Bill is making waves on a rope with one end attached to a wall. When he starts moving the rope quickly, the frequency of the waves created on the rope doubles. What happens to the energy that is transferred to the wall when the frequency of the waves doubles?

Question

A.
The energy transferred to the wall is halved.
B.
The energy transferred to the wall is tripled.
C.
The energy transferred to the wall is doubled.
D.
The energy transferred to the wall is quadrupled.

Answer 1

The energy of a wave is related to both its frequency and amplitude. The energy transported by a wave is proportional to the square of its amplitude and the square of its frequency. Hence, if only the frequency of the wave doubles (while the amplitude remains constant), the energy transported by the wave increases.

Specifically, if the frequency \( f \) doubles, the energy \( E \) transferred by the wave can be expressed as:

\[ E \propto f^2 \]

So when the frequency doubles:

\[ E' = k(2f)^2 = k \cdot 4f^2 = 4E \]

Thus, the energy transferred to the wall is quadrupled when the frequency of the waves doubles.

Therefore, the correct answer is:

D. The energy transferred to the wall is quadrupled.