Phys 2214 hw4 Solutions - Phys 2214 Homework#4 Solutions 1...

Phys 2214 Homework #4 Solutions
September 30, 2011
1. For a long, thin rod, the compressions created by a longitudinal wave
traveling along the rod will change the length to a greater extent than
the diameter. So, we can replace the equation for the bulk modulus, p =
- B (Δ V/V ), by the equation for Young’s modulus, F/A = Y (Δ x/x ).
s(x+Δx,t)
x+Δx x
s(x,t)
(a) In general, the change in length of the segment is
Δ L = [ x + Δ x + s ( x + Δ x, t )] - [ x + s ( x, t )] - Δ x
= s ( x + Δ x, t ) + x + Δ x - x - Δ x - s ( x, t )
= s ( x + Δ x, t ) - s ( x, t ) .
1

(b) The linear stress-strain relation is
F
A
= Y
Δ L
L
= Y
s ( x, t ) - s ( x + Δ x, t )
Δ x
,
where the fact that Δ L is negative has been used and the original
length is Δ x .
(c) The force is given by
F ( x, t ) = lim
Δ x → 0
Y A
s ( x, t ) - s ( x + Δ x, t )
Δ x
= - Y A
∂s
∂x
.
(d) The net force acting on the segment is F ( x, t ) - F ( x + Δ x, t ).
Then Newton’s 2 nd law gives
- Y A
"
∂s ( x, t )
∂x
-
∂s ( x + Δ x, t )
∂x
#
= ρ
0
A Δ x
∂ 2 s
∂t
2
.
Dividing both sides by Δ x and taking the limit as Δ x → 0 gives
lim
Δ x → 0
h
∂s ( x,t )
∂x
-
∂s ( x +Δ x,t )
∂x
i
Δ x
= -
∂ 2 s
∂x
2
.
We then get the wave equation
∂ 2 s
∂x
2
=
ρ
0
Y
∂ 2 s
∂t
2
,
which, upon comparison with the general wave equation, gives
v =
q
Y/ρ
0
.
2. The fundamental frequency is given as f
1
= 150 Hz .