Physics GRE - #18

A string consists of two parts attached together.

The right part of the string has mass $\mu_r$ per unit length and the left part of the string has mass $\mu_l$ per unit length. If a wave of unit amplitude travels along the left part of the string, what is the amplitude of the wave that is transmitted to the right part of the string?

1. $1$
2. $\dfrac{2}{1+\sqrt{\mu_l / \mu_r}}$
3. $\dfrac{2\sqrt{\mu_l / \mu_r}}{1+\sqrt{\mu_l / \mu_r}}$
4. $\dfrac{\sqrt{\mu_l / \mu_r}-1}{\sqrt{\mu_l / \mu_r}+1}$
5. $0$

Solution :

The correct answer is choice 3.

To solve the problem we can look at some nice limiting cases.
First, suppose $\mu_l =\mu_r$, then there should be no difference in amplitude as the wave travels (i.e. amplitude is 1 under this condition). Choices 1, 2, and 3 satisfy this.


Second, suppose $\mu_r\rightarrow\infty$. Then the wave should be completely reflected at the part where the two strings join (imagine a string attached to a wall). Thus, we should have amplitude equal to 0 under this condition. Only choice 3 satisfy this.