Let's analyze the problem step by step:
1. Wave Speed Formula:
The speed [tex]\( v \)[/tex] of a wave on a stretched string is given by the formula:
[tex]\[
v = \sqrt{\frac{T}{\mu}}
\][/tex]
where [tex]\( T \)[/tex] is the tension in the string, and [tex]\( \mu \)[/tex] is the linear mass density (mass per unit length of the string).
2. Initial Conditions:
Initially, the string has:
- Length [tex]\( L \)[/tex]
- Linear mass density [tex]\( \mu \)[/tex]
- Tension [tex]\( T \)[/tex]
Therefore, the initial speed of the wave is:
[tex]\[
v = \sqrt{\frac{T}{\mu}}
\][/tex]
3. Change in Length:
The length of the string is increased to [tex]\( 2L \)[/tex], with no change in the total mass or the tension.
4. Effect on Linear Mass Density:
Since the mass of the string stays the same but the length doubles, the new linear mass density [tex]\( \mu_{\text{new}} \)[/tex] becomes:
[tex]\[
\mu_{\text{new}} = \frac{\text{Total mass}}{2L} = \frac{\mu L}{2L} = \frac{\mu}{2}
\][/tex]
5. New Speed of the Wave:
The new speed [tex]\( v_{\text{new}} \)[/tex] can be calculated using the new linear mass density:
[tex]\[
v_{\text{new}} = \sqrt{\frac{T}{\mu_{\text{new}}}} = \sqrt{\frac{T}{\mu / 2}} = \sqrt{\frac{2T}{\mu}} = \sqrt{2} \sqrt{\frac{T}{\mu}} = v \sqrt{2}
\][/tex]
Putting it all together, the new speed of the wave when the length of the string is increased to [tex]\( 2L \)[/tex] is:
[tex]\[
v_{\text{new}} = v \sqrt{2}
\][/tex]
Hence, the correct answer is:
[tex]\[
v \sqrt{2}
\][/tex]
