In the context of transverse waves on a stretched string, which expression correctly gives the wave speed in terms of the string tension and linear mass density?

• A. $v=\frac{T}{\mu }$
• B. $v=T-\mu$
• C. $v=\sqrt{\frac{T}{\mu }}$
• D. $v=\sqrt{\frac{\mu }{T}}$

Answer: (C)

Verified step by step guidance

1. Identify the physical quantities involved: the tension in the string, denoted by \(T\), and the linear mass density of the string, denoted by \(\mu\) (mass per unit length).
2. Recall that the wave speed \(v\) on a stretched string depends on how tightly the string is stretched (tension) and how much mass is distributed along its length (linear mass density).
3. Use the fundamental formula for the speed of a transverse wave on a string, which is derived from Newton's second law applied to a small segment of the string:
4. \[v = \sqrt{\frac{T}{\mu}}\]
5. Understand that this formula means the wave speed increases with greater tension and decreases with greater mass density, reflecting the physical intuition that a tighter and lighter string allows waves to travel faster.
6. Verify that the other given expressions do not have the correct form: for example, \(v = \frac{T}{\mu}\) lacks the square root, and \(v = T - \mu\) is dimensionally inconsistent and physically incorrect.