How can an opera singer be heard over the much louder orchestra?

John R. Smith, a physicist at the University of New South Wales in Sydney, Australia, explains:

Opera singers are able to maximize their sound output in frequencies where the orchestra is less powerful and to which the ear is more sensitive.
In both speech and singing, we produce sustained vowel sounds by using vibrations of our vocal folds—small flaps of mucous membrane in our voice box—that periodically interrupt the airflow from the lungs. The folds vibrate at a fundamental frequency, which determines the pitch: typically between 100 to 220 hertz (Hz), or vibrations per second, for normal speech and 50 to 1,500 Hz for singing. Speech and singing also contain a series of harmonics, which are basically multiples of that frequency. Singers, especially sopranos, can learn to tune the resonances of their vocal tract to match the fundamental frequency, providing a dramatic increase in acoustic power.
An orchestra is typically loudest around 500 Hz, with the sound level dropping off quickly at higher frequencies; the ear is most sensitive around 3,000 to 4,000 Hz. Many opera singers learn to increase the power in the harmonics at frequencies above 2,000 Hz, which helps to distinguish their voices. Finally, opera singers often use much more vibrato —a slow, cyclic variation, or “wobble,” in pitch — than orchestral musicians do. This effect aids the signal processing within our auditory system in distinguishing the voice of a singer as something quite different from the accompaniment of the orchestra.