# Normal modes of a string thought experiment

• Nikitin
In summary, the conversation discusses the conditions for a standing wave to appear on a stretched string and whether it is possible to create a superposition of normal modes by moving the string with a different frequency. It is mentioned that driving a string in this manner can result in a "driven harmonic oscillator" with complex solutions that can be represented as a superposition of normal modes. The possibility of real strings being more complicated is also mentioned.
Nikitin
Hey! So If I have a stretched string of length L fastened at one end, and I am moving the other end sinusoidally, will a standing wave appear ONLY if I move the other end with one of the normal-mode frequencies of the string? If not, will the resulting wave be a moving wave which is a superposition of the string's normal modes?

Last edited:
Do it and see :)

Shaking the string from one end is a bit different from plucking it.
You get a node a little distance x from your fingers - and you will have excited a normal mode of a string length L-x which is fixed at both ends. Otherwise you have to move your hand with the frequency of the normal modes of your string (work them out for a string length L with one end free).

When you drive a string like that - the system is a "driven harmonic oscillator" and the solutions can get quite complicated. You can represent the result as a superposition of normal modes - just as the normal modes can be represented as a superposition of traveling waves.

Real strings can get more complicated still.

## 1. What is a normal mode of a string?

A normal mode of a string refers to the specific pattern of vibration that a string will exhibit when it is plucked or struck at a certain point. It is determined by the length, tension, and mass of the string, and can be represented mathematically as a standing wave pattern.

## 2. How are normal modes of a string studied in a thought experiment?

In a thought experiment, normal modes of a string are studied by imagining a string fixed at both ends and then exploring how it would vibrate in different scenarios. This allows for the observation and analysis of different normal modes without the limitations and complexities of conducting an actual physical experiment.

## 3. What are the factors that affect the normal modes of a string?

The normal modes of a string are affected by the length, tension, and mass of the string. These factors can be altered in a thought experiment to observe how they influence the pattern of vibration. Additionally, the type of material and the shape of the string can also play a role in the normal modes.

## 4. How do normal modes of a string relate to real-life musical instruments?

Normal modes of a string are crucial for understanding and creating musical instruments. The different normal modes of a string produce different pitches, and by manipulating these modes, musicians can create a variety of sounds and tones. This is seen in string instruments such as guitars, violins, and pianos.

## 5. Why are normal modes of a string important in physics?

Normal modes of a string are important in physics because they demonstrate the concept of resonance and the relationship between frequency and wavelength. They also provide a practical application for the study of waves and vibrations in physics and can be used to analyze and understand other systems that exhibit similar behaviors, such as sound waves and electromagnetic waves.

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