Transversal Waves: Solving Vibrating String Equations | Easy Guide

In summary, the conversation is about transversal waves and specifically the vibrating string. The person needs help determining which mathematical representations can be solutions to the vibrating string wave equation and how to write them in the D’Alembert form. They are also trying to confirm if their solution is correct and if it makes sense that it is imaginary. The expert advises them to use a trigonometric identity to simplify their solution.
  • #1
Feynmanfan
129
0
Dear friends,

I need some help with transversal waves, to be precise: the vibrating string. I’ve been given many mathematical representations of what can be a wave (e.g: 10(x^2-v^2*t^2)
or this one, 5 Sinx cosv*t)

I have to argue which of them can be a solution of the vibrating string wave equation. And if it’s a solution I’ve been asked to write it in the D’Alembert form (that’s f(x-vt)+g(x+vt)).

Just tell me if what I think is correct: I insert the possible solution in the wave equation to see if both sides of the equation match (is that all or am I missing something?).

Thanks
 
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  • #2
Yeah,it needs to satisfy 1D d'Alembert's equation.

Daniel.
 
  • #3
Thanks Daniel.

Now I have a more specific question on this vibrating string problem. Given u(x,t)=5Senx*Cos(vt), I've proved that it's a solution. However, while trying to write it in the D'Alembert form I get this: 5i cos(x+vt)cos(x-vt)

Does it make sense that it is imaginary? Aren't string waves supposed to be real. I don't know if I'm mixing up things.
 
  • #4
It can't be complex (with a nonzero imaginary part,that is).U should use a trigonometric identity

[tex] \sin x\cos y\equiv \frac{1}{2}\left[\sin\left(x+y\right)+\sin\left(x-y\right)\right] [/tex]

Daniel.
 
Last edited:

1. What is a vibrating string?

A vibrating string is a physical phenomenon where a string is set into motion, causing it to move back and forth rapidly. This is due to the tension and elasticity of the string, which allows it to vibrate at different frequencies.

2. How does a vibrating string produce sound?

When a string vibrates, it produces sound waves that travel through the air. These waves can be perceived by our ears as sound. The frequency of the string's vibration determines the pitch of the sound produced.

3. What factors affect the vibration of a string?

The tension, length, and thickness of a string all affect its vibration. A string with higher tension, shorter length, and smaller thickness will vibrate at a higher frequency and produce a higher pitch sound. Additionally, the material and level of elasticity of the string can also impact its vibration.

4. How is the vibration of a string measured?

The vibration of a string is typically measured by its frequency, which is the number of times the string vibrates back and forth in one second. This is typically measured in Hertz (Hz). The higher the frequency, the higher the pitch of the sound produced by the string.

5. What are some real-life applications of vibrating strings?

Vibrating strings have a wide range of applications in various fields. In music, they are used in instruments such as guitars, violins, and pianos to produce different pitches and create music. In science, they are used in experiments to study sound and vibrations. They are also used in engineering for applications such as measuring tension and detecting cracks in structures.

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