vibrate and produce standing waves. These standing waves have specific frequencies, also known as harmonics or modes. The fundamental frequency (first harmonic) is the lowest frequency at which the wire vibrates in one complete cycle. The higher harmonics have frequencies that are integer multiples of the fundamental frequency. The formula for the frequencies of the standing waves in a wire is:
f(n) = n * (v/2L)
Where:
- f(n) is the frequency of the nth harmonic
- n is the harmonic number (1, 2, 3, ...)
- v is the speed of the wave on the wire
- L is the length of the wire
2. Similarities and Differences between Vibrating Strings and Resonating Pipes:
Both vibrating strings and resonating pipes can produce standing waves and have similar behavior in terms of harmonics and frequencies. However, they differ in their shape and geometry.
- Vibrating Strings: Strings, like those on musical instruments, are typically long and thin. The string vibrates transversely, meaning it moves back and forth perpendicular to its length. The standing waves on a vibrating string create distinct nodes (points of no vibration) and antinodes (points of maximum vibration). The fundamental frequency is determined by the length, tension, and mass per unit length of the string.
- Resonating Pipes: Resonating pipes, such as organ pipes or flutes, are typically cylindrical tubes with one or both ends open or closed. Air resonates inside the pipe, producing standing waves, and the pipe itself does not physically vibrate. The standing waves in a resonating pipe have distinct nodes and antinodes, similar to vibrating strings. The fundamental frequency is determined by the length and speed of sound in the pipe. The end conditions (open or closed) also affect the harmonic series.
3. Influence of Wire Diameter on Standing Waves:
The diameter of the wire does have a minor influence on the frequencies of the standing waves. Generally, thinner wires will have higher frequencies for a given tension and length. However, this effect is not as significant as other factors like tension, length, and speed of the wave on the wire. Therefore, the wire diameter's impact on the frequencies of the standing waves is often considered negligible for practical purposes.
4. Influence of Pipe Diameter on Resonating Pipes:
The diameter of a resonating pipe does affect its frequencies and harmonics. Smaller diameter pipes have higher frequencies for a given length and speed of sound. Similarly to wires, this effect is not as pronounced as other factors (e.g., length, speed of sound), but it is significant. The diameter affects the boundary conditions and the formation of standing waves within the pipe.