Q9DQ - University Physics with Modern Physics

Question

The four strings on a violin have different thicknesses but are all under approximately the same tension. Do waves travel faster on the thick strings or the thin strings? Why? How does the fundamental vibration frequency compare for the thick versus the thin strings?

Step-by-Step Solution

Answer

The speed of the wave will be larger in the thinnest string and the fundamental frequency is inversely proportional to the thickness and it decreases as the thickness increases.

Step 1: Concept of the speed of the wave

The speed of a wave is given by the following equation. 

  $\mathbf{v}\mathbf{=}\sqrt{\frac{\mathbf{T}}{\mathbf{\mu }}}$

Here, v is the speed of a wave, T is the tension, and $\mathit{\mu }$ is the mass density.  

Given that all four strings have the same tension. As a result, the linear density of the string will be the only factor that affects the wave's speed. According to the equation above, the wave speed is inversely proportional to linear density. Thus, the thicker the string more the mass density hence the slower the velocity of sound on the string. Thus, waves will travel faster in thin strings.

Step 2: Fundamental frequency of the wave.

The fundamental frequency is given by, 

$$\begin{gathered} \mathrm{f}=\frac{\mathrm{v}}{2\mathrm{L}} \\ =\frac{1}{2\mathrm{L}}\sqrt{\frac{\mathrm{T}}{\mathrm{\mu }}} \end{gathered}$$   

Here, L is the length of the string. 

So, it is clear that frequency is inversely proportional to the mass density. Hence, thinner strings will have a higher frequency or thicker strings will have lower notes.