How does sympathetic vibration occur within a system?

Sympathetic vibration occurs when an external force or vibration frequency matches the natural frequency of a system. When this happens, the system is able to absorb energy efficiently, resulting in an increase in amplitude of the oscillating system. This phenomenon is often observed in musical instruments, where one vibrating string can cause another string with the same natural frequency to vibrate as well, even if it is not directly struck.

The concept of natural frequency is critical because it represents the frequency at which a system naturally oscillates. When the frequencies align, the system reaches resonance, leading to significant energy transfer and enhanced vibration. This is why the choice highlighting equal frequencies to the natural frequency is correct; it precisely describes the condition under which sympathetic vibration occurs.

On the other hand, varying frequencies below the natural frequency, fixed amplitudes regardless of frequency, and multiple harmonics do not align with the conditions necessary for sympathetic vibration. Those factors might influence other types of resonance or wave behaviors, but they do not create the specific effect of sympathetic vibration.