Explaining Wave Equation Solution for Vibrating Strings

In summary: Secondly, it would be more difficult to find the equilibrium position of the string since it would not be a simple equation.
  • #1
mahdert
15
0
In deriving the governing equation for a vibrating string, there are several assumptions that are made. One of the assumptions that I had a hard time understanding was the following.

Once the string is split into n particles, the force of tension on each particle from the particles in the right and the left is assumed to be proportional to the ratio of the vertical displacement to the horizontal displacement.

Could you please explain to me how this assumption is correct. What are the reasons behind it. Thanks.
 
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  • #2
mahdert said:
Once the string is split into n particles, the force of tension on each particle from the particles in the right and the left is assumed to be proportional to the ratio of the vertical displacement to the horizontal displacement.
Generally, one assumes that the tension is constant throughout the string. You need the transverse component of the tension--which is the restoring force tending to pull the string back to its equilibrium position. At any point, the string makes some angle θ. The transverse component of the tension = T sinθ, which for small angles ≈ T tanθ = T Δy/Δx.

(One should derive this, as above, not just assume it.)
 
  • #3
I see. I can only suppose that this follows the assumption that the string is of uniform density. What if this is not the case? How would one proceed.
 
  • #4
I don't think the string density would affect the assumption of uniform tension.
 
  • #5
So what is the justification for assuming uniform tension across the string.
 
  • #6
You could assume otherwise, but why? If uniformity gives simple solutions that match reality, isn't all you need? It's a hypothesis that works out to be correct, an example of successful science.

Assuming non-uniform tension would be the next step, in the case that the solutions didn't match reality. It would also complicate the math tremendously. First in that you would have to make another guess as how the tension behaves (which function T(x) ?).
 

What is the wave equation for vibrating strings?

The wave equation for vibrating strings is a partial differential equation that describes the motion of a string under tension. It is given by the formula: 2u/∂t2 = c2(∂2u/∂x2), where u is the displacement of the string, t is time, x is the position along the string, and c is the speed of the wave.

What is the general solution to the wave equation for vibrating strings?

The general solution to the wave equation for vibrating strings is given by the formula: u(x,t) = f(x + ct) + g(x - ct), where f and g are arbitrary functions that represent the initial position and velocity of the string, respectively.

How does the wave equation for vibrating strings relate to music?

The wave equation for vibrating strings is the fundamental equation that governs the behavior of musical instruments such as stringed instruments (e.g. guitar, violin) and keyboard instruments (e.g. piano). It helps us understand how the vibrations of a string produce musical notes and how changing the tension, length, and mass of a string can affect the pitch and timbre of a musical sound.

What are the boundary conditions for the wave equation for vibrating strings?

The boundary conditions for the wave equation for vibrating strings depend on the specific setup of the string. In general, the boundary conditions specify the behavior of the string at its endpoints, which can be fixed, free, or forced to vibrate in a certain way. These conditions are essential for solving the wave equation and determining the specific solution for a given string.

What are some real-life applications of the wave equation for vibrating strings?

The wave equation for vibrating strings has many practical applications in fields such as engineering, physics, and music. It is used to understand the behavior of structures like bridges and buildings under seismic waves, the dynamics of vibrating strings in musical instruments, and the propagation of electromagnetic waves in transmission lines. It is also used in mathematical models for studying ocean waves, earthquakes, and other natural phenomena.

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