7/29/2019 wp_hw_3_04
1/3
1.138J/2.062J/18.376J, WAVE PROPAGATION
Fall, 2004 MIT
Homework set no 3, Due Oct 21,2004
1 Dispersion of waves in a beam
Read notes Section 1.9, Chapter 1, Notes on beams. Start from
equation (9.5) and
assume that k ,E,I , are constants andp(x, t) = 0 .
Examine a simple progressive wave
V= Aeikxit (H.1.1)
and find the dispersion relation between and k. Sketch the
dependence of, the phase
and group velocities, on k.
Solve the inital-value problem by using exponential Fourier
transform subject to
V(x, 0) = f(x), Vt(x, 0) = 0, (H.1.2)
where f(x) is a given even function ofx and vanishing for large
x.
Get the inverse transform formally, and then use the method of
stationary phase to
find the asymptotic behavior for fixed x/t but large t. Describ
e the physics by sketching
several snapshots for different t.
2 Identity relating the reflection and and transmis-
sion coifficients in a rod
Let the rod cross section S(x) vary from one constant thorugh
some general profile to
another constant, i.e.,
(H.2.3)S(x) S(x) S2, xS1, x ;
where S1, S2 are two different constants. Consider monochromatic
wave of fixed fre-
quency so that
U(x)eitu(x, t) = [ ] (H.2.4)
1
