Section 9.1 - Wave Equation
Transcript of Section 9.1 - Wave Equation
Section 9.1 - Wave Equation
* enough already with the string equation* 3-d waves - another application of stress and strain tensor ~ Strain tensor: formed from changes in the displacement field along different directions
~ Newton’s law: stress causes transfer of momentum or acceleration
~ Hooke’s law: relation between stress (force) and strain (stretch)
~ combine these to derive the wave equation for u(x,t)
* P-wave:
* S-wave:
* mode-conversion: Zoeppritz eq’s (mech. equivalent of Frenel eq’s)
NOTE: no relation whatsoever to H-atom S (l=0) and P (l=1) -waves
Lamé’s 1st param (~pres.)Lamé’s 2nd param (sheer)Poisson’s ratioP-wave (longitudinal) modulusS-Wave (sheer,trans) modulusYoung’s modulusBulk Modulus
Elastic moduli (homogeneous, isotropic):
for fluids
air: v=343 m/s @20degCwater: v=1482 m/ssteel: v=5960 m/s
(Wikipedia)
Solutions to the Wave Equation
* separation of variables of wave equation to form Helmholtz equation
* rectangular coords
~ plane wave - now can be oscillating in all three dimensions, vs. Laplace: * cylindrical coords
* spherical coords
solutions of Bessel equation:integer order Bessel functions(circular wave functions)
spherical Bessel equationspherical Bessel functions (1/2 integer order)
solutions toLaplace eq’n
solutions toLaplace eq’n
spherical harmonics(associated Legendre fn’s)
d’Alembertian
phase velocityeigenfunctioneigenfunction
spatial freq.temporal freq.eigenvalueoperator
Helmholtz Eq. (wave eq. in frequency domain)
(Cylindrical) Bessel FunctionsSpherical Bessel Functions
from Wolfram Mathworld