State space of a standing wave?

In summary, the conversation discusses the concept of a state being defined by independent variables to describe a system, such as (q,p)-phase-space for a point mass and Hilbert-space for quantum-states. The question arises whether a 'state-space' can be defined for frequencies of a standing wave, and the connection between this concept and the original concept of phase space from wave and harmonic oscillator motions is explored. The answer is interpreted as possibly being yes, and a suggestion is made to consider writing a governing equation for the motion of the standing wave in a string.
  • #1
pliep2000
5
1
Lets say a state is defined by the minimal amount of independent variables to completely describe a system.
One would come up with the (q,p)-phase-space for a point mass and as another example the Hilbert-space for quantum-states.

Consider the very simple case of a standing wave in string where f1, f2 etc are the fundamental and the harmonics.

Question: Could one define a 'state-space' of the frequencies f1, f2 etc.?
 
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  • #2
Doesn't the original concept behind "phase space" pretty much come from wave/harmonic oscillator motions?
 
  • #3
Do i interpret your answer correctly as: yes probably?
 
  • #4
Well, maybe you can think of it this way:
Suppose you were to write down a governing equation for the motion of the standing wave in a string. What would such an equation look like?
 

1. What is a standing wave and how is it formed?

A standing wave is a type of wave that occurs when two waves with the same amplitude and frequency traveling in opposite directions interfere with each other. This interference results in a stationary pattern of nodes (points of no displacement) and antinodes (points of maximum displacement). Standing waves can be formed in a variety of systems, such as musical instruments, electromagnetic fields, and water waves.

2. How is the state space of a standing wave defined?

The state space of a standing wave is a mathematical representation of the possible states the wave can take on at any given time. It is defined by the amplitude, frequency, and phase of the wave, as well as the boundary conditions of the system. The state space allows us to visualize and analyze the behavior of the standing wave.

3. What is the significance of the state space in understanding standing waves?

The state space is crucial in understanding standing waves as it provides a visual representation of the wave's behavior. By analyzing the state space, we can determine important characteristics of the standing wave, such as its nodes and antinodes, and how they change over time. Additionally, the state space helps us understand the energy and momentum of the wave.

4. How does the state space change over time for a standing wave?

The state space of a standing wave remains constant over time, as the wave is in a stationary state. However, the individual points on the state space will oscillate back and forth between maximum and minimum values as the wave propagates. This results in a pattern of nodes and antinodes that do not move but have changing amplitudes.

5. Can the state space of a standing wave be affected by external factors?

Yes, the state space of a standing wave can be altered by external factors such as changes in temperature, pressure, or the addition of a boundary or obstacle. These factors can change the boundary conditions of the system, which will affect the state space of the standing wave. Additionally, the state space may also change if the amplitude or frequency of the wave is altered.

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