Normal Modes Normal Modes Normal modes of a

Normal Modes 简正模式

Normal Modes ß ß Normal modes of a string Normal modes of a sphere Þ Þ Spherical harmonics Radial variation � � Þ Þ ß Fundamental modes Overtone modes Torsional modes Spheroidal modes Observations of normal modes Þ Þ Overview 2004 Sumatra earthquake

Normal Modes of a String The boundaries make the modes! No boundaries, no discrete modes! (an infinitely long string would have continuous modes at all frequencies)

Propagating and Standing Waves Point source at position 8. Displacement of the string at time 1. 5.

Normal Modes of a Sphere (1) r: radial q: colatitude f: longitude

Generated by the great June 9, 1994 deep focus Bolivia earthquake. Recorded at Pasadena, California.

Spherical harmonics

Spherical harmonics

Spherical harmonics

Miaki Ishii

Radial Variations, Fundamental and Overtone Modes

Free oscillations of the earth (online animations) ß http: //lucien. saviot. free. fr/terre/index. en. html

Free oscillations m=0; n=0; l=0, Period ~20 minutes m=+-2; n=0; l=3, Period~36 minutes

Torsional (Toroidal) Modes

Torsional (Toroidal) Modes

Torsional (Toroidal) Modes

For torsional mode, the number of nodal planes = l - 1

For torsional mode, the number of nodal planes = l - 1

For torsional mode, the number of nodal planes = l - 1

For spheriodal mode, the number of nodal planes = l

For spheriodal mode, the number of nodal planes = l

Observation of Normal Modes 1882 – Lamb – fundamental mode of Earth (as steel ball), 78 minutes 1911 – Love – included self-gravitation – fundamental mode period of 60 minutes 1952 – Kamchatka EQ is first to reveal Earth’s normal modes 1960 – Chile earthquake reveals over 40 modes

Free oscillation observed after the 2004 Sumatra earthquake

2004 Sumata Earthquake

Chandler wobble Earth is not perfect ellipsoid Predicted by Leonhard Euler Observed by

Summary ß ß Normal modes of a string Normal modes of a sphere Þ Þ Spherical harmonics Radial variation � � Þ Þ ß Torsional modes Spheroidal modes Observations of normal modes Þ Þ ¢ Fundamental modes Overtone modes Overview The 2004 Sumatra earthquake The Chandler Wobble