360
8 Waves
Thus the resultant wave = standing wave + travelling wave in the negative
direction.
The amplitudes are (a) 2A (b) 2A
8.5 (a) The wave number k =
2π
λ =
2π
2 = π/m
Frequency f =
v
λ =
8
2 = 4 Hz
Angular frequency ω = 2π f = (2π)(4) = 8π rad/s
(b) y = A sin(kx − ωt) = A sin π(x − 8t)
8.6 Let y = A sin(kx − ωt + φ)
At x = 0, t = 0, the wave has the maximum displacement and y = A:
A = A sin(0 − 0 + φ)
or sin φ = 1 → φ =
π
2
∴ y = A sin
kx − ωt +
1
2
π
= A cos(kx − ωt)
∴ y = 0.2 cos(3x − 20t)
8.7 y = 2A sin kx cos ωt (standing wave)
∂ y
∂t
= −2Aω sin kx sin ωt
Acceleration, a =
∂ 2 y
∂t 2 = −ω 2 2A sin kx cos ωt = −ω 2 y.
This is the defining equation for the SHM.
8.8 f N =
N v
2L
f 1 =
1 × 120
2 × 2
= 30 Hz
f 2 =
2 × 120
2 × 2
= 60 Hz
f 3 =
3 × 120
2 × 2
= 90 Hz
f 4 =
4 × 120
2 × 2
= 120 Hz