364
8 Waves
8.16 General equation for a progressive wave in the negative x-direction is
y = A sin(kx + ωt)
ω = 2π f = 2π × 170 = 340π rad/s
k =
ω
v
=
340π
340
= π/m
∴ y = 0.01 sin π(x + 340t)
8.17 (a) y 1 = A sin(kx − ωt)
y 2 = A sin(kx + ωt)
y = y 1 + y 2 = 2A sin kx cos ωt
where we have used the identity stated in prob. (8.4).
(b) The nodes are formed when kx = nπ or
2π
λ
x = nπ
or x =
nλ
2
x = 0,
λ
2
, λ, . . .
The antinodes are formed when kx =
nπ
2
or x =
nλ
4
x =
1
4
,
3
4
,
5
4
. . .
8.18 y 1 = A sin(kx − ωt)
y 2 = A sin(kx − ωt + δ)
y = y 1 + y 2 = A[sin(kx − ωt) + sin(kx − ωt + δ)]
= 2A cos
1
2
δ sin
kx − ωt +
δ
2
Thus the amplitude of the resultant wave is 2A cos
1
2
δ.
For A = 6 cm and δ =
π
2
, the amplitude of the resultant wave will be 2 ×
6 cos
π
4
or 6
√
2 cm.
For 2 A cos
1
2
δ = 6
cos
1
2
δ =
6
2A
=
6
2 × 6
=
1
2
= cos
π
3
∴
1
2
δ =
π
3
or δ =
2π
3