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