Physics CS 33 Read
Solve Set #2 -­‐ for Wednesday April 16 Spring 2014 RHK Ch. 19 (Tuesday lecture will help you to solve problems marked red) From RHK
Ch. 19
Exercises 2, 9, 24, 27
From K&K
Ch. 5
Problem 5.8 Problems 3, 5, 12 Problem 1. Consider a square drumhead vibraMng in the degenerate (12)-­‐(21) mode with frequency ω12 (= ω21). Assume that this vibraMon mode has the form '
! 2π x $ ! π y $
! π x $ ! 2π y $*
z(x,
y,
t)
=
Asin
sin
+
Bsin
#
& # &
# & sin #
&, cos(ω12 t)
)
"
%
"
%
"
%
"
a
a
a
a %+
(
Find the nodal lines (lines of zero displacement: z = 0) for the four special cases A = 0; B = 0; A − B = 0; A + B = 0. The length of the side of the square is a. That is, for each case you set z = 0 and see what relaMonship you get for x and y. Plot result on x-y plane These are the nodal lines. a
Problem 2. Find the lowest frequency of a drumhead in the shape of an isosceles triangle with sides a, a, a 2
a
a 2
Go to page 2
Physics CS 33 Set #2 -­‐ for Wednesday April 16 Spring 2014 Problem 3. Use index notaMons to show the results ! i( k⋅! r!−ωt ) ! i( k⋅! r!−ωt )
! ! i( k⋅! r!−ωt )
! ! i( k⋅! r!−ωt )
(a) ∇Ae
= ikAe
(c) ∇ × Ae
=i k ×A e
! ! i( k⋅! r!−ωt )
! ! i( k⋅! r!−ωt )
! i( k⋅! r!−ωt )
! 2 ! i( k⋅! r!−ωt )
2
(b) ∇ ⋅ Ae
=i k⋅A e
(d) ∇ Ae
= −k Ae
!!
!
i( k⋅r −ω t )
Problem 4. Verify by direct subsMtuMon that p(
r, t) = Ae
! , where A is a constant, is a soluMon 2
1 ∂ p
of the 3-­‐D wave equaMon ∇ 2 p − 2 2 = 0, provided ω = c k .
c ∂t
Problem 5. Show that for waves with spherical symmetry: p = p(r, t), (a) The wave equaMon can be wri]en as 1 ∂ " 2 ∂p % 1 ∂2 p
= 0.
$r
'−
r 2 ∂r # ∂r & c 2 ∂t 2
(b) Further 1 ∂2
1 ∂2 p
rp) − 2 2 = 0.
2 (
r ∂r
c ∂t
! !
A
(c) Show that p = sin
( k ⋅ r − ω t ) with ω=ck is a soluMon of (b) above, and that in general r
1
1
p(r,
t)
=
f
kr
−
ω
t
+
g ( kr + ω t )
(
)
r
r
Hint: Let ψ
(r,
t)
= rp(r,
t)
and solve the wave equaMon saMsfied by ψ (r, t).
Good Luck, Dr. B
(
)
(
)