Frequencies, periods, and resonance

Understanding ground shaking during earthquakes requires understanding the concepts of frequencies (or periods) of shaking, and resonance.

The frequency of shaking is the number of cycles (back and forth) per second, and the period is the length of time it takes for one back and forth cycle. The two are the inverse of each other: frequency = 1/period.

A clock pendulum swings at a frequency of one vibration per second (1 Hz), or a period of 1 sec. When playing a piano, the middle C string vibrates 128 times per second (128 Hz; 0.0078125 second period). A 10-story building, if you pull the top to one side and release it, would sway back and forth about once per second (1 Hz; 1 second period); a 100-story building sways back and forth about once every 10 sec (0.1 Hz; 10 second period).

Earthquakes shake the ground at many frequencies at the same time, but mostly at frequencies between 0.1 Hz and 10 Hz (periods of 10 seconds to 0.1 seconds). The shaking at some frequencies is stronger than at others (graph at left), however, and each earthquake has different strengths of shaking at a given frequency.

Resonance is when motion at a given frequency is amplified by waves of that same frequency. For example, when a child is being pushed on a swing, the swinging is increased by a push being applied at the right time (at the correct frequency) during each swing. Resonance can be damaging to a building during an earthquake, as when one seismic wave pushes the building and the next one arrives just as the building finishes one swing. After repeated swings, the motion can be increased to the point of damaging or destroying the building. If the frequency of the seismic waves (the pushes) is different than the frequency at which the building sways, the motion will not be amplified. Similarly, if the child on the swing is pushed at different times during the swing, the pushing will decrease rather than increase the swinging.

Damage to buildings during earthquakes is most likely to occur when the frequency of the ground shaking is the same as the frequency at which a building shakes. A general rule of thumb is that a building’s frequency is about 10 Hz divided by the number of floors (see chart). This implies that a 30 story building will be most sensitive to ground shaking at a frequency of about 0.3 Hz (3.33 sec period).

number of floors	frequency (Hz)	period (seconds)
1	10.0	0.1
2	5.0	0.2
5	2.0	0.5
10	1.0	1.0
30	0.3	3.3
100	0.1	10.0