Vibration of Strings & Air Columns

Vibration in Strings

The vibration of strings is a key concept in understanding how musical instruments produce sound. The frequency of vibration depends on several factors:

• Length of the string: Shorter strings vibrate faster, producing higher frequencies.
• Tension in the string: Greater tension results in higher frequencies.
• Mass per unit length: Heavier strings vibrate slower, producing lower frequencies.

A sonometer is an instrument used to demonstrate these dependencies by varying the length, tension, and mass of a string.

Frequency Formula for Strings

The frequency $f$ of a vibrating string is given by:

$$f = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$$

• $L$ is the length of the string
• $T$ is the tension in the string
• $\mu$ is the mass per unit length

Vibration of Air in Pipes

Air columns can vibrate in pipes, producing sound waves. There are two types of pipes:

• Open pipes: Both ends are open, allowing air to move freely.
• Closed pipes: One end is closed, restricting air movement.

Frequency Formula for Open Pipes

The fundamental frequency $f$ of an open pipe is given by:

$$f = \frac{v}{2L}$$

• $v$ is the speed of sound in air
• $L$ is the length of the pipe

Forced Vibration and Resonance

Forced vibration occurs when an external force causes an object to vibrate at a frequency other than its natural frequency. Resonance happens when the frequency of the external force matches the object's natural frequency, resulting in a large amplitude of vibration.

• Spiral spring: Can demonstrate resonance when a periodic force is applied.
• Bifilar suspension: Used to study oscillations and resonance.